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\begin{document}
\title[Your comment to [email protected]]{Biholomorphic mapping on the boundary I} \maketitle \author{Won K. Park} \address{Department of Mathematics, University of Seoul} \email{[email protected]}
\begin{abstract} We present a new proof of Chern-Ji's mapping theorem on a strongly pseudoconvex domain with differentiable spherical boundary. We show that a proper holomorphic self mapping of a strongly pseudoconvex domain with the real analytic boundary is biholomorphic. \end{abstract}
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\section{Introduction and Preliminaries}
We shall show that a bounded domain $D$ is biholomorphic to an open ball $ B^{n+1}$ whenever the boundary $bD$ is locally biholomorphic to the boundary of an open ball $B^{n+1}.$
\begin{theorem} \label{first}Let $D$ be a simply connected bounded domain in $\Bbb{C}^{n+1}$ with differentiable spherical boundary $bD.$ Suppose that there is a biholomorphic mapping \begin{equation*} \phi \in H\left( U\cap D\right) \cap C^{1}\left( U\cap \overline{D}\right) \end{equation*} for a connected open neighborhood $U$ of a point $p\in bD$ satisfying \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Then the mapping $\phi $ is analytically continued to a biholomorphic mapping from $D$ onto $B^{n+1}.$ \end{theorem}
Our result is a new proof of a weaker version of Chern-Ji's mapping theorem \cite{CJ}. The main steps of our proof come as follows: We show that the inverse mapping $\phi ^{-1}$ is analytically continued on the unit ball $ B^{n+1}$ to be a locally biholomorphic mapping \begin{equation*} \varphi :B^{n+1}\rightarrow D. \end{equation*} We show that the mapping $\varphi $ is a proper holomorphic mapping onto a universal covering Riemann domain over $D.$ Thus the mapping $\varphi $ is a biholomorphic mapping whenever $D$ is simply connected.
We shall study on a proper holomorphic mapping $\phi $ between strongly pseudoconvex bounded domains $D,D^{\prime }$ with real analytic boundaries $ bD,bD^{\prime }.$
\begin{theorem} \label{second}Let $D,D^{\prime }$ be strongly pseudoconvex bounded domains in $\Bbb{C}^{n+1}$ with real analytic boundaries $bD,bD^{\prime }$ and $\phi :D\rightarrow D^{\prime }$ be a proper holomorphic mapping. Then the mapping $\phi $ is locally biholomorphic. If $D=D^{\prime }$, then the mapping $\phi $ is a biholomorphic self mapping. \end{theorem}
Our result is a new proof of a weaker version of Pinchuk's mapping theorem \cite{Pi}. The main steps of our proof come as follows: We show that the mapping $\phi $ is analytically continued along any path on $bD$ as a locally biholomorphic mapping when $bD$ is nonspherical so that the mapping $ \phi :D\rightarrow D^{\prime }$ are locally biholomorphic. From the study of Theorem \ref{first}, we show that the same is true when $bD$ is spherical so that the mapping $\phi :D\rightarrow D^{\prime }$ are locally biholomorphic. For the case of $D=D^{\prime },$ we show that the boundary $bD$ is necessarily spherical whenever the claim is not true. Then we show that there is a sequence of automorphisms $\phi _{j}\in Aut\left( D\right) $ and a sequence of points $p_{j}$ on a compact subset $K\subset \subset D$ such that \begin{equation*} \phi _{j}\left( p_{j}\right) \rightarrow bD \end{equation*} whenever the boundary $bD$ is spherical and the claim is not true. We apply Wong-Rosay Theorem so that the domain $D$ is biholomorphic to an open ball $ B^{n+1}.$ Then we obtain a contradiction that the mapping $\phi $ induces a nonautomorphic proper self mapping of an open ball $B^{n+1}$ whenever the boundary $bD$ is spherical and the claim is not true.
We remark that Theorem \ref{first} is a weaker version of the following theorem:
\begin{theorem}[cf. Chern-Ji [CJ]] Let $D$ be a simply connected bounded domain in $\Bbb{C}^{n+1}$ with continuous spherical boundary $bD.$ Suppose that there is a biholomorphic mapping \begin{equation*} \phi \in H\left( U\cap D\right) \cap C\left( U\cap \overline{D}\right) \end{equation*} for a connected open neighborhood $U$ of a point $p\in bD$ satisfying \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Then the mapping $\phi $ is analytically continued to a biholomorphic mapping from $D$ onto $B^{n+1}.$ \end{theorem}
We remark that Theorem \ref{second} is a weaker version of the following theorem:
\begin{theorem}[cf. Pinchuk [Pi]] Let $D,D^{\prime }$ be strongly pseudoconvex bounded domains in $\Bbb{C} ^{n+1}$ with the boundaries $bD,bD^{\prime }$ of class $C^{2}$ and $\phi :D\rightarrow D^{\prime }$ be a proper holomorphic mapping. Then the mapping $\phi $ is locally biholomorphic. If $D=D^{\prime }$, then the mapping $\phi $ is a biholomorphic self mapping. \end{theorem}
We presented parts of this article in the spring meeting of Korean Mathematical Society in 2000. This article is a preliminary version. Any comment shall be greatly appreciated. Send your comments to [email protected].
\subsection{Canonical normalizing mapping}
Let $M$ be a nondegenerate analytic real hypersurface in $\Bbb{C}^{n+1}$. For each point $p\in M,$ there is a complex tangent hyperplane $H_{p}\subset T_{p}M$ so that there is a unit tangent vector $v_{p}\in T_{p}M$ perpendicular to the complex tangent hyperplane $H_{p}$ with respect to the usual riemannian metric in $\Bbb{C}^{n+1}=\Bbb{R}^{2n+2}.$ Then we can take a unique distinguished chain $\gamma _{p}$ tangential to the direction $ v_{p} $ and passing through the complex tangent hyperplane $H_{p}$ at the point $p$ on $M$(cf. \cite{Pa3}). Further, there is a distinguished normal parametrization on the chain $\gamma _{p}$ having the same values up to order $2$ of the straight real line to the direction $v_{p}$ with the usual euclidean parametrization.. Therefore, we can take a distinguished normalizing mapping $\mu _{p}$ to Moser normal form \begin{equation*} \mu _{p}:M\rightarrow \mu _{p}\left( M\right) \end{equation*} sending the germ $M$ at the point $p$ to a normal form such that $\mu _{p}(p) $ is the origin and $\mu _{p}\left( \gamma _{p}\right) $ is on the straightened chain of the normal form.
We take a local orientation near the point $p\in M$ so that the tangent vector $v_{p}$ extends to a smooth unit vector field $v$ on $M\cap U$ for an open neighborhood $U$ of the point $p$ such that $v_{q}\in T_{q}M$ is a unit tangent vector perpendicular to the complex tangent hyperplane $H_{q}$ for each $q\in M\cap U.$ Then we obtain a family of the distinguished normalizing mapping $\mu _{q}$ for $q\in M\cap U$ : \begin{equation*} \mu _{q}:M\rightarrow \mu _{q}\left( M\right) \end{equation*} associated the unit tangent vector $v_{q}\in T_{q}M\backslash H_{q}.$ The distinguished normalizing mapping $\mu _{q}$ shall be called the canonical normalizing mapping associated Moser normal form.
\begin{lemma} \label{canonicalmapping}Let $p_{j}\in M$ be a sequence of points converging to a point $p\in M.$ Then there is a positive real number $\delta >0$ such that
\begin{enumerate} \item the mapping $\mu _{p_{j}}$ and its inverse $\mu _{p_{j}}^{-1}$ are analytically continued respectively on \begin{equation*} B\left( p_{j};\delta \right) \quad \text{and}\quad B\left( 0;\delta \right) \end{equation*} as a biholomorphic mapping,
\item the real hypersurface $\mu _{p_{j}}\left( M\right) $ is analytically continued on $B\left( 0;\delta \right) $ by its defining equation,
\item the sequence $\mu _{p_{j}}^{-1}$ uniformly converges to $\mu _{p}^{-1} $ on $B\left( 0;\delta \right) $ as a biholomorphic mapping,
\item the sequence $\mu _{p_{j}}\left( M\right) $ uniformly converges on $ B\left( 0;\delta \right) $ to the real hypersurface $\mu _{p}\left( M\right) .$ \end{enumerate} \end{lemma}
Let $H$ be the local automorphism group at the origin of the real hyperquadric \begin{equation*} v=\langle z,z\rangle . \end{equation*} The isotropy subgroup $Aut_{p}\left( M\right) $ is naturally identified to the isotropy subgroup $Aut_{0}\left( \mu _{p}\left( M\right) \right) $ by the following relation: \begin{equation*} \mu _{p}\circ \phi \circ \mu _{p}^{-1}\in Aut_{0}\left( \mu _{p}\left( M\right) \right) \quad \text{for}\quad \phi \in Aut_{p}\left( M\right) . \end{equation*} Note that every biholomorphic mapping $\varphi $ between real hypersurfaces in normal form is faithfully represented by a natural group action of the isotropy subgroup $H$(cf. \cite{Pa3}) such that \begin{equation*} \varphi =N_{e}\quad \text{for}\quad e\in H. \end{equation*} Because $\mu _{p}\left( M\right) $ is in normal form, there is a natural identification of a local automorphism $\phi \in Aut_{p}\left( M\right) $ to an element \begin{equation*} \left( U_{\phi },a_{\phi },\rho _{\phi },r_{\phi }\right) \in H, \end{equation*} where $U_{\phi },a_{\phi },\rho _{\phi },r_{\phi }$ are the normalizing parameters(cf. \cite{Pa3}) of the mapping \begin{equation*} \mu _{p}\circ \phi \circ \mu _{p}^{-1}\in Aut_{0}\left( \mu _{p}\left( M\right) \right) . \end{equation*}
\begin{lemma} If the isotropy subgroup $Aut_{p}\left( M\right) $ is compact, then $ Aut_{p}\left( M\right) $ is isomorphic to the subgroup \begin{equation*} \left\{ \left( U_{\phi },a_{\phi },\rho _{\phi },r_{\phi }\right) \in H:\phi \in Aut_{p}\left( M\right) \right\} \end{equation*} as a Lie group. \end{lemma}
\subsection{Preliminary Lemmas}
We have lemmas on the automorphism of $B^{n+1}.$
\begin{lemma} Let $p,q$ be two distinct points on $bB^{n+1}$ and $\phi \in Aut\left( bB^{n+1}\right) $ be a local automorphism of $bB^{n+1}$ such that \begin{equation*} \phi \left( p\right) \neq q. \end{equation*} Then there is a unique decomposition \begin{equation*} \phi =\psi \circ \varphi \end{equation*} where \begin{equation*} \varphi \in Aut_{p}\left( bB^{n+1}\right) ,\quad \psi \in Aut_{q}\left( bB^{n+1}\right) \end{equation*} and the local automorphism $\psi $ acts trivially on the complex tangent hyperplane of $bB^{n+1}$ at the fixed point $q.$ \end{lemma}
\proof
Note that the isotropy subgroup $Aut_{q}\left( bB^{n+1}\right) $ acts on $ bB^{n+1}\backslash q$ transitively. Further, there is a unique element $\psi \in Aut_{q}\left( bB^{n+1}\right) $ for each point displacement on $ bB^{n+1}\backslash q$ by requiring the element $\psi $ acts trivially on the complex tangent hyperplane of $bB^{n+1}$ at the point $q$(cf. \cite{Pa1})$.$
Let's put $p^{\prime }=\phi \left( p\right) .$ Since $p^{\prime }\neq q,$ we take a unique automorphism $\psi \in Aut_{q}\left( bB^{n+1}\right) $ such that \begin{equation*} \psi \left( p^{\prime }\right) =p. \end{equation*} Then $\varphi \equiv \psi \circ \phi \in Aut_{p}\left( bB^{n+1}\right) $ so that \begin{equation*} \phi =\psi ^{-1}\circ \varphi . \end{equation*} This completes the proof.
\endproof
\begin{lemma} \label{decomposition}Let $p,q$ be two distinct points on $bB^{n+1}$ and $ \phi \in Aut\left( bB^{n+1}\right) $ be a local automorphism of $bB^{n+1}$ such that \begin{equation*} p^{\prime }\equiv \phi \left( p\right) \neq q. \end{equation*} Then there is a unique decomposition \begin{equation*} \phi =\varphi \circ \psi \end{equation*} where \begin{equation*} \varphi \in Aut_{p^{\prime }}\left( bB^{n+1}\right) ,\quad \psi \in Aut_{q}\left( bB^{n+1}\right) \end{equation*} and the local automorphism $\psi $ acts trivially on the complex tangent hyperplane of $bB^{n+1}$ at the fixed point $q.$ \end{lemma}
\proof
By Lemma \ref{decomposition}, there is a decomposition \begin{equation*} \phi ^{-1}=\psi \circ \varphi \end{equation*} where \begin{equation*} \varphi \in Aut_{p^{\prime }}\left( bB^{n+1}\right) ,\quad \psi \in Aut_{q}\left( bB^{n+1}\right) \end{equation*} and the local automorphism $\psi $ acts trivially on the complex tangent hyperplane of $bB^{n+1}$ at the fixed point $q.$ Hence we obtain \begin{equation*} \phi =\varphi ^{-1}\circ \psi ^{-1} \end{equation*} where \begin{equation*} \varphi ^{-1}\in Aut_{p^{\prime }}\left( bB^{n+1}\right) ,\quad \psi ^{-1}\in Aut_{q}\left( bB^{n+1}\right) . \end{equation*} This completes the proof.
\endproof
\begin{lemma} \label{either}Let $\phi _{j}$ be a sequence of automorphisms of $B^{n+1}$. Suppose that the sequence $\phi _{j}$ converges to a holomorphic mapping $ \lambda $ uniformly on every compact subset of $B^{n+1}.$ Then the mapping $ \lambda $ is either a constant mapping or an automorphism of $B^{n+1}.$ \end{lemma}
\proof
Note that the mapping $\lambda $ satisfies \begin{equation} \lambda \left( B^{n+1}\right) \subset \overline{B^{n+1}} \label{boundarypoint} \end{equation} and a complex line is mapped to a complex line under the biholomorphic automorphism of the unit ball $B^{n+1}.$
Suppose that there is a complex line $\pi $ such that \begin{equation*} \pi \cap B^{n+1}\neq \emptyset \quad \text{and}\quad \lambda \left( \pi \cap B^{n+1}\right) =\emptyset . \end{equation*} Note that $bB^{n+1}$ is strongly pseudoconvex so that, by the condition \ref {boundarypoint}, there is a point $q\in bB^{n+1}$ satisfying \begin{equation*} q=\lambda \left( \pi \right) \cap \overline{B^{n+1}}. \end{equation*} Then we obtain \begin{equation*}
\left. \lambda \right| _{\pi \cap B^{n+1}}=q. \end{equation*} Let $x$ be an interior point of $\pi \cap B^{n+1}$ and $p^{\prime }$ be an arbitrary point of $B^{n+1}$ so that we take a complex line $\pi ^{\prime }$ passes through $x$ and $p^{\prime }.$ Since $B^{n+1}$ is strongly pseudoconvex, the maximum modulus theorem of one complex variable yields \begin{equation*}
\left. \lambda \right| _{\pi ^{\prime }\cap B^{n+1}}=q \end{equation*} so that the mapping $\lambda $ is a constant mapping.
Suppose that the mapping $\lambda $ is not a constant mapping. Then, for a complex line $\pi $ satisfying \begin{equation*} \pi \cap B^{n+1}\neq \emptyset , \end{equation*} there is a real number $\varepsilon >0$ such that \begin{equation}
\left| \phi _{j}\left( \pi \cap B^{n+1}\right) \right| \geq \varepsilon \label{finitesize} \end{equation}
where $\left| \phi _{j}\left( \pi \cap B^{n+1}\right) \right| $ is the area of the analytic disk \begin{equation*} \phi _{j}\left( \pi \cap B^{n+1}\right) . \end{equation*} We take a point $p\in \pi \cap bB^{n+1}$ so that \begin{equation} \phi _{j}\left( p\right) \rightarrow p^{\prime }\in bB^{n+1}. \label{approaching} \end{equation} Then we take a point $p^{\prime \prime }\in bB^{n+1}$ such that \begin{equation*} p^{\prime \prime }\notin \overline{\left\{ \phi _{j}\left( p\right) :j\in \Bbb{N}^{+}\right\} }, \end{equation*} if necessary, passing to a subsequence. We have the following decomposition \begin{equation*} \phi _{j}=\varphi _{j}\circ \psi _{j} \end{equation*} where \begin{equation*} \varphi _{j}\in Aut_{p^{\prime }}\left( bB^{n+1}\right) \quad \text{and} \quad \psi _{j}\in Aut_{p^{\prime \prime }}\left( bB^{n+1}\right) \end{equation*} where the automorphisms $\psi _{j}$ act trivially on the complex tangent hyperplane at the fixed point $p^{\prime \prime }.$ Then we obtain \begin{equation*} U_{\psi _{j}}=id_{n\times n},\quad \rho _{\psi _{j}}=1. \end{equation*} By the condition \ref{approaching}, there is a real number $e>0$ such that \begin{equation*}
\left| a_{\psi _{j}}\right| \leq e,\quad \left| r_{\psi _{j}}\right| \leq e. \end{equation*} By the condition \ref{finitesize}, there is a real number $e>0,$ if necessary, increasing $e,$ such that \begin{equation*}
\left| a_{\varphi _{j}}\right| \leq e. \end{equation*} Since the mapping $\lambda $ is not a constant mapping, there is a real number $e>0,$ if necessary, increasing $e,$ such that \begin{equation*}
e^{-1}\leq \left| \rho _{\varphi _{j}}\right| \leq e,\quad \left| r_{\varphi _{j}}\right| \leq e. \end{equation*} Since $bB^{n+1}$ is strongly pseudoconvex, we have \begin{equation*}
\left| U_{\varphi _{j}}\right| =1. \end{equation*} Then the Jacobian determinant $\det \varphi _{j}^{\prime }$ is uniformly bounded from the zero on an open neighborhood of the point $p\in bB^{n+1}.$ By Hurwitz theorem, the mapping $\lambda $ is locally biholomorphic and, further, the mapping $\lambda $ is one-to-one. Hence the mapping $\lambda $ is an automorphism of the unit ball $B^{n+1}.$ This completes the proof.
\endproof
We have lemmas on the chain of $bB^{n+1}.$
\begin{lemma} Let $\gamma :[0,1]\rightarrow bB^{n+1}$ be a chain-segment on $bB^{n+1}$. Then there is a complex line $\pi $ such that \begin{equation*} \gamma [0,1]\subset \pi \cap bB^{n+1}. \end{equation*} \end{lemma}
\proof
We take a point $p\in \gamma [0,1].$ Then the chain-segment $\mu _{p}\circ \gamma [0,1]$ on $\mu _{p}\left( bB^{n+1}\right) $ is on a complex line $\pi ^{\prime }$(cf. \cite{Pa3}). Since the canonical normalizing mapping $\mu _{p}$ of the sphere $bB^{n+1}$ is a fractional linear mapping(cf. \cite{Pa3} ), we take \begin{equation*} \pi =\mu _{p}^{-1}\left( \pi ^{\prime }\right) \end{equation*} so that \begin{equation*} \gamma [0,1]\subset \pi \cap bB^{n+1}. \end{equation*} This completes the proof.
\endproof
\begin{lemma} Let $\pi $ be a complex line such that \begin{equation*} \pi \cap B^{n+1}\neq \emptyset . \end{equation*} Then the circle $\pi \cap bB^{n+1}$ is a chain on $bB^{n+1}.$ \end{lemma}
\proof
We take a point $p\in \pi \cap bB^{n+1}.$ Then we obtain \begin{equation*} \mu _{p}\left( \pi \right) \cap \mu _{p}\left( bB^{n+1}\right) =\mu _{p}\left( \pi \cap bB^{n+1}\right) \neq \emptyset . \end{equation*} Since the canonical normalizing mapping $\mu _{p}$ of the sphere $bB^{n+1}$ is a fractional linear mapping(cf. \cite{Pa3}), $\mu _{p}\left( \pi \right) $ is a complex line so that $\mu _{p}\left( \pi \right) \cap \mu _{p}\left( bB^{n+1}\right) $ is a chain(cf. \cite{Pa3}). Thus the circle \begin{equation*} \pi \cap bB^{n+1}=\mu _{p}^{-1}\left( \mu _{p}\left( \pi \right) \cap \mu _{p}\left( bB^{n+1}\right) \right) \end{equation*} is a chain as well. This completes the proof.
\endproof
\begin{lemma} Let $\gamma $ be a chain passing through a point $p\in bB^{n+1}$ and $\delta _{\gamma }$ be an analytic disk such that \begin{equation*} \gamma =\pi \cap bB^{n+1}\quad \text{and}\quad \delta _{\gamma }=\pi \cap B^{n+1} \end{equation*} where $\pi $ is a complex line. Let $\theta _{\gamma }$ be the angle between the tangent vector of $\gamma $ at the point $p$ and a unit vector $v_{p}$ perpendicular to the complex tangent hyperplane at the point $p\in bB^{n+1}$
and $\left| \delta _{\gamma }\right| $ be the area of the analytic disk $ \delta _{\gamma }$. Then \begin{equation*}
\left| a_{\gamma }\right| \equiv \left| \tan \theta _{\gamma }\right| \rightarrow \infty \end{equation*} if and only if \begin{equation*}
\left| \delta _{\gamma }\right| \rightarrow 0. \end{equation*} \end{lemma}
\proof
We easily see that \begin{equation*} \theta _{\gamma }\rightarrow \frac{\pi }{2} \end{equation*} if and only if \begin{equation*}
\left| a_{\gamma }\right| \equiv \left| \tan \theta _{\gamma }\right| \rightarrow \infty . \end{equation*} Note that the complex line $\pi $ would be on the complex tangent hyperplane of $bB^{n+1}$ at the point $p$ if \begin{equation*} \theta _{\gamma }=\frac{\pi }{2}. \end{equation*} Since $bB^{n+1}$ is strongly pseudoconvex and $B^{n+1}$ is strongly convex, the complex tangent hyperplane of $bB^{n+1}$ at the point $p$ has no intersection to $B^{n+1}.$ Thus we easily see \begin{equation*} \theta _{\gamma }\rightarrow \frac{\pi }{2} \end{equation*} if and only if \begin{equation*}
\left| \delta _{\gamma }\right| \rightarrow 0. \end{equation*} This completes the proof.
\endproof
We may require the following well-known results in this article(cf. \cite{Kr} , \cite{Ra}, \cite{Bo}).
\begin{lemma}[Lewy, Pinchuk] \label{Lewy-Pinchuk}Let $D,D^{\prime }$ be domains with strongly pseudoconvex real analytic boundaries $bD,bD^{\prime }$ and $U$ be a connected open neighborhood of a point $p\in bD.$ Suppose that there is a holomorphic mapping $\phi $ on $U\cap D$ such that \begin{equation*} \phi \in H\left( U\cap D\right) \cap C^{1}\left( U\cap \overline{D}\right) ,\quad \phi \left( U\cap bD\right) \subset bD^{\prime } \end{equation*} and the induced mapping $\phi :U\cap bD\rightarrow bD^{\prime }$ is CR diffeomorphic. Then the mapping $\phi $ is analytically continued on $U,$ if necessary, shrinking $U.$ \end{lemma}
\begin{lemma}[Lewy] \label{localhull}Let $D$ be a domain with a strongly pseudoconvex boundary $ bD$ and $U$ be an open connected neighborhood of a point $p\in bD.$ Suppose that there is a holomorphic mapping $\phi $ on $U\cap bD.$ Then there is an open neighborhood $V$ of the point $p$ such that the mapping $\phi $ is analytically continued onto \begin{equation*} V\cap \overline{D}. \end{equation*} \end{lemma}
\begin{lemma}[Wong, Rosay] \label{Wong-Rosay}Let $D$ be a strongly pseudoconvex bounded domain. Suppose that there is a compact set $K\subset \subset D$ and a sequence $p_{j}\in K$ and automorphisms $\phi _{j}\in Aut\left( D\right) $ such that \begin{equation*} \phi _{j}\left( p_{j}\right) \rightarrow bD. \end{equation*} Then the domain $D$ is biholomorphic to an open unit ball $B^{n+1}.$ \end{lemma}
\begin{lemma}[Bell-Catlin, Diederich-Fornaess] \label{b-regularity}Let $D,D^{\prime }$ be strongly pseudoconvex bounded domains with the boundaries $bD,bD^{\prime }$ of class $C^{\infty }$ and $ \phi :D\rightarrow D^{\prime }$ be a proper holomorphic mapping. Then $\phi :D\rightarrow D^{\prime }$ is a locally biholomorphic mapping and the induced mapping $\phi :bD\rightarrow bD^{\prime }$ is a locally CR diffeomorphism. \end{lemma}
\section{Analytic Continuation on a Sphere}
\subsection{Analytic continuation with finiteness}
Let $D$ be a domain in $\Bbb{C}^{n+1},n\geq 1,$ with real analytic boundary $ bD.$ The boundary $bD$ shall be called spherical if, for each point $p\in bD, $ there is a connected open neighborhood $U$ of the point $p$ and a biholomorphic mapping $\phi $ on $U$ such that \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Note that a domain $D$ with spherical real analytic boundary is necessarily strongly pseudoconvex.
\begin{lemma} \label{sphere}Let $p$ be a point of $bB^{n+1}$ and $U$ be a connected open neighborhood of the point $p$. Suppose that there is a biholomorphic mapping $\phi $ on $U$ such that \begin{equation} \phi \left( U\cap bB^{n+1}\right) \subset bB^{n+1}. \label{sph} \end{equation} Then the mapping $\phi $ is analytically continued on an open neighborhood of the closed ball $\overline{B^{n+1}}.$ \end{lemma}
\proof
Note that each local automorphism $\varphi \in Aut_{p}\left( bB^{n+1}\right) $ for any point $p\in bB^{n+1}$ is necessarily birational such that $\varphi $ is analytically continued on an open neighborhood of $\overline{B^{n+1}}$ as a biholomorphic mapping(cf.\cite{Pa1})$.$
Let's put $q=\phi \left( p\right) .$ We take a point $r\in bB^{n+1}$ such that $r\neq p,$ $r\neq q.$ Then we take an automorphism \begin{equation*} \psi \in Aut_{r}\left( bB^{n+1}\right) \end{equation*} satisfying \begin{equation*} \psi \left( q\right) =p \end{equation*} so that \begin{equation*} \varphi \equiv \psi \circ \phi \in Aut_{p}\left( bB^{n+1}\right) . \end{equation*} Then the mapping $\psi ^{-1}\circ \varphi $ is an automorphism of $B^{n+1}$ and an analytic continuation of the mapping $\phi $ such that \begin{equation*} \phi =\psi ^{-1}\circ \varphi \quad \text{on}\quad U. \end{equation*} This completes the proof.
\endproof
\begin{theorem} \label{anypath}Let $D$ be a domain in $\Bbb{C}^{n+1}$ with spherical real analytic boundary $bD.$ Suppose that there is a connected open neighborhood $ U$ of a point $p\in bD$ and a biholomorphic mapping $\phi $ on $U$ such that $\phi \left( U\cap bD\right) \subset bB^{n+1}.$ Then the mapping $\phi $ is analytically continued along any path on $bD$ as a local biholomorphic mapping. \end{theorem}
\proof
Suppose that the assertion is not true. Then there would be a path $\gamma :[0,1]\rightarrow bD$ such that $\gamma \left( 0\right) \in U\cap bD$ and the germ of a biholomorphic mapping $\phi $ at the point $\gamma \left( 0\right) $ is analytically continued along the subpath $\gamma [0,\tau ]$ with all $\tau <1$ as a local biholomorphic mapping, but not the whole path $ \gamma [0,1].$
Since $bD$ is spherical, by definition, there exist a connected open neighborhood $V$ of the point $\gamma (1)$ and a biholomorphic mapping $ \varphi $ on $V$ such that \begin{equation*} \varphi (V\cap bD)\subset bB^{n+1}. \end{equation*} We take $\lambda \in [0,1)$ such that \begin{equation*} \gamma (\tau )\in V\cap bD\quad \text{for all }\tau \in [\lambda ,1] \end{equation*} and we take a sufficiently small connected open neighborhood $W$ of the point $\gamma \left( \lambda \right) $ such that $\phi $ is analytically continued on $W\subset V$ along the path $\gamma [0,\lambda ]$ and \begin{equation*} \varphi \left( W\right) \cap bB^{n+1}\neq \emptyset . \end{equation*} Then we have \begin{equation*} \psi \left( \varphi \left( W\right) \cap bB^{n+1}\right) \subset bB^{n+1} \end{equation*} where \begin{equation*} \psi =\phi \circ \varphi ^{-1}. \end{equation*} By Lemma \ref{sphere}, $\psi $ is analytically continued on an open neighborhood of $bB^{n+1}$ as a local biholomorphic mapping. By abuse of notation, the mapping $\psi \circ \varphi $ is biholomorphic on the open set $V$ such that \begin{equation*} \psi \circ \varphi =\phi \quad \text{on }W. \end{equation*} Thus the germ $\psi \circ \varphi $ at the point $\gamma (1)$ is an analytic continuation of the germ $\phi $ at the point $\gamma \left( 0\right) $ along the path $\gamma [0,1]$ as a local biholomorphic mapping. This contradiction completes the proof.
\endproof
\begin{lemma} \label{boundary}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical real analytic boundary $bD$ such that the fundamental group $\pi _{1}\left( bD\right) $ is finite. Suppose that there is a connected open neighborhood $U$ of a point $p\in bD$ and a biholomorphic mapping $\phi $ on $U$ such that $\phi \left( U\cap bD\right) \subset bB^{n+1}.$ Then $\phi $ is analytically continued to a biholomorphic mapping from $D$ onto $B^{n+1}.$ \end{lemma}
\proof
By Lemma \ref{anypath}, the mapping $\phi $ is analytically continued along any path on $bD$ as a local biholomorphic mapping. Let $E$ be the path space of $bD$ pointed at the point $p$ mod homotopy so that $E$ is a universal covering of $bD$ with a natural CR structure and a CR projection $\varphi :E\rightarrow bD.$ Then there is a unique CR lift $\psi :E\rightarrow bB^{n+1}$ as the analytic continuation of the biholomorphic mapping $\phi $. Note that $\psi :E\rightarrow bB^{n+1}$ is an open mapping because $\phi $ and $\varphi $ are both locally CR diffeomorphisms.
Since $bD$ is finitely connected, $E$ is necessarily compact so that the mapping $\psi :E\rightarrow bB^{n+1}$ is surjective. Further, the mapping $ \psi :E\rightarrow bB^{n+1}$ is a simple covering map because $\psi $ is locally a CR diffeomorphism and the sphere $bB^{n+1}$ is simply connected. Hence there exists a locally biholomorphic mapping $\lambda :bB^{n+1}\rightarrow bD$ defined by \begin{equation*} \lambda =\varphi \circ \psi ^{-1}\quad \text{on}\quad bB^{n+1}. \end{equation*}
By Hartogs extension theorem, the mapping $\lambda $ uniquely extends to the open ball $B^{n+1}$ as a local biholomorphic mapping and, further, the extension is smooth up to the boundary. Hence $\lambda $ is well defined as a locally biholomorphic mapping on an open neighborhood of $\overline{B^{n+1} }$ by Lemma \ref{Lewy-Pinchuk}.
We obtain a proper mapping $\lambda :B^{n+1}\rightarrow D$ so that $\lambda $ is a globally branched covering and the branched locus of $\lambda $ cannot be bounded by $bD.$ Since $\lambda ^{-1}=\phi $ is locally biholomorphic on $ bD$, $\lambda :B^{n+1}\rightarrow D$ and $\lambda :bB^{n+1}\rightarrow bD$ are finite coverings respectively of $D$ and $bD.$ Since the closed ball $ \overline{B^{n+1}}$ has the fixed point property, the mapping $\lambda :B^{n+1}\rightarrow D$ is globally one-to-one. Otherwise, there would be a nontrivial deck transform of $\overline{B^{n+1}}$ which is continuous on $ \overline{B^{n+1}}$ without a fixed point. Hence the mapping $\lambda ^{-1}:D\rightarrow B^{n+1}$ is biholomorphic with $\lambda ^{-1}=\phi $ on $ bD.$ This completes the proof.
\endproof
\begin{theorem} \label{germs}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with a connected spherical real analytic boundary $bD.$ Suppose that there is a biholomorphic mapping $\phi $ on a connected open neighborhood $U$ of a point $p\in bD$ satisfying \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1} \end{equation*} such that the analytic continuation of $\phi $ on the boundary $bD$ yields finitely many germs at each point on $bD.$ Then $D$ is necessarily simply connected and the mapping $\phi $ is analytically continued to a biholomorphic mapping from $D$ onto $B^{n+1}.$ \end{theorem}
\proof
We claim that there is a finite covering space $E_{1}$ of $bD$ with a natural CR structure and a CR projection $\varphi _{1}:E_{1}\rightarrow bD$ and a local CR diffeomorphism $\psi _{1}:E_{1}\rightarrow bB^{n+1}$ satisfying the relation $\psi _{1}=\phi \circ \varphi _{1}.$ Then the desired result follows from this claim by the same argument in the proof of Lemma \ref{boundary}.
Let $E$ be the path space of $bD$ pointed at the point $p\in bD$ mod homotopy so that $E$ is a universal covering of $bD$ with a natural CR structure and a CR projection $\varphi :E\rightarrow bD.$ Then there is a unique CR lift $\psi :E\rightarrow bB^{n+1}:$ \begin{equation*} \begin{array}{lll} E & & \\ \downarrow \varphi & \overset{\psi }{\searrow } & \\ bD & \overset{\phi }{\longrightarrow } & bB^{n+1} \end{array} \end{equation*} satisfying the relation $\psi =\phi \circ \varphi $. Note that $\psi :E\rightarrow bB^{n+1}$ is an open mapping because $\phi $ and $\varphi $ are both local CR diffeomorphisms. Let $F$ be the image of the mapping $\psi $ such that $F=\psi \left( E\right) .$ Then $F$ is an open subset of $bD.$
Suppose that $bF\neq \emptyset .$ Then we take a point $p\in bF$ and a sequence $p_{j}\in F$ such that $p_{j}\rightarrow p.$ Thus there exists a point $q_{j}\in bD$ and germs of biholomorphic mappings $\phi _{j}$ such that \begin{equation*} \phi _{j}\left( q_{j}\right) \rightarrow p \end{equation*} where $\phi _{j}$ are analytic continuations of the mapping $\phi $ on $bD.$ Since $bD$ is compact, there exist a point $q\in bD$ and a subsequence $ q_{m_{j}}$ of $q_{j}$ such that $q_{m_{j}}\rightarrow q.$ Further, by passing to a subsequence, if necessary, we may assume that \begin{equation*} \phi ^{*}=\phi _{m_{j}}\quad \text{for all }m_{j} \end{equation*} because the analytic continuation of the mapping $\phi $ yields only finitely many germs at the point $q\in bD.$ Hence we obtain \begin{equation*} \phi ^{*}\left( q\right) =\lim_{j\rightarrow \infty }\phi _{m_{j}}\left( q_{m_{j}}\right) =p\in F. \end{equation*} This contradiction implies that $bF=\emptyset ,$ i.e., $\psi \left( E\right) =bB^{n+1}.$
For each point $q\in bB^{n+1},$ there is a subset $X_{q}\subset E$ such that \begin{equation*} X_{q}=\left\{ p\in E:\psi \left( p\right) =q\right\} . \end{equation*} Since $\psi $ is a CR diffeomorphism, $X_{q}$ is necessarily a discrete set on $E.$ Then we define a subset $Y_{q}\subset bD$ such that \begin{equation*} Y_{q}=\left\{ \varphi \left( p\right) \in bD:p\in X_{q}\right\} . \end{equation*} Suppose that $Y_{q}$ has an accumulation point $y\in bD.$ Then there is a sequence of point $p_{j}\in bD$ satisfying \begin{equation*} p_{j}\rightarrow y\quad \text{and}\quad p_{j}\neq y, \end{equation*} and biholomorphic mappings $\phi _{j}$ such that \begin{equation*} \phi _{j}\left( p_{j}\right) =q \end{equation*} where the mapping $\phi $ at the point $p_{j}$ is the analytic continuation of the mapping $\phi .$ Because the analytic continuation of $\phi $ yields only finitely many germs at the point $y,$ we can take a subsequence of $ \phi $ and a biholomorphic mapping $\phi ^{*}$ such that $\phi ^{*}=\phi _{m_{j}}.$ Then we have \begin{equation*} \phi ^{*}\left( y\right) =\lim_{j\rightarrow \infty }\phi _{m_{j}}\left( p_{m_{j}}\right) =q. \end{equation*} Since $\phi ^{*}$ is locally biholomorphic, it is impossible that $\phi ^{*}\left( p_{m_{j}}\right) =q=\phi ^{*}\left( y\right) $ and $ p_{m_{j}}\rightarrow y,p_{m_{j}}\neq y$ at the same time. Thus we find that the set $Y_{q}$ is finite.
Therefore, the analytic continuation of the mapping $\phi $ on $bD$ is mapped to a point of $bB^{n+1}$ only at finitely many points of $bD.$ Since the analytic continuation of the mapping $\phi $ yields finitely many germs at each point on $bD,$ only finitely many germs of the analytic continuation of the mapping $\phi $ on $bD$ are mapped to each point of $bB^{n+1}.$ Then, by the compactness of $bB^{n+1},$ we obtain a finite covering space $E_{1}$ of $bD$ satisfying all conditions in the claim. This completes the proof.
\endproof
\subsection{First Dogginal Lemma}
\begin{lemma}[First Scaling Lemma] \label{scaling}Let $p$ be a point of the boundary $bB^{n+1}$ and $p_{j},$ $ j\in \Bbb{N}^{+},$ be a sequence of points of $bB^{n+1}$ such that $ p_{j}\neq p$ for all $j$ and $p_{j}\rightarrow p$ as $j\rightarrow \infty $ to a direction transversal to the complex tangent hyperplane at the point $ p\in bB^{n+1}.$ Let $\varepsilon _{j}$ be the euclidean distance between the two points $p_{j}$ and $p,$ and $\delta _{j}$ be the analytic disk \begin{equation*} \delta _{j}=\pi _{j}\cap B^{n+1} \end{equation*} where $\pi _{j}$ is the complex line passing through the two points $p_{j}$ and $p.$ Suppose that there is a sequence $p_{j}^{\prime }$ of points of $ bB^{n+1}$ satisfying \begin{equation*} p_{j}^{\prime }\rightarrow p^{\prime }\in bB^{n+1}, \end{equation*} and a sequence of biholomorphic automorphisms $\phi _{j}\in Aut\left( B^{n+1}\right) $ satisfying \begin{equation*} \phi _{j}\left( p_{j}^{\prime }\right) =p_{j} \end{equation*}
such that the sequence $\phi _{j}$ converges to a constant mapping and the area $\left| \phi _{j}^{-1}\left( \delta _{j}\right) \right| $ of the analytic disks \begin{equation*} \phi _{j}^{-1}\left( \delta _{j}\right) \end{equation*} is bounded from the below, i.e., there is a real number $c>0$ satisfying \begin{equation*}
\left| \phi _{j}^{-1}\left( \delta _{j}\right) \right| \geq c. \end{equation*} Then there is a subsequence $\phi _{m_{j}}$ and a sequence of local automorphisms $\sigma _{j}\in Aut_{p_{m_{j}}}\left( B^{n+1}\right) $ such that \begin{equation*} U_{\sigma _{j}}=id_{n\times n},\quad \rho _{\sigma _{j}}=\varepsilon _{m_{j}},\quad a_{\sigma _{j}}=0,\quad r_{\sigma _{j}}=0 \end{equation*} and the composition \begin{equation*} \sigma _{j}^{-1}\circ \phi _{m_{j}}:B^{n+1}\rightarrow B^{n+1} \end{equation*} uniformly converges to an automorphism of the unit ball $B^{n+1}.$ \end{lemma}
\proof
Note that there is a subsequence $\phi _{m_{j}}$ which converges to the point $p\in bB^{n+1}$ uniformly on every compact subset of the unit ball $ B^{n+1}.$ We take a point $p^{\prime }\in bB^{n+1}$ such that $p^{\prime }\neq p$ and \begin{equation*} p^{\prime }\notin \overline{\left\{ p_{m_{j}}:j\in \Bbb{N}^{+}\right\} }, \end{equation*} if necessary, passing to a subsequence.. Then, by Lemma \ref{decomposition}, there is a unique decomposition of the automorphism $\phi _{m_{j}}$ such that \begin{equation*} \phi _{m_{j}}=\varphi _{j}\circ \psi _{j} \end{equation*} where \begin{equation*} \varphi _{j}\in Aut_{p_{m_{j}}}\left( bB^{n+1}\right) ,\quad \psi _{j}\in Aut_{p^{\prime }}\left( bB^{n+1}\right) \end{equation*} and the local automorphism $\psi _{j}$ acts trivially on the complex tangent hyperplane of $bB^{n+1}$ at the fixed point $p^{\prime }.$
Let $U_{\psi _{j}},\rho _{\psi _{j}},a_{\psi _{j}},r_{\psi _{j}}$ be the normalizing parameters of the local automorphism $\psi _{j}.$ Since the local automorphism $\psi _{j}$ acts trivially on the complex tangent hyperplane of $bB^{n+1}$ at the fixed point $p^{\prime },$ we obtain \begin{equation*} U_{\psi _{j}}=id_{n\times n},\quad \rho _{\psi _{j}}=1. \end{equation*} Since the sequence $\phi _{m_{j}}$ uniformly converges to the point $p,$ there is a real number $e>0$ such that \begin{equation}
\left| a_{\psi _{j}}\right| \leq e,\quad \left| r_{\psi _{j}}\right| \leq e. \label{estimate1} \end{equation}
Let $\pi _{j}$ be the complex line passing through the two points $p$ and $ p_{m_{j}}.$ Since the points $p_{j}$ converges to the point $p$ to a direction transversal to the complex tangent hyperplane at the point $p\in bB^{n+1},$ the analytic disks \begin{equation*} \delta _{j}=\pi _{j}\cap B^{n+1} \end{equation*} is uniformly bounded in their area from the below by a positive number.
Let $U_{\varphi _{j}},\rho _{\varphi _{j}},a_{\varphi _{j}},r_{\varphi _{j}}$ be the normalizing parameters of the local automorphism $\varphi _{j}.$ Note that the analytic disk \begin{equation*} \phi _{m_{j}}^{-1}\left( \delta _{j}\right) \end{equation*} is mapped by $\phi _{m_{j}}$ onto the analytic disk $\delta _{j},$ where the areas of the analytic disks in both classes \begin{equation*}
\left| \phi _{m_{j}}^{-1}\left( \delta _{j}\right) \right| \quad \text{and}
\quad \left| \delta _{j}\right| \end{equation*} are bounded from the below. Since the chain $\phi _{m_{j}}^{-1}\left( b\delta _{j}\right) $ is mapped by $\phi _{m_{j}}$ to the chain $b\delta _{j},$ there is a real number $e>0,$ if necessary, increasing $e,$ such that \begin{equation}
\left| a_{\varphi _{j}}\right| \leq e. \label{estimate2} \end{equation} Since the area of the analytic disks $\delta _{j}$ is bounded by a positive number, the point $p$ is attracted to the center of the local automorphism $ Aut_{p_{m_{j}}}\left( bB^{n+1}\right) .$ Then, by passing to a subsequence, if necessary and increasing $e$, there is a real number $e>0$ such that \begin{equation}
\left| r_{\varphi _{j}}\right| \leq e \label{estimate3} \end{equation} and \begin{equation*} \rho _{\varphi _{j}}\rightarrow 0\quad \text{as }j\rightarrow \infty . \end{equation*} Since $bB^{n+1}$ is strongly pseudoconvex, we obtain \begin{equation*}
\left| U_{\varphi _{j}}\right| =1. \end{equation*}
Let $\eta _{j}$ be a local automorphism in $Aut_{p_{m_{j}}}\left( bB^{n+1}\right) $ defined by the normalizing parameters \begin{equation*} U_{\eta _{j}}=id_{n\times n},\quad \rho _{\eta _{j}}=\rho _{\varphi _{j}},\quad a_{\eta _{j}}=0,\quad r_{\eta _{j}}=0. \end{equation*} Then the composition \begin{equation*} \varphi _{j}^{\prime }\equiv \eta _{j}^{-1}\circ \varphi _{j}\in Aut_{p_{m_{j}}}\left( bB^{n+1}\right) \end{equation*} has the same normalizing parameters of the mapping $\varphi _{j}$ except for $\rho _{\varphi _{j}^{\prime }}=1,$ i.e., \begin{equation*} U_{\varphi _{j}^{\prime }}=U_{\varphi _{j}},\quad \rho _{\varphi _{j}^{\prime }}=1,\quad a_{\varphi _{j}^{\prime }}=a_{\varphi _{j}},\quad r_{\varphi _{j}^{\prime }}=r_{\varphi _{j}}. \end{equation*} Therefore, by passing to a subsequence, if necessary, the sequence \begin{equation*} \tau _{j}\equiv \eta _{j}^{-1}\circ \phi _{m_{j}}:B^{n+1}\rightarrow B^{n+1} \end{equation*} converges by Hurwitz theorem to a locally biholomorphic mapping. Since $\tau _{j}$ are automorphism of $B^{n+1},$ the sequence $\tau _{j}$ converges to an automorphism of $B^{n+1}.$
Let $q\in \overline{B^{n+1}}$ be the limit point of the sequence \begin{equation*} \tau _{j}\left( p_{m_{j}}^{\prime }\right) \rightarrow q\in \overline{B^{n+1} }. \end{equation*} We set \begin{equation*} \lambda _{j}=\frac{\varepsilon _{m_{j}}}{\rho _{\varphi _{j}}}. \end{equation*} The sequence $\lambda _{j}$ converges to the distance between the two distinct points $q$ and $p\in bB^{n+1}$ so that there is a real number $e>0$ such that \begin{equation}
e^{-1}\leq \left| \lambda _{j}\right| \leq e, \label{estimate4} \end{equation} if necessary, increasing $e.$
Let $\sigma _{j}$ be a local automorphism in $Aut_{p_{m_{j}}}\left( bB^{n+1}\right) $ defined by the normalizing parameters \begin{equation*} U_{\sigma _{j}}=id_{n\times n},\quad \rho _{\sigma _{j}}=\varepsilon _{m_{j}},\quad a_{\sigma _{j}}=0,\quad r_{\sigma _{j}}=0. \end{equation*} Then the composition \begin{equation*} \varphi _{j}^{\prime \prime }\equiv \sigma _{j}^{-1}\circ \varphi _{j}\in Aut_{p_{m_{j}}}\left( bB^{n+1}\right) \end{equation*} has the same normalizing parameters of the mapping $\varphi _{j}$ except for $\rho _{\varphi _{j}^{\prime \prime }}=\lambda _{j}^{-1},$ i.e., \begin{equation*} U_{\varphi _{j}^{\prime \prime }}=U_{\varphi _{j}},\quad \rho _{\varphi _{j}^{\prime \prime }}=\lambda _{j}^{-1},\quad a_{\varphi _{j}^{\prime \prime }}=a_{\varphi _{j}},\quad r_{\varphi _{j}^{\prime \prime }}=r_{\varphi _{j}}. \end{equation*} Therefore, by the estimates \ref{estimate1}, \ref{estimate2}, \ref{estimate3} , \ref{estimate4}, the sequence \begin{equation*} \tau _{j}^{\prime }\equiv \sigma _{j}^{-1}\circ \phi _{m_{j}}:B^{n+1}\rightarrow B^{n+1} \end{equation*} uniformly converges to an automorphism of $B^{n+1}.$ This completes the proof.
\endproof
\begin{lemma} \label{alongchain}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical real analytic boundary $bD.$ Suppose that there is a connected open neighborhood $U$ of a point $p\in bD$ and a biholomorphic mapping $\phi $ on $U$ such that $\phi \left( U\cap bD\right) \subset bB^{n+1}$ and the inverse mapping $\phi ^{-1}$ on $bB^{n+1}$ is analytically continued along every chain of $bB^{n+1}.$ Then the mapping $\phi $ is analytically continued to a biholomorphic mapping from $D$ onto $B^{n+1}.$ \end{lemma}
\proof
The chain on $bB^{n+1}$ is characterized to be the intersection of a complex line on $bB^{n+1}$. Thus the chains on $bB^{n+1}$ form a continuous family so that the analytic continuity of the inverse mapping $\phi ^{-1}$ along every chain on $bB^{n+1}$ is equivalent to the analytic continuity along any path on $bB^{n+1}$.
Note that the inverse mapping $\phi ^{-1}$ is analytically continued along any path on $bB^{n+1}$ and, by Hartogs extension theorem, the branching locus of a proper mapping cannot be bounded by $bD.$ Since $bB^{n+1}$ is simply connected, by the monodromy theorem, the mapping $\phi ^{-1}$ is analytically continued to, by abuse of notation, a locally biholomorphic proper mapping $\phi ^{-1}:B^{n+1}\rightarrow D$ such that $\phi ^{-1}\left( bB^{n+1}\right) =bD.$ By the fixed point property of the closed ball $ \overline{B^{n+1}},$ the proper mapping $\phi ^{-1}:B^{n+1}\rightarrow D$ is globally one-to-one so that the mapping $\phi ^{-1}:B^{n+1}\rightarrow D$ is biholomorphic. This completes the proof.
\endproof
\begin{lemma} \label{circle}Let $D$ be a domain in $\Bbb{C}^{n+1}$ with spherical real analytic boundary $bD$ and $U$ be a connected open neighborhood of a point $
p\in bD.$ For a chain $\gamma $ on $bD$ passing through the point $p$ and tangential to the direction with an angle $\theta _{\gamma }$ with respect to a unit vector $v_{p}$ at the point $p$ perpendicular to the complex tangent hyperplane, we denote $\left| a_{\gamma }\right| \equiv \left| \tan
\theta _{\gamma }\right| $. Then there is a real number $e>0$ such that every chain $\gamma $ passing through the point $p$ with $\left| a_{\gamma
}\right| \geq e$ is the boundary of a nonsingular analytic disk $\delta _{\gamma }\subset U\cap D$ such that \begin{equation*} \gamma =\overline{\delta _{\gamma }}\cap bD. \end{equation*} \end{lemma}
\proof
We take a biholomorphic mapping $\phi $ on the open set $U,$ if necessary, shrinking $U$ such that \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Every chain $\lambda $ on $bB^{n+1}$ is an intersection with a complex line $ \pi _{\lambda }$ such that \begin{equation*} \lambda =\pi _{\lambda }\cap bB^{n+1}. \end{equation*} For a sufficiently large real number $e>0,$ each chain $\gamma $ with $
\left| a_{\gamma }\right| \geq e$ is obtain by the relation \begin{equation*} \gamma =\phi ^{-1}\left( \lambda \right) \end{equation*} where the chain $\lambda $ on $bB^{n+1}$ satisfies the condition \begin{equation*} \pi _{\lambda }\cap B^{n+1}\subset \phi \left( U\cap D\right) . \end{equation*} Then we take \begin{equation*} \delta _{\gamma }=\phi ^{-1}\left( \pi _{\lambda }\cap B^{n+1}\right) . \end{equation*} This completes the proof.
\endproof
\begin{lemma}[First Dogginal Lemma] \label{dogginal}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical real analytic boundary $bD$ and $\phi $ be a biholomorphic mapping on a connected open neighborhood $U$ of a point $p\in bB^{n+1}$ satisfying \begin{equation*} \phi \left( U\cap bB^{n+1}\right) \subset bD. \end{equation*} Suppose that there is a chain-segment $\gamma :[0,1]\rightarrow bB^{n+1}$ such that $\gamma \left( 0\right) \in U\cap bB^{n+1}$ and the mapping $\phi $ is analytically continued along the subpath $\gamma [0,\tau ]$ for all $\tau <1,$ but not the whole path $\gamma [0,1]$ as a local biholomorphic mapping. Let $\pi $ be the complex line containing the chain-segment $\gamma [0,1].$ Then there is an open neighborhood $V$ along the path $\gamma [0,1]$ such that
\begin{enumerate} \item $\gamma [0,\tau ]\subset V\quad $for all $\tau <1,$
\item $bV\cap \pi \cap B\left( \gamma \left( 1\right) ;\delta \right) $ is an angle for a sufficient small $\delta >0,$ which contains the chain-segment $\gamma [0,1],$
\item $bV\cap bB^{n+1}\cap B\left( \gamma \left( 1\right) ;\delta \right) $ is slanted paraboloid for a sufficiently small $\delta >0,$ which smoothly touches the complex tangent hyperplane at the point $\gamma \left( 1\right) , $
\item the mapping $\phi $ is analytically continued on $V$ as a local biholomorphic mapping. \end{enumerate} \end{lemma}
\proof
By the analytic continuation of the mapping $\phi $ along the subpath $ \gamma [0,\tau ]$ for all $\tau <1$, there is a path $\phi \circ \gamma :[0,1)\rightarrow bD.$ Then we consider the following sequences \begin{eqnarray*} p_{j} &=&\gamma \left( 1-\frac{1}{j}\right) ,\quad \text{for }j\in \Bbb{N} ^{+}, \\ p_{j}^{\prime } &=&\phi \circ \gamma \left( 1-\frac{1}{j}\right) ,\quad \text{for }j\in \Bbb{N}^{+}. \end{eqnarray*} Since $bD$ is compact, there is a subsequence $p_{m_{j}}^{\prime }$ and a point $p^{\prime }\in bD$ such that \begin{equation*} p_{m_{j}}^{\prime }\rightarrow p^{\prime }. \end{equation*} By Theorem \ref{anypath}, the mapping $\phi ^{-1}$ is analytically continued along the path $\phi \circ \gamma [0,1)\subset bD.$
Let $\varphi _{j}$ be the analytic continuation of the mapping $\phi ^{-1}$ at the point $p_{m_{j}}^{\prime }$ along the path $\phi \circ \gamma [0,1)\subset bD.$ By Theorem \ref{anypath}, there is an open neighborhood $W$ of the point $p^{\prime }$ such that $\varphi _{j}$ is locally biholomorphic on an open neighborhood of $W\cap bD.$ By Lemma \ref{localhull}, we may assume that $\varphi _{j}$ is holomorphic on $W\cap D,$ if necessary, shrinking $W.$
Since $bD$ is spherical, there is an open neighborhood $W$ of the point $ p^{\prime },$ if necessary, shrinking $W,$ and a biholomorphic mapping $ \varphi $ on $W$ such that \begin{equation*} \varphi \left( W\cap bD\right) \subset bB^{n+1}. \end{equation*} Then, by Lemma \ref{sphere}, the compositions \begin{equation*} \phi _{j}\equiv \varphi _{j}\circ \varphi ^{-1}:\varphi \left( W\right) \cap bB^{n+1}\subset bB^{n+1} \end{equation*} are analytically continued, by abuse of notation, to automorphisms $\phi _{j} $ of the unit ball $B^{n+1}.$ Without loss of generality, we may assume that the sequence $\phi _{j}$ converges to a holomorphic mapping uniformly on every compact subset of $B^{n+1}.$
We set \begin{equation*} p_{m_{j}}^{\prime \prime }\equiv \varphi \left( p_{m_{j}}^{\prime }\right) \in bB^{n+1}\quad \text{and\quad }p^{\prime \prime }\equiv \varphi \left( p^{\prime }\right) \end{equation*} so that \begin{equation*} p_{m_{j}}^{\prime \prime }\rightarrow p^{\prime \prime }. \end{equation*} Hence the relation \begin{equation*} \phi _{j}\left( p_{m_{j}}^{\prime \prime }\right) =p_{m_{j}} \end{equation*} yields \begin{equation*} \phi _{j}\left( p_{m_{j}}^{\prime \prime }\right) \rightarrow p. \end{equation*}
By Lemma \ref{either}, the mapping $\phi _{j}$ converges to either a constant mapping or an automorphism of $B^{n+1}.$ We claim that the sequence $\phi _{j}$ converges to the point $p$ uniformly on every compact subset of $ B^{n+1}.$ Otherwise, the sequence $\phi _{j}$ converges to an automorphism of $B^{n+1}$ so that the sequence \begin{equation*} \varphi _{j}=\phi _{j}\circ \varphi \end{equation*} converges to a biholomorphic mapping on an open neighborhood $W$ of the point $p^{\prime }.$ Then there is a real number $\delta $ such that the mapping $\varphi _{j}$ and its inverse $\varphi _{j}^{-1}$ are analytically continued respectively on \begin{equation*} B\left( p_{m_{j}}^{\prime };\delta \right) \quad \text{and}\quad B\left( p_{m_{j}};\delta \right) \end{equation*} as a locally biholomorphic mapping. Thus the mapping $\phi =\varphi _{j}^{-1} $ is biholomorphic on $B\left( p_{m_{j}};\delta \right) $ for every point $p_{m_{j}}\rightarrow p.$ This is impossible by the hypothesis on the mapping $\phi .$ Thus, by Lemma \ref{either}, the sequence $\phi _{j}$ converges to the point $p$ on every compact subset of $B^{n+1}.$
Let $\pi _{j}$ be the complex line passing through the two points $p$ and $ p_{m_{j}},$ and $\delta _{j}$ be the analytic disk \begin{equation*} \delta _{j}=\pi _{j}\cap B^{n+1}. \end{equation*} Let $\varepsilon _{j}$ be the euclidean length between the two points $p$
and $p_{m_{j}}.$ Note that the area $\left| \phi _{j}^{-1}\left( \delta _{j}\right) \right| $ of the analytic disk \begin{equation*} \phi _{j}^{-1}\left( \delta _{j}\right) \end{equation*} is bounded from the below. Otherwise, by Lemma \ref{circle}, the mapping $ \phi =\varphi _{j}^{-1}$ is analytically continued over the point $p$ as a locally biholomorphic mapping. Therefore, by First Scaling Lemma, there is a sequence $\sigma _{j}\in Aut_{p_{m_{j}}}\left( bB^{n+1}\right) $ such that \begin{equation*} U_{\sigma _{j}}=id_{n\times n},\quad \rho _{\sigma _{j}}=\varepsilon _{j},\quad a_{\sigma _{j}}=0,\quad r_{\sigma _{j}}=0 \end{equation*} and the sequence \begin{equation*} \psi _{j}\equiv \sigma _{j}^{-1}\circ \phi _{j} \end{equation*} converges to an automorphism of $B^{n+1}.$
Note that there is a real number $\delta >0$ such that the mapping $\varphi $ and its inverse $\varphi ^{-1}$ are biholomorphically continued respectively on \begin{equation*} B\left( p_{m_{j}}^{\prime };\delta \right) \quad \text{and}\quad B\left( p_{m_{j}}^{\prime \prime };\delta \right) , \end{equation*} if necessary, passing to a subsequence. Further, there is a real number $ \delta >0$ such that the mapping $\psi _{j}$ and its inverse $\psi _{j}^{-1}$ are biholomorphically continued respectively to \begin{equation*} B\left( p_{m_{j}}^{\prime };\delta \right) \quad \text{and}\quad B\left( p_{m_{j}};\delta \right) , \end{equation*} if necessary, shrinking $\delta .$ Then the mapping \begin{eqnarray*} \phi &=&\varphi _{j}^{-1} \\ &=&\varphi ^{-1}\circ \psi _{j}^{-1}\circ \sigma _{j}^{-1} \end{eqnarray*} is biholomorphically continued on the open neighborhood \begin{equation*} \sigma _{j}\left( B\left( p_{m_{j}};\delta \right) \right) . \end{equation*}
For a canonical normalizing mapping $\mu _{p_{m_{j}}},$ we obtain \begin{equation*} \sigma _{j}^{\prime }\equiv \mu _{p_{m_{j}}}\circ \sigma _{j}\circ \mu _{p_{m_{j}}}^{-1}:\left\{ \begin{array}{l} z^{*}=\sqrt{\varepsilon _{j}}z \\ w^{*}=\varepsilon _{j}w \end{array} \right. . \end{equation*} Since $p_{m_{j}}\rightarrow p,$ by Lemma \ref{canonicalmapping}, there is a real number $\delta >0$ such that the mapping $\mu _{p_{m_{j}}}$ and its inverse $\mu _{p_{m_{j}}}^{-1}$ are biholomorphically continued respectively to \begin{equation*} B\left( p_{m_{j}};\delta \right) \quad \text{and}\quad B\left( 0;\delta \right) . \end{equation*} Hence the mapping \begin{equation*} \phi \circ \mu _{p_{m_{j}}}^{-1}=\varphi ^{-1}\circ \psi _{j}^{-1}\circ \mu _{p_{m_{j}}}^{-1}\circ \sigma _{j}^{\prime -1} \end{equation*} is biholomorphically continued on \begin{equation*} \sigma _{j}^{\prime }\left( B\left( 0;\delta \right) \right) . \end{equation*} Since $p_{m_{j}}\rightarrow p,$ by Lemma \ref{canonicalmapping}, the canonical normalizing mapping $\mu _{p_{m_{j}}}$ uniformly converges to the mapping $\mu _{p}$ so that the mapping $\phi $ is biholomorphically continued near the point $p_{m_{j}}$ on \begin{equation*} \mu _{p_{m_{j}}}^{-1}\circ \sigma _{j}^{\prime }\left( B\left( 0;\delta \right) \right) . \end{equation*} Therefore, the analytically continued region of the mapping $\phi $ along the chain $\gamma [0,1)\subset bB^{n+1}$ contains an open set along the chain $\gamma [0,1]$ which touches to the point $\gamma \left( 1\right) $ by an edge shape transversal to $bB^{n+1}$ and by a slanted paraboloid shape on $bB^{n+1}.$ This completes the proof.
\endproof
\subsection{Doggaebi variety on a sphere}
Let $\phi $ be a biholomorphic mapping on an open neighborhood $U$ of a point $p\in bD$ satisfying $\phi \left( U\cap bD\right) \subset bB^{n+1}.$ Let $L\subset bB^{n+1}$ be the singular locus of the analytic continuation of the mapping $\phi ^{-1}$, which shall be called the Doggaebi variety associated to the mapping $\phi .$
\begin{lemma}[Second Scaling Lemma] Let $p$ be a point of the boundary $bB^{n+1}$ and $p_{j},$ $j\in \Bbb{N} ^{+}, $ be a sequence of points of $bB^{n+1}$ such that $p_{j}\neq p$ for all $j$ and $p_{j}\rightarrow p$ as $j\rightarrow \infty $ to a direction tangential to the complex tangent hyperplane at the point $p\in bB^{n+1}.$ Let $\varepsilon _{j}$ be the euclidean distance between the two points $ p_{j}$ and $p,$ and $\delta _{j}$ be the analytic disk \begin{equation*} \delta _{j}=\pi _{j}\cap B^{n+1} \end{equation*} where $\pi _{j}$ is the complex line passing through the two points $p_{j}$ and $p.$ Suppose that there is a sequence $p_{j}^{\prime }$ of points of $ bB^{n+1}$ satisfying \begin{equation*} p_{j}^{\prime }\rightarrow p^{\prime }\in bB^{n+1}, \end{equation*} and a sequence of biholomorphic automorphisms $\phi _{j}\in Aut\left( B^{n+1}\right) $ satisfying \begin{equation*} \phi _{j}\left( p_{j}^{\prime }\right) =p_{j} \end{equation*}
such that the area $\left| \phi _{j}^{-1}\left( \delta _{j}\right) \right| $ of the analytic disks \begin{equation*} \phi _{j}^{-1}\left( \delta _{j}\right) \end{equation*} is bounded from the below, i.e., there is a real number $c>0$ satisfying \begin{equation*}
\left| \phi _{j}^{-1}\left( \delta _{j}\right) \right| \geq c. \end{equation*} Then there is a subsequence $\phi _{m_{j}}$ and a sequence of local automorphisms $\sigma _{j}\in Aut_{p_{m_{j}}}\left( bB^{n+1}\right) $ such that \begin{equation*} U_{\sigma _{j}}=id_{n\times n},\quad \rho _{\sigma _{j}}=\varepsilon _{m_{j}}^{2},\quad a_{\sigma _{j}}=0,\quad r_{\sigma _{j}}=0 \end{equation*} and the composition \begin{equation*} \sigma _{j}^{-1}\circ \phi _{m_{j}}:B^{n+1}\rightarrow B^{n+1} \end{equation*} uniformly converges to an automorphism of the unit ball $B^{n+1}.$ \end{lemma}
\proof
Note that there is a subsequence $\phi _{m_{j}}$ which converges to the point $p\in bB^{n+1}$ uniformly on an open neighborhood of the closed ball $ \overline{B^{n+1}}.$ We take a point $p^{\prime \prime }\in bB^{n+1}$ such that $p^{\prime \prime }\neq p$ and \begin{equation*} p^{\prime \prime }\notin \overline{\left\{ p_{m_{j}}:j\in \Bbb{N} ^{+}\right\} }, \end{equation*} if necessary, passing to a subsequence.. Then, by Corollary \ref {decomposition}, there is a unique decomposition of the automorphism $\phi _{m_{j}}$ such that \begin{equation*} \phi _{m_{j}}=\varphi _{j}\circ \psi _{j} \end{equation*} where \begin{equation*} \varphi _{j}\in Aut_{p_{m_{j}}}\left( bB^{n+1}\right) ,\quad \psi _{j}\in Aut_{p^{\prime \prime }}\left( bB^{n+1}\right) \end{equation*} and the local automorphism $\psi _{j}$ acts trivially on the tangent hyperplane of $bB^{n+1}$ at the fixed point $p^{\prime }.$
Let $U_{\psi _{j}},\rho _{\psi _{j}},a_{\psi _{j}},r_{\psi _{j}}$ be the normalizing parameters of the local automorphism $\psi _{j}.$ Since the local automorphism $\psi _{j}$ acts trivially on the tangent hyperplane of $ bB^{n+1}$ at the fixed point $p^{\prime \prime },$ we obtain \begin{equation*} U_{\psi _{j}}=id_{n\times n},\quad \rho _{\psi _{j}}=1. \end{equation*} Since the sequence $\phi _{m_{j}}$ uniformly converges to the point $p,$ there is a real number $e>0$ such that \begin{equation}
\left| a_{\psi _{j}}\right| \leq e,\quad \left| r_{\psi _{j}}\right| \leq e. \label{estimate5} \end{equation}
Let $\pi _{j}$ be the complex line passing through the two points $p$ and $ p_{m_{j}},$ and $\delta _{j}$ be the analytic disks \begin{equation*} \delta _{j}=\pi _{j}\cap B^{n+1}. \end{equation*} Let $\sigma _{j}$ be a local automorphism in $Aut_{p_{m_{j}}}\left( bB^{n+1}\right) $ defined by the normalizing parameters \begin{equation*} U_{\sigma _{j}}=id_{n\times n},\quad \rho _{\sigma _{j}}=\varepsilon _{m_{j}}^{2},\quad a_{\sigma _{j}}=0,\quad r_{\sigma _{j}}=0. \end{equation*} Since the points $p_{j}$ converges to the point $p$ to a direction tangential to the complex tangent hyperplane at the point $p\in bB^{n+1},$
the area $\left| \sigma _{j}^{-1}\left( \delta _{j}\right) \right| $ of the analytic disk \begin{equation*} \sigma _{j}^{-1}\left( \delta _{j}\right) \end{equation*} is bounded from below.
Let $U_{\varphi _{j}},\rho _{\varphi _{j}},a_{\varphi _{j}},r_{\varphi _{j}}$ be the normalizing parameters of the local automorphism $\varphi _{j}.$ Note that the analytic disk \begin{equation*} \phi _{m_{j}}^{-1}\left( \delta _{j}\right) \end{equation*} is mapped by $\sigma _{j}^{-1}\circ \phi _{m_{j}}$ onto the analytic disk \begin{equation*} \sigma _{j}^{-1}\left( \delta _{j}\right) \end{equation*} where the areas of the analytic disks in both classes are bounded from below. Further, the normalizing parameters $a_{\varphi _{j}^{\prime }},r_{\varphi _{j}^{\prime }}$ of the composition $\varphi _{j}^{\prime }=\sigma _{j}^{-1}\circ \varphi _{_{j}}$ is the same value of $a_{\varphi _{j}},r_{\varphi _{j}},$ i.e., \begin{equation*} a_{\varphi _{j}^{\prime }}=a_{\varphi _{j}},\quad r_{\varphi _{j}^{\prime }}=r_{\varphi _{j}} \end{equation*} so that there is a real number $e>0,$ if necessary, increasing $e,$ such that \begin{equation}
\left| a_{\varphi _{j}}\right| \leq e. \label{estimate6} \end{equation} Since the area of the analytic disks $\sigma _{j}^{-1}\left( \delta _{j}\right) $ is bounded by a positive number, the point $p$ should be attracted to the center of the local automorphism $Aut_{p_{m_{j}}}\left( bB^{n+1}\right) .$ Hence, by passing to a subsequence, if necessary and increasing $e$, there is a real number $e>0$ such that \begin{equation}
\left| r_{\varphi _{j}}\right| \leq e \label{estimate7} \end{equation} and \begin{equation*} \rho _{\varphi _{j}}\rightarrow 0\quad \text{as }j\rightarrow \infty . \end{equation*} Since $bB^{n+1}$ is strongly pseudoconvex, we obtain \begin{equation*}
\left| U_{\varphi _{j}}\right| =1. \end{equation*}
We set \begin{equation*} \lambda _{j}=\frac{\varepsilon _{m_{j}}^{2}}{\rho _{\varphi _{j}}} \end{equation*} so that there is a real number $e>0$ such that \begin{equation}
e^{-1}\leq \left| \lambda _{j}\right| \leq e, \label{estimate8} \end{equation} if necessary, increasing $e.$ Then the composition \begin{equation*} \varphi _{j}^{\prime }\equiv \sigma _{j}^{-1}\circ \varphi _{j}\in Aut_{p_{m_{j}}}\left( bB^{n+1}\right) \end{equation*} has the same normalizing parameters of the mapping $\varphi _{j}$ except for $\rho _{\varphi _{j}^{\prime }}=\lambda _{j}^{-1},$ i.e., \begin{equation*} U_{\varphi _{j}^{\prime }}=U_{\varphi _{j}},\quad \rho _{\varphi _{j}^{\prime }}=\lambda _{j}^{-1},\quad a_{\varphi _{j}^{\prime }}=a_{\varphi _{j}},\quad r_{\varphi _{j}^{\prime }}=r_{\varphi _{j}}. \end{equation*} Therefore, by the estimates \ref{estimate5}, \ref{estimate6}, \ref{estimate7} , \ref{estimate8}, the sequence \begin{equation*} \tau _{j}\equiv \sigma _{j}^{-1}\circ \phi _{m_{j}}:B^{n+1}\rightarrow B^{n+1} \end{equation*} uniformly converges to an automorphism of $B^{n+1}.$ This completes the proof.
\endproof
\begin{theorem} \label{finite}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical real analytic boundary $bD.$ Suppose that there is a biholomorphic mapping $ \phi $ on a connected open neighborhood $U$ of a point $p\in bD$ satisfying \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Then the Doggaebi variety $L$ associated to the mapping $\phi $ is a finite subset of $bB^{n+1}$ such that the inverse mapping $\phi ^{-1}$ is analytically continued along any piecewise chain curve on $ bB^{n+1}\backslash L$ as a locally biholomorphic mapping. \end{theorem}
\proof
By First Dogginal Lemma, the singular locus of the analytic continuation of the inverse mapping $\phi ^{-1}$ on $bB^{n+1}$ is an integral manifold of the subdistribution of the complex tangent hyperplanes on $bB^{n+1}$ in its smooth part. Thus the mapping $\phi ^{-1}$ cannot be branched on $bB^{n+1}$ by a branching locus passing through the boundary $bB^{n+1}.$ Otherwise, the intersection of the nontrivial branch of the mapping $\phi ^{-1}$ to the boundary $bB^{n+1}$ would be a nontrivial complex submanifold on $bB^{n+1}.$ Hence the singular locus of the analytic continuation of the mapping $\phi ^{-1}$ is well defined so that the inverse mapping $\phi ^{-1}$ is analytically continued on $bB^{n+1}\backslash L$ as a locally biholomorphic mapping.
We take a chain-segment $\gamma :[0,1]\rightarrow bB^{n+1}$ with $\gamma \left( 1\right) =p\in L$ such that a germ of the mapping $\phi ^{-1}$ at the point $\gamma \left( 0\right) $ is analytically continued along all subpath $ \gamma [0,\tau ]$ with $\tau <1,$ but not the whole path $\gamma [0,1].$ Note that the distribution of the complex tangent hyperplanes on $bB^{n+1}$ is maximally nonintegrable. Thus, by First Dogginal Lemma, the singular locus $L$ of the analytic continuation of the mapping $\phi ^{-1}$ cannot bound the open region of the analytic continuation of the mapping $\phi ^{-1}.$ Then the mapping $\phi ^{-1}$ is analytically continued on the opposite side of the complex tangent hyperplane $H_{p}$ at the point $p\in bB^{n+1}.$ Thus there is a complex line $\pi $ passing through the point $p$ and an open neighborhood $U$ of the point $p$ such that \begin{equation*} \pi \cap bB^{n+1} \end{equation*} is a chain on $bB^{n+1}$ satisfying \begin{equation*} L\cap U\cap \pi \cap bB^{n+1}=\left\{ p\right\} . \end{equation*}
We claim that, if necessary, shrinking $U,$ \begin{equation*} L\cap U\cap bB^{n+1}=\left\{ p\right\} \end{equation*} so that the singular locus $L$ is a finite set on $bB^{n+1}.$ By First Dogginal Lemma, there is a sequence $p_{j}\in bB^{n+1}\backslash L$ such that $p_{j}\rightarrow p$ and the sequence $p_{j}$ converges to the point $p$ to a direction tangential to the complex tangent hyperplane at the point $ p\in L\subset bB^{n+1}.$ Since $bD$ is compact, there is a subsequence $ p_{m_{j}}$ and a point $p^{\prime }\in bD$ such that \begin{equation*} p_{m_{j}}^{\prime }\equiv \phi ^{-1}\left( p_{m_{j}}\right) \rightarrow p^{\prime }. \end{equation*} Let $\phi _{j}$ be the germ of the mapping $\phi $ at the point $ p_{m_{j}}^{\prime }\in bD$ such that \begin{equation*} \phi _{j}\left( p_{m_{j}}^{\prime }\right) =p_{m_{j}}. \end{equation*} Since $bD$ is spherical, there is an open neighborhood $W$ of the point $ p^{\prime }$ and a biholomorphic mapping $\psi $ on $W$ such that \begin{equation*} \psi \left( W\cap bD\right) \subset bB^{n+1}. \end{equation*} Then the compositions $\varphi _{j}\equiv \phi _{j}\circ \psi ^{-1}$ satisfy \begin{equation*} \varphi _{j}\left( \psi \left( W\right) \cap bB^{n+1}\right) \subset bB^{n+1}, \end{equation*} if necessary, shrinking $W$ so that, by abuse of notation, the mapping $ \varphi _{j}$ is an automorphism of the unit ball $B^{n+1}.$
Let $\pi _{j}$ be the complex line passing through the points $p_{m_{j}}$ and $p,$ and $\delta _{j}$ be the analytic disk \begin{equation*} \delta _{j}=\pi _{j}\cap B^{n+1}. \end{equation*}
Note that the area $\left| \varphi _{j}^{-1}\left( \delta _{j}\right)
\right| $ of the analytic disk $\varphi _{j}^{-1}\left( \delta _{j}\right) $ is bounded from the below. Otherwise, $p\in bB^{n+1}\backslash L.$ Further, the sequence $\varphi _{j}$ converges to the point $p$ uniformly on every compact subset of $B^{n+1},$ if necessary, passing to a subsequence. Otherwise, the sequence $\varphi _{j}$ converges to an automorphism of the unit ball $B^{n+1}$ so that $p\in bB^{n+1}\backslash L.$ Let $\varepsilon _{j}$ be the euclidean length between the two points $p$ and $p_{m_{j}}.$ Then, by Second Scaling Lemma, there is a sequence of local automorphisms $ \sigma _{j}\in Aut_{p_{m_{j}}}\left( bB^{n+1}\right) $ such that \begin{equation*} U_{\sigma _{j}}=id_{n\times n},\quad \rho _{\sigma _{j}}=\varepsilon _{j}^{2},\quad a_{\sigma _{j}}=0,\quad r_{\sigma _{j}}=0 \end{equation*} and the composition \begin{equation*} \tau _{j}\equiv \sigma _{j}^{-1}\circ \varphi _{j}:B^{n+1}\rightarrow B^{n+1} \end{equation*} uniformly converges to an automorphism of the unit ball $B^{n+1}.$ Thus there is a positive real number $\delta >0$ such that the mapping $\tau _{j}$ and its inverse $\tau _{j}^{-1}$ are analytically continued respectively on \begin{equation*} B\left( p_{m_{j}}^{\prime \prime };\delta \right) \quad \text{and}\quad B\left( p_{m_{j}};\delta \right) \end{equation*} where \begin{equation*} p_{m_{j}}^{\prime \prime }=\psi \left( p_{m_{j}}^{\prime }\right) . \end{equation*} Therefore the mapping $\phi ^{-1}$ at the point $p_{m_{j}}:$ \begin{eqnarray*} \phi ^{-1} &=&\phi _{j}^{-1} \\ &=&\psi ^{-1}\circ \tau _{j}^{-1}\circ \sigma _{j}^{-1} \end{eqnarray*} is analytically continued to the open neighborhood \begin{equation*} \sigma _{j}\left( B\left( p_{m_{j}};\delta \right) \right) . \end{equation*} By the canonical normalizing mapping $\mu _{p_{m_{j}}},$ we obtain \begin{equation*} \sigma _{j}^{\prime }\equiv \mu _{p_{m_{j}}}\circ \sigma _{j}\circ \mu _{p_{m_{j}}}^{-1}:\left\{ \begin{array}{l} z^{*}=\varepsilon _{j}z \\ w^{*}=\varepsilon _{j}^{2}w \end{array} .\right. \end{equation*} Hence the mapping $\phi ^{-1}$ at the point $p_{m_{j}}$ is analytically continued to the open neighborhood \begin{equation*} \mu _{p_{m_{j}}}^{-1}\circ \sigma _{j}^{\prime }\left( B\left( 0;\delta \right) \right) , \end{equation*} if necessary, shrinking $\delta .$ Since $\varepsilon _{j}$ is the euclidean length between the two points $p$ and $p_{m_{j}},$ the mapping $\phi ^{-1}$ is analytically continued on an open region which touches to the point $p$ to a converging direction to the sequence $p_{j}$ by an edge shape tangential to the complex tangent hyperplane $H_{p}$ at the point $p$ and a $
\sqrt{\left| x\right| }$ curve shape normal to $H_{p}$ on $bB^{n+1}.$
Therefore, the singular locus $L$ of the analytic continuation of the mapping $\phi ^{-1}$ is isolated to the direction of the complex tangent hyperplane at each point of $L.$ Since the boundary $bB^{n+1}$ is compact, the singular locus $L$ is a finite subset of $bB^{n+1}.$ This completes the proof.
\endproof
\begin{lemma} \label{length}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical real analytic boundary $bD.$ Then every chain on $bD$ is extended each direction infinitely in its euclidean length. \end{lemma}
\proof
Suppose that the assertion is not true. Then there would be a path $\gamma :[0,1]\rightarrow bD$ such that the subpath $\gamma [0,\tau ]$ is a chain-segment for all $\tau <1$ but the whole path $\gamma [0,1]$ is not a chain segment.
Since $bD$ is spherical, by definition, there exist a connected open neighborhood $U$ of the point $\gamma \left( 1\right) $ and a biholomorphic mapping $\phi $ on $U$ such that $\phi \left( U\cap bD\right) \subset bB^{n+1}$. There is a unique closed circle $\chi $ on $bB^{n+1}$ such that $ \chi $ is a chain on $bB^{n+1}$ and $\phi \circ \gamma [0,\tau ]\subset \chi $ for all $\tau <1$(cf. \cite{Pa3}).
Then the inverse image $\phi ^{-1}\left( \chi \cap \phi \left( U\right) \right) $ under the biholomorphic mapping $\phi $ is a chain on $bB^{n+1}$ (cf. \cite{Pa3}) such that \begin{equation*} \gamma [0,1]\cap U\subset \phi ^{-1}\left( \chi \cap \phi \left( U\right) \right) . \end{equation*} This is a contradiction. This completes the proof.
\endproof
\begin{theorem} \label{creq}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical real analytic boundary $bD.$ Suppose that there is a biholomorphic mapping $ \phi $ on a connected open neighborhood $U$ of a point $p\in bD$ satisfying \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Let $E$ be the path space of $bD$ pointed at the point $p\in bD$ mod homotopic relation so that $E$ be a universal covering of $bD$ with a natural CR structure and a natural CR covering map $\psi :E\rightarrow bD$. Then there is a unique CR equivalence $\varphi :E\rightarrow bB^{n+1}\backslash L$ commuting the diagram \begin{equation*} \begin{array}{lll} E & & \\ \downarrow \psi & \overset{\varphi }{\searrow } & \\ bD & \overset{\phi }{\rightarrow } & bB^{n+1}\backslash L \end{array} \end{equation*} where $L\subset bB^{n+1}$ is the Doggaebi variety associated to the mapping $ \phi .$ \end{theorem}
\proof
We obtain the set $bB^{n+1}\backslash L$ when the mapping $\phi $ is analytically continued along chains on $bD.$ Since the mapping $\varphi :E\rightarrow bB^{n+1}$ is naturally induced by the analytic continuation of the mapping $\phi $ on $bD$(cf. Theorem \ref{anypath}), we obtain \begin{equation*} bB^{n+1}\backslash L\subset \varphi \left( E\right) \end{equation*}
Let $\gamma :[0,1]\rightarrow bD$ be a path such that $\gamma \left( 0\right) =p.$ Then, for a sufficiently small $\varepsilon >0,$ there is an $ \varepsilon $ open neighborhood $V$ of the path $\gamma [0,1]$ such that the mapping $\phi $ is analytically continued on $V$ as a local biholomorphic mapping. Then we take a piecewise chain curve $\eta :[0,1]\rightarrow bD$ (cf. \cite{Pa3}) such that \begin{equation*} \eta \left( 0\right) =\gamma \left( 0\right) ,\quad \eta \left( 1\right) =\gamma \left( 1\right) \end{equation*} and \begin{equation*} \eta [0,1]\subset V. \end{equation*} Further, we may take a continuous function $\Gamma :[0,1]\times [0,1]\rightarrow V$ such that \begin{equation*} \Gamma \left( 0,\tau \right) =\gamma \left( \tau \right) ,\quad \Gamma \left( 1,\tau \right) =\eta \left( \tau \right) \quad \text{for }\tau \in [0,1] \end{equation*} and $\Gamma \left( \cdot ,\tau \right) :[0,1]\rightarrow V$ is a chain-segment for all $\tau \in [0,1].$ Note that every chain-segment in this construction is finite in its euclidean length. By Lemma \ref{length}, the mapping $\phi $ is analytically continued along the whole path $\gamma [0,1].$
Then the image of $E$ under the mapping $\varphi $ satisfies \begin{equation*} \varphi \left( E\right) \subset bB^{n+1}\backslash L \end{equation*} which yields \begin{equation*} \varphi \left( E\right) =bB^{n+1}\backslash L. \end{equation*} Since the mapping $\phi ^{-1}$ is analytically continued on $ bB^{n+1}\backslash L,$ the mapping $\varphi :E\rightarrow bB^{n+1}$ is a CR equivalence. This completes the proof.
\endproof
\begin{corollary} Let $D$ be a bounded domain with spherical real analytic boundary $bD.$ Then there is a natural embedding \begin{equation*} Aut\left( D\right) \subset Aut\left( B^{n+1}\right) . \end{equation*} \end{corollary}
\proof
Let $E$ be the path space of $bD$ pointed at the point $p\in bD$ mod homotopy so that $E$ be a universal covering of $bD$ with a natural CR structure and a natural CR covering map $\psi :E\rightarrow bD$. Then we take a biholomorphic mapping $\phi $ on an open neighborhood $U$ of a point $ p\in bD$ such that \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Let $L$ be the Doggaebi variety associated to the mapping $\phi .$ Then, by Theorem \ref{creq}, the analytic continuation of the mapping $\phi $ yields a natural CR equivalence \begin{equation*} Aut\left( E\right) \simeq Aut\left( bB^{n+1}\backslash L\right) . \end{equation*} Note that \begin{equation*} Aut\left( D\right) =Aut(bD)\subset Aut\left( E\right) \simeq Aut\left( bB^{n+1}\backslash L\right) \subset Aut\left( bB^{n+1}\right) =Aut\left( B^{n+1}\right) . \end{equation*} This completes the proof.
\endproof
\begin{lemma} \label{sph-proper}Let $D,D^{\prime }$ be bounded domains in $\Bbb{C}^{n+1}$ with spherical real analytic boundaries $bD,bD^{\prime }$, and $\phi :D\rightarrow D^{\prime }$ be a proper holomorphic mapping. Suppose that there is an open neighborhood $U$ of a point $p\in bD$ such that the mapping $\phi $ is analytically continued on $U.$ Then the mapping $\phi $ is analytically continued along any path on $bD$ as a locally biholomorphic mapping so that $\phi :D\rightarrow D^{\prime }$ is locally biholomorphic. \end{lemma}
\proof
We may obtain the result by the boundary regularity of Lemma \ref {b-regularity} and the transformation formula for a complex Monge-Ampere equation. Here we may give an independent proof. Note that \begin{equation*} \phi \left( U\cap bD\right) \subset bD^{\prime } \end{equation*} so that, by shrinking $U$, if necessary, the mapping $\phi $ is biholomorphic on $U.$ By shrinking $U$, if necessary, we take biholomorphic mappings $\varphi ,\varphi ^{\prime }$ respectively on $U,\phi \left( U\right) $ such that \begin{eqnarray*} \varphi \left( U\cap bD\right) &\subset &bB^{n+1} \\ \varphi ^{\prime }\left( \phi \left( U\right) \cap bD^{\prime }\right) &\subset &bB^{n+1}. \end{eqnarray*} Then there are Doggaebi varieties $L,L^{\prime }$ on $bB^{n+1}$ such that the inverses $\varphi ^{-1},\varphi ^{\prime -1}$ are analytically continued respectively on $bB^{n+1}\backslash L$ and $bB^{n+1}\backslash L^{\prime }$ as a local biholomorphic mapping. Note that the composition $\psi =\varphi ^{\prime }\circ \phi \circ \varphi ^{-1}$ satisfies the relation \begin{equation*} \psi \left( \varphi \left( U\right) \cap bB^{n+1}\right) \subset bB^{n+1}. \end{equation*} By abuse of notation, the mapping $\psi $ is an automorphism of $bB^{n+1}$ which comments the following diagram: \begin{equation*} \begin{array}{lll} bB^{n+1}\backslash L & \overset{\psi }{\longrightarrow } & bB^{n+1}\backslash L^{\prime } \\ \downarrow \kappa & & \downarrow \kappa ^{\prime } \\ bD & \overset{\phi }{\longrightarrow } & bD^{\prime } \end{array} . \end{equation*} where $\kappa :bB^{n+1}\backslash L\rightarrow bD$ is the analytic continuation of the mapping $\varphi ^{-1}$ and $\kappa ^{\prime }:bB^{n+1}\backslash L^{\prime }\rightarrow bD^{\prime }$ is the analytic continuation of the mapping $\varphi ^{\prime -1}$ on the boundary $ bB^{n+1}. $
We claim \begin{equation*} L^{\prime }\subset \psi \left( L\right) \end{equation*} so that the mapping $\phi $ is analytically continued along any path on $bD$ as a locally biholomorphic mapping. Otherwise, there is a point $q^{\prime }\in L^{\prime }$ such that \begin{equation*} q^{\prime }\in L^{\prime }\backslash \psi \left( L\right) \end{equation*} From the proof of First Scaling Lemma, there is a sequence $p_{j}^{\prime }\in bD^{\prime }$ with $p_{j}^{\prime }\rightarrow p^{\prime }\in bD^{\prime }$ and a sequence $\varphi _{j}^{\prime }$ of the analytic continuation of the mapping $\varphi ^{\prime }$ at the point $p_{j}^{\prime }$ with $q_{j}^{\prime }\equiv \varphi _{j}^{\prime }\left( p_{j}^{\prime }\right) \rightarrow q^{\prime }\in L^{\prime }\backslash \psi \left( L\right) $ such that there is an open neighborhood $V$ of the point $ p^{\prime }$ such that the mapping $\varphi _{j}^{\prime }$ is biholomorphic on $V\cap D^{\prime }$ and the sequence $\varphi _{j}^{\prime }$ converges to the point $r$ uniformly on every compact subset of $V\cap D^{\prime }.$ Since $\psi $ is an automorphism of the unit ball $B^{n+1},$ we obtain \begin{equation*} \lambda _{j}\equiv \kappa \circ \psi ^{-1}\circ \varphi _{j}^{\prime }:V\cap D^{\prime }\rightarrow D \end{equation*} and \begin{equation*} \phi \circ \lambda _{j}=id\quad \text{on }V\cap D^{\prime }. \end{equation*} We set \begin{equation*} p_{j}\equiv \lambda _{j}\left( p_{j}^{\prime }\right) \in bD. \end{equation*} Since $bD$ is compact, there is a subsequence $p_{m_{j}}\in bD$ and a point $ p\in bD$ such that \begin{equation*} p_{m_{j}}\rightarrow p. \end{equation*} Since $\phi $ is a proper mapping, the mapping $\phi $ is a globally finite covering so that there is an open neighborhood $W$ of the point $p$ such that \begin{equation*} \lambda _{m_{j}}=\phi ^{-1}\quad \text{on }\phi \left( W\right) \cap D. \end{equation*} Hence we obtain \begin{equation*} \psi =\varphi _{m_{j}}^{\prime }\circ \phi \circ \kappa \end{equation*}
Since the mapping $\psi $ is an automorphism of the unit ball $B^{n+1},$ there is a sequence $q_{j}\in bB^{n+1}$ such that \begin{equation*} \psi \left( q_{j}\right) =q_{j}^{\prime }. \end{equation*} Hence we obtain \begin{equation*} \kappa \left( q_{m_{j}}\right) =p_{m_{j}}. \end{equation*} Note that there is a real number $\delta >0$ such that \begin{equation*} B\left( r;2\delta \right) \cap bB^{n+1}\cap \psi \left( L\right) =\emptyset . \end{equation*} Passing to a subsequence, if necessary, we may assume \begin{equation*} q_{m_{j}}\in \psi ^{-1}\left( B\left( r;\delta \right) \right) \cap bB^{n+1} \end{equation*} so that there is a point $q\in bB^{n+1}\backslash L$ satisfying \begin{equation*} q_{m_{j}}\rightarrow q. \end{equation*} Hence we set \begin{equation*} \chi _{m_{j}}\equiv \psi ^{-1}\circ \varphi _{m_{j}}^{\prime }\circ \lambda _{m_{j}}^{-1} \end{equation*} and \begin{equation*} \kappa \circ \chi _{m_{j}}=id. \end{equation*} Finally, we obtain \begin{equation*} \varphi _{m_{j}}^{\prime }=\psi \circ \chi _{m_{j}}\circ \lambda _{m_{j}}. \end{equation*}
Note that there is a real number $\delta >0$ that the mappings $\lambda _{m_{j}}$ and the inverse mappings $\lambda _{m_{j}}^{-1}$ are analytically continued respectively on \begin{equation*} B\left( p_{m_{j}}^{\prime };\delta \right) \quad \text{and}\quad B\left( p_{m_{j}};\delta \right) , \end{equation*} and the mappings $\chi _{m_{j}}$ and the inverse mappings $\chi _{m_{j}}^{-1} $ are analytically continued respectively on \begin{equation*} B\left( p_{m_{j}};\delta \right) \quad \text{and}\quad B\left( q_{m_{j}};\delta \right) , \end{equation*} and the mapping $\psi $ and the inverse mapping $\psi ^{-1}$ are analytically continued respectively on \begin{equation*} B\left( q_{m_{j}};\delta \right) \quad \text{and}\quad B\left( q_{m_{j}}^{\prime };\delta \right) , \end{equation*} Hence the mapping $\varphi _{m_{j}}^{\prime }$ and the inverse mappings $ \varphi _{m_{j}}^{\prime -1}=\kappa ^{\prime }$ are analytically continued respectively on \begin{equation*} B\left( p_{m_{j}}^{\prime };\delta \right) \quad \text{and}\quad B\left( q_{m_{j}}^{\prime };\delta \right) , \end{equation*} as a locally biholomorphic mapping. Since $q_{m_{j}}^{\prime }\rightarrow q^{\prime },$ the mapping $\kappa ^{\prime }$ is analytically continued to the point $q^{\prime }$ as a locally biholomorphic mapping so that \begin{equation*} q^{\prime }\in bB^{n+1}\backslash L^{\prime }. \end{equation*} This is a contradiction so that \begin{equation*} L^{\prime }\subset \psi \left( L\right) . \end{equation*} This completes the proof.
\endproof
\section{Bounded Domains with Spherical Boundaries}
\subsection{Differentiable spherical boundary}
Let $D$ be a domain in $\Bbb{C}^{n+1},n\geq 1,$ with a differentiable boundary $bD.$ The boundary $bD$ shall be called spherical if, for each point $p\in bD,$ there is a connected open neighborhood $U$ of the point $p$ and a biholomorphic mapping $\phi $ on $U\cap D$ such that \begin{equation} \phi \in H\left( U\cap D\right) \cap C^{1}\left( U\cap \overline{D}\right) ,\quad \phi \left( U\cap bD\right) \subset bB^{n+1} \label{maptosphere} \end{equation} and the induced mapping $\phi :U\cap bD\rightarrow bB^{n+1}$ is CR diffeomorphic.
\begin{lemma} \label{onsphere}Let $U$ be a connected open neighborhood of a point $p\in bB^{n+1}$ and $\phi $ be a biholomorphic mapping on $U\cap B^{n+1}$ such that \begin{equation*} \phi \in H\left( U\cap B^{n+1}\right) \cap C^{1}\left( U\cap \overline{ B^{n+1}}\right) ,\quad \phi \left( U\cap bB^{n+1}\right) \subset bB^{n+1} \end{equation*} and the induced mapping $\phi :U\cap bB^{n+1}\rightarrow bB^{n+1}$ is CR diffeomorphic. Then the mapping $\phi $ is analytically continued to an automorphism of the unit ball $B^{n+1}.$ \end{lemma}
\proof
By Lemma \ref{Lewy-Pinchuk}, the mapping $\phi $ is biholomorphic on $U,$ if necessary, shrinking $U.$ Then, by Lemma \ref{sphere}, we obtain the desired result. This completes the proof.
\endproof
The chain on $bB^{n+1}$ is defined to be the intersection on $bB^{n+1}$ by a complex line. The family of chains on $bB^{n+1}$ leaves invariant under the action of biholomorphic automorphisms of $B^{n+1}.$ We define the chain on the spherical differentiable boundary $bD$ of a domain $D$ to be the inverse image of the chain on $bB^{n+1}$ under the mapping \ref{maptosphere}. By Lemma \ref{onsphere}, the chain on the spherical differentiable boundary $bD$ is well defined so that a chain of a spherical differentiable boundary is mapped to a chain of another spherical differentiable boundary under any induced CR diffeomorphism.
\begin{lemma} Let $D$ be a domain in $\Bbb{C}^{n+1}$ with spherical differentiable boundary $bD.$ Suppose that there is a connected open neighborhood $U$ of a point $p\in bD$ and a biholomorphic mapping $\phi $ on $U\cap D$ such that \begin{equation*} \phi \in H\left( U\cap D\right) \cap C^{1}\left( U\cap \overline{D}\right) ,\quad \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Then the mapping $\phi $ is analytically continued along any path on $bD$ as a local biholomorphic mapping. \end{lemma}
\proof
Suppose that the assertion is not true. Then there would be a path $\gamma :[0,1]\rightarrow bD$ such that $\gamma \left( 0\right) \in U\cap bD$ and the germ of a biholomorphic mapping $\phi $ at the point $\gamma \left( 0\right) $ is analytically continued along the subpath $\gamma [0,\tau ]$ with all $\tau <1$ as a local biholomorphic mapping, but not the whole path $ \gamma [0,1].$
Thus we reduce the proof to a local problem near the point $\gamma \left( 1\right) \in bD.$ Then, by the definition, there is a connected open neighborhood $V$ of the point $\gamma \left( 1\right) $ and a biholomorphic mapping $\varphi $ on $V\cap D$ such that \begin{equation*} \varphi \in H\left( V\cap D\right) \cap C^{1}\left( V\cap \overline{D} \right) ,\quad \varphi \left( V\cap bD\right) \subset bB^{n+1} \end{equation*} and the induced mapping $\varphi :U\cap bD\rightarrow bB^{n+1}$ is CR diffeomorphic.
Then we consider the mapping \begin{equation*} \phi \circ \varphi ^{-1}\in H\left( \varphi \left( V\cap D\right) \right) \cap C^{1}\left( \varphi \left( V\cap \overline{D}\right) \right) ,\quad \phi \circ \varphi ^{-1}\left( \varphi \left( V\cap bD\right) \right) \subset bB^{n+1} \end{equation*} and the curve \begin{equation*} \varphi \circ \gamma [0,1]\cap \varphi \left( V\cap bD\right) \subset bB^{n+1}. \end{equation*} By Lemma \ref{Lewy-Pinchuk}, the remaining part of the proof repeats the proof of Lemma \ref{anypath}. This completes the proof.
\endproof
\begin{lemma}[First Dogginal Lemma] \label{dogginal2}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical differentiable boundary $bD$ and $\phi $ be a biholomorphic mapping on $U\cap D$ for a connected open neighborhood $U$ of a point $p\in bB^{n+1}$ satisfying \begin{equation*} \phi \in H\left( U\cap B^{n+1}\right) \cap C^{1}\left( U\cap \overline{ B^{n+1}}\right) ,\quad \phi \left( U\cap bB^{n+1}\right) \subset bD. \end{equation*} Suppose that there is an injective path $\gamma :[0,1]\rightarrow bB^{n+1}$ such that $\gamma [0,1]\subset bB^{n+1}$ is a chain-segment satisfying \begin{equation*} \gamma \left( 0\right) \in U\cap bB^{n+1} \end{equation*} and the mapping $\phi $ is analytically continued along the subpath $\gamma [0,\tau ]$ for all $\tau <1,$ but not the whole path $\gamma [0,1]$ as a local biholomorphic mapping. Let $\pi $ be the complex line containing the chain-segment $\gamma [0,1].$ Then there is an open neighborhood $V$ along the path $\gamma [0,1]$ such that
\begin{enumerate} \item $\gamma [0,\tau ]\subset V\quad $for all $\tau <1,$
\item $bV\cap \pi \cap B\left( \gamma \left( 1\right) ;\delta \right) $ is an angle for a sufficient small $\delta >0,$ which contains the chain-segment $\gamma [0,1],$
\item $bV\cap bB^{n+1}\cap B\left( \gamma \left( 1\right) ;\delta \right) $ is paraboloid for a sufficiently small $\delta >0,$ which smoothly touches the complex tangent hyperplane at the point $\gamma \left( 1\right) ,$
\item the mapping $\phi $ is analytically continued on $V\cap B^{n+1}$ as a local biholomorphic mapping \begin{equation*} \phi \in H\left( V\cap B^{n+1}\right) \cap C^{1}\left( V\cap \overline{ B^{n+1}}\right) . \end{equation*} \end{enumerate} \end{lemma}
\proof
By the analytic continuation of the mapping $\phi $ along the subpath $ \gamma [0,\tau ]$ for all $\tau <1$, there is a path $\phi \circ \gamma :[0,1)\rightarrow bD.$ Then we consider the following sequences \begin{eqnarray*} p_{j} &=&\gamma \left( 1-\frac{1}{j}\right) ,\quad \text{for }j\in \Bbb{N} ^{+}, \\ p_{j}^{\prime } &=&\phi \circ \gamma \left( 1-\frac{1}{j}\right) ,\quad \text{for }j\in \Bbb{N}^{+}. \end{eqnarray*} Since $bD$ is compact, there is a subsequence $p_{m_{j}}^{\prime }$ and a point $p^{\prime }\in bD$ such that \begin{equation*} p_{m_{j}}^{\prime }\rightarrow p^{\prime }. \end{equation*}
Thus we reduce the proof to a local problem near the point $p^{\prime }\in bD.$ Then, by the definition, there is a connected open neighborhood $W$ of the point $p^{\prime }$ and a biholomorphic mapping $\varphi $ on $W\cap D$ such that \begin{equation*} \varphi \in H\left( W\cap D\right) \cap C^{1}\left( W\cap \overline{D} \right) ,\quad \varphi \left( W\cap bD\right) \subset bB^{n+1} \end{equation*} and the induced mapping $\varphi :U\cap bD\rightarrow bB^{n+1}$ is CR diffeomorphic.
Then we consider the mapping \begin{equation*} \phi ^{-1}\circ \varphi ^{-1}\in H\left( \varphi \left( W\cap D\right) \right) \cap C^{1}\left( \varphi \left( W\cap \overline{D}\right) \right) ,\quad \phi ^{-1}\circ \varphi ^{-1}\left( \varphi \left( W\cap bD\right) \right) \subset bB^{n+1}. \end{equation*} By Lemma \ref{Lewy-Pinchuk}, the remaining part of the proof repeats the proof of Lemma \ref{dogginal}. This completes the proof.
\endproof
By Lemma \ref{onsphere} and Lemma \ref{dogginal2}, we obtain the following result by the same argument of the proof of Theorem \ref{creq}.
\begin{theorem} \label{covering}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical differentiable boundary $bD.$ Suppose that there is a biholomorphic mapping $\phi $ on $U\cap D$ for a connected open neighborhood $U$ of a point $p\in bD$ satisfying \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Let $E$ be the path space of $bD$ pointed at the point $p\in bD$ mod homotopy so that $E$ be a universal covering of $bD$ with a natural CR structure and a natural CR covering map $\psi :E\rightarrow bD$. Then there is a unique CR equivalence $\varphi :E\rightarrow bB^{n+1}\backslash L$ commuting the diagram \begin{equation*} \begin{array}{lll} E & & \\ \downarrow \psi & \overset{\varphi }{\searrow } & \\ bD & \overset{\phi }{\rightarrow } & bB^{n+1}\backslash L \end{array} \end{equation*} where $L\subset bB^{n+1}$ is the Doggaebi variety associated to the mapping $ \phi .$ \end{theorem}
\subsection{Second Dogginal Lemma}
We have examined the analytic continuation of a biholomorphic mapping $\phi $ from the spherical differentiable boundary $bD$ of a bounded domain $D$ into the boundary $bB^{n+1}$ of the unit ball $B^{n+1}.$ From now on, we shall examine the analytic continuation of the mapping $\phi $ into the domain $D.$
\begin{lemma} \label{branched}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical differentiable boundary $bD.$ Suppose that there is a biholomorphic mapping $\phi $ on a connected open neighborhood $U$ of a point $p\in bD$ satisfying \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Then the inverse mapping $\phi ^{-1}$ is analytically continued to a locally biholomorphic mapping from $B^{n+1}$ into $D.$ \end{lemma}
\proof
By Theorem \ref{covering}, there is a Doggaebi variety $L$ on $bB^{n+1}$ such that the mapping $\phi ^{-1}$ is uniquely analytically continued on the set $bB^{n+1}\backslash L.$ Then the mapping $\phi ^{-1}$ is analytically continued to a holomorphic mapping on the unit ball $B^{n+1}.$ Hence we have a holomorphic mapping $\varphi :B^{n+1}\rightarrow D$ which extends to the mapping $\phi ^{-1}:bB^{n+1}\backslash L\rightarrow bD.$ Note that the zero set of the determinant of the Jacobian matrix $\varphi ^{\prime }$ of the mapping $\varphi $ cannot be located on the set $L$ on the boundary $ bB^{n+1}.$ Thus the mapping $\varphi :B^{n+1}\rightarrow D$ is locally biholomorphic. This completes the proof.
\endproof
\begin{lemma} \label{transversally}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical differentiable boundary $bD$ and $\phi $ be a biholomorphic mapping on an open neighborhood $U$ of a point $r\in bD$ satisfying \begin{equation*} \phi \in H\left( U\cap D\right) \cap C^{1}\left( U\cap \overline{D}\right) \quad \text{and}\quad \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Let $L$ be the Doggaebi variety associated to the mapping $\phi $ and $ \varphi :B^{n+1}\rightarrow D$ be a locally biholomorphic mapping to be an analytic continuation of the mapping $\phi ^{-1}$. Suppose that there is a line segment $\gamma :[0,1]\rightarrow D$ with \begin{equation*} p=\gamma \left( 1\right) \quad \text{and}\quad p_{j}=\gamma \left( 1-\frac{1 }{j}\right) \end{equation*} and the germ of a locally biholomorphic mapping $\phi =\varphi ^{-1}$ at the point $p_{1}\equiv \gamma \left( 0\right) $ is analytically continued along the subpath $\gamma [0,\tau ]$ with all $\tau <1$ as a locally biholomorphic mapping. Let $p_{j}^{\prime }\equiv \phi \left( p_{j}\right) \in B^{n+1}$ be the sequence obtained by the analytic continuation of the mapping $\phi =\varphi ^{-1}$ along the path $\gamma [0,1).$ Then there is a subsequence $ p_{m_{j}}^{\prime }$ and a point $p^{\prime }\in L\subset bB^{n+1}$ such that the subsequence $p_{m_{j}}^{\prime }$ converges to the point $p^{\prime }$ to a direction transversal to the complex tangent hyperplane at the point $p^{\prime }\in bB^{n+1}$. \end{lemma}
\proof
Since the closed ball $\overline{B^{n+1}}$ is compact, there is a subsequence $p_{m_{j}}^{\prime }$ which converges to a point $p^{\prime }\in \overline{B^{n+1}}.$ Since the mapping $\varphi $ is locally biholomorphic on $B^{n+1}$, the point $p^{\prime }$ should be on the boundary $bB^{n+1}.$ Further, since $\varphi =\phi ^{-1}:bB^{n+1}\backslash L\rightarrow bD$ is locally biholomorphic, we obtain \begin{equation*} p_{m_{j}}^{\prime }\rightarrow p^{\prime }\in L. \end{equation*} Let $\pi _{p}$ be the complex line containing the line segment $\gamma [0,1]. $ Since $p_{m_{j}}^{\prime }\equiv \phi \left( p_{m_{j}}\right) ,$ the analytic continuation of the mapping $\phi =\varphi ^{-1}$ on the complex line $\pi _{p}$ yields a complex curve $\varphi ^{-1}\left( \pi _{p}\cap D\right) $ which touches the point $p^{\prime }\in L\subset bB^{n+1}.$ Since the mapping $\varphi :B^{n+1}\rightarrow D$ is locally biholomorphic up to the boundary subset $bB^{n+1}\backslash L$ and the boundary $bB^{n+1}$ is strongly pseudoconvex, the extension \begin{equation*} \varphi ^{-1}\left( \pi _{p}\cap \overline{D}\right) \end{equation*} of the complex curve $\varphi ^{-1}\left( \pi _{p}\cap D\right) $ is transversal to the boundary $bB^{n+1}$ near the point $p^{\prime }\in L\subset bB^{n+1}.$
We claim that the extension $\varphi ^{-1}\left( \pi _{p}\cap \overline{D} \right) $ is transversal to the complex tangent hyperplane at the point $ p^{\prime }\in L.$ We take a path $\lambda :[0,1]\rightarrow bB^{n+1}$ such that $p^{\prime }=\lambda (1)$ and \begin{equation*} \lambda [0,1]\subset \overline{\varphi ^{-1}\left( \pi _{p}\cap bD\right) } \cap bB^{n+1}. \end{equation*} Note that there are finitely many closed paths $\gamma _{j}$ on $bD$ such that \begin{equation*} \pi _{p}\cap bD=\bigcup_{j}\gamma _{j}. \end{equation*} Since $bD$ is strongly pseudoconvex, each path $\gamma _{j}$ is transversal to the complex tangent hyperplane on $bD$ at each point.
We consider the following sequence \begin{equation*} q_{j}=\varphi \circ \lambda \left( 1-\frac{1}{j}\right) \in \pi _{p}\cap bD,\quad q_{j}^{\prime }=\lambda \left( 1-\frac{1}{j}\right) \quad \text{for }j\in \Bbb{N}^{+} \end{equation*} such that \begin{equation*} q_{j}^{\prime }\rightarrow p^{\prime }\quad \text{as }j\rightarrow \infty . \end{equation*} Since the set $\pi _{p}\cap bD$ is compact, there is a subsequence $ q_{m_{j}} $ and a point $q\in \pi _{p}\cap bD$ such that \begin{equation*} q_{m_{j}}\rightarrow q\quad \text{as }j\rightarrow \infty . \end{equation*} Let $\phi _{m_{j}}$ be the germ of the analytic continuation of the mapping $ \phi =\varphi ^{-1}$ at the point $q_{m_{j}}.$ By Theorem \ref{anypath} and Lemma \ref{localhull}, we may assume that there is an open neighborhood $W$ of the point $q$ such that the mapping $\phi _{m_{j}}$ is holomorphic on $ W\cap D.$
Since $bD$ is spherical, there is a biholomorphic mapping $\psi $ on $W,$ if necessary, shrinking $W,$ such that \begin{equation*} \psi \in H\left( W\cap D\right) \cap C^{1}\left( W\cap \overline{D}\right) \quad \text{and}\quad \psi \left( W\cap bD\right) \subset bB^{n+1}. \end{equation*} We may assume that \begin{equation*} q^{\prime \prime }\equiv \psi \left( q\right) \neq p^{\prime }\quad \text{and }\quad q_{m_{j}}^{\prime \prime }\equiv \psi \left( q_{m_{j}}\right) . \end{equation*} Then, by Lemma \ref{sphere}, the composition \begin{equation*} \eta _{j}\equiv \phi _{m_{j}}\circ \psi ^{-1}\in Aut\left( B^{n+1}\right) \end{equation*} are automorphisms of the unit ball $B^{n+1}.$ Note that we have the condition \begin{equation*} \eta _{j}\left( q_{m_{j}}^{\prime \prime }\right) =q_{m_{j}}^{\prime }\rightarrow p^{\prime } \end{equation*} and the sequence $\eta _{j}$ converges to the point $p^{\prime }$ uniformly on every compact subset of $B^{n+1},$ if necessary, passing to a subsequence. Otherwise, $p^{\prime }\notin L.$
Let $\pi _{j}^{\prime }$ be the complex line passing through the two points $ p^{\prime }$ and $q_{m_{j}}^{\prime },$ and $\delta _{j}^{\prime }$ be the analytic disk \begin{equation*} \delta _{j}^{\prime }=\pi _{j}^{\prime }\cap B^{n+1}. \end{equation*}
The area $\left| \eta _{j}^{-1}\left( \delta _{j}^{\prime }\right) \right| $ of the analytic disk $\eta _{j}^{-1}\left( \delta _{j}^{\prime }\right) $ is bounded from the below. Otherwise, we would have $p^{\prime }\notin L.$ Then we decompose the mapping $\eta _{j}$ as follows \begin{equation*} \eta _{j}=\mu _{j}\circ \nu _{j} \end{equation*} where \begin{equation*} \mu _{j}\in Aut_{q_{m_{j}}^{\prime }}\left( bB^{n+1}\right) ,\quad \nu _{j}\in Aut_{p^{\prime }}\left( bB^{n+1}\right) . \end{equation*} Then, by First and Second Scaling Lemmas, we take a sequence of local automorphisms $\sigma _{j}\in Aut_{q_{m_{j}}^{\prime }}\left( bB^{n+1}\right) $ defined by the normalizing parameters \begin{equation*} U_{\sigma _{j}}=id_{n\times n},\quad \rho _{\sigma _{j}}=\rho _{\mu _{j}},\quad a_{\sigma _{j}}=0,\quad r_{\sigma _{j}}=0. \end{equation*} Then the following sequence \begin{equation*} \beta _{j}=\sigma _{j}^{-1}\circ \eta _{j} \end{equation*} converges to an automorphism of the unit ball $B^{n+1},$ if necessary, passing to a subsequence.
Let $\pi _{j}$ be the complex line containing the tangent vector of the path \begin{equation*} \psi \left( \varphi \circ \lambda [0,1]\cap W\right) \subset bB^{n+1} \end{equation*} at the point $q_{m_{j}}^{\prime \prime },$ and $\delta _{j}$ be the analytic disk \begin{equation*} \delta _{j}=\pi _{j}\cap B^{n+1}. \end{equation*}
Clearly, the area $\left| \delta _{j}\right| $ of the analytic disk $\delta _{j}$ is bounded from the below. Note that, by the mapping $\sigma _{j}\in Aut_{q_{m_{j}}^{\prime }}\left( bB^{n+1}\right) ,$ the area $\left| \sigma _{j}^{-1}\left( \delta _{j}^{\prime \prime }\right) \right| $ of the analytic disk $\sigma _{j}\left( \delta _{j}^{\prime \prime }\right) $ is bounded from the below whenever the area $\left| \delta _{j}^{\prime \prime
}\right| $ of the analytic disk $\delta _{j}^{\prime \prime }$ is bounded from the below. Thus the area $\left| \eta _{j}\left( \delta _{j}\right)
\right| $ of the analytic disk \begin{equation*} \eta _{j}\left( \delta _{j}\right) =\sigma _{j}\circ \beta _{j}\left( \delta _{j}\right) \end{equation*} is bounded from the below. Since the analytic disk $\eta _{j}\left( \delta _{j}\right) $ is the intersection of $B^{n+1}$ and the complex line containing the tangent vector of the path $\lambda [0,1]$ at the point \begin{equation*} q_{m_{j}}^{\prime }=\lambda \left( 1-\frac{1}{m_{j}}\right) . \end{equation*} Thus the path $\lambda [0,1]$ is transversal to the complex tangent hyperplane at the point $p^{\prime }=\lambda \left( 1\right) $. Therefore, the point $p_{j}^{\prime }$ approaches to the point $p^{\prime }\in L$ to a direction transversal to the complex tangent hyperplane at the point $ p^{\prime }\in bB^{n+1}.$ This completes the proof.
\endproof
\begin{lemma}[Second Dogginal Lemma] \label{doggi}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical differentiable boundary $bD$ and $\phi $ be a biholomorphic mapping on an open neighborhood $U$ of a point $r\in bD$ satisfying \begin{equation*} \phi \in H\left( U\cap D\right) \cap C^{1}\left( U\cap \overline{D}\right) \quad \text{and}\quad \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Let $L$ be the Doggaebi variety associated to the mapping $\phi $ and $ \varphi :B^{n+1}\rightarrow D$ be a locally biholomorphic mapping to be an analytic continuation of the mapping $\phi ^{-1}$. Suppose that there is a line segment $\gamma :[0,1]\rightarrow D$ with \begin{equation*} p=\gamma \left( 1\right) \quad \text{and}\quad p_{j}=\gamma \left( 1-\frac{1 }{j}\right) \end{equation*} and the germ of a locally biholomorphic mapping $\phi =\varphi ^{-1}$ at the point $p_{1}\equiv \gamma \left( 0\right) $ is analytically continued along the subpath $\gamma [0,\tau ]$ with all $\tau <1$ as a locally biholomorphic mapping. Let $p_{j}^{\prime }\equiv \phi \left( p_{j}\right) \in B^{n+1}$ be the sequence obtained by the analytic continuation of the mapping $\phi =\varphi ^{-1}$ along the path $\gamma [0,1).$ Suppose that there is a point $p^{\prime }\in L\subset bB^{n+1}$ such that the sequence $p_{j}^{\prime }$ converges to the point $p^{\prime }$ to a direction transversal to the complex tangent hyperplane at the point $p^{\prime }\in bB^{n+1}$. Let $\pi _{p}$ be the complex line containing the line segment $\gamma [0,1]$. Then there is a distinguished complex hyperplane $H_{p}\subset T_{p}\Bbb{C}^{n+1}$ at the point $p\in D$ satisfying \begin{equation*} T_{p}\Bbb{C}^{n+1}=H_{p}\oplus \pi _{p} \end{equation*} and an open neighborhood $V$ of the line segment $\gamma [0,1)$ such that
\begin{enumerate} \item $bV\cap \pi _{p}\cap B\left( p;\delta \right) $ is an angle for a sufficient small $\delta >0,$ which contains the path $\gamma [0,1],$
\item $bV\cap R_{p}\cap B\left( p;\delta \right) $ is a slanted paraboloid for a sufficiently small $\delta >0,$ which smoothly touches the complex hyperplane $H_{p}$ at the point $p,$
\item the germ of a locally biholomorphic mapping $\phi =\varphi ^{-1}$ at the point $p_{1}\equiv \gamma \left( 0\right) $ is analytically continued on $V$ as a locally biholomorphic mapping, \end{enumerate}
\noindent where $R_{p}$ is a real hyperplane containing the line segment $ \gamma [0,1]$ and \begin{equation*} H_{p}\subset R_{p}. \end{equation*} \end{lemma}
\proof
We take a sequence of automorphisms $\phi _{j}\in Aut\left( B^{n+1}\right) $ such that \begin{equation*} \phi _{j}\left( p_{1}^{\prime }\right) =p_{j}^{\prime }\rightarrow p^{\prime }\in L. \end{equation*} Then the composition $\tau _{j}\equiv \varphi \circ \phi _{j}:B^{n+1}\rightarrow D$ forms a normal family so that there is a subsequence $\tau _{m_{j}}\equiv \varphi \circ \phi _{m_{j}}$ which converges to a holomorphic mapping $\tau :B^{n+1}\rightarrow D$ uniformly on every compact subset of $B^{n+1}.$
We claim that $\tau $ is a locally biholomorphic mapping. We take a point $ q\in bB^{n+1}\backslash L$ and $q_{m_{j}}^{\prime }=\tau _{m_{j}}\left( q\right) \in bD$ such that \begin{equation*} q_{m_{j}}^{\prime }=\tau _{m_{j}}\left( q\right) \rightarrow q^{\prime }\in bD, \end{equation*} if necessary, passing to a subsequence. Since $bD$ is spherical, there is an open neighborhood $W$ of the point $q^{\prime }$ and a biholomorphic mapping $\psi $ on $W\cap D$ satisfying \begin{equation*} \psi \in H\left( W\cap D\right) \cap C^{1}\left( W\cap \overline{D}\right) \quad \text{and}\quad \psi \left( W\cap bD\right) \subset bB^{n+1}. \end{equation*} Then we consider the composition $\chi _{m_{j}}\equiv \psi \circ \varphi \circ \phi _{m_{j}}$ so that, by Lemma \ref{sphere} and by abuse of notation, \begin{equation*} \chi _{m_{j}}\in Aut\left( B^{n+1}\right) . \end{equation*} Suppose that the assertion is not true. Then the sequence $\chi _{m_{j}}$ converges to a boundary point $\psi \left( q^{\prime }\right) \in bB^{n+1}$ uniformly on every compact subset of $B^{n+1},$ if necessary, passing to a subsequence. Hence the sequence $\tau _{m_{j}}\equiv \varphi \circ \phi _{m_{j}}$ would converge to a boundary point $q^{\prime }\in bD$ uniformly on every compact subset of $B^{n+1}$. By the way, note that \begin{equation*} \tau _{m_{j}}\left( p_{1}^{\prime }\right) =p_{m_{j}}\rightarrow p\in D. \end{equation*} This is a contradiction. Hence the holomorphic mapping $\tau $ is locally biholomorphic. Thus there is a real number $\delta >0$ such that the mapping $\tau _{m_{j}}$ and the inverse $\tau _{m_{j}}^{-1}$ are analytically continued to a biholomorphic mapping respectively on \begin{equation*} B\left( p_{m_{j}};\delta \right) ,\quad B\left( p_{1}^{\prime };\delta \right) . \end{equation*}
Let $\varepsilon _{j}$ be the euclidean length between the two points $ p^{\prime }$ and $p_{j}^{\prime }$. By First Scaling Lemma, there is a sequence of automorphisms \begin{equation*} \sigma _{j}\in Aut_{p^{\prime }}\left( bB^{n+1}\right) \end{equation*} such that, for a subsequence $\phi _{m_{j}},$ \begin{equation*} U_{\sigma _{j}}=id_{n\times n},\quad \rho _{\sigma _{j}}=\varepsilon _{m_{j}},\quad a_{\sigma _{j}}=0,\quad r_{\sigma _{j}}=0 \end{equation*} and the composition \begin{equation*} \eta _{j}\equiv \sigma _{j}^{-1}\circ \phi _{m_{j}} \end{equation*} uniformly converges to an automorphism of the unit ball $B^{n+1}.$ Then there is a real number $\delta >0$ such that the mapping $\eta _{j}$ and the inverse $\eta _{j}^{-1}$ are analytically continued respectively on \begin{equation*} B\left( p_{1}^{\prime };\delta \right) ,\quad B\left( p_{m_{j}}^{\prime \prime };\delta \right) \end{equation*} where \begin{equation*} p_{m_{j}}^{\prime \prime }=\eta _{j}\left( p_{m_{j}}^{\prime }\right) . \end{equation*}
Then we obtain \begin{equation*} \varphi =\tau _{m_{j}}\circ \eta _{j}^{-1}\circ \sigma _{j}^{-1}. \end{equation*} Thus the mapping $\phi =\varphi ^{-1}$ is analytically continued onto the open set \begin{equation*} \tau _{m_{j}}\circ \eta _{j}^{-1}\left( \sigma _{j}\left( B\left( p_{m_{j}}^{\prime \prime };\delta \right) \right) \right) \end{equation*} which is centered at the point $p_{m_{j}}.$ For a canonical normalizing mapping $\mu _{p^{\prime }},$ we obtain \begin{equation*} \sigma _{j}^{\prime }\equiv \mu _{p^{\prime }}\circ \sigma _{j}\circ \mu _{p^{\prime }}^{-1}:\left\{ \begin{array}{l} z^{*}=\sqrt{\varepsilon _{m_{j}}}z \\ w^{*}=\varepsilon _{m_{j}}w \end{array} \right. . \end{equation*} We set \begin{equation*} \mu _{p^{\prime }}\left( p_{m_{j}}^{\prime \prime }\right) \rightarrow p^{\prime \prime \prime }\in \mu _{p^{\prime }}\left( B^{n+1}\right) \end{equation*} so that, shrinking $\delta >0,$ if necessary, the mapping $\phi =\varphi ^{-1}$ is analytically continued onto the open set \begin{equation*} \tau _{m_{j}}\circ \eta _{j}^{-1}\circ \mu _{p^{\prime }}\left( \sigma _{j}^{\prime }\left( B\left( p^{\prime \prime \prime };\delta \right) \right) \right) \end{equation*} which is centered at the point $p_{m_{j}}.$
Since $\varepsilon _{m_{j}}$ is the euclidean distance between the two points $p$ and $p_{m_{j}},$ the analytically continued region of the mapping $\phi =\varphi ^{-1}$ along the line segment $\gamma [0,1]\subset D$ contains an open set along the line segment $\gamma [0,1]$ which touches to the point $p=\gamma \left( 1\right) $ by an edge shape on the complex line $ \pi _{p}$ and by a slanted paraboloid shape on the real hyperplane $R_{p}.$ This completes the proof.
\endproof
\begin{lemma} Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical differentiable boundary $bD$ and $\phi $ be a biholomorphic mapping on an open neighborhood $U$ of a point $r\in bD$ satisfying \begin{equation*} \phi \in H\left( U\cap D\right) \cap C^{1}\left( U\cap \overline{D}\right) \quad \text{and}\quad \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Let $L$ be the Doggaebi variety associated to the mapping $\phi $ and $ \varphi :B^{n+1}\rightarrow D$ be a locally biholomorphic mapping to be an analytic continuation of the mapping $\phi ^{-1}$. Let $H_{p}$ be the complex hyperplane at the point $p\in D$ in Lemma \ref{doggi} and $\pi _{p}$ be a complex line satisfying \begin{equation*} T_{p}\Bbb{C}^{n+1}=H_{p}\oplus \pi _{p}. \end{equation*} Suppose that, along a line segment $\gamma :[0,1]\rightarrow \pi _{p}\cap D$ with $p=\gamma \left( 1\right) ,$ the germ of a locally biholomorphic mapping $\phi =\varphi ^{-1}$ at the point $\gamma \left( 0\right) $ is analytically continued along the subpath $\gamma [0,\tau ]$ with all $\tau <1 $ as a locally biholomorphic mapping such that the limit point \begin{equation*} \lim_{\tau \rightarrow 1}\phi \circ \gamma \left( \tau \right) \end{equation*} is the point $p^{\prime }\in L\subset bB^{n+1}$ in Lemma \ref{doggi}. Then there is an open neighborhood $V$ of the line segment $\gamma [0,1)$ such that
\begin{enumerate} \item $bV\cap \pi _{p}\cap B\left( p;\delta \right) $ is an angle for a sufficient small $\delta >0,$ which contains the path $\gamma [0,1],$
\item $bV\cap R_{p}\cap B\left( p;\delta \right) $ is a slanted paraboloid for a sufficiently small $\delta >0,$ which smoothly touches the complex hyperplane $H_{p}$ at the point $p,$
\item the germ of a locally biholomorphic mapping $\phi =\varphi ^{-1}$ at the point $\gamma \left( 0\right) $ is analytically continued on $V$ as a locally biholomorphic mapping, \end{enumerate}
\noindent where $R_{p}$ is a real hyperplane containing the line segment $ \gamma [0,1]$ and \begin{equation*} H_{p}\subset R_{p}. \end{equation*} \end{lemma}
\proof
By Lemma \ref{transversally}, $\phi \circ \gamma \left( \tau \right) $ approaches to the point $p^{\prime }\in L$ to a direction transversal to the complex tangent hyperplane at the point $p^{\prime }$ as $\tau \rightarrow 1. $ Then there is a complex hyperplane $H_{p}^{\prime }$ satisfying the conditions in Lemma \ref{doggi}.
Note that the complex hyperplane $H_{p}$ is determined by the mapping $ \varphi :B^{n+1}\rightarrow D$ and the complex tangent hyperplane $ H_{p^{\prime }}$ at the point $p^{\prime }\in L\subset bB^{n+1},$ but $H_{p}$ does not depend on the approaching direction of $\phi \circ \gamma \left( \tau \right) \rightarrow p^{\prime }$ as $\tau \rightarrow 1.$ Thus $ H_{p}^{\prime }=H_{p}$. This completes the proof.
\endproof
\subsection{Biholomorphic equivalence}
\begin{lemma} \label{lifting}Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical differentiable boundary $bD$ and $G$ be the path space of $D$ mod homotopic relation so that $G$ is a universal covering Riemann domain with a natural complex structure and a natural holomorphic covering map $\kappa :G\rightarrow D.$ Suppose that there is a biholomorphic mapping \begin{equation*} \phi \in H\left( U\cap D\right) \cap C^{1}\left( U\cap \overline{D}\right) \end{equation*} for a connected open neighborhood $U$ of a point $p\in bD$ satisfying \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Then the analytic continuation of the inverse mapping $\phi ^{-1}$ is lifted to a holomorphic mapping $\psi $ from $B^{n+1}$ onto $G$ such that $\psi :B^{n+1}\rightarrow G$ is a proper locally biholomorphic mapping, i.e., a finite covering of $G,$ satisfying the following relation: \begin{equation*} \begin{array}{lcl} & & G \\ & \overset{\psi }{\nearrow } & \downarrow \kappa \\ B^{n+1} & \overset{\varphi }{\longrightarrow } & D \end{array} . \end{equation*} \end{lemma}
\proof
By Theorem \ref{branched}, there is a locally biholomorphic mapping \begin{equation*} \varphi :B^{n+1}\rightarrow D \end{equation*} such that $\varphi $ is the analytic continuation of the mapping $\phi ^{-1}$ and \begin{equation*} \varphi =\phi ^{-1}\quad \text{on}\quad bB^{n+1}\backslash L \end{equation*} where $L$ is the Doggaebi variety associated to the mapping $\phi .$
A piecewise line segment curve is a path $\gamma :[0,1]\rightarrow D$ consisting of finitely many line segments. Fix a point $b\in U\cap D.$ We shall show that the mapping $\phi $ is analytically continued along any piecewise line segment curve. Suppose that $\gamma :[0,1]\rightarrow D$ be a piecewise line segment curve with $\gamma \left( 0\right) =b$ such that the germ of the mapping $\phi $ at the point $\gamma \left( 0\right) $ is analytically continued along all subpath $\gamma [0,\tau ]$ with $\tau <1,$ but not the whole path $\gamma [0,1].$ Let $\pi _{p}$ be a complex line satisfying, for sufficiently small $\varepsilon >0,$ \begin{equation*} \gamma [1-\varepsilon ,1]\subset \pi _{p}. \end{equation*} Then, by Second Dogginal Lemma, there is a distinguished complex hyperplane $ H_{p}$ at the point $p=\gamma \left( 1\right) $ satisfying \begin{equation*} T_{p}\Bbb{C}^{n+1}=H_{p}\oplus \pi _{p} \end{equation*} and an open neighborhood $V$ containing the line segment $\gamma [0,\tau ]$ with all $\tau <1$ such that
\begin{enumerate} \item $bV\cap \pi _{p}\cap B\left( p;\varepsilon \right) $ is an angle for a sufficient small $\varepsilon >0,$ which contains the path $\gamma [0,1],$
\item $bV\cap R_{p}\cap B\left( p;\varepsilon \right) $ is a slanted paraboloid for a sufficiently small $\varepsilon >0,$ which smoothly touches the complex hyperplane $H_{p}$ at the point $p,$
\item the germ of a locally biholomorphic mapping $\phi =\varphi ^{-1}$ at the point $\gamma \left( 0\right) $ is analytically continued on $V$ as a locally biholomorphic mapping, \end{enumerate}
\noindent where $R_{p}$ is a real hyperplane satisfying, for sufficiently small $\varepsilon >0,$ \begin{equation*} \gamma [1-\varepsilon ,1]\subset R_{p}\quad \text{and}\quad H_{p}\subset R_{p}. \end{equation*}
Thus we can find a sequence of points $p_{j}\in V\cap B\left( p;\varepsilon \right) $ and a sequence of germs $\phi _{j}$ of the analytic continuation of the mapping $\phi $ at the point $p_{j}$ such that the sequence $p_{j}$ converges to the point $p$ to a direction tangential to the complex hyperplane $H_{p}$ at the point $p.$ We set \begin{equation*} p_{j}^{\prime }=\phi _{j}\left( p_{j}\right) \in B^{n+1}. \end{equation*} Then there is a point $p^{\prime }\in L\subset bB^{n+1}$ such that $ p_{j}^{\prime }\rightarrow p^{\prime }$ to a direction tangential to the complex tangent hyperplane at the point $p^{\prime }\in bB^{n+1}.$ We take a sequence of automorphisms $\varphi _{j}\in Aut\left( B^{n+1}\right) $ such that \begin{equation*} \varphi _{j}\left( p_{1}^{\prime }\right) =p_{j}^{\prime }\rightarrow p^{\prime }\in L. \end{equation*} Then the composition $\tau _{j}\equiv \varphi \circ \varphi _{j}:B^{n+1}\rightarrow D$ forms a normal family so that there is a subsequence $\tau _{m_{j}}\equiv \varphi \circ \varphi _{m_{j}}$ which converges to a holomorphic mapping $\tau :B^{n+1}\rightarrow D$ uniformly on every compact subset of $B^{n+1}.$
We claim that $\tau $ is a locally biholomorphic mapping. We take a point $ q\in bB^{n+1}\backslash L$ and $q_{m_{j}}^{\prime }=\tau _{m_{j}}\left( q\right) \in bD$ such that \begin{equation*} q_{m_{j}}^{\prime }=\tau _{m_{j}}\left( q\right) \rightarrow q^{\prime }\in bD, \end{equation*} if necessary, passing to a subsequence. Since $bD$ is spherical, there is an open neighborhood $W$ of the point $q^{\prime }$ and a biholomorphic mapping $\psi $ on $W\cap D$ satisfying \begin{equation*} \psi \in H\left( W\cap D\right) \cap C^{1}\left( W\cap \overline{D}\right) \quad \text{and}\quad \psi \left( W\cap bD\right) \subset bB^{n+1}. \end{equation*} Then we consider the composition $\chi _{m_{j}}\equiv \psi \circ \varphi \circ \varphi _{m_{j}}$ so that, by Lemma \ref{sphere} and by abuse of notation, \begin{equation*} \chi _{m_{j}}\in Aut\left( B^{n+1}\right) . \end{equation*} Suppose that the assertion is not true. Then the sequence $\chi _{m_{j}}$ converges to a boundary point $\psi \left( q^{\prime }\right) \in bB^{n+1}$ uniformly on every compact subset of $B^{n+1},$ if necessary, passing to a subsequence. Hence the sequence $\tau _{m_{j}}\equiv \varphi \circ \varphi _{m_{j}}$ would converge to a boundary point $q^{\prime }\in bD$ uniformly on every compact subset of $B^{n+1}$. By the way, note that \begin{equation*} \tau _{m_{j}}\left( p_{1}^{\prime }\right) =p_{m_{j}}\rightarrow p\in D. \end{equation*} This is a contradiction. Hence the holomorphic mapping $\tau $ is locally biholomorphic. Thus there is a real number $\delta >0$ such that the mapping $\tau _{m_{j}}$ and the inverse $\tau _{m_{j}}^{-1}$ are analytically continued to a biholomorphic mapping respectively on \begin{equation*} B\left( p_{m_{j}};\delta \right) ,\quad B\left( p_{1}^{\prime };\delta \right) . \end{equation*}
Let $\varepsilon _{j}$ be the euclidean length between the two points $ p^{\prime }$ and $p_{j}^{\prime }$. By Second Scaling Lemma, there is a sequence of automorphisms \begin{equation*} \sigma _{j}\in Aut_{p^{\prime }}\left( bB^{n+1}\right) \end{equation*} such that, for a subsequence $\varphi _{m_{j}},$ \begin{equation*} U_{\sigma _{j}}=id_{n\times n},\quad \rho _{\sigma _{j}}=\varepsilon _{m_{j}}^{2},\quad a_{\sigma _{j}}=0,\quad r_{\sigma _{j}}=0 \end{equation*} and the composition \begin{equation*} \eta _{j}\equiv \sigma _{j}^{-1}\circ \varphi _{m_{j}} \end{equation*} uniformly converges to an automorphism of the unit ball $B^{n+1}.$ Then there is a real number $\delta >0$ such that the mapping $\eta _{j}$ and the inverse $\eta _{j}^{-1}$ are analytically continued respectively on \begin{equation*} B\left( p_{1}^{\prime };\delta \right) ,\quad B\left( p_{m_{j}}^{\prime \prime };\delta \right) \end{equation*} where \begin{equation*} p_{m_{j}}^{\prime \prime }=\eta _{j}\left( p_{m_{j}}^{\prime }\right) . \end{equation*}
Then we obtain \begin{equation*} \varphi =\tau _{m_{j}}\circ \eta _{j}^{-1}\circ \sigma _{j}^{-1}. \end{equation*} Thus the mapping $\phi =\varphi ^{-1}$ is analytically continued onto the open set \begin{equation*} \tau _{m_{j}}\circ \eta _{j}^{-1}\left( \sigma _{j}\left( B\left( p_{m_{j}}^{\prime \prime };\delta \right) \right) \right) \end{equation*} which is centered at the point $p_{m_{j}}.$ For a canonical normalizing mapping $\mu _{p^{\prime }},$ we obtain \begin{equation*} \sigma _{j}^{\prime }\equiv \mu _{p^{\prime }}\circ \sigma _{j}\circ \mu _{p^{\prime }}^{-1}:\left\{ \begin{array}{l} z^{*}=\varepsilon _{m_{j}}z \\ w^{*}=\varepsilon _{m_{j}}^{2}w \end{array} \right. . \end{equation*} We set \begin{equation*} \mu _{p^{\prime }}\left( p_{m_{j}}^{\prime \prime }\right) \rightarrow p^{\prime \prime \prime }\in \mu _{p^{\prime }}\left( B^{n+1}\right) \end{equation*} so that, shrinking $\delta >0,$ if necessary, the mapping $\phi =\varphi ^{-1}$ is analytically continued onto the open set \begin{equation*} \tau _{m_{j}}\circ \eta _{j}^{-1}\circ \mu _{p^{\prime }}\left( \sigma _{j}^{\prime }\left( B\left( p^{\prime \prime \prime };\delta \right) \right) \right) \end{equation*} which is centered at the point $p_{m_{j}}.$
Since the sequence $p_{m_{j}}$ converges to $p$ to a direction tangential to the complex hyperplane $H_{p}$ at the point $p,$ and $\varepsilon _{m_{j}}$
is the euclidean distance between the two points $p$ and $p_{m_{j}},$ the analytically continued region of the mapping $\phi =\varphi ^{-1}$ along the sequence $p_{m_{j}}\in V\cap D$ contains an open set along the converging direction of the sequence $p_{m_{j}}$ by an edge shape tangential to the complex hyperplane $H_{p}$ at the point $p$ and a $\sqrt{\left| x\right| }$ curve shape normal to $H_{p}.$
Thus there is a complex line $\pi _{p}^{\prime }$ satisfying \begin{equation*} \pi _{p}^{\prime }\subset H_{p}\quad \text{and}\quad p\in \pi _{p}^{\prime }, \end{equation*} and an open neighborhood $W$ of the point $p$ such that the mapping $\phi $ is analytically continued on \begin{equation*} W\cap \pi _{p}^{\prime }. \end{equation*} Then we take a line segment $\gamma :[0,1]\rightarrow \pi _{p}^{\prime }\subset H_{p}$ such that \begin{equation*} \gamma :[0,1)\subset W\cap \left( \pi _{p}^{\prime }\backslash p\right) \quad \text{and}\quad \gamma \left( 1\right) =p. \end{equation*} Note that \begin{equation*} \lim_{\tau \rightarrow 1}\phi \circ \gamma \left( \tau \right) =p^{\prime }\in L\subset bB^{n+1}. \end{equation*} By Second Dogginal Lemma, the distinguished complex tangent hyperplane $ H_{p} $ satisfies \begin{equation*} T_{p}\Bbb{C}^{n+1}=H_{p}\oplus \pi _{p}^{\prime } \end{equation*} so that \begin{equation*} \pi _{p}^{\prime }\cap H_{p}=\left\{ p\right\} . \end{equation*} This is a contradiction so that the germ of the mapping $\phi $ at the point $b\in U\cap D$ is analytically continued along the whole path $\gamma [0,1].$ Hence the mapping $\phi $ is analytically continued along any piecewise line segment curve on $D$ as a locally biholomorphic mapping.
We claim that the mapping $\phi :U\cap D\rightarrow B^{n+1}$ is analytically continued along any path on $D$ as a locally biholomorphic mapping. Let $ \gamma :[0,1]\rightarrow D$ be a path on $D.$ Then we take a piecewise line segment curve $\lambda :[0,1]$ and a continuous function $\Gamma :[0,1]\times [0,1]\rightarrow D$ such that \begin{eqnarray*} \gamma \left( 0\right) &=&\lambda \left( 0\right) ,\quad \gamma \left( 1\right) =\lambda \left( 1\right) \\ \Gamma \left( 0,\tau \right) &=&\gamma \left( \tau \right) ,\quad \Gamma \left( 1,\tau \right) =\lambda \left( \tau \right) \quad \text{for all }\tau \in [0,1] \end{eqnarray*} and the path $\Gamma \left( \cdot ,\tau \right) :[0,1]\rightarrow D$ for each $\tau \in [0,1]$ is a piecewise line segment curve on $D.$ Clearly, the mapping $\phi $ is analytically continued along the whole path $\gamma [0,1]$ so that the mapping $\phi $ is analytically continued along any path on $D$ as a locally biholomorphic mapping.
Let $G$ be the path space of $D$ pointed by a point of $U\cap D$ mod homotopic relation so that $G$ is a universal covering of $D$ with a natural complex structure and a natural holomorphic covering map $\kappa :G\rightarrow D.$ By the analytic continuation of the mapping $\phi :U\cap D\rightarrow B^{n+1}$ on $D,$ the mapping $\varphi :B^{n+1}\rightarrow D$ has its natural proper locally biholomorphic lift $\psi :B^{n+1}\rightarrow G.$ Since the mapping $\psi :B^{n+1}\rightarrow G$ is proper and locally biholomorphic, the mapping $\psi $ is a finite covering of $G$. This completes the proof.
\endproof
\begin{theorem} Let $D$ be a bounded domain in $\Bbb{C}^{n+1}$ with spherical differentiable boundary $bD$ such that the fundamental group $\pi _{1}(D)$ is finite. Suppose that there is a biholomorphic mapping \begin{equation*} \phi \in H\left( U\cap D\right) \cap C^{1}\left( U\cap \overline{D}\right) \end{equation*} for a connected open neighborhood $U$ of a point $p\in bD$ satisfying \begin{equation*} \phi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Then $D$ is necessarily simply connected and the mapping $\phi $ is analytically continued to a biholomorphic mapping from $D$ onto $B^{n+1}.$ \end{theorem}
\proof
Let $G$ be the path space of $D$ pointed by a point of $U\cap D$ mod homotopic relation with a natural complex structure and a natural locally biholomorphic covering map $\kappa :G\rightarrow D.$ Then, by Lemma \ref {lifting}, there is a locally biholomorphic finite covering lift $\psi :B^{n+1}\rightarrow G$ satisfying \begin{equation*} \varphi =\kappa \circ \psi \end{equation*} where $\varphi :B^{n+1}\rightarrow D$ is an analytic continuation of the inverse mapping $\phi ^{-1}.$
Since the fundamental group $\pi _{1}\left( D\right) $ is finite, the mapping $\kappa :G\rightarrow D$ is a finite covering. Thus the mapping \begin{equation*} \varphi =\kappa \circ \psi :B^{n+1}\rightarrow D \end{equation*} is a locally biholomorphic finite covering. Therefore, the analytic continuation of the inverse mapping $\phi ^{-1}$ on the boundary $bB^{n+1}$ yields finitely many germs at each point of $bB^{n+1}.$ By Lemma \ref{germs} , the Doggaebi variety $L$ associated to the mapping $\phi $ is empty so that the mapping \begin{equation*} \varphi =\phi ^{-1}:bB^{n+1}\rightarrow bD \end{equation*} is also a locally biholomorphic finite covering. Thus the mapping \begin{equation*} \varphi :\overline{B^{n+1}}\rightarrow \overline{D} \end{equation*} is a finite covering mapping. Then the fixed point property of the close ball $\overline{B^{n+1}}$ implies that the mapping $\varphi :\overline{ B^{n+1}}\rightarrow \overline{D}$ is a simple cover. Otherwise, a nontrivial deck transformation of the closed ball $\overline{B^{n+1}}$ yields a continuous function on $\overline{B^{n+1}}$ with no fixed point. Therefore, the analytic continuation of the mapping $\phi :U\cap D\rightarrow B^{n+1}$ is analytically continued to a biholomorphic mapping \begin{equation*} \varphi ^{-1}:D\rightarrow B^{n+1}. \end{equation*} This completes the proof.
\endproof
\section{Locally Biholomorphic Mappings}
\subsection{Estimates of normalizing parameters}
\begin{lemma} \label{est1}Let $M$ be a nonspherical analytic real hypersurface in $\Bbb{C} ^{n+1}$ and $Aut_{p}(M)$ be the isotropy subgroup at a point $p\in M.$ Suppose that there is a real number $c\geq 1$ satisfying \begin{equation*}
\sup_{\varphi \in Aut_{p}(M)}\left| U_{\varphi }\right| \leq c<\infty . \end{equation*} Then there is a real number $e\geq 1$ satisfying \begin{equation*}
\left| a_{\phi }\right| \leq e,\quad e^{-1}\leq \left| \rho _{\phi }\right|
\leq e,\quad \left| r_{\phi }\right| \leq e \end{equation*} for every local automorphism $\phi \in Aut_{p}(M).$ \end{lemma}
\proof
The real hypersurface $\mu _{p}\left( M\right) $ in normal form is expanded as follows: \begin{equation*} v=\langle z,z\rangle +\sum_{k=4}^{\infty }F_{k}\left( z,\overline{z},u\right) \end{equation*} where \begin{equation*} F_{k}\left( \nu z,\nu \overline{z},\nu ^{2}u\right) =\nu ^{k}F_{k}\left( z, \overline{z},u\right) . \end{equation*} Since $M$ is nonspherical, not all $F_{k}\left( z,\overline{z},u\right) $ are zero. Then we make an estimate of the normalizing parameters $a_{\phi },\rho _{\phi },r_{\phi }$ for $\phi \in Aut_{p}(M)$(cf.\cite{Pa3}). This completes the proof.
\endproof
\begin{theorem} \label{strongly}Let $M$ be a nonspherical strongly pseudoconvex analytic real hypersurface in a complex manifold. Then the local isotropy subgroup $ Aut_{p}\left( M\right) $ is compact for every point $p\in M.$ \end{theorem}
\proof
Because the situation is local, we may assume by taking a coordinate chart, if necessary, that the complex manifold is $\Bbb{C}^{n+1}.$ Then the pseudoconvexity of $M$ leads to \begin{equation*}
\left| U_{\varphi }\right| =1\quad \text{for}\quad \varphi \in Aut_{p}\left( M\right) . \end{equation*} By Lemma \ref{est1}, we obtain the desired result. This completes the proof.
\endproof
\begin{lemma} \label{germ-biho}Let $M,$ $M^{\prime }$ be nonspherical analytic real hypersurfaces in $\Bbb{C}^{n+1}$ and $p,p^{\prime }$ be points respectively of $M,M^{\prime }$ such that the two germs $M$ at $p$ and $M^{\prime }$ at $ p^{\prime }$ are biholomorphically equivalent. Suppose that the isotropy subgroup $Aut_{p}\left( M\right) $ is compact. Then there is a real number $ \delta _{p}>0$ such that each germ of a biholomorphic mapping $\phi $ sending the germ $M$ at $p$ to the germ $M^{\prime }$ at $p^{\prime }$ is analytically continued to the open ball $B(p;\delta _{p}).$ \end{lemma}
\proof
We take a biholomorphic mapping $\phi $ on a connected open neighborhood $U$ of the point $p\in M$ such that \begin{equation*} \phi \left( U\cap M\right) \subset M^{\prime }. \end{equation*} Then every germ of a biholomorphic mapping sending the germ $M$ at $p$ to the germ $M^{\prime }$ at $p^{\prime }$ is one of the following \begin{equation*} \phi \circ \varphi \quad \text{for}\quad \varphi \in Aut_{p}\left( M\right) . \end{equation*} Then the compactness of the group $Aut_{p}\left( M\right) $ leads to the desired result. This completes the proof.
\endproof
\begin{lemma} \label{uniform1}Let $M$ be a nonspherical analytic real hypersurface in $ \Bbb{C}^{n+1}$ such that the isotropy subgroup $Aut_{p}(M)$ is compact at every point $p\in M.$ Then, for each compact subset $K\subset \subset M,$ there is a real number $e\geq 1$ satisfying \begin{equation*}
e^{-1}\leq \left| U_{\phi }\right| \leq e,\quad \left| a_{\phi }\right| \leq e,\quad e^{-1}\leq \left| \rho _{\phi }\right| \leq e,\quad \left| r_{\phi
}\right| \leq e \end{equation*} for every point $p\in K$ and every local automorphism $\phi \in Aut_{p}(M).$ \end{lemma}
\proof
The real hypersurface $\mu _{p}\left( M\right) $ in normal form is expanded as follows: \begin{equation*}
v=\langle z,z\rangle +\sum_{\left| I\right| ,\left| J\right| \geq 2,k\geq 1}\lambda _{IJk}\left( p\right) z^{I}\overline{z}^{J}u^{k}. \end{equation*} Suppose that the normalization $N_{e},$ $e=\left( U,a,\rho ,r\right) \in H,$ transforms the real hypersurface $\mu _{p}\left( M\right) $ to a real hypersurfaces expanded as follows: \begin{equation*}
v=\langle z,z\rangle +\sum_{\left| I\right| ,\left| J\right| \geq 2,k\geq 1}\eta _{IJk}\left( p;U,a,\rho ,r\right) z^{I}\overline{z}^{J}u^{k}. \end{equation*} Note that the element $\left( U_{\phi },a_{\phi },\rho _{\phi },r_{\phi }\right) $ for $\phi \in Aut_{p}\left( M\right) $ is characterized by the following equalities: \begin{equation*} \lambda _{IJk}\left( p\right) =\eta _{IJk}\left( p;U,a,\rho ,r\right) ,\quad
\left| I\right| ,\left| J\right| \geq 2,k\geq 1. \end{equation*} Then, since the algebraic subset of $\left( U,a,\rho ,r\right) $ is characterized by finitely many equalities, there is an integer $K$ such that the following equalities \begin{equation} \lambda _{IJk}\left( p\right) =\eta _{IJk}\left( p;U,a,\rho ,r\right) ,\quad
\left| I\right| ,\left| J\right| ,k\leq K \label{variety} \end{equation} characterize the element $\left( U_{\phi },a_{\phi },\rho _{\phi },r_{\phi }\right) $ for $\phi \in Aut_{p}\left( M\right) .$
Since the isotropy subgroup $Aut_{p}\left( M\right) $ is compact, there is a real number $e_{p}\geq 1$ such that \begin{equation*}
e_{p}^{-1}\leq \left| U_{\phi }\right| \leq e_{p},\quad \left| a_{\phi
}\right| \leq e_{p},\quad e_{p}^{-1}\leq \left| \rho _{\phi }\right| \leq e_{p},\quad \left| r_{\phi }\right| \leq e_{p} \end{equation*} for every element $\phi \in Aut_{p}\left( M\right) .$ Thus the algebraic set defined by the equalities in \ref{variety} is bounded. Note that the boundedness of the algebraic set is preserved on an open neighborhood of the point $p$ so that there are a real number $\delta _{p}>0$ and a real number $ c_{p}\geq 1$ satisfying \begin{equation*}
c_{p}^{-1}\leq \left| U_{\phi }\right| \leq c_{p},\quad \left| a_{\phi
}\right| \leq c_{p},\quad c_{p}^{-1}\leq \left| \rho _{\phi }\right| \leq c_{p},\quad \left| r_{\phi }\right| \leq c_{p} \end{equation*} for every point $p\in B\left( p;\delta _{p}\right) \cap M$ and every local automorphism $\phi \in Aut_{p}(M).$
Since the subset $K$ is compact, there are finitely many points $ p_{1},\cdots ,p_{l}$ such that \begin{equation*} K\subset \bigcup_{k=1}^{l}B\left( p_{k};\delta _{p_{k}}\right) \cap M. \end{equation*} Then we take \begin{equation*} e=\max \left\{ e_{p_{k}}:1\leq k\leq l\right\} . \end{equation*} This completes the proof.
\endproof
\begin{lemma} \label{compact}Let $M$ be nonspherical analytic real hypersurfaces in $\Bbb{C }^{n+1}$ such that the isotropy subgroup $Aut_{p}\left( M\right) $ is compact at every point $p\in M.$ Then, for each compact subset $K\subset \subset M,$ there is a real number $\delta >0$ such that each germ of a local automorphism $\phi \in Aut_{p}\left( M\right) ,$ $p\in K,$ is analytically continued to the open ball $B(p;\delta ).$ \end{lemma}
\proof
By the construction of the normalizing map $\mu _{p},$ the real number $ \delta _{p}$ depend only on the point $p\in M.$ Hence there is a real number $\delta _{p}>0$ for each $p\in M$ such that, for every $q\in B\left( p;\delta _{p}\right) \cap M,$ the mapping $\mu _{q}$ is biholomorphic on $ B\left( q;\delta _{p}\right) $ and the inverse $\mu _{q}^{-1}$ is biholomorphic on $B\left( 0;\delta _{p}\right) .$ Since $K$ is a compact subset, there are finitely many points $p_{1},\cdots ,p_{l}$ such that \begin{equation*} K\subset \bigcup_{k=1}^{l}B\left( p_{k};\delta _{p_{k}}\right) \cap M. \end{equation*} Then we take \begin{equation*} \delta _{1}=\max \left\{ \delta _{p_{k}}:1\leq k\leq l\right\} \end{equation*} so that $\mu _{p}$ is biholomorphic on $B\left( p;\delta _{1}\right) $ and $ \mu _{p}^{-1}$ is biholomorphic on $B\left( 0;\delta _{1}\right) $ for every $p\in K.$
By Lemma \ref{uniform1}, there is a real number $\delta _{2}$ such that every local automorphism $\varphi \in Aut_{0}\left( \mu _{p}\left( M\right) \right) $ for every $p\in K$ is biholomorphically continued to the neighborhood $B\left( 0;\delta _{2}\right) $. Then we take \begin{equation*} \delta =\min \left\{ \delta _{1},\delta _{2}\right\} . \end{equation*} This completes the proof.
\endproof
\begin{lemma} \label{continuation}Let $M$ be a nonspherical analytic real hypersurface and $\gamma :[0,1]\rightarrow M$ be a chain-segment. Let $M^{\prime }$ be a nonspherical analytic real hypersurface in Moser-Vitushkin normal form(cf. \cite{Pa1}). Suppose that there is a connected open neighborhood $U$ of the point $\gamma \left( 0\right) $ and a biholomorphic mapping $\phi $ on $U$ such that \begin{equation*} \phi \left( U\cap M\right) \subset M^{\prime } \end{equation*} and the image $\phi \left( U\cap \gamma [0,1]\right) $ is on the straightened chain of $M^{\prime }.$ Then the mapping $\phi $ is analytically continued along the whole chain-segment $\gamma [0,1]$ as a local biholomorphic mapping. \end{lemma}
\proof
Suppose that the assertion is not true. Then there is a real number $\lambda ,$ $0<\lambda \leq 1,$ such that the mapping $\phi $ is analytically continued along all the subpath $\gamma [0,\tau ],\tau <\lambda ,$ but not the whole path $\gamma [0,\lambda ],$ as a local biholomorphic mapping$.$
Since $\gamma [0,1]$ is a chain-segment, there is a biholomorphic mapping $ \varphi $ on an open neighborhood $V$ of the point $\gamma \left( \lambda \right) $ such that the image $\varphi \left( V\cap \gamma [0,1]\right) $ is on the straightened chain of $M^{\prime }.$ Then we take a point $p\in \varphi \left( V\cap \gamma [0,\lambda )\right) $ and an open neighborhood $ W $ of the point $p$ such that the mapping $\phi $ is analytically continued on $W$ along the chain-segment $\gamma [0,1]$ and $W\subset V$ so that \begin{equation*} \phi \circ \varphi ^{-1}\left( \varphi \left( W\right) \cap M^{\prime }\right) \subset M^{\prime } \end{equation*} and the mapping $\phi \circ \varphi ^{-1}$ maps the straightened chain $ \varphi \left( W\cap \gamma [0,1]\right) $ of $M^{\prime }$ onto the straightened chain of $M^{\prime }.$
Note that the composition $\psi =\phi \circ \varphi ^{-1}$ is necessarily analytically continued along the whole straightened chain of $M^{\prime }$ (cf. \cite{Pa3}). Then, by abuse of notation, the composition $\psi \circ \varphi $ is the analytic continuation of the mapping $\phi $ on the point $ \gamma \left( \lambda \right) $ along the chain-segment $\gamma [0,1].$ This is a contradiction. This completes the proof.
\endproof
\subsection{Analytic continuation on a real hypersurface}
We defined a canonical normalization $\nu _{p}$ of a nondegenerate analytic real hypersurface $M$ at a point $p\in M$ to Moser-Vitushkin normal form(cf. \cite{Pa1}) by the same way of the canonical normalization $\mu _{p}$ to Moser normal form.
\begin{lemma} \label{chainsegment}Let $M,M^{\prime }$ be nonspherical analytic real hypersurfaces in complex manifolds such that $M^{\prime }$ is compact and the isotropy subgroup $Aut_{p}\left( M\right) $ is compact at every point $ p\in M.$ Let $\gamma :[0,1]\rightarrow M$ be a chain-segment and $\phi $ be a biholomorphic mapping on a connected open set $U$ of the point $\gamma \left( 0\right) $ such that \begin{equation*} \phi \left( U\cap M\right) \subset M^{\prime }. \end{equation*} Then the mapping $\phi $ is analytically continued along the whole chain-segment $\gamma [0,1]$ as a local biholomorphic mapping. \end{lemma}
\proof
Suppose that the assertion is not true. Then, without loss of generality, we may assume that the mapping $\phi $ is analytically continued along all the subpath $\gamma [0,\tau ],\tau <1,$ but not the whole chain-segment $\gamma [0,1]$ as a local biholomorphic mapping. By the analytic continuation of the mapping $\phi ,$ there is a path $\phi \circ \gamma :[0,1)\rightarrow M^{\prime }.$ Then we consider the following sequences \begin{eqnarray*} p_{j} &\equiv &\gamma \left( 1-\frac{1}{j}\right) ,\quad j\in \Bbb{N}^{+}, \\ p_{j}^{\prime } &\equiv &\phi \circ \gamma \left( 1-\frac{1}{j}\right) ,\quad j\in \Bbb{N}^{+}. \end{eqnarray*} Since $M^{\prime }$ is compact, passing to a subsequence, say $m_{j},$ there is a point $p^{\prime }\in M^{\prime }$ such that $p_{m_{j}}^{\prime }\rightarrow p^{\prime }$ as $j\rightarrow \infty $ so that the closure \begin{equation*} \overline{\left\{ p_{m_{j}}^{\prime }\equiv \phi \circ \gamma \left( 1-\frac{ 1}{m_{j}}\right) ,\quad j\in \Bbb{N}^{+}\right\} } \end{equation*} is a compact subset in a coordinate chart of the complex manifold.
Without loss of generality, we may assume that the chain $\gamma [0,1]\subset M$ in a coordinate chart of the complex manifold. By abuse of notation, we assume that $M,M^{\prime }$ are in $\Bbb{C}^{n+1}.$ Let $\nu _{p}$ be the canonical normalization of a nondegenerate real hypersurface $M$ at a point $p\in M$ to Moser-Vitushkin normal form.
Then the mapping $\phi $ yields the following germs of a biholomorphic mapping:
\begin{equation*} \varphi _{j}\equiv \nu _{p_{m_{j}}^{\prime }}\circ \phi \circ \nu _{p_{m_{j}}}^{-1}:\nu _{p_{m_{j}}}\left( M\right) \rightarrow \nu _{p_{m_{j}}^{\prime }}\left( M^{\prime }\right) . \end{equation*} Since $\gamma :[0,1]\rightarrow M$ is a chain-segment, there is a sequence of local automorphisms \begin{equation*} \sigma _{j}\in Aut_{p_{m_{j}}}\left( M\right) \end{equation*} such that the mapping $\nu _{p_{m_{j}}^{\prime }}\circ \phi \circ \sigma _{j} $ sends the germ of the chain $\gamma $ at the point $p_{m_{j}}$ onto the straightened chain of the real hypersurface $\mu _{p_{m_{j}}^{\prime }}\left( M^{\prime }\right) $ in Moser-Vitushkin normal form. By Lemma \ref {continuation}, the composition $\psi _{j}=\nu _{p_{m_{j}}^{\prime }}\circ \phi \circ \sigma _{j}$ is analytically continued along the whole chain-segment $\gamma [0,1].$ Therefore, there is a real number $\delta >0$ such that, by abuse of notation, the mapping $\psi _{j}$ is biholomorphic on the open neighborhood $B\left( p_{m_{j}};\delta \right) .$
By Lemma \ref{compact}, the set $\left\{ \nu _{p_{m_{j}}}\circ \sigma _{j}\circ \nu _{p_{m_{j}}}^{-1}:j\in \Bbb{N}^{+}\right\} $ is relatively compact.. Thus, by passing to a subsequence and shrinking $\delta >0$, if necessary, we may assume that the composition $\tau _{j}=\nu _{p_{m_{j}}}\circ \sigma _{j}\circ \nu _{p_{m_{j}}}^{-1}$ and its converse $ \tau _{j}^{-1}$ are biholomorphically continued on the open neighborhood $ B\left( 0;\delta \right) .$
Since $p_{m_{j}}\rightarrow \gamma \left( 1\right) $ and $p_{m_{j}}^{\prime }\rightarrow p^{\prime },$ by passing to a subsequence and shrinking $\delta >0$, if necessary, we may assume that the canonical normalizations $\nu _{p_{m_{j}}},\nu _{p_{m_{j}}^{\prime }}$ and their inverses $\nu _{p_{m_{j}}}^{-1},\nu _{p_{m_{j}}^{\prime }}^{-1}$ are biholomorphically continued respectively on the open neighborhoods \begin{equation*} B\left( p_{m_{j}};\delta \right) ,\quad B\left( p_{m_{j}}^{\prime };\delta \right) ,\quad B\left( 0;\delta \right) ,\quad B\left( 0;\delta \right) . \end{equation*} Then, by abuse of notation, the composition \begin{equation*} \chi _{j}=\nu _{p_{m_{j}}^{\prime }}^{-1}\circ \psi _{j}\circ \nu _{p_{m_{j}}}^{-1}\circ \tau _{j}^{-1}\circ \nu _{p_{m_{j}}} \end{equation*} is biholomorphic on the open neighborhood $B\left( p_{m_{j}};\delta \right) , $ if necessary, by shrinking $\delta >0.$
Note that the mapping $\chi _{j}$ is a local biholomorphic continuation of the germ of the mapping $\phi $ at the point $p_{m_{j}}$ for each $j\in \Bbb{ N}^{+}.$ Thus we take an integer $K$ such that \begin{equation*} \gamma \left( 1\right) \in B\left( p_{m_{K}};\frac{\delta }{2}\right) \end{equation*} so that the mapping $\chi _{K}$ is an analytic continuation of the mapping $ \phi $ on the point $\gamma \left( 1\right) $ along the chain-segment $ \gamma [0,1].$ This contradiction completes the proof.
\endproof
\begin{lemma} \label{alongpath}Let $M,M^{\prime }$ be nonspherical analytic real hypersurfaces in complex manifolds such that $M^{\prime }$ is compact and the isotropy subgroup $Aut_{p}\left( M\right) $ is compact at every point $ p\in M.$ Suppose that there is a biholomorphic mapping $\phi $ on a connected open set $U$ of a point $p\in M$ such that \begin{equation*} \phi \left( U\cap M\right) \subset M^{\prime }. \end{equation*} Then the biholomorphic mapping $\phi $ is analytically continued along any path on $M$ as a local biholomorphic mapping. \end{lemma}
\proof
For each point $p\in M,$ we make a biholomorphically equivalent deformation of the real hypersurfaces $\mu _{p}\left( M\right) $ in normal form continuously to a real hyperquadric by using the scaling mapping \begin{equation*} \left\{ \begin{array}{l} z^{*}=\lambda z \\ w^{*}=\lambda ^{2}w \end{array} \right. ,\quad \lambda \in \Bbb{R}^{+}. \end{equation*} Because the chain is characterized by an order differential equation, the continuous family of chains on the real hyperquadric is continuously deformed by the parameter $\lambda $ on a real hypersurface biholomorphic to $M$ near the point $p$(cf. \cite{Pa3}).
Let $\gamma :[0,1]\rightarrow M$ be a path on $M$ such that $\gamma \left( 0\right) \in U\cap M.$ Then, for each $\tau \in [0,1],$ there is a real number $\varepsilon _{\tau }>0,$ a point $p_{\tau }\in M$ and a continuous function \begin{equation*} \Gamma _{\tau }:[0,1]\times \left( [0,1]\cap (\tau -\varepsilon _{\tau },\tau +\varepsilon _{\tau })\right) \rightarrow M \end{equation*} such that
\begin{enumerate} \item $\Gamma _{\tau }\left( \cdot ,\sigma \right) :[0,1]\rightarrow M$ is a chain-segment for each $\sigma \in [0,1]\cap (\tau -\varepsilon _{\tau },\tau +\varepsilon _{\tau }),$
\item $\Gamma _{\tau }\left( 0,\sigma \right) =\gamma \left( \sigma \right) $ for each $\sigma \in [0,1]\cap (\tau -\varepsilon _{\tau },\tau +\varepsilon _{\tau }),$
\item $\Gamma _{\tau }\left( 1,\sigma \right) =p_{\tau }$ for all $\sigma \in [0,1]\cap (\tau -\varepsilon _{\tau },\tau +\varepsilon _{\tau }).$ \end{enumerate}
Note that the family $\left\{ [0,1]\cap (\tau -\varepsilon _{\tau },\tau +\varepsilon _{\tau }):\tau \in [0,1]\right\} $ is an open covering of the compact set $[0,1].$ Thus there is a finite subcover \begin{equation*} \left\{ \lbrack 0,1]\cap (\tau _{j}-\varepsilon _{\tau _{j}},\tau _{j}+\varepsilon _{\tau _{j}}):\tau _{j}\in [0,1],\quad j=1,\cdots ,m\right\} . \end{equation*} Then, by Lemma \ref{chainsegment}, the biholomorphic mapping $\phi $ is analytically continued along the whole path $\gamma [0,1]$ as a local biholomorphic mapping. This completes the proof.
\endproof
\subsection{Holomorphic mapping on the boundary}
\begin{lemma} \label{polynomial}Let $Q$ be a real hyperquadric defined by \begin{equation*} v=\langle z,z\rangle \equiv z^{1}\overline{z}^{1}+\cdots +z^{n}\overline{z} ^{n} \end{equation*} and $\phi $ be a polynomial mapping as follows: \begin{equation*} \phi :\left\{ \begin{array}{l} z^{*}=f(z,w) \\ w^{*}=g(z,w) \end{array} \right. \end{equation*} where, for $m\geq 2,$ \begin{eqnarray} f(\mu z,\mu ^{2}w) &=&\mu ^{m}f(z,w) \notag \\ g(\mu z,\mu ^{2}w) &=&\mu ^{2m}g(z,w). \label{condi} \end{eqnarray} Suppose that \begin{equation*} \phi \left( Q\right) \subset Q. \end{equation*} Then $\phi \equiv 0.$ \end{lemma}
\proof
The mapping $\phi =(f,g)$ yields the identity \begin{equation} \Im g(z,u+i\langle z,z\rangle )=\langle f(z,u+i\langle z,z\rangle ),f(z,u+i\langle z,z\rangle )\rangle . \label{identity} \end{equation} Suppose that $m$ is even, i.e., $m=2k\geq 2.$ We may consider $z,\overline{z} ,u$ as independent variables in the identity \ref{identity}. Taking $ \overline{z}=0$ in the equality \ref{identity} yields \begin{equation*} g(z,u)-\overline{g}(0,u)=2i\langle f(z,u),f(0,u)\rangle . \end{equation*} Thus we can put \begin{equation} g(z,u)=2i\langle f(z,u),f(0,u)\rangle +\Re g(0,u)-i\langle f(0,u),f(0,u)\rangle . \label{identity2} \end{equation} Note that \begin{equation*} f(0,u)=f(0,1)u^{\frac{m}{2}},\quad g(0,u)=g(0,1)u^{m}. \end{equation*} Then the identity \ref{identity} yields \begin{eqnarray*} &&2\Re \left\{ \langle f(z,u+i\langle z,z\rangle ),f(0,1)\rangle (u+i\langle z,z\rangle )^{\frac{m}{2}}\right\} \\ &&+\Re g(0,1)\Im (u+i\langle z,z\rangle )^{m} \\ &&-\langle f(0,1),f(0,1)\rangle \Re (u+i\langle z,z\rangle )^{m} \\ &=&\langle f(z,u+i\langle z,z\rangle ),f(z,u+i\langle z,z\rangle )\rangle . \end{eqnarray*}
Let $p\left( z,\overline{z}\right) $ be a polynomial of the variables $z, \overline{z}$ satisfying \begin{equation*} p\left( \mu z,\nu \overline{z}\right) =\mu ^{l}\nu ^{m}p\left( z,\overline{z} \right) . \end{equation*} Then the polynomial $p\left( z,\overline{z}\right) $ is said to be of type $ (l,m).$ Collecting the terms of type $(l,1),$ $l=0,1,\cdots ,m-1,$ yields \begin{equation*} miu^{-1}\langle z,z\rangle \left\{ \langle f(z,u),f(0,u)\rangle -\langle f(0,u),f(0,u)\rangle +\Re g(0,u)\right\} =0. \end{equation*} Hence we obtain \begin{equation*} \langle f(z,u),f(0,u)\rangle -\langle f(0,u),f(0,u)\rangle +\Re g(0,u)=0, \end{equation*} so that \begin{eqnarray*} \langle f(z,u),f(0,u)\rangle &=&\langle f(0,u),f(0,u)\rangle \\ \Re g(0,u) &=&0. \end{eqnarray*} Thus, from the identity \ref{identity2}, we obtain \begin{equation*} g(z,u)=i\langle f(0,u),f(0,u)\rangle \end{equation*} by which the identity \ref{identity} yields \begin{equation*} \langle f(0,1),f(0,1)\rangle \Re (u+i\langle z,z\rangle )^{m}=\langle f(z,u+i\langle z,z\rangle ),f(z,u+i\langle z,z\rangle )\rangle . \end{equation*} Note that \begin{equation*} \langle f(z,u+i\langle z,z\rangle ),f(z,u+i\langle z,z\rangle )\rangle \geq 0, \end{equation*} but \begin{equation*} \Re (u+i\langle z,z\rangle )^{m}=u^{m}-\frac{m(m-1)}{2}u^{m-2}\langle z,z\rangle ^{2}+\cdots . \end{equation*} Thus collecting terms of type $(2,2)$ yields \begin{eqnarray*} &&-\frac{m(m-1)}{2}\langle f(0,1),f(0,1)\rangle u^{m-2}\langle z,z\rangle ^{2} \\
&=&\left| \sum_{\alpha ,\beta }\frac{z^{\alpha }z^{\beta }}{2}\left( \frac{
\partial ^{2}f}{\partial z^{\alpha }\partial z^{\beta }}\right) (0,u)\right| ^{2}+\frac{mu^{-2}}{2}\langle z,z\rangle ^{2}\langle f(0,u),f(0,u)\rangle \\ &\geq &0. \end{eqnarray*} Since $m\geq 2,$ we obtain \begin{equation*} \langle f(0,1),f(0,1)\rangle =0 \end{equation*} which yields \begin{equation*} \langle f(z,u+i\langle z,z\rangle ),f(z,u+i\langle z,z\rangle )\rangle =0, \end{equation*} i.e., \begin{equation*} f(z,w)=g(z,w)=0. \end{equation*}
Suppose that $m$ is odd, i.e., $m=2k+1\geq 3.$ We may consider $z,\overline{z },u$ as independent variables in the identity \ref{identity}. Then, by the condition \ref{condi}, taking $\overline{z}=0$ in the equality \ref{identity} yields \begin{equation*} g(z,u)=\overline{g}(0,u) \end{equation*} since \begin{equation*} f(0,u)=0. \end{equation*} Thus we can put \begin{equation*} g(z,u)=g(0,u),\quad g(0,1)\in \Bbb{R}. \end{equation*} Then the identity \ref{identity} yields \begin{equation} g(0,1)\Im (u+i\langle z,z\rangle )^{m}=\langle f(z,u+i\langle z,z\rangle ),f(z,u+i\langle z,z\rangle )\rangle . \label{identity3} \end{equation} Note that \begin{equation*} \langle f(z,u+i\langle z,z\rangle ),f(z,u+i\langle z,z\rangle )\rangle \geq 0, \end{equation*} but \begin{eqnarray*} \Im (u+i\langle z,z\rangle )^{m} &=&mu^{m-1}\langle z,z\rangle \\ &&-\frac{m(m-1)(m-2)}{2}u^{m-3}\langle z,z\rangle ^{3}+\cdots . \end{eqnarray*} Thus collecting terms of type $(1,1)$ yields \begin{eqnarray*} &&mg(0,1)u^{m-1}\langle z,z\rangle \\
&=&\left| \sum_{\alpha }z^{\alpha }\left( \frac{\partial f}{\partial z^{\alpha }}\right) (0,u)\right| ^{2}\geq 0. \end{eqnarray*} Collecting terms of type $(3,3)$ yields \begin{eqnarray*} &&-\frac{m(m-1)(m-2)}{2}g(0,1)u^{m-3}\langle z,z\rangle ^{3} \\
&=&\left| \sum_{\alpha ,\beta ,\gamma }\frac{z^{\alpha }z^{\beta }z^{\gamma } }{6}\left( \frac{\partial ^{3}f}{\partial z^{\alpha }\partial z^{\beta
}\partial z^{\gamma }}\right) (0,u)\right| ^{2} \\
&&+\frac{m-1}{2}\left| \sum_{\alpha }z^{\alpha }\left( \frac{\partial f}{
\partial z^{\alpha }}\right) (0,u)\right| ^{2}u^{-2}\langle z,z\rangle ^{2} \\ &\geq &0. \end{eqnarray*} Since $m\geq 3,$ we obtain \begin{equation*} g(0,1)\langle z,z\rangle =0 \end{equation*} by which the identity \ref{identity3} yields \begin{equation*} \langle f(z,u+i\langle z,z\rangle ),f(z,u+i\langle z,z\rangle )\rangle =0, \end{equation*} i.e., \begin{equation*} f(z,w)=g(z,w)=0. \end{equation*} This completes the proof.
\endproof
\begin{lemma} \label{nonconstant}Let $M,M^{\prime }$ be strongly pseudoconvex analytic real hypersurfaces in $\Bbb{C}^{n+1}.$ Let $\phi $ be a holomorphic mapping on an open neighborhood $U$ of a point $p\in M$ such that \begin{equation*} \phi (U\cap M)\subset M^{\prime }. \end{equation*} Then the mapping $\phi $ is either a constant mapping or a biholomorphic mapping on $U,$ if necessary, shrinking $U.$ \end{lemma}
\proof
We take $q=\phi \left( p\right) $ so that \begin{equation*} \varphi \equiv \mu _{q}\circ \phi \circ \mu _{p}^{-1}:\mu _{p}\left( M\right) \rightarrow \mu _{q}\left( M^{\prime }\right) . \end{equation*} Then the mapping $\varphi $ is a holomorphic mapping on an open neighborhood $V$ of the origin satisfying \begin{equation*} \varphi \left( \mu _{p}\left( M\right) \cap V\right) \subset \mu _{q}\left( M^{\prime }\right) . \end{equation*}
The mapping $\varphi =(f,g)$ in $\Bbb{C}^{n}\times \Bbb{C}$ is decomposed as follows: \begin{equation*} f(z,w)=\sum_{k=1}^{\infty }f_{k}(z,w),\mathrm{\quad }g(z,w)=\sum_{k=1}^{ \infty }g_{k}(z,w), \end{equation*} where \begin{equation*} f_{m}(\mu z,\mu ^{2}w)=\mu ^{m}f_{m}(z,w),\mathrm{\quad }g_{m}(\mu z,\mu ^{2}w)=\mu ^{m}g_{m}(z,w). \end{equation*} We assume that $\mu _{p}\left( M\right) ,\mu _{q}\left( M^{\prime }\right) $ are defined respectively by the equations \begin{equation*} v=F(z,\overline{z},u),\mathrm{\quad }v=\langle z,z\rangle +F^{*}(z,\overline{ z},u) \end{equation*} where \begin{equation*}
F(z,\overline{z},u)=\langle z,z\rangle +O\left( \left| z\right| ^{4}\right) ,
\mathrm{\quad }F^{*}(z,\overline{z},u)=O\left( \left| z\right| ^{4}\right) . \end{equation*}
Then we obtain the following identity near the origin \begin{eqnarray} &&\Im g(z,u+iF(z,\overline{z},u)) \notag \\ &=&\langle f(z,u+iF(z,\overline{z},u)),f(z,u+iF(z,\overline{z},u))\rangle \notag \\ &&+F^{*}(f(z,u+iF(z,\overline{z},u)),\overline{f(z,u+iF(z,\overline{z},u))} ,\Re g(z,u+iF(z,\overline{z},u))). \label{iden} \end{eqnarray} Then, up to weight $2,$ we obtain \begin{align} \Im g_{1}(z,0)& =0 \notag \\ \Im g_{2}(z,u+i\langle z,z\rangle )& =\langle f_{1}(z,0),f_{1}(z,0)\rangle \notag \end{align} which yields \begin{equation*} \langle f_{1}(z,0),f_{1}(z,0)\rangle =\langle z,z\rangle \Re g_{2}(0,1) \end{equation*} and \begin{eqnarray*} g_{1}(z,0) &=&0 \\ g_{2}(z,w) &=&g_{2}(0,1)w,\mathrm{\quad }g_{2}(0,1)\in \Bbb{R}. \end{eqnarray*} Note that $g_{2}(0,1)\neq 0$ if and only if the mapping $\varphi $ is a biholomorphic mapping at the origin.
Suppose that the assertion is not true. Then we have \begin{equation*} f_{1}(z,0)=g_{1}(z,0)=g_{2}(z,w)=0. \end{equation*} As inductive hypothesis, suppose that, for $m\geq 2,$ \begin{eqnarray} f_{l}(z,w) &=&0,\quad l=1,\cdots ,m-1, \notag \\ g_{l}(z,w) &=&0,\quad l=1,\cdots ,2m-2. \label{hypo} \end{eqnarray} By the condition \begin{equation*}
F(z,\overline{z},u)=\langle z,z\rangle +O\left( \left| z\right| ^{4}\right)
,\quad F^{*}(z,\overline{z},u)=o\left( \left| z\right| ^{4}\right) , \end{equation*} the identity \ref{iden} yields \begin{equation*} \Im g_{2m-1}(z,u+i\langle z,z\rangle )=0 \end{equation*} Here we may consider $z,\overline{z},u$ as independent variables so that taking $\overline{z}=0$ yields \begin{equation*} g_{2m-1}(z,u+i\langle z,z\rangle )=\overline{g_{2m-1}}(0,u-i\langle z,z\rangle )=0. \end{equation*} Hence the hypothesis \ref{hypo} necessarily comes to \begin{eqnarray*} f_{l}(z,w) &=&0,\quad l=1,\cdots ,m-1, \\ g_{l}(z,w) &=&0,\quad l=1,\cdots ,2m-1. \end{eqnarray*} Then the identity \ref{iden} yields \begin{equation*} \Im g_{2m}(z,u+i\langle z,z\rangle )=\langle f_{m}(z,u+i\langle z,z\rangle ),f_{m}(z,u+i\langle z,z\rangle )\rangle . \end{equation*} Note that the polynomial mapping $\varphi _{m}\equiv (f_{m},g_{2m}),$ $m\geq 2,$ satisfies \begin{equation*} \varphi _{m}\left( Q\right) \subset Q. \end{equation*} By Lemma \ref{polynomial}, $\varphi _{m}\equiv 0$ so that \begin{eqnarray*} f_{l}(z,w) &=&0,\quad l=1,\cdots ,m, \\ g_{l}(z,w) &=&0,\quad l=1,\cdots ,2m. \end{eqnarray*} This completes the induction so that $\varphi \equiv 0,$ i.e., the mapping $ \varphi $ is a constant mapping. This completes the proof.
\endproof
\subsection{Proper holomorphic mappings}
\begin{lemma} Let $D,D^{\prime }$ be strongly pseudoconvex bounded domains with nonspherical real analytic boundaries $bD,bD^{\prime }$ such that the boundaries $bD,bD^{\prime }$ are both simply connected. Suppose that there is a biholomorphic mapping $\phi $ on a connected open neighborhood $U$ of a point $p\in bD$ such that \begin{equation*} \phi \left( U\cap bD\right) \subset bD^{\prime }. \end{equation*} Then the mapping $\phi $ is analytically continued to a biholomorphic mapping from $D$ onto $D^{\prime }.$ \end{lemma}
\proof
Since $bD,bD^{\prime }$ are simply connected, by Lemma \ref{alongpath}, the mappings $\phi ,\phi ^{-1}$ are both analytically continued, by abuse of notation, respectively to a biholomorphic mapping \begin{equation*} \phi :D\rightarrow D \end{equation*} and \begin{equation*} \phi ^{-1}:D^{\prime }\rightarrow D. \end{equation*} Then, by the identity theorem, the mapping $\phi $ is biholomorphic. This completes the proof.
\endproof
\begin{lemma} Let $D,D^{\prime }$ be strongly pseudoconvex bounded domains with nonspherical real analytic boundaries $bD,bD^{\prime }$ such the boundary $ bD^{\prime }$ is simply connected and the closed set $\overline{D^{\prime }}$ satisfies the fixed point property. Suppose that there is a biholomorphic mapping $\phi $ on a connected open neighborhood $U$ of a point $p\in bD$ such that \begin{equation*} \phi \left( U\cap bD\right) \subset bD^{\prime }. \end{equation*} Then the mapping $\phi $ is analytically continued to a biholomorphic mapping from $D$ onto $D^{\prime }.$ \end{lemma}
\proof
Since $bD^{\prime }$ is simply connected, by Lemma \ref{alongpath}, the inverse mapping $\phi ^{-1}$ is analytically continued to a locally biholomorphic proper mapping \begin{equation*} \varphi :D^{\prime }\rightarrow D. \end{equation*} Note that the mappings $\varphi :D^{\prime }\rightarrow D$ and $\varphi :bD^{\prime }\rightarrow bD$ are both covering maps. We claim that the covering is simple. Otherwise, there is a nontrivial deck transformation of $ \overline{D^{\prime }}$ yields a continuous self mapping of $\overline{ D^{\prime }}$ with no fixed point. This is a contradiction. Therefore, the mapping $\varphi :D^{\prime }\rightarrow D$ is biholomorphic. This completes the proof.
\endproof
\begin{lemma} \label{nonspheric}Let $D,D^{\prime }$ be strongly pseudoconvex bounded domains with nonspherical real analytic boundaries $bD,bD^{\prime }$ such that the domain $D$ and the boundary $bD^{\prime }$ are both simply connected. Suppose that there is a biholomorphic mapping $\phi $ on a connected open neighborhood $U$ of a point $p\in bD$ such that \begin{equation*} \phi \left( U\cap bD\right) \subset bD^{\prime }. \end{equation*} Then the mapping $\phi $ is analytically continued to a biholomorphic mapping from $D$ onto $D^{\prime }.$ \end{lemma}
\proof
Since $bD^{\prime }$ is simply connected, by Lemma \ref{alongpath}, the inverse mapping $\phi ^{-1}$ is analytically continued to a locally biholomorphic proper mapping \begin{equation*} \varphi :D^{\prime }\rightarrow D. \end{equation*} Note that the mapping $\varphi :D^{\prime }\rightarrow D$ is a covering map. Since $D$ is simply connected, the covering is simple. Therefore, the mapping $\varphi :D^{\prime }\rightarrow D$ is biholomorphic. This completes the proof.
\endproof
\begin{theorem} \label{convex}Let $D,D^{\prime }$ be strongly convex bounded domains with real analytic boundaries $bD,bD^{\prime }$. Suppose that there is a biholomorphic mapping $\phi $ on a connected open neighborhood $U$ of a point $p\in bD$ such that \begin{equation*} \phi \left( U\cap bD\right) \subset bD^{\prime }. \end{equation*} Then the mapping $\phi $ is analytically continued to a biholomorphic mapping from $D$ onto $D^{\prime }.$ \end{theorem}
\proof
Note that a strongly convex bounded domain is homeomorphic to an open ball $ B^{n+1}.$ Suppose that the boundaries $bD,bD^{\prime }$ are spherical. Then we take a biholomorphic mapping $\varphi $ on $U,$ if necessary, shrinking $ U,$ such that \begin{equation*} \varphi \left( U\cap bD\right) \subset bB^{n+1}. \end{equation*} Then, by Lemma \ref{boundary}, the mapping $\varphi $ and the composition $ \psi \equiv \varphi \circ \phi ^{-1}$ are analytically continued, by abuse of notation, to biholomorphic mappings as follows: \begin{equation*} \varphi :D\rightarrow B^{n+1}\quad \text{and}\quad \psi :D^{\prime }\rightarrow B^{n+1}. \end{equation*} Thus the composition $\psi ^{-1}\circ \varphi :D\rightarrow D^{\prime }$ is a biholomorphic mapping and the analytic continuation of the mapping $\phi .$
Suppose that the boundaries $bD,bD^{\prime }$ are nonspherical. Then, by Lemma \ref{nonspheric}, the mapping $\phi $ is analytically continued to a biholomorphic mapping from $D$ onto $D^{\prime }.$ This completes the proof.
\endproof
\begin{lemma} \label{proper}Let $D,D^{\prime }$ be strongly pseudoconvex bounded domains with real analytic boundaries $bD,bD^{\prime }.$ Suppose that there is a proper holomorphic mapping $\phi :D\rightarrow D^{\prime }.$ Then there is an open neighborhood $U$ of a point $p\in bD$ such that the mapping $\phi $ is analytically continued on $U$ and \begin{equation*} \phi \left( U\cap bD\right) \subset bD^{\prime }. \end{equation*} \end{lemma}
\proof
We may apply the boundary regularity of Lemma \ref{b-regularity} so that the mapping $\varphi :D\rightarrow D^{\prime }$ is holomorphic on an open neighborhood of $\overline{D}$. This completes the proof.
\endproof
\begin{lemma} \label{Alexander}Let $\phi :B^{n+1}\rightarrow B^{n+1}$ be a proper holomorphic mapping. Then $\phi \in Aut\left( B^{n+1}\right) .$ \end{lemma}
\proof
By Lemma \ref{proper}, there is an open neighborhood $U$ of a point $p\in bB^{n+1}$ such that $\phi $ is analytically continued on $U$ and \begin{equation*} \phi \left( U\cap bB^{n+1}\right) \subset bB^{n+1}. \end{equation*} By Lemma \ref{sphere}, $\phi \in Aut\left( B^{n+1}\right) .$ This completes the proof.
\endproof
\begin{lemma} \label{equiv}Let $D,D^{\prime }$ be strongly pseudoconvex bounded domains in $\Bbb{C}^{n+1}$ with real analytic boundaries $bD,bD^{\prime }.$ Suppose that there is a proper holomorphic mapping $\phi :D\rightarrow D^{\prime }$. Then the mapping $\phi :D\rightarrow D^{\prime }$ is a locally biholomorphic mapping so that $\phi :\overline{D}\rightarrow \overline{D^{\prime }}$ is a covering map. \end{lemma}
\proof
By Lemma \ref{proper}, there is a point $p\in bD$ and an open neighborhood $ U $ of the point $p$ such that the mapping $\phi $ is analytically continued to $U$ and \begin{equation*} \phi \left( U\cap bD\right) \subset bD^{\prime }. \end{equation*} By Lemma \ref{nonconstant}, the mapping $\phi $ is biholomorphic on $U,$ if necessary, shrinking $U.$
Suppose that the boundaries $bD,bD^{\prime }$ are spherical. Then, by Lemma \ref{sph-proper}, the mapping $\phi :D\rightarrow D^{\prime }$ is a locally biholomorphic mapping.
Suppose that the boundary $bD,bD^{\prime }$ are nonspherical. Then, by Lemma \ref{alongpath}, the mapping $\phi :D\rightarrow D^{\prime }$ is analytically continued on an open neighborhood of the boundary $bD$ to be locally biholomorphic. Thus the mapping $\phi $ is a locally biholomorphic mapping. This completes the proof.
\endproof
\begin{theorem} Let $D$ be a strongly pseudoconvex bounded domain with real analytic boundary $bD.$ Suppose that there is a proper holomorphic self mapping $\phi :D\rightarrow D.$ Then $\phi $ is a biholomorphic automorphism of $D$. \end{theorem}
\proof
By Lemma \ref{equiv}, the mapping $\phi :\overline{D}\rightarrow \overline{D} $ is a self covering map. We claim that $\phi :D\rightarrow D$ is a simple covering. Otherwise, there would be a integer $m>1$ such that, for every $ p\in \overline{D},$ $m$ is the order of the set \begin{equation*} \left\{ q\in \overline{D}:\phi \left( q\right) =p\right\} . \end{equation*} Then we define the $k$ times composition $\phi ^{k}$ of the mapping $\phi $ as follows \begin{equation*} \phi ^{k}\equiv \underbrace{\phi \circ \cdots \circ \phi }_{k}:\overline{D} \rightarrow \overline{D}. \end{equation*} Note that the inverse image of each point $p\in \overline{D}$ under the mapping $\phi ^{k}:\overline{D}\rightarrow \overline{D},$ $k\in \Bbb{N}^{+},$ is a set of order $m^{k}.$
We claim that, for a given real number $\delta >0,$ there is a point $q\in bD $ and a compact subset $K\subset \subset D$ and a subsequence $\phi ^{m_{k}}$ such that \begin{equation*} \phi ^{-m_{k}}\left( B\left( q;\delta \right) \cap D\right) \cap K\neq \emptyset \quad \text{for all }k\in \Bbb{N}^{+}. \end{equation*} Otherwise, for every compact subset $K\subset \subset D,$ there is an integer $l$ such that \begin{equation*} \phi ^{-k}\left( D\backslash K\right) \subset D\backslash K\quad \text{for all }k\geq l. \end{equation*} Since $\phi ^{k}:D\rightarrow D$ is a finite covering map, we have \begin{equation*} D=\phi ^{-k}\left( K\cup D\backslash K\right) =\phi ^{-k}\left( K\right) \cup \phi ^{-k}\left( D\backslash K\right) \end{equation*} so that \begin{equation*} K\subset \phi ^{-k}\left( K\right) \quad \text{for all }k\geq l. \end{equation*} Note that the inverse image of $\phi ^{-k}\left( K\right) $ is a union of $ m^{k}$ disconnected compact subsets. We may assume that $K$ is connected so that $K$ is in a compact subset of the union $\phi ^{-k}\left( K\right) .$ This is impossible. Thus, with such a point $q\in bD$, we take an accumulation point $q^{\prime }\in D^{\prime }$ of the set \begin{equation*} \lim_{k\rightarrow \infty }\phi ^{-m_{k}}\left( \left\{ q\right\} \right) =\lim_{k\rightarrow \infty }\left\{ p\in bD:\phi ^{m_{k}}\left( p\right) =q\right\} . \end{equation*}
Suppose that $bD$ is spherical. Then there are open neighborhoods $ U,U^{\prime }$ respectively of $q,q^{\prime }$ and biholomorphic mappings $ \psi ,\psi ^{\prime }$ respectively on $U,U^{\prime }$ such that \begin{eqnarray*} \psi \left( U\cap bD\right) &\subset &bB^{n+1} \\ \psi ^{\prime }\left( U^{\prime }\cap bD\right) &\subset &bB^{n+1}. \end{eqnarray*} By Lemma \ref{sphere}, the compositions \begin{equation*} \varphi _{k}\equiv \psi ^{\prime }\circ \phi ^{-k}\circ \psi ^{-1}:\psi \left( U\cap D\right) \rightarrow \psi ^{\prime }\left( U^{\prime }\cap D\right) \end{equation*} is analytically continued to, by abuse of notation, automorphisms $\varphi _{k}\in Aut\left( B^{n+1}\right) .$ By the construction, a subsequence $ \varphi _{m_{k}}$ must converges to the point $\psi ^{\prime }\left( q^{\prime }\right) $ uniformly on every compact subset of $B^{n+1}$. Thus there is a sequence of real numbers $\delta _{k}\searrow 0$ such that \begin{equation*} \phi ^{m_{k}}:B\left( q^{\prime };\delta _{k}\right) \cap D\rightarrow U\cap D \end{equation*} and \begin{equation*} \phi ^{m_{k}}=\psi ^{-1}\circ \varphi _{m_{k}}^{-1}\circ \psi ^{\prime }\quad \text{on }B\left( q^{\prime };\delta _{k}\right) \cap D. \end{equation*} Then we take a point $p\in U\cap D$ such that there is a sequence $q_{k}$ satisfying \begin{equation*} q_{k}\in \phi ^{-m_{k}}\left( \left\{ p\right\} \right) \cap B\left( q^{\prime };\delta _{k}\right) \end{equation*} and, for a compact subset $K\subset \subset D,$ there is a sequence $p_{k}$ satisfying \begin{equation*} p_{k}\in \phi ^{-m_{k}}\left( \left\{ p\right\} \right) \cap K. \end{equation*}
Therefore, there is a sequence of deck transformations $\phi _{k}\in Aut\left( D\right) $ of the covering map $\phi ^{m_{k}}:D\rightarrow D$ such that \begin{equation*} \phi _{k}\left( p_{k}\right) =q_{k}\rightarrow q^{\prime }\in bD. \end{equation*} Since $p_{k}\in K,$ by Lemma \ref{Wong-Rosay}, there is a biholomorphic mapping \begin{equation*} \sigma :D\rightarrow B^{n+1}. \end{equation*} Then the composition \begin{equation*} \iota \equiv \sigma \circ \phi \circ \sigma ^{-1}:B^{n+1}\rightarrow B^{n+1} \end{equation*} would be a proper self mapping, but not an automorphism of the unit ball $ B^{n+1}.$ This is a contradiction to Lemma \ref{Alexander} that every proper self mapping of the unit ball $B^{n+1}$ is an automorphism of the unit ball $ B^{n+1}.$
Suppose that $bD$ is nonspherical. We take a sequence $p_{k}\in bD$ such that \begin{equation*} p_{k}\in \phi ^{-k}\left( \left\{ q\right\} \right) \subset bD \end{equation*} and \begin{equation*}
\left| p_{k}-q^{\prime }\right| =\min \left\{ \left| p-q^{\prime }\right| :p\in \phi ^{-k}\left( \left\{ q\right\} \right) \right\} . \end{equation*} Let $\mu _{p_{k}}$ be the canonical normalizing mapping at the point $ p_{k}\in bD.$ Since $p_{k}\rightarrow q^{\prime },$ there is an open neighborhood $W$ of the point $q^{\prime }$ such that the mapping $\mu _{p_{k}}$ is analytically continued on $W.$ Then we define a function $ \varepsilon _{k}\left( p\right) $ for a sufficiently large $k$ and a point $ p\in W\cap bD$ such that \begin{equation*}
\varepsilon _{k}\left( p\right) \equiv \sum_{j=1}^{n}\left| z_{j}\circ \mu _{p_{k}}\left( p\right) \right| ^{2}+\left| w\circ \mu _{p_{k}}\left(
p\right) \right| \end{equation*} where $z_{j},$ $j=1,\cdots ,n,$ $w$ are the coordinate functions of $\Bbb{C} ^{n+1}$ such that $z_{j},$ $j=1,\cdots ,n,$ are for the complex tangent hyperplane and $w$ are for the complex line normal to the complex tangent hyperplane of $\mu _{p_{k}}\left( W\cap bD\right) $ at the origin. Then we take a sequence $q_{k}$ such that \begin{equation*} q_{k}\in \phi ^{-k}\left( \left\{ q\right\} \right) \subset bD \end{equation*} and \begin{equation*} \varepsilon _{k}\left( q_{k}\right) =\min \left\{ \varepsilon _{k}\left( p\right) :p\in \phi ^{-k}\left( \left\{ q\right\} \right) \right\} . \end{equation*} Let $\pi _{k}$ be a complex line containing $p_{k}$ and $q_{k}.$ Then we take a subsequence $\pi _{m_{k}}$ so that $\pi _{m_{k}}$ converges to a complex line $\pi _{q^{\prime }}\subset T_{q^{\prime }}\Bbb{C}^{n+1}.$ In other words, the a subsequence $q_{m_{k}}$ converges to the point $q^{\prime }$ to a direction. Then we obtain \begin{equation*} \varphi _{k}\equiv \mu _{p_{k}}\circ \phi ^{-k}\circ \mu _{q}^{-1}:\mu _{q}\left( bD\right) \rightarrow \mu _{p_{k}}\left( bD\right) . \end{equation*} Then we define a biholomorphic mapping \begin{equation*} \sigma _{k}:\left\{ \begin{array}{l} z^{*}=\sqrt{\varepsilon _{m_{k}}}z \\ w^{*}=\varepsilon _{m_{k}}w \end{array} \right. \end{equation*} where \begin{equation*} \varepsilon _{m_{k}}=\varepsilon _{m_{k}}\left( q_{m_{k}}\right) . \end{equation*} Then we take the composition \begin{equation*} \kappa _{k}\equiv \sigma _{k}^{-1}\circ \varphi _{m_{k}}:\mu _{q}\left( bD\right) \rightarrow \sigma _{k}^{-1}\circ \mu _{p_{m_{k}}}\left( bD\right) . \end{equation*} Note that there is a real number $\delta >0$ such that the germs of real hypersurfaces $\mu _{q}\left( bD\right) $ and $\sigma _{k}^{-1}\circ \mu _{p_{m_{k}}}\left( bD\right) $ are analytically continued to the open neighborhood $B\left( 0;\delta \right) .$ Since the boundary $bD$ is nonspherical, by Lemma \ref{compact}, the mapping $\kappa _{k}$ is biholomorphic on $B\left( 0;\delta \right) ,$ if necessary, shrinking $ \delta >0.$ Further, by the construction, the limit of the sequence $\kappa _{k}$ cannot be a constant mapping on $B\left( 0;\delta \right) $ so that, by Theorem \ref{nonconstant}, the limit of the sequence $\kappa _{k}$ would be biholomorphic on $B\left( 0;\delta \right) .$
By the way, the sequence of real hypersurfaces $\sigma _{k}^{-1}\circ \mu _{p_{m_{k}}}\left( bD\right) $ converges to a real hyperquadric uniformly on an open neighborhood of the origin. Therefore, the germ of the boundary $bD$ at the point $q^{\prime }$ is spherical so that the boundary $bD$ is spherical(cf. \cite{Pa3}). This is a contradiction to the fact that the boundary $bD$ is nonspherical. This completes the proof.
\endproof
\subsection{Locally realizable CR manifolds}
Let $M$ be a CR manifold of CR dimension $n$ and CR codimension $1$ with a CR structure $(D,I)$ where $D$ is $2n$ dimensional smooth subbundle of the tangent bundle $TM$ and $I$ is an automorphism on $D$ such that \begin{equation*} I^{2}V=-V\quad \text{for}\quad V\in \Gamma D. \end{equation*} For each point $p\in M,$ there is a local coordinate chart $\left( U,\varphi \right) $ such that \begin{equation*} \varphi \left( U\right) \subset \Bbb{R}^{2n+1}. \end{equation*} Then $M$ shall be called locally realizable CR manifold if there is an open neighborhood $U$ of each point $p\in M$ and CR functions $f_{1},\cdots ,f_{n+1}$ on $\varphi \left( U\right) $ satisfying \begin{equation*} df_{1}\wedge \cdots \wedge df_{n+1}\neq 0. \end{equation*} Let $z_{j},$ $j=1,\cdots ,n+1$, be the coordinate functions of $\Bbb{C} ^{n+1}.$ Then we obtain a local embedding $\sigma $ of $U\subset M$ into $ \Bbb{C}^{n+1}$ such that \begin{equation*} f_{j}\equiv z_{j}\circ \sigma \circ \varphi ^{-1}. \end{equation*} The CR manifold $M$ shall be called a locally analytically realizable CR manifold when $\sigma \left( U\right) $ is real analytic by a local embedding $\sigma $ on an open neighborhood $U$ of each point $p\in M.$ Note that $M$ is either spherical or nonspherical(cf. \cite{Pa3}).
\begin{lemma} Let $M$ be a connected locally analytically realizable CR manifold. Suppose that there is a nontrivial CR mapping $\varphi $ on an open neighborhood $U$ of a point $p\in M$ such that \begin{equation*} \varphi \left( U\right) \subset bB^{n+1}. \end{equation*} Then the mapping $\varphi $ is CR continued along any path on $M$ as a locally CR diffeomorphic mapping. \end{lemma}
\proof
Since $M$ is connected, $M$ is necessarily spherical(cf. \cite{Pa3}). Suppose that the assertion is not true. Then there is a path $\gamma :[0,1]\rightarrow M$ with $\gamma \left( 0\right) =p$ such that the mapping $ \varphi $ is CR continued along all subpath $\gamma [0,\tau ]$ with $\tau <1, $ but not the whole path $\gamma [0,1].$ Then we take a CR embedding $ \sigma \left( V\right) \subset \Bbb{C}^{n+1}$ of an open neighborhood $V$ of the point $\gamma \left( 1\right) $ such that $\sigma \left( V\right) $ is a spherical analytic real hypersurface and \begin{equation*} \varphi \circ \sigma ^{-1}:\sigma \left( V\right) \rightarrow bB^{n+1} \end{equation*} is a locally CR diffeomorphism. Without loss of generality, we may assume that $\gamma [0,1]\subset V.$ Hence there is an open neighborhood $W$ of the point $\sigma \left( \gamma \left( 0\right) \right) $ and a biholomorphic mapping $\phi $ on $W$ such that \begin{equation*} \phi =\varphi \circ \sigma ^{-1}\quad \text{on }\sigma \left( V\right) \cap W. \end{equation*} Then the mapping $\phi $ is analytically continued along the whole path $ \sigma \left( \gamma [0,1]\right) .$ From, by abuse of notation, the mapping $\phi $ at an open neighborhood $W^{\prime }$ of the point $\sigma \left( \gamma \left( 1\right) \right) $ and a CR embedding $\sigma $ of an open neighborhood $V^{\prime }$ of the point $\gamma \left( 1\right) $, we obtain the CR mapping \begin{equation*} \phi \circ \sigma :\sigma ^{-1}\left( W^{\prime }\cap \sigma \left( V^{\prime }\right) \right) \rightarrow bB^{n+1} \end{equation*} which is a CR continuation of the CR mapping $\varphi .$ This completes the proof.
\endproof
\begin{theorem} Let $M$ be a connected locally analytically realizable CR manifold. Suppose that $M$ is compact and there is a nontrivial CR mapping $\varphi $ on an open neighborhood $U$ of a point $p\in M$ such that \begin{equation*} \varphi \left( U\right) \subset bB^{n+1}. \end{equation*} Then there is a finite subset $L\subset bB^{n+1}$ such that the inverse CR mapping $\varphi ^{-1}$ is CR continued along any path on $ bB^{n+1}\backslash L$ as a locally CR diffeomorphic mapping. \end{theorem}
\proof
Since $M$ is connected, $M$ is necessarily spherical(cf. \cite{Pa3}). Let $ q_{j}\in bB^{n+1}$ be a sequence converging to a point $q\in bB^{n+1}$ and $ \phi _{j}$ be a sequence of the CR continuation of the inverse mapping $ \varphi ^{-1}$ at the point $q_{j}$. We set \begin{equation*} q_{j}^{\prime }=\phi _{j}\left( q_{j}\right) \in M. \end{equation*} Since $M$ is compact, there is a subsequence $q_{m_{j}}^{\prime }$ and a point $q^{\prime }\in M$ such that \begin{equation*} q_{m_{j}}^{\prime }\rightarrow q^{\prime }. \end{equation*} Then we take a CR embedding $\sigma $ of an open neighborhood $V$ of the point $q^{\prime }\in M$ such that $\sigma \left( V\right) \subset \Bbb{C} ^{n+1}$ is a spherical analytic real hypersurface. Then we apply the same argument as in the previous sections so that the singular locus $L$ is a finite subset of $bB^{n+1}$ and the inverse CR mapping $\varphi ^{-1}$ is CR continued along any path on $bB^{n+1}\backslash L.$ This completes the proof.
\endproof
\begin{lemma} \label{cranypath}Let $M,M^{\prime }$ be connected locally analytically realizable CR manifolds with positive definite Levi form. Suppose that $ M,M^{\prime }$ are nonspherical and $M^{\prime }$ is compact, and there is a nontrivial CR mapping $\varphi $ on an open neighborhood $U$ of a point $ p\in M$ such that \begin{equation*} \varphi \left( U\right) \subset M^{\prime }. \end{equation*} Then the mapping $\varphi $ is CR continued along any path on $M$ as a locally CR diffeomorphic mapping. \end{lemma}
\proof
Since $M,M^{\prime }$ are locally analytically realizable and the Levi forms of $M,M^{\prime }$ are positive definite, the isotropy subgroups $ Aut_{p}\left( M\right) ,Aut_{p^{\prime }}\left( M^{\prime }\right) $ are compact for every point $p\in M,p^{\prime }\in M^{\prime }.$
Let $q_{j}\in M$ be a sequence converging to a point $q\in M$ and $\varphi _{j}$ be a sequence of the CR continuation of the mapping $\varphi $ at the point $q_{j}$. We set \begin{equation*} q_{j}^{\prime }=\varphi _{j}\left( q_{j}\right) \in M^{\prime }. \end{equation*} Since $M^{\prime }$ is compact, there is a subsequence $q_{m_{j}}^{\prime }$ and a point $q^{\prime }\in M^{\prime }$ such that \begin{equation*} q_{m_{j}}^{\prime }\rightarrow q^{\prime }. \end{equation*} Then we take CR embeddings $\sigma ,\sigma ^{\prime }$ respectively of open neighborhoods $V,V^{\prime }$ respectively of the points $q\in M,q^{\prime }\in M^{\prime }$ such that $\sigma \left( V\right) ,\sigma ^{\prime }\left( V^{\prime }\right) \subset \Bbb{C}^{n+1}$ are nonspherical analytic real hypersurface. Then we apply the same argument as in the previous subsection. This completes the proof.
\endproof
\begin{theorem} Let $M,M^{\prime }$ be connected locally analytically realizable CR manifolds with positive definite Levi form. Suppose that $M,M^{\prime }$ are compact and nonspherical, and there is a nontrivial CR mapping $\varphi $ on an open neighborhood $U$ of a point $p\in M$ such that \begin{equation*} \varphi \left( U\right) \subset M^{\prime }. \end{equation*} Then the maximal CR extension of the mapping $\varphi $ is a CR equivalence between the natural universal covering spaces of $M,M^{\prime }$ to be the pointed path spaces respectively of $M,M^{\prime }$ mod homotopic relation. \end{theorem}
\proof
By Lemma \ref{cranypath}, the mapping $\varphi $ is CR continued along any path on $M$ as a locally CR diffeomorphic mapping. Note that $M$ is compact and \begin{equation*} \varphi ^{-1}\left( \varphi \left( U\right) \right) \subset M \end{equation*} so that we apply Lemma \ref{cranypath} to the inverse mapping $\varphi ^{-1}. $ Thus the inverse mapping $\varphi ^{-1}$ is CR continued along any path on $M^{\prime }$ as a locally CR diffeomorphic mapping. Thus the CR continuation of the mapping $\phi $ induces a CR equivalence between the natural universal coverings of $M,M^{\prime },$ which are the path spaces mod homotopic relation respectively over $M,M^{\prime }$ with the natural CR structure. This completes the proof.
\endproof
\end{document} | arXiv |
Effect size
In statistics, an effect size is a value measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the value of a parameter for a hypothetical population, or to the equation that operationalizes how statistics or parameters lead to the effect size value.[1] Examples of effect sizes include the correlation between two variables,[2] the regression coefficient in a regression, the mean difference, or the risk of a particular event (such as a heart attack) happening. Effect sizes complement statistical hypothesis testing, and play an important role in power analyses, sample size planning, and in meta-analyses. The cluster of data-analysis methods concerning effect sizes is referred to as estimation statistics.
Effect size is an essential component when evaluating the strength of a statistical claim, and it is the first item (magnitude) in the MAGIC criteria. The standard deviation of the effect size is of critical importance, since it indicates how much uncertainty is included in the measurement. A standard deviation that is too large will make the measurement nearly meaningless. In meta-analysis, where the purpose is to combine multiple effect sizes, the uncertainty in the effect size is used to weigh effect sizes, so that large studies are considered more important than small studies. The uncertainty in the effect size is calculated differently for each type of effect size, but generally only requires knowing the study's sample size (N), or the number of observations (n) in each group.
Reporting effect sizes or estimates thereof (effect estimate [EE], estimate of effect) is considered good practice when presenting empirical research findings in many fields.[3][4] The reporting of effect sizes facilitates the interpretation of the importance of a research result, in contrast to its statistical significance.[5] Effect sizes are particularly prominent in social science and in medical research (where size of treatment effect is important).
Effect sizes may be measured in relative or absolute terms. In relative effect sizes, two groups are directly compared with each other, as in odds ratios and relative risks. For absolute effect sizes, a larger absolute value always indicates a stronger effect. Many types of measurements can be expressed as either absolute or relative, and these can be used together because they convey different information. A prominent task force in the psychology research community made the following recommendation:
Always present effect sizes for primary outcomes...If the units of measurement are meaningful on a practical level (e.g., number of cigarettes smoked per day), then we usually prefer an unstandardized measure (regression coefficient or mean difference) to a standardized measure (r or d).[3]
Overview
Population and sample effect sizes
As in statistical estimation, the true effect size is distinguished from the observed effect size, e.g. to measure the risk of disease in a population (the population effect size) one can measure the risk within a sample of that population (the sample effect size). Conventions for describing true and observed effect sizes follow standard statistical practices—one common approach is to use Greek letters like ρ [rho] to denote population parameters and Latin letters like r to denote the corresponding statistic. Alternatively, a "hat" can be placed over the population parameter to denote the statistic, e.g. with ${\hat {\rho }}$ being the estimate of the parameter $\rho $.
As in any statistical setting, effect sizes are estimated with sampling error, and may be biased unless the effect size estimator that is used is appropriate for the manner in which the data were sampled and the manner in which the measurements were made. An example of this is publication bias, which occurs when scientists report results only when the estimated effect sizes are large or are statistically significant. As a result, if many researchers carry out studies with low statistical power, the reported effect sizes will tend to be larger than the true (population) effects, if any.[6] Another example where effect sizes may be distorted is in a multiple-trial experiment, where the effect size calculation is based on the averaged or aggregated response across the trials.[7]
Smaller studies sometimes show different, often larger, effect sizes than larger studies. This phenomenon is known as the small-study effect, which may signal publication bias.[8]
Relationship to test statistics
Sample-based effect sizes are distinguished from test statistics used in hypothesis testing, in that they estimate the strength (magnitude) of, for example, an apparent relationship, rather than assigning a significance level reflecting whether the magnitude of the relationship observed could be due to chance. The effect size does not directly determine the significance level, or vice versa. Given a sufficiently large sample size, a non-null statistical comparison will always show a statistically significant result unless the population effect size is exactly zero (and even there it will show statistical significance at the rate of the Type I error used). For example, a sample Pearson correlation coefficient of 0.01 is statistically significant if the sample size is 1000. Reporting only the significant p-value from this analysis could be misleading if a correlation of 0.01 is too small to be of interest in a particular application.
Standardized and unstandardized effect sizes
The term effect size can refer to a standardized measure of effect (such as r, Cohen's d, or the odds ratio), or to an unstandardized measure (e.g., the difference between group means or the unstandardized regression coefficients). Standardized effect size measures are typically used when:
• the metrics of variables being studied do not have intrinsic meaning (e.g., a score on a personality test on an arbitrary scale),
• results from multiple studies are being combined,
• some or all of the studies use different scales, or
• it is desired to convey the size of an effect relative to the variability in the population.
In meta-analyses, standardized effect sizes are used as a common measure that can be calculated for different studies and then combined into an overall summary.
Interpretation
Whether an effect size should be interpreted as small, medium, or large depends on its substantive context and its operational definition. Cohen's conventional criteria small, medium, or big[9] are near ubiquitous across many fields, although Cohen[9] cautioned:
"The terms 'small,' 'medium,' and 'large' are relative, not only to each other, but to the area of behavioral science or even more particularly to the specific content and research method being employed in any given investigation....In the face of this relativity, there is a certain risk inherent in offering conventional operational definitions for these terms for use in power analysis in as diverse a field of inquiry as behavioral science. This risk is nevertheless accepted in the belief that more is to be gained than lost by supplying a common conventional frame of reference which is recommended for use only when no better basis for estimating the ES index is available." (p. 25)
In the two sample layout, Sawilowsky [10] concluded "Based on current research findings in the applied literature, it seems appropriate to revise the rules of thumb for effect sizes," keeping in mind Cohen's cautions, and expanded the descriptions to include very small, very large, and huge. The same de facto standards could be developed for other layouts.
Lenth [11] noted for a "medium" effect size, "you'll choose the same n regardless of the accuracy or reliability of your instrument, or the narrowness or diversity of your subjects. Clearly, important considerations are being ignored here. Researchers should interpret the substantive significance of their results by grounding them in a meaningful context or by quantifying their contribution to knowledge, and Cohen's effect size descriptions can be helpful as a starting point."[5] Similarly, a U.S. Dept of Education sponsored report said "The widespread indiscriminate use of Cohen’s generic small, medium, and large effect size values to characterize effect sizes in domains to which his normative values do not apply is thus likewise inappropriate and misleading."[12]
They suggested that "appropriate norms are those based on distributions of effect sizes for comparable outcome measures from comparable interventions targeted on comparable samples." Thus if a study in a field where most interventions are tiny yielded a small effect (by Cohen's criteria), these new criteria would call it "large". In a related point, see Abelson's paradox and Sawilowsky's paradox.[13][14][15]
Types
About 50 to 100 different measures of effect size are known. Many effect sizes of different types can be converted to other types, as many estimate the separation of two distributions, so are mathematically related. For example, a correlation coefficient can be converted to a Cohen's d and vice versa.
Correlation family: Effect sizes based on "variance explained"
These effect sizes estimate the amount of the variance within an experiment that is "explained" or "accounted for" by the experiment's model (Explained variation).
Pearson r or correlation coefficient
Pearson's correlation, often denoted r and introduced by Karl Pearson, is widely used as an effect size when paired quantitative data are available; for instance if one were studying the relationship between birth weight and longevity. The correlation coefficient can also be used when the data are binary. Pearson's r can vary in magnitude from −1 to 1, with −1 indicating a perfect negative linear relation, 1 indicating a perfect positive linear relation, and 0 indicating no linear relation between two variables. Cohen gives the following guidelines for the social sciences:[9][16]
Effect sizer
Small0.10
Medium0.30
Large0.50
Coefficient of determination (r2 or R2)
A related effect size is r2, the coefficient of determination (also referred to as R2 or "r-squared"), calculated as the square of the Pearson correlation r. In the case of paired data, this is a measure of the proportion of variance shared by the two variables, and varies from 0 to 1. For example, with an r of 0.21 the coefficient of determination is 0.0441, meaning that 4.4% of the variance of either variable is shared with the other variable. The r2 is always positive, so does not convey the direction of the correlation between the two variables.
Eta-squared (η2)
Eta-squared describes the ratio of variance explained in the dependent variable by a predictor while controlling for other predictors, making it analogous to the r2. Eta-squared is a biased estimator of the variance explained by the model in the population (it estimates only the effect size in the sample). This estimate shares the weakness with r2 that each additional variable will automatically increase the value of η2. In addition, it measures the variance explained of the sample, not the population, meaning that it will always overestimate the effect size, although the bias grows smaller as the sample grows larger.
$\eta ^{2}={\frac {SS_{\text{Treatment}}}{SS_{\text{Total}}}}.$
Omega-squared (ω2)
See also: Adjusted R2
A less biased estimator of the variance explained in the population is ω2[17]
$\omega ^{2}={\frac {{\text{SS}}_{\text{treatment}}-df_{\text{treatment}}\cdot {\text{MS}}_{\text{error}}}{{\text{SS}}_{\text{total}}+{\text{MS}}_{\text{error}}}}.$
This form of the formula is limited to between-subjects analysis with equal sample sizes in all cells.[17] Since it is less biased (although not unbiased), ω2 is preferable to η2; however, it can be more inconvenient to calculate for complex analyses. A generalized form of the estimator has been published for between-subjects and within-subjects analysis, repeated measure, mixed design, and randomized block design experiments.[18] In addition, methods to calculate partial ω2 for individual factors and combined factors in designs with up to three independent variables have been published.[18]
Cohen's f2
Cohen's f2 is one of several effect size measures to use in the context of an F-test for ANOVA or multiple regression. Its amount of bias (overestimation of the effect size for the ANOVA) depends on the bias of its underlying measurement of variance explained (e.g., R2, η2, ω2).
The f2 effect size measure for multiple regression is defined as:
$f^{2}={R^{2} \over 1-R^{2}}$
where R2 is the squared multiple correlation.
Likewise, f2 can be defined as:
$f^{2}={\eta ^{2} \over 1-\eta ^{2}}$
or
$f^{2}={\omega ^{2} \over 1-\omega ^{2}}$
for models described by those effect size measures.[19]
The $f^{2}$ effect size measure for sequential multiple regression and also common for PLS modeling[20] is defined as:
$f^{2}={R_{AB}^{2}-R_{A}^{2} \over 1-R_{AB}^{2}}$
where R2A is the variance accounted for by a set of one or more independent variables A, and R2AB is the combined variance accounted for by A and another set of one or more independent variables of interest B. By convention, f2 effect sizes of $0.1^{2}$, $0.25^{2}$, and $0.4^{2}$ are termed small, medium, and large, respectively.[9]
Cohen's ${\hat {f}}$ can also be found for factorial analysis of variance (ANOVA) working backwards, using:
${\hat {f}}_{\text{effect}}={\sqrt {(F_{\text{effect}}df_{\text{effect}}/N)}}.$
In a balanced design (equivalent sample sizes across groups) of ANOVA, the corresponding population parameter of $f^{2}$ is
${SS(\mu _{1},\mu _{2},\dots ,\mu _{K})} \over {K\times \sigma ^{2}},$
wherein μj denotes the population mean within the jth group of the total K groups, and σ the equivalent population standard deviations within each groups. SS is the sum of squares in ANOVA.
Cohen's q
Another measure that is used with correlation differences is Cohen's q. This is the difference between two Fisher transformed Pearson regression coefficients. In symbols this is
$q={\frac {1}{2}}\log {\frac {1+r_{1}}{1-r_{1}}}-{\frac {1}{2}}\log {\frac {1+r_{2}}{1-r_{2}}}$
where r1 and r2 are the regressions being compared. The expected value of q is zero and its variance is
$\operatorname {var} (q)={\frac {1}{N_{1}-3}}+{\frac {1}{N_{2}-3}}$
where N1 and N2 are the number of data points in the first and second regression respectively.
Difference family: Effect sizes based on differences between means
The raw effect size pertaining to a comparison of two groups is inherently calculated as the differences between the two means. However, to facilitate interpretation it is common to standardise the effect size; various conventions for statistical standardisation are presented below.
Standardized mean difference
A (population) effect size θ based on means usually considers the standardized mean difference (SMD) between two populations[21]: 78
$\theta ={\frac {\mu _{1}-\mu _{2}}{\sigma }},$
where μ1 is the mean for one population, μ2 is the mean for the other population, and σ is a standard deviation based on either or both populations.
In the practical setting the population values are typically not known and must be estimated from sample statistics. The several versions of effect sizes based on means differ with respect to which statistics are used.
This form for the effect size resembles the computation for a t-test statistic, with the critical difference that the t-test statistic includes a factor of ${\sqrt {n}}$. This means that for a given effect size, the significance level increases with the sample size. Unlike the t-test statistic, the effect size aims to estimate a population parameter and is not affected by the sample size.
SMD values of 0.2 to 0.5 are considered small, 0.5 to 0.8 are considered medium, and greater than 0.8 are considered large.[22]
Cohen's d
Cohen's d is defined as the difference between two means divided by a standard deviation for the data, i.e.
$d={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s}}.$
Jacob Cohen defined s, the pooled standard deviation, as (for two independent samples):[9]: 67
$s={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}$
where the variance for one of the groups is defined as
$s_{1}^{2}={\frac {1}{n_{1}-1}}\sum _{i=1}^{n_{1}}(x_{1,i}-{\bar {x}}_{1})^{2},$
and similarly for the other group.
The table below contains descriptors for magnitudes of d = 0.01 to 2.0, as initially suggested by Cohen and expanded by Sawilowsky.[10]
Effect sizedReference
Very small0.01[10]
Small0.20[9]
Medium0.50[9]
Large0.80[9]
Very large1.20[10]
Huge2.0[10]
Other authors choose a slightly different computation of the standard deviation when referring to "Cohen's d" where the denominator is without "-2"[23][24]: 14
$s={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}}}}$
This definition of "Cohen's d" is termed the maximum likelihood estimator by Hedges and Olkin,[21] and it is related to Hedges' g by a scaling factor (see below).
With two paired samples, we look at the distribution of the difference scores. In that case, s is the standard deviation of this distribution of difference scores. This creates the following relationship between the t-statistic to test for a difference in the means of the two groups and Cohen's d:
$t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\text{SE}}}={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\frac {\text{SD}}{\sqrt {N}}}}={\frac {{\sqrt {N}}({\bar {X}}_{1}-{\bar {X}}_{2})}{SD}}$
and
$d={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\text{SD}}}={\frac {t}{\sqrt {N}}}$
Cohen's d is frequently used in estimating sample sizes for statistical testing. A lower Cohen's d indicates the necessity of larger sample sizes, and vice versa, as can subsequently be determined together with the additional parameters of desired significance level and statistical power.[25]
For paired samples Cohen suggests that the d calculated is actually a d', which doesn't provide the correct answer to obtain the power of the test, and that before looking the values up in the tables provided, it should be corrected for r as in the following formula:[26]
$d={\frac {d'}{\sqrt {1-r}}}$
Glass' Δ
In 1976, Gene V. Glass proposed an estimator of the effect size that uses only the standard deviation of the second group[21]: 78
$\Delta ={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s_{2}}}$
The second group may be regarded as a control group, and Glass argued that if several treatments were compared to the control group it would be better to use just the standard deviation computed from the control group, so that effect sizes would not differ under equal means and different variances.
Under a correct assumption of equal population variances a pooled estimate for σ is more precise.
Hedges' g
Hedges' g, suggested by Larry Hedges in 1981,[27] is like the other measures based on a standardized difference[21]: 79
$g={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{s^{*}}}$
where the pooled standard deviation $s^{*}$ is computed as:
$s^{*}={\sqrt {\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}.$
However, as an estimator for the population effect size θ it is biased. Nevertheless, this bias can be approximately corrected through multiplication by a factor
$g^{*}=J(n_{1}+n_{2}-2)\,\,g\,\approx \,\left(1-{\frac {3}{4(n_{1}+n_{2})-9}}\right)\,\,g$
Hedges and Olkin refer to this less-biased estimator $g^{*}$ as d,[21] but it is not the same as Cohen's d. The exact form for the correction factor J() involves the gamma function[21]: 104
$J(a)={\frac {\Gamma (a/2)}{{\sqrt {a/2\,}}\,\Gamma ((a-1)/2)}}.$
Ψ, root-mean-square standardized effect
A similar effect size estimator for multiple comparisons (e.g., ANOVA) is the Ψ root-mean-square standardized effect:[19]
$\Psi ={\sqrt {{\frac {1}{k-1}}\cdot \sum _{j=1}^{k}\left({\frac {\mu _{j}-\mu }{\sigma }}\right)^{2}}}$
where k is the number of groups in the comparisons.
This essentially presents the omnibus difference of the entire model adjusted by the root mean square, analogous to d or g.
In addition, a generalization for multi-factorial designs has been provided.[19]
Distribution of effect sizes based on means
Provided that the data is Gaussian distributed a scaled Hedges' g, $ {\sqrt {n_{1}n_{2}/(n_{1}+n_{2})}}\,g$, follows a noncentral t-distribution with the noncentrality parameter $ {\sqrt {n_{1}n_{2}/(n_{1}+n_{2})}}\theta $ and (n1 + n2 − 2) degrees of freedom. Likewise, the scaled Glass' Δ is distributed with n2 − 1 degrees of freedom.
From the distribution it is possible to compute the expectation and variance of the effect sizes.
In some cases large sample approximations for the variance are used. One suggestion for the variance of Hedges' unbiased estimator is[21] : 86
${\hat {\sigma }}^{2}(g^{*})={\frac {n_{1}+n_{2}}{n_{1}n_{2}}}+{\frac {(g^{*})^{2}}{2(n_{1}+n_{2})}}.$
Other metrics
Mahalanobis distance (D) is a multivariate generalization of Cohen's d, which takes into account the relationships between the variables.[28]
Categorical family: Effect sizes for associations among categorical variables
$\varphi ={\sqrt {\frac {\chi ^{2}}{N}}}$
$\varphi _{c}={\sqrt {\frac {\chi ^{2}}{N(k-1)}}}$
Phi (φ) Cramér's V (φc)
Commonly used measures of association for the chi-squared test are the Phi coefficient and Cramér's V (sometimes referred to as Cramér's phi and denoted as φc). Phi is related to the point-biserial correlation coefficient and Cohen's d and estimates the extent of the relationship between two variables (2 × 2).[29] Cramér's V may be used with variables having more than two levels.
Phi can be computed by finding the square root of the chi-squared statistic divided by the sample size.
Similarly, Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the length of the minimum dimension (k is the smaller of the number of rows r or columns c).
φc is the intercorrelation of the two discrete variables[30] and may be computed for any value of r or c. However, as chi-squared values tend to increase with the number of cells, the greater the difference between r and c, the more likely V will tend to 1 without strong evidence of a meaningful correlation.
Cohen's omega (ω)
Another measure of effect size used for chi-squared tests is Cohen's omega ($\omega $). This is defined as
$\omega ={\sqrt {\sum _{i=1}^{m}{\frac {(p_{1i}-p_{0i})^{2}}{p_{0i}}}}}$
where p0i is the proportion of the ith cell under H0, p1i is the proportion of the ith cell under H1 and m is the number of cells.
In Statistical Power Analysis for the Behavioral Sciences (1988, pp.224-225), Cohen gives the following general guideline for interpreting omega (see table below), but warns against its "possible inaptness in any given substantive context" and advises to use context-relevant judgment instead.
Effect Size$\omega $
Small0.10
Medium0.30
Large0.50
Odds ratio
The odds ratio (OR) is another useful effect size. It is appropriate when the research question focuses on the degree of association between two binary variables. For example, consider a study of spelling ability. In a control group, two students pass the class for every one who fails, so the odds of passing are two to one (or 2/1 = 2). In the treatment group, six students pass for every one who fails, so the odds of passing are six to one (or 6/1 = 6). The effect size can be computed by noting that the odds of passing in the treatment group are three times higher than in the control group (because 6 divided by 2 is 3). Therefore, the odds ratio is 3. Odds ratio statistics are on a different scale than Cohen's d, so this '3' is not comparable to a Cohen's d of 3.
Relative risk
The relative risk (RR), also called risk ratio, is simply the risk (probability) of an event relative to some independent variable. This measure of effect size differs from the odds ratio in that it compares probabilities instead of odds, but asymptotically approaches the latter for small probabilities. Using the example above, the probabilities for those in the control group and treatment group passing is 2/3 (or 0.67) and 6/7 (or 0.86), respectively. The effect size can be computed the same as above, but using the probabilities instead. Therefore, the relative risk is 1.28. Since rather large probabilities of passing were used, there is a large difference between relative risk and odds ratio. Had failure (a smaller probability) been used as the event (rather than passing), the difference between the two measures of effect size would not be so great.
While both measures are useful, they have different statistical uses. In medical research, the odds ratio is commonly used for case-control studies, as odds, but not probabilities, are usually estimated.[31] Relative risk is commonly used in randomized controlled trials and cohort studies, but relative risk contributes to overestimations of the effectiveness of interventions.[32]
Risk difference
The risk difference (RD), sometimes called absolute risk reduction, is simply the difference in risk (probability) of an event between two groups. It is a useful measure in experimental research, since RD tells you the extent to which an experimental interventions changes the probability of an event or outcome. Using the example above, the probabilities for those in the control group and treatment group passing is 2/3 (or 0.67) and 6/7 (or 0.86), respectively, and so the RD effect size is 0.86 − 0.67 = 0.19 (or 19%). RD is the superior measure for assessing effectiveness of interventions.[32]
Cohen's h
One measure used in power analysis when comparing two independent proportions is Cohen's h. This is defined as follows
$h=2(\arcsin {\sqrt {p_{1}}}-\arcsin {\sqrt {p_{2}}})$
where p1 and p2 are the proportions of the two samples being compared and arcsin is the arcsine transformation.
Common language effect size
To more easily describe the meaning of an effect size to people outside statistics, the common language effect size, as the name implies, was designed to communicate it in plain English. It is used to describe a difference between two groups and was proposed, as well as named, by Kenneth McGraw and S. P. Wong in 1992.[33] They used the following example (about heights of men and women): "in any random pairing of young adult males and females, the probability of the male being taller than the female is .92, or in simpler terms yet, in 92 out of 100 blind dates among young adults, the male will be taller than the female",[33] when describing the population value of the common language effect size.
The population value, for the common language effect size, is often reported like this, in terms of pairs randomly chosen from the population. Kerby (2014) notes that a pair, defined as a score in one group paired with a score in another group, is a core concept of the common language effect size.[34]
As another example, consider a scientific study (maybe of a treatment for some chronic disease, such as arthritis) with ten people in the treatment group and ten people in a control group. If everyone in the treatment group is compared to everyone in the control group, then there are (10×10=) 100 pairs. At the end of the study, the outcome is rated into a score, for each individual (for example on a scale of mobility and pain, in the case of an arthritis study), and then all the scores are compared between the pairs. The result, as the percent of pairs that support the hypothesis, is the common language effect size. In the example study it could be (let's say) .80, if 80 out of the 100 comparison pairs show a better outcome for the treatment group than the control group, and the report may read as follows: "When a patient in the treatment group was compared to a patient in the control group, in 80 of 100 pairs the treated patient showed a better treatment outcome." The sample value, in for example a study like this, is an unbiased estimator of the population value.[35]
Vargha and Delaney generalized the common language effect size (Vargha-Delaney A), to cover ordinal level data.[36]
Rank-biserial correlation
An effect size related to the common language effect size is the rank-biserial correlation. This measure was introduced by Cureton as an effect size for the Mann–Whitney U test.[37] That is, there are two groups, and scores for the groups have been converted to ranks. The Kerby simple difference formula computes the rank-biserial correlation from the common language effect size.[34] Letting f be the proportion of pairs favorable to the hypothesis (the common language effect size), and letting u be the proportion of pairs not favorable, the rank-biserial r is the simple difference between the two proportions: r = f − u. In other words, the correlation is the difference between the common language effect size and its complement. For example, if the common language effect size is 60%, then the rank-biserial r equals 60% minus 40%, or r = 0.20. The Kerby formula is directional, with positive values indicating that the results support the hypothesis.
A non-directional formula for the rank-biserial correlation was provided by Wendt, such that the correlation is always positive.[38] The advantage of the Wendt formula is that it can be computed with information that is readily available in published papers. The formula uses only the test value of U from the Mann-Whitney U test, and the sample sizes of the two groups: r = 1 – (2U)/(n1 n2). Note that U is defined here according to the classic definition as the smaller of the two U values which can be computed from the data. This ensures that 2U < n1n2, as n1n2 is the maximum value of the U statistics.
An example can illustrate the use of the two formulas. Consider a health study of twenty older adults, with ten in the treatment group and ten in the control group; hence, there are ten times ten or 100 pairs. The health program uses diet, exercise, and supplements to improve memory, and memory is measured by a standardized test. A Mann-Whitney U test shows that the adult in the treatment group had the better memory in 70 of the 100 pairs, and the poorer memory in 30 pairs. The Mann-Whitney U is the smaller of 70 and 30, so U = 30. The correlation between memory and treatment performance by the Kerby simple difference formula is r = (70/100) − (30/100) = 0.40. The correlation by the Wendt formula is r = 1 − (2·30)/(10·10) = 0.40.
Effect size for ordinal data
Cliff's delta or $d$, originally developed by Norman Cliff for use with ordinal data,[39] is a measure of how often the values in one distribution are larger than the values in a second distribution. Crucially, it does not require any assumptions about the shape or spread of the two distributions.
The sample estimate $d$ is given by:
$d={\frac {\sum _{i,j}[x_{i}>x_{j}]-[x_{i}<x_{j}]}{mn}}$
where the two distributions are of size $n$ and $m$ with items $x_{i}$ and $x_{j}$, respectively, and $[\cdot ]$ is the Iverson bracket, which is 1 when the contents are true and 0 when false.
$d$ is linearly related to the Mann–Whitney U statistic; however, it captures the direction of the difference in its sign. Given the Mann–Whitney $U$, $d$ is:
$d={\frac {2U}{mn}}-1$
Confidence intervals by means of noncentrality parameters
Confidence intervals of standardized effect sizes, especially Cohen's ${d}$ and $f^{2}$, rely on the calculation of confidence intervals of noncentrality parameters (ncp). A common approach to construct the confidence interval of ncp is to find the critical ncp values to fit the observed statistic to tail quantiles α/2 and (1 − α/2). The SAS and R-package MBESS provides functions to find critical values of ncp.
t-test for mean difference of single group or two related groups
For a single group, M denotes the sample mean, μ the population mean, SD the sample's standard deviation, σ the population's standard deviation, and n is the sample size of the group. The t value is used to test the hypothesis on the difference between the mean and a baseline μbaseline. Usually, μbaseline is zero. In the case of two related groups, the single group is constructed by the differences in pair of samples, while SD and σ denote the sample's and population's standard deviations of differences rather than within original two groups.
$t:={\frac {M-\mu _{\text{baseline}}}{\text{SE}}}={\frac {M-\mu _{\text{baseline}}}{{\text{SD}}/{\sqrt {n}}}}={\frac {{\sqrt {n}}\left({\frac {M-\mu }{\sigma }}\right)+{\sqrt {n}}\left({\frac {\mu -\mu _{\text{baseline}}}{\sigma }}\right)}{\frac {\text{SD}}{\sigma }}}$
$ncp={\sqrt {n}}\left({\frac {\mu -\mu _{\text{baseline}}}{\sigma }}\right)$
and Cohen's
$d:={\frac {M-\mu _{\text{baseline}}}{\text{SD}}}$
is the point estimate of
${\frac {\mu -\mu _{\text{baseline}}}{\sigma }}.$
So,
${\tilde {d}}={\frac {ncp}{\sqrt {n}}}.$
t-test for mean difference between two independent groups
n1 or n2 are the respective sample sizes.
$t:={\frac {M_{1}-M_{2}}{{\text{SD}}_{\text{within}}/{\sqrt {\frac {2*n_{1}n_{2}}{n_{1}+n_{2}}}}}},$
wherein
${\text{SD}}_{\text{within}}:={\sqrt {\frac {{\text{SS}}_{\text{within}}}{{\text{df}}_{\text{within}}}}}={\sqrt {\frac {(n_{1}-1){\text{SD}}_{1}^{2}+(n_{2}-1){\text{SD}}_{2}^{2}}{n_{1}+n_{2}-2}}}.$
$ncp={\sqrt {\frac {n_{1}n_{2}}{n_{1}+n_{2}}}}{\frac {\mu _{1}-\mu _{2}}{\sigma }}$
and Cohen's
$d:={\frac {M_{1}-M_{2}}{SD_{\text{within}}}}$
is the point estimate of ${\frac {\mu _{1}-\mu _{2}}{\sigma }}.$
So,
${\tilde {d}}={\frac {ncp}{\sqrt {\frac {n_{1}n_{2}}{n_{1}+n_{2}}}}}.$
One-way ANOVA test for mean difference across multiple independent groups
One-way ANOVA test applies noncentral F distribution. While with a given population standard deviation $\sigma $, the same test question applies noncentral chi-squared distribution.
$F:={\frac {{\frac {{\text{SS}}_{\text{between}}}{\sigma ^{2}}}/{\text{df}}_{\text{between}}}{{\frac {{\text{SS}}_{\text{within}}}{\sigma ^{2}}}/{\text{df}}_{\text{within}}}}$
For each j-th sample within i-th group Xi,j, denote
$M_{i}(X_{i,j}):={\frac {\sum _{w=1}^{n_{i}}X_{i,w}}{n_{i}}};\;\mu _{i}(X_{i,j}):=\mu _{i}.$
While,
${\begin{aligned}{\text{SS}}_{\text{between}}/\sigma ^{2}&={\frac {{\text{SS}}\left(M_{i}(X_{i,j});i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)}{\sigma ^{2}}}\\&={\text{SS}}\left({\frac {M_{i}(X_{i,j}-\mu _{i})}{\sigma }}+{\frac {\mu _{i}}{\sigma }};i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)\\&\sim \chi ^{2}\left({\text{df}}=K-1,\;ncp=SS\left({\frac {\mu _{i}(X_{i,j})}{\sigma }};i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)\right)\end{aligned}}$
So, both ncp(s) of F and $\chi ^{2}$ equate
${\text{SS}}\left(\mu _{i}(X_{i,j})/\sigma ;i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right).$
In case of $n:=n_{1}=n_{2}=\cdots =n_{K}$ for K independent groups of same size, the total sample size is N := n·K.
${\text{Cohens }}{\tilde {f}}^{2}:={\frac {{\text{SS}}(\mu _{1},\mu _{2},\dots ,\mu _{K})}{K\cdot \sigma ^{2}}}={\frac {{\text{SS}}\left(\mu _{i}(X_{i,j})/\sigma ;i=1,2,\dots ,K,\;j=1,2,\dots ,n_{i}\right)}{n\cdot K}}={\frac {ncp}{n\cdot K}}={\frac {ncp}{N}}.$
The t-test for a pair of independent groups is a special case of one-way ANOVA. Note that the noncentrality parameter $ncp_{F}$ of F is not comparable to the noncentrality parameter $ncp_{t}$ of the corresponding t. Actually, $ncp_{F}=ncp_{t}^{2}$, and ${\tilde {f}}=\left|{\frac {\tilde {d}}{2}}\right|$.
See also
• Estimation statistics
• Statistical significance
• Z-factor, an alternative measure of effect size
References
1. Kelley, Ken; Preacher, Kristopher J. (2012). "On Effect Size". Psychological Methods. 17 (2): 137–152. doi:10.1037/a0028086. PMID 22545595. S2CID 34152884.
2. Rosenthal, Robert, H. Cooper, and L. Hedges. "Parametric measures of effect size." The handbook of research synthesis 621 (1994): 231–244. ISBN 978-0871541635
3. Wilkinson, Leland (1999). "Statistical methods in psychology journals: Guidelines and explanations". American Psychologist. 54 (8): 594–604. doi:10.1037/0003-066X.54.8.594. S2CID 428023.
4. Nakagawa, Shinichi; Cuthill, Innes C (2007). "Effect size, confidence interval and statistical significance: a practical guide for biologists". Biological Reviews of the Cambridge Philosophical Society. 82 (4): 591–605. doi:10.1111/j.1469-185X.2007.00027.x. PMID 17944619. S2CID 615371.
5. Ellis, Paul D. (2010). The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results. Cambridge University Press. ISBN 978-0-521-14246-5.
6. Brand A, Bradley MT, Best LA, Stoica G (2008). "Accuracy of effect size estimates from published psychological research" (PDF). Perceptual and Motor Skills. 106 (2): 645–649. doi:10.2466/PMS.106.2.645-649. PMID 18556917. S2CID 14340449. Archived from the original (PDF) on 2008-12-17. Retrieved 2008-10-31.
7. Brand A, Bradley MT, Best LA, Stoica G (2011). "Multiple trials may yield exaggerated effect size estimates" (PDF). The Journal of General Psychology. 138 (1): 1–11. doi:10.1080/00221309.2010.520360. PMID 21404946. S2CID 932324.
8. Sterne, Jonathan A. C.; Gavaghan, David; Egger, Matthias (2000-11-01). "Publication and related bias in meta-analysis: Power of statistical tests and prevalence in the literature". Journal of Clinical Epidemiology. 53 (11): 1119–1129. doi:10.1016/S0895-4356(00)00242-0. ISSN 0895-4356. PMID 11106885.
9. Cohen, Jacob (1988). Statistical Power Analysis for the Behavioral Sciences. Routledge. ISBN 978-1-134-74270-7.
10. Sawilowsky, S (2009). "New effect size rules of thumb". Journal of Modern Applied Statistical Methods. 8 (2): 467–474. doi:10.22237/jmasm/1257035100. http://digitalcommons.wayne.edu/jmasm/vol8/iss2/26/
11. Russell V. Lenth. "Java applets for power and sample size". Division of Mathematical Sciences, the College of Liberal Arts or The University of Iowa. Retrieved 2008-10-08.
12. Lipsey, M.W.; et al. (2012). Translating the Statistical Representation of the Effects of Education Interventions Into More Readily Interpretable Forms (PDF). United States: U.S. Dept of Education, National Center for Special Education Research, Institute of Education Sciences, NCSER 2013–3000.
13. Sawilowsky, S. S. (2005). "Abelson's paradox and the Michelson-Morley experiment". Journal of Modern Applied Statistical Methods. 4 (1): 352. doi:10.22237/jmasm/1114907520.
14. Sawilowsky, S.; Sawilowsky, J.; Grissom, R. J. (2010). "Effect Size". In Lovric, M. (ed.). International Encyclopedia of Statistical Science. Springer.
15. Sawilowsky, S. (2003). "Deconstructing Arguments from the Case Against Hypothesis Testing". Journal of Modern Applied Statistical Methods. 2 (2): 467–474. doi:10.22237/jmasm/1067645940.
16. Cohen, J (1992). "A power primer". Psychological Bulletin. 112 (1): 155–159. doi:10.1037/0033-2909.112.1.155. PMID 19565683.
17. Tabachnick, B.G. & Fidell, L.S. (2007). Chapter 4: "Cleaning up your act. Screening data prior to analysis", p. 55 In B.G. Tabachnick & L.S. Fidell (Eds.), Using Multivariate Statistics, Fifth Edition. Boston: Pearson Education, Inc. / Allyn and Bacon.
18. Olejnik, S.; Algina, J. (2003). "Generalized Eta and Omega Squared Statistics: Measures of Effect Size for Some Common Research Designs" (PDF). Psychological Methods. 8 (4): 434–447. doi:10.1037/1082-989x.8.4.434. PMID 14664681. S2CID 6931663.
19. Steiger, J. H. (2004). "Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis" (PDF). Psychological Methods. 9 (2): 164–182. doi:10.1037/1082-989x.9.2.164. PMID 15137887.
20. Hair, J.; Hult, T. M.; Ringle, C. M. and Sarstedt, M. (2014) A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM), Sage, pp. 177–178. ISBN 1452217440
21. Larry V. Hedges & Ingram Olkin (1985). Statistical Methods for Meta-Analysis. Orlando: Academic Press. ISBN 978-0-12-336380-0.
22. Andrade, Chittaranjan (22 September 2020). "Mean Difference, Standardized Mean Difference (SMD), and Their Use in Meta-Analysis". The Journal of Clinical Psychiatry. 81 (5). doi:10.4088/JCP.20f13681. eISSN 1555-2101. PMID 32965803. S2CID 221865130. SMD values of 0.2-0.5 are considered small, values of 0.5-0.8 are considered medium, and values > 0.8 are considered large. In psychopharmacology studies that compare independent groups, SMDs that are statistically significant are almost always in the small to medium range. It is rare for large SMDs to be obtained.
23. Robert E. McGrath; Gregory J. Meyer (2006). "When Effect Sizes Disagree: The Case of r and d" (PDF). Psychological Methods. 11 (4): 386–401. CiteSeerX 10.1.1.503.754. doi:10.1037/1082-989x.11.4.386. PMID 17154753. Archived from the original (PDF) on 2013-10-08. Retrieved 2014-07-30.
24. Hartung, Joachim; Knapp, Guido; Sinha, Bimal K. (2008). Statistical Meta-Analysis with Applications. John Wiley & Sons. ISBN 978-1-118-21096-3.
25. Kenny, David A. (1987). "Chapter 13" (PDF). Statistics for the Social and Behavioral Sciences. Little, Brown. ISBN 978-0-316-48915-7.
26. Cohen 1988, p. 49.
27. Larry V. Hedges (1981). "Distribution theory for Glass' estimator of effect size and related estimators". Journal of Educational Statistics. 6 (2): 107–128. doi:10.3102/10769986006002107. S2CID 121719955.
28. Del Giudice, Marco (2013-07-18). "Multivariate Misgivings: Is D a Valid Measure of Group and Sex Differences?". Evolutionary Psychology. 11 (5): 147470491301100. doi:10.1177/147470491301100511.
29. Aaron, B., Kromrey, J. D., & Ferron, J. M. (1998, November). Equating r-based and d-based effect-size indices: Problems with a commonly recommended formula. Paper presented at the annual meeting of the Florida Educational Research Association, Orlando, FL. (ERIC Document Reproduction Service No. ED433353)
30. Sheskin, David J. (2003). Handbook of Parametric and Nonparametric Statistical Procedures (Third ed.). CRC Press. ISBN 978-1-4200-3626-8.
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32. Stegenga, J. (2015). "Measuring Effectiveness". Studies in History and Philosophy of Biological and Biomedical Sciences. 54: 62–71. doi:10.1016/j.shpsc.2015.06.003. PMID 26199055.
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Further reading
• Aaron, B., Kromrey, J. D., & Ferron, J. M. (1998, November). Equating r-based and d-based effect-size indices: Problems with a commonly recommended formula. Paper presented at the annual meeting of the Florida Educational Research Association, Orlando, FL. (ERIC Document Reproduction Service No. ED433353)
• Bonett, D. G. (2008). "Confidence intervals for standardized linear contrasts of means". Psychological Methods. 13 (2): 99–109. doi:10.1037/1082-989x.13.2.99. PMID 18557680.
• Bonett, D. G. (2009). "Estimating standardized linear contrasts of means with desired precision". Psychological Methods. 14 (1): 1–5. doi:10.1037/a0014270. PMID 19271844.
• Brooks, M.E.; Dalal, D.K.; Nolan, K.P. (2013). "Are common language effect sizes easier to understand than traditional effect sizes?". Journal of Applied Psychology. 99 (2): 332–340. doi:10.1037/a0034745. PMID 24188393.
• Cumming, G.; Finch, S. (2001). "A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions". Educational and Psychological Measurement. 61 (4): 530–572. doi:10.1177/0013164401614002. S2CID 120672914.
• Kelley, K (2007). "Confidence intervals for standardized effect sizes: Theory, application, and implementation". Journal of Statistical Software. 20 (8): 1–24. doi:10.18637/jss.v020.i08.
• Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage: Thousand Oaks, CA.
External links
Wikiversity has learning resources about Effect size
Further explanations
• Effect Size (ES)
• EffectSizeFAQ.com
• EstimationStats.com Web app for generating effect-size plots.
• Measuring Effect Size
• Computing and Interpreting Effect size Measures with ViSta
• effsize package for the R Project for Statistical Computing
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These days, nootropics are beginning to take their rightful place as a particularly powerful tool in the Neurohacker's toolbox. After all, biochemistry is deeply foundational to neural function. Whether you are trying to fix the damage that is done to your nervous system by a stressful and toxic environment or support and enhance your neural functioning, getting the chemistry right is table-stakes. And we are starting to get good at getting it right. What's changed?
Smart drugs offer significant memory enhancing benefits. Clinical studies of the best memory pills have shown gains to focus and memory. Individuals seek the best quality supplements to perform better for higher grades in college courses or become more efficient, productive, and focused at work for career advancement. It is important to choose a high quality supplement to get the results you want.
And in his followup work, An opportunity cost model of subjective effort and task performance (discussion). Kurzban seems to have successfully refuted the blood-glucose theory, with few dissenters from commenting researchers. The more recent opinion seems to be that the sugar interventions serve more as a reward-signal indicating more effort is a good idea, not refueling the engine of the brain (which would seem to fit well with research on procrastination).↩
Because executive functions tend to work in concert with one another, these three categories are somewhat overlapping. For example, tasks that require working memory also require a degree of cognitive control to prevent current stimuli from interfering with the contents of working memory, and tasks that require planning, fluency, and reasoning require working memory to hold the task goals in mind. The assignment of studies to sections was based on best fit, according to the aspects of executive function most heavily taxed by the task, rather than exclusive category membership. Within each section, studies are further grouped according to the type of task and specific type of learning, working memory, cognitive control, or other executive function being assessed.
After my rudimentary stacking efforts flamed out in unspectacular fashion, I tried a few ready-made stacks—brand-name nootropic cocktails that offer to eliminate the guesswork for newbies. They were just as useful. And a lot more expensive. Goop's Braindust turned water into tea-flavored chalk. But it did make my face feel hot for 45 minutes. Then there were the two pills of Brain Force Plus, a supplement hawked relentlessly by Alex Jones of InfoWars infamy. The only result of those was the lingering guilt of knowing that I had willingly put $19.95 in the jorts pocket of a dipshit conspiracy theorist.
The resurgent popularity of nootropics—an umbrella term for supplements that purport to boost creativity, memory, and cognitive ability—has more than a little to do with the recent Silicon Valley-induced obsession with disrupting literally everything, up to and including our own brains. But most of the appeal of smart drugs lies in the simplicity of their age-old premise: Take the right pill and you can become a better, smarter, as-yet-unrealized version of yourself—a person that you know exists, if only the less capable you could get out of your own way.
It looks like the overall picture is that nicotine is absorbed well in the intestines and the colon, but not so well in the stomach; this might be the explanation for the lack of effect, except on the other hand, the specific estimates I see are that 10-20% of the nicotine will be bioavailable in the stomach (as compared to 50%+ for mouth or lungs)… so any of my doses of >5ml should have overcome the poorer bioavailability! But on the gripping hand, these papers are mentioning something about the liver metabolizing nicotine when absorbed through the stomach, so…
When I worked on the Bulletproof Diet book, I wanted to verify that the effects I was getting from Bulletproof Coffee were not coming from modafinil, so I stopped using it and measured my cognitive performance while I was off of it. What I found was that on Bulletproof Coffee and the Bulletproof Diet, my mental performance was almost identical to my performance on modafinil. I still travel with modafinil, and I'll take it on occasion, but while living a Bulletproof lifestyle I rarely feel the need.
As with other nootropics, the way it works is still partially a mystery, but most research points to it acting as a weak dopamine reuptake inhibitor. Put simply, it increases your dopamine levels the same way cocaine does, but in a much less extreme fashion. The enhanced reward system it creates in the brain, however, makes it what Patel considers to be the most potent cognitive enhancer available; and he notes that some people go from sloth to superman within an hour or two of taking it.
One idea I've been musing about is the connections between IQ, Conscientiousness, and testosterone. IQ and Conscientiousness do not correlate to a remarkable degree - even though one would expect IQ to at least somewhat enable a long-term perspective, self-discipline, metacognition, etc! There are indications in studies of gifted youth that they have lower testosterone levels. The studies I've read on testosterone indicate no improvements to raw ability. So, could there be a self-sabotaging aspect to human intelligence whereby greater intelligence depends on lack of testosterone, but this same lack also holds back Conscientiousness (despite one's expectation that intelligence would produce greater self-discipline and planning), undermining the utility of greater intelligence? Could cases of high IQ types who suddenly stop slacking and accomplish great things sometimes be due to changes in testosterone? Studies on the correlations between IQ, testosterone, Conscientiousness, and various measures of accomplishment are confusing and don't always support this theory, but it's an idea to keep in mind.
Supplements, medications, and coffee certainly might play a role in keeping our brains running smoothly at work or when we're trying to remember where we left our keys. But the long-term effects of basic lifestyle practices can't be ignored. "For good brain health across the life span, you should keep your brain active," Sahakian says. "There is good evidence for 'use it or lose it.'" She suggests brain-training apps to improve memory, as well as physical exercise. "You should ensure you have a healthy diet and not overeat. It is also important to have good-quality sleep. Finally, having a good work-life balance is important for well-being." Try these 8 ways to get smarter while you sleep.
Took pill 12:11 PM. I am not certain. While I do get some things accomplished (a fair amount of work on the Silk Road article and its submission to places), I also have some difficulty reading through a fiction book (Sum) and I seem kind of twitchy and constantly shifting windows. I am weakly inclined to think this is Adderall (say, 60%). It's not my normal feeling. Next morning - it was Adderall.
I never watch SNL. I just happen to know about every skit, every line of dialogue because I'm a stable genius.Hey Donnie, perhaps you are unaware that:1) The only Republican who is continually obsessed with how he or she is portrayed on SNL is YOU.2) SNL has always been laden with political satire.3) There is something called the First Amendment that would undermine your quest for retribution.
With just 16 predictions, I can't simply bin the predictions and say yep, that looks good. Instead, we can treat each prediction as equivalent to a bet and see what my winnings (or losses) were; the standard such proper scoring rule is the logarithmic rule which pretty simple: you earn the logarithm of the probability if you were right, and the logarithm of the negation if you were wrong; he who racks up the fewest negative points wins. We feed in a list and get back a number:
Smart drugs act within the brain speeding up chemical transfers, acting as neurotransmitters, or otherwise altering the exchange of brain chemicals. There are typically very few side effects, and they are considered generally safe when used as indicated. Special care should be used by those who have underlying health conditions, are on other medications, pregnant women, and children, as there is no long-term data on the use and effects of nootropics in these groups.
The advantage of adrafinil is that it is legal & over-the-counter in the USA, so one removes the small legal risk of ordering & possessing modafinil without a prescription, and the retailers may be more reliable because they are not operating in a niche of dubious legality. Based on comments from others, the liver problem may have been overblown, and modafinil vendors post-2012 seem to have become more unstable, so I may give adrafinil (from another source than Antiaging Central) a shot when my modafinil/armodafinil run out.
In sum, the evidence concerning stimulant effects of working memory is mixed, with some findings of enhancement and some null results, although no findings of overall performance impairment. A few studies showed greater enhancement for less able participants, including two studies reporting overall null results. When significant effects have been found, their sizes vary from small to large, as shown in Table 4. Taken together, these results suggest that stimulants probably do enhance working memory, at least for some individuals in some task contexts, although the effects are not so large or reliable as to be observable in all or even most working memory studies.
Nicotine absorption through the stomach is variable and relatively reduced in comparison with absorption via the buccal cavity and the small intestine. Drinking, eating, and swallowing of tobacco smoke by South American Indians have frequently been reported. Tenetehara shamans reach a state of tobacco narcosis through large swallows of smoke, and Tapirape shams are said to eat smoke by forcing down large gulps of smoke only to expel it again in a rapid sequence of belches. In general, swallowing of tobacco smoke is quite frequently likened to drinking. However, although the amounts of nicotine swallowed in this way - or in the form of saturated saliva or pipe juice - may be large enough to be behaviorally significant at normal levels of gastric pH, nicotine, like other weak bases, is not significantly absorbed.
I do recommend a few things, like modafinil or melatonin, to many adults, albeit with misgivings about any attempt to generalize like that. (It's also often a good idea to get powders, see the appendix.) Some of those people are helped; some have told me that they tried and the suggestion did little or nothing. I view nootropics as akin to a biological lottery; one good discovery pays for all. I forge on in the hopes of further striking gold in my particular biology. Your mileage will vary. All you have to do, all you can do is to just try it. Most of my experiences were in my 20s as a right-handed 5'11 white male weighing 190-220lbs, fitness varying over time from not-so-fit to fairly fit. In rough order of personal effectiveness weighted by costs+side-effects, I rank them as follows:
The FDA has approved the first smart pill for use in the United States. Called Abilify MyCite, the pill contains a drug and an ingestible sensor that is activated when it comes into contact with stomach fluid to detect when the pill has been taken. The pill then transmits this data to a wearable patch that subsequently transfers the information to an app on a paired smartphone. From that point, with a patient's consent, the data can be accessed by the patient's doctors or caregivers via a web portal.
It may also be necessary to ask not just whether a drug enhances cognition, but in whom. Researchers at the University of Sussex have found that nicotine improved performance on memory tests in young adults who carried one variant of a particular gene but not in those with a different version. In addition, there are already hints that the smarter you are, the less smart drugs will do for you. One study found that modafinil improved performance in a group of students whose mean IQ was 106, but not in a group with an average of 115.
When Giurgea coined the word nootropic (combining the Greek words for mind and bending) in the 1970s, he was focused on a drug he had synthesized called piracetam. Although it is approved in many countries, it isn't categorized as a prescription drug in the United States. That means it can be purchased online, along with a number of newer formulations in the same drug family (including aniracetam, phenylpiracetam, and oxiracetam). Some studies have shown beneficial effects, including one in the 1990s that indicated possible improvement in the hippocampal membranes in Alzheimer's patients. But long-term studies haven't yet borne out the hype.
Let's start with the basics of what smart drugs are and what they aren't. The field of cosmetic psychopharmacology is still in its infancy, but the use of smart drugs is primed to explode during our lifetimes, as researchers gain increasing understanding of which substances affect the brain and how they do so. For many people, the movie Limitless was a first glimpse into the possibility of "a pill that can make you smarter," and while that fiction is a long way from reality, the possibilities - in fact, present-day certainties visible in the daily news - are nevertheless extremely exciting.
Since dietary supplements do not require double-blind, placebo-controlled, pharmaceutical-style human studies before going to market, there is little incentive for companies to really prove that something does what they say it does. This means that, in practice, nootropics may not live up to all the grandiose, exuberant promises advertised on the bottle in which they come. The flip side, though? There's no need to procure a prescription in order to try them out. Good news for aspiring biohackers—and for people who have no aspirations to become biohackers, but still want to be Bradley Cooper in Limitless (me).
Probably most significantly, use of the term "drug" has a significant negative connotation in our culture. "Drugs" are bad: So proclaimed Richard Nixon in the War on Drugs, and Nancy "No to Drugs" Reagan decades later, and other leaders continuing to present day. The legitimate demonization of the worst forms of recreational drugs has resulted in a general bias against the elective use of any chemical to alter the body's processes. Drug enhancement of athletes is considered cheating – despite the fact that many of these physiological shortcuts obviously work. University students and professionals seeking mental enhancements by taking smart drugs are now facing similar scrutiny.
There are seven primary classes used to categorize smart drugs: Racetams, Stimulants, Adaptogens, Cholinergics, Serotonergics, Dopaminergics, and Metabolic Function Smart Drugs. Despite considerable overlap and no clear border in the brain and body's responses to these substances, each class manifests its effects through a different chemical pathway within the body.
Like caffeine, nicotine tolerates rapidly and addiction can develop, after which the apparent performance boosts may only represent a return to baseline after withdrawal; so nicotine as a stimulant should be used judiciously, perhaps roughly as frequent as modafinil. Another problem is that nicotine has a half-life of merely 1-2 hours, making regular dosing a requirement. There is also some elevated heart-rate/blood-pressure often associated with nicotine, which may be a concern. (Possible alternatives to nicotine include cytisine, 2'-methylnicotine, GTS-21, galantamine, Varenicline, WAY-317,538, EVP-6124, and Wellbutrin, but none have emerged as clearly superior.)
This would be a very time-consuming experiment. Any attempt to combine this with other experiments by ANOVA would probably push the end-date out by months, and one would start to be seriously concerned that changes caused by aging or environmental factors would contaminate the results. A 5-year experiment with 7-month intervals will probably eat up 5+ hours to prepare <12,000 pills (active & placebo); each switch and test of mental functioning will probably eat up another hour for 32 hours. (And what test maintains validity with no practice effects over 5 years? Dual n-back would be unusable because of improvements to WM over that period.) Add in an hour for analysis & writeup, that suggests >38 hours of work, and 38 \times 7.25 = 275.5. 12,000 pills is roughly $12.80 per thousand or $154; 120 potassium iodide pills is ~$9, so \frac{365.25}{120} \times 9 \times 5 = 137.
My general impression is positive; it does seem to help with endurance and extended the effect of piracetam+choline, but is not as effective as that combo. At $20 for 30g (bought from Smart Powders), I'm not sure it's worthwhile, but I think at $10-15 it would probably be worthwhile. Sulbutiamine seems to affect my sleep negatively, like caffeine. I bought 2 or 3 canisters for my third batch of pills along with the theanine. For a few nights in a row, I slept terribly and stayed awake thinking until the wee hours of the morning; eventually I realized it was because I was taking the theanine pills along with the sleep-mix pills, and the only ingredient that was a stimulant in the batch was - sulbutiamine. I cut out the theanine pills at night, and my sleep went back to normal. (While very annoying, this, like the creatine & taekwondo example, does tend to prove to me that sulbutiamine was doing something and it is not pure placebo effect.)
70 pairs is 140 blocks; we can drop to 36 pairs or 72 blocks if we accept a power of 0.5/50% chance of reaching significance. (Or we could economize by hoping that the effect size is not 3.5 but maybe twice the pessimistic guess; a d=0.5 at 50% power requires only 12 pairs of 24 blocks.) 70 pairs of blocks of 2 weeks, with 2 pills a day requires (70 \times 2) \times (2 \times 7) \times 2 = 3920 pills. I don't even have that many empty pills! I have <500; 500 would supply 250 days, which would yield 18 2-week blocks which could give 9 pairs. 9 pairs would give me a power of:
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Research | Open | Open Peer Review | Published: 28 December 2017
A general approach to risk modeling using partial surrogate markers with application to perioperative acute kidney injury
Derek K. Smith1,
Loren E. Smith2,
Frederic T. Billings IV2 &
Jeffrey D. Blume1
Surrogate outcomes are often utilized when disease outcomes are difficult to directly measure. When a biological threshold effect exists, surrogate outcomes may only represent disease in specific subpopulations. We refer to these outcomes as "partial surrogate outcomes." We hypothesized that risk models of partial surrogate outcomes would perform poorly if they fail to account for this population heterogeneity. We developed criteria for predictive model development using partial surrogate outcomes and demonstrate their importance in model selection and evaluation within the clinical example of serum creatinine, a partial surrogate outcome for acute kidney injury.
Data from 4737 patients who underwent cardiac surgery at a major academic center were obtained. Linear and mixture models were fit on maximum 2-day serum creatinine change as a surrogate for estimated glomerular filtration rate at 90 days after surgery (eGFR90), adjusted for known AKI risk factors. The AUC for eGFR90 decline and Spearman's rho were calculated to compare model discrimination between the linear model and a single component of the mixture model deemed to represent the informative subpopulation. Simulation studies based on the clinical data were conducted to further demonstrate the consistency and limitations of the procedure.
The mixture model was highly favored over the linear model with BICs of 2131.3 and 5034.3, respectively. When model discrimination was evaluated with respect to the partial surrogate, the linear model displays superior performance (p < 0.001); however, when it was evaluated with respect to the target outcome, the mixture model approach displays superior performance (AUC difference p = 0.002; Spearman's difference p = 0.020). Simulation studies demonstrate that the nature of the heterogeneity determines the magnitude of any advantage the mixture model.
Partial surrogate outcomes add complexity and limitations to risk score modeling, including the potential for the usual metrics of discrimination to be misleading. Partial surrogacy can be potentially uncovered and appropriately accounted for using a mixture model approach. Serum creatinine behaved as a partial surrogate outcome consistent with two patient subpopulations, one representing patients whose injury did not exceed their renal functional reserve and a second population representing patients whose injury did exceed renal functional reserve.
Patient level clinical risk score development and associated decision support applications are vitally important to modern personalized medicine. For many pathologies, this process is straightforward. First, the disease process of interest is defined and the data about relevant covariates are collected. This information is used to develop a statistical model that meets desired performance measures. When the disease process is difficult to directly measure, however, surrogate measurements are often used which prevent the use of simple risk score modeling methodology.
In 1989, Prentice defined necessary surrogate outcome criteria to ensure valid hypothesis testing [1]. Further work on surrogate outcome criteria has focused on the preservation of type I error rates for inference [2]. Surrogate outcome criteria for the development of risk scores, however, remain undefined. These criteria will be developed in Section 2 of this work.
After delineating criteria for the use of surrogate markers in risk score development, Section 3 will examine the increased modeling complexity associated with partial surrogacy situations. Partial surrogates are a class of markers that behave differently in different patient subpopulations. In one subpopulation, partial surrogates may display a high association with the outcome of interest, while in others, they may display no association. In this study, we examine how serum creatinine change for the assessment of acute kidney injury behaves in this manner. However, there are many other commonly used clinical markers that scientific considerations make suspect for partial surrogacy. For example, liver enzymes are often used to measure acute liver injury in the same way serum creatinine change is used to measure acute kidney injury. Unlike serum creatinine, liver enzymes are a direct biomarker for liver damage entering the blood stream as a direct result of liver cell death. Although it might seem like that would preclude partial surrogate behavior, there are subpopulations where liver enzymes behave differently than in health people. When used as a surrogate for acute liver injury in patients with cirrhosis, the utility of the marker is greatly reduced because in this subpopulation liver damage often resulting in little to no enzyme production. Fitting a predictive model meant to quantify acute liver injury in a population that contains both healthy patients and patients with cirrhosis would likely demonstrate the same type of partial surrogate behavior noted in the AKI example, but instead of arising from a threshold which the injury must overcome to be detectable, it would likely demonstrate a ceiling effect which chronic liver disease patients may have exceeded. Some other examples of markers for which scientific considerations might imply partial surrogacy include alveolar bone loss for the assessment of periodontal disease severity and ST elevations for the assessment of myocardial infarct in a population containing patients with left bundle branch block.
The heterogeneity displayed by partial surrogates between relevant subpopulations influences their ability to satisfy Prentice's criteria for hypothesis testing. We demonstrate that partial surrogate outcomes also complicate our proposed surrogate criteria for risk score prediction. Additionally, evaluating risk scores using a partial surrogate is complicated by the observation that the model which provides optimum discrimination for the surrogate outcome does not necessarily discriminate the target outcome. The implications of this observation for model selection and evaluation of likely clinical benefit will be described. Finally, Section 4 of this manuscript will explore analytical challenges introduced by partial surrogacy theoretically and computationally.
In the course of this work, an analysis of perioperative acute kidney injury (AKI) will be performed to emphasize the clinical importance of our surrogate criteria for risk score modeling and to demonstrate the limitations and special considerations associated with partial surrogates in an applied analysis context.
Section 2: surrogate outcomes in risk score models
Clinical risk scores are commonly assessed in two ways. The first way in which they are assessed is by model discrimination, the degree to which a risk score is ordered similarly to the disease marker of interest. Second, they are assessed by model calibration, a comparison of the magnitude of the risk score and the magnitude of the disease marker of interest. Risk scores that are well calibrated are simpler to implement and are traditionally considered ideal due to the observation that good calibration generally implies good discrimination. Unfortunately, there is no reason to expect that a risk model built on a surrogate measure will be well calibrated as it is designed to predict the surrogate and not the target outcome. For this reason, the risk scores developed here will be evaluated on measures of pure discrimination (area under the receiver operating curve for binary measures and Spearman's rho for continuous ones) as opposed to more traditional measures of predictive performance such as mean square error, which are sensitive to calibration.
Suppose that we are interested in developing a risk score, R, for a true clinical outcome, T, where R is any one-dimensional summary of a patient's data that is intended to help quantify a patient's disease state disposition. Next, suppose that we are unable to measure T itself in the timeframe necessary to develop a useful decision support tool. Finally, suppose a surrogate outcome, S, is readily measurable and related to T either as a mediator or a consequence which presents more readily.
While developing R, our goal will be to obtain a one-dimensional summary of the data that discriminates well and maintains some interpretability of the model coefficients as these are often used to generate hypotheses about potential mechanisms. A score, R, should provide higher scores for higher risk or more severely diseased patients uniformly over the entire range of plausible scores. We will refer to this last property as being clinically useful. Ideally, clinical utility should be consistent over the entire range of potential risk scores. Otherwise, R's discriminatory ability might look favorable when examined over the entire population, despite R preforming poorly for a particular subset of patients. This could result in a net-benefit to the population at the expense of a particular group of individuals, raising questions about the ethical implementation of R for generalized patient care.
What criteria of S which make the resulting risk score more likely to be clinically useful? Although Prentice's criteria have been criticized for being overly stringent for practical application, we will use them here to aid in the development of a less restrictive set of criteria for the evaluation of surrogates for risk stratification. Following the pattern of Prentice's first and second criteria for valid hypothesis testing [1], surrogate endpoints must display a relationship between the suspected risk factors to be included in the model, Z, and both the surrogate and target outcomes, S and T, respectively. Stated more formally, the conditional distributions of S and T on Z must not be equal to the marginal distributions over Z. For clarity, Prentice's criteria will be labeled with P, and the prediction criteria will be labeled with an R.
P 1. The proposed risk factor is related to the surrogate f(S| Z) ≠ f(S).
P 2. The proposed risk factor is related to the target outcome f(T| Z) ≠ f(T).
The necessity of these two criteria, which when applied to risk score procedures will be referred to as R 1 and R 2, respectively, is fairly evident. A failure of criterion R 1 suggests that the covariates included in the predictive model contain no information about the distribution of the surrogate. As such, models built on the surrogate would display little variation in the risk score R|Z and any variation observed would be random. A failure of criterion R 2 suggests that the covariates are not related in any way to the distribution of the target outcome, and although R|Z may display a rich variation, it would be expected that f(T| R, Z) = f(T).
For risk score development, a third, less restrictive relationship between variables is necessary in order to obtain good discrimination and produce a clinically useful model. It is desirable that the distribution of R ∣ T be changing to favor more extreme values as T increases. Therefore, for some T 1 < T 2 corresponding to risk scores R 1 |T 1 and R 2 |T 2 , we have that
R 3. P(R 1 < R 2 | T 1, T 2) > 0.5.
The R 3 criterion promotes variation in R over different values of T. This ensures that, on average, the risk score is producing more extreme values when T is more extreme.
Ideally, the probability described in R 3 would be large. This occurs when the locational shift in the distribution of R|T as T changes is large relative to its variance. Although not a strict requirement, having a risk score that is precise will naturally enhance its value.
In summary, our criteria for the development of a surrogate outcome-based risk score are:
R 1. The proposed risk factor is related to the surrogate f(S| Z) ≠ f(S).
R 2. The proposed risk factor is related to the target outcome f(T| Z) ≠ f(T)
R 3. The distribution of the risk score conditional on T needs to be shifting toward more extreme values amongst those at highest risk for disease
$$ P\left({R}_1\left\langle {R}_2\right|{T}_1,{T}_2\right)>0.5,{T}_1<{T}_2, $$
with Prentice's 3rd and 4th criteria and the magnitude of the variance of R ∣ T relative to its distributional shift being unnecessary but playing roles in determining the value of the resultant score.
These criteria encompass a surrogate outcome's minimum requirements to produce a valid risk score. In the next section, we will begin to examine partial surrogates, and how the failure of some of these criteria in patient subpopulations can negatively impact risk score performance.
Section 3: theoretical considerations regarding partial surrogacy and risk score modeling
In the ideal situation, R 1–R 3 would hold in every subpopulation on which a risk score model is to be trained. In other words, it is beneficial if the phenotype defined by the relationship between Z, S, and T is homogenous throughout a population, P. However, if there are subpopulations demonstrating differing phenotypes, extra care is required to maximize the benefit of risk score models and provide valid estimation procedures. When these heterogeneous subpopulations exist, we will redefine S to be a "partial surrogate."
As an example, suppose you have collected data from P which is composed of two subpopulations V and I, defined by a latent indicator variable, l. In subpopulation V, R 1–R 3 hold, suggesting subpopulation V might produce a valuable risk score model. In subpopulation I, however, only R 2 holds. This suggests that in subpopulation I, S is not meaningfully related to Z or T and is therefore unlikely to result in a profitable risk score in this subpopulation.
The ideal method for risk score development when faced with a partial surrogate is not immediately apparent. One method is to use traditional modeling strategies in the full training dataset. In cases where the full dataset satisfies R 1–R 3, this approach is likely to result in valuable models. If l was known, an analyst might reasonably decide to use only the data from subpopulation V for model development and then generalize the model to the entire population as appropriate. This second method relies on the relationship between T and Z being homogeneous over P. Homogeneity will occur if the subpopulations were defined completely at random. Alternatively, in cases where l is unknown, a latent variable mixture model can be used to produce a similar result. For the duration of this manuscript, l is assumed to be latent.
Given these two approaches, the analyst is forced to choose between the full-data approach and the mixture model approach. For inference and estimation, the choice is clear. Since failing to account for the partial nature of the surrogate will result in a violation of Prentice's 4th criterion, the mixture model is preferable. For example, consider a very simple partial surrogate where
$$ T=S\left|V+{\varepsilon}_{T\mid S}\ \mathrm{and}\ S\right|I={\varepsilon}_{S\mid I},{\varepsilon}_i\sim N\left(0,{\sigma}_i\right) $$
$$ T\mid Z={\beta}_{T\mid Z}+{\beta}_1Z+{\varepsilon}_{T\mid Z}. $$
In this situation, the surrogate is equal to truth plus error when a patient belongs to subpopulation V, but it is a random deviate when the patient is from subpopulation I. The relationship between T and Z is consistent across the entire population. Thus, we have
$$ E\left[T\left|S\right.\right]=P(V)\ S+P(I)\ E\left[T\right]=P(V)S+\left(1-P(V)\right)E\left[T\right] $$
and also that
$$ E\left[T|S,Z\right]=P(V)S+P(I)E\left[T|Z\right]=P(V)S+\left(1-P(V)\right)\left({\beta}_{T\mid Z}+{\beta}_1Z\right). $$
P 4 requires that the distribution of T|S be the same as the distribution of T|S, Z, but even this simple partial surrogate violates that criterion as evidenced by the differing expectations.
However, for risk score modeling, the decision is less clear. Using the full dataset and not accounting for the partial nature of the surrogate generally results in risk scores with lower variance due to higher effective sample size but higher bias due to the inclusion of training data from population I. The mixture model approach generally boasts reduced bias by correctly accounting for heterogeneous subpopulations but suffers higher variance due to diminished training set sample size. There are several aspects unique to a given partial surrogate situation that should affect the analyst's decision regarding these modeling strategies.
When making the decision between using a traditional model or a mixture model, the first consideration is whether the added complexity of the mixture model approach is likely to be beneficial. The mixture model's primary purpose is to estimate covariate/outcome relationships in the subpopulations separately. In order for this to practically improve the risk score's discrimination, it needs to result in a different rank ordering of subjects compared to the traditional approach. This is likely to occur whenever the phenotype expressed in subpopulation I is substantively different than that in subpopulation V in terms of the relative magnitude of the associations between the covariates and outcome. This distinctness of subpopulation phenotypes simultaneously allows the expectation maximization (EM) algorithm used for model fitting to achieve adequate subpopulation separation while achieving a more appropriate ordering of predictions with respect to T.
In order to apply the mixture modeling approach to a clinical problem, it is necessary to decide how many components the model should have. In the case of partial surrogates, this is the number of subpopulations that are present. In many cases, the number of subpopulations may be strongly suspected based on clinical considerations, but in cases where the number is less certain, there is a large literature that describes various methods for identifying the proper number of components and the consequences of selecting the wrong number [3,4,5,6].
The EM algorithm involves beginning with a prior probability of group assignment, fitting a model that is weighted by the prior probability to assess the likelihood of subpopulation membership, and calculating a posterior probability of subpopulation membership based on the prior and the likelihood. This is repeated until convergence is achieved with the posterior probabilities being used to generate the prior probabilities for the next iteration [7]. Substantial separation between subgroup phenotypes results in the EM algorithm calculating final posterior probabilities of subpopulation membership that are close to zero and one, suggesting there is good evidence in the data to direct each patient's subpopulation assignment. When the subpopulations cannot be effectively separated, mixture model variance will be magnified, detracting from its utility and favoring the traditional modeling approach.
A second consideration affecting the development of partial surrogate-based risk scores is how generalizable a subpopulation model based on V will be to the entire population P. If separation into subpopulations I and V is completely random, then any result obtained from subpopulation V should be fully generalizable. If subpopulations I and V are generated by a non-random process, however, neither modeling technique considered above is guaranteed to result in a clinically beneficial risk score, and additional external verification would be necessary to allow generalization.
The last major consideration that influences whether the mixture model approach is viable for risk score development with partial surrogates is the mixing proportion of the population. It is necessary to estimate what proportion of observations is from V versus the proportion from I. If the training data are composed almost entirely of data from V, the mixture model adds little benefit over the traditional model which ignores subpopulations. In contrast, if the data are almost entirely from I, there may not be enough information in the data to accurately fit a model for subpopulation V, which embodies the clinically relevant covariate/outcome relationship. In both of the situations described here, partial surrogate-based risk score models are unlikely to provide a benefit over the traditional modeling approach because the available dataset does not contain enough information regarding the true relationship between covariates and the outcome of interest.
In summary, there is no universal solution to measuring pathology with partial surrogate outcomes. The mixture model approach provides great benefits in some situations, but in others, the mixture model approach fails to adequately fit the data and will lead to inferior performance compared to a more traditional, non-mixture approach.
Section 4: examination of developed risk score criteria and partial surrogate guidelines through clinical analysis and simulation studies of perioperative AKI
Biological background
Ten to 40% of patients develop AKI following major inpatient surgical procedures [8]. Perioperative AKI has been associated with increased short and long-term mortality, increased hospital length of stay, increased risk of developing chronic kidney disease (CKD), and increased risk of developing dialysis dependence [8, 9]. Unlike other perioperative injuries, such as myocardial infarction, there is currently no direct biomarker of kidney injury or cell death that accurately and consistently reflects AKI. Current consensus guidelines for AKI diagnosis use changes in serum concentrations of creatinine to diagnose AKI [10]. Creatinine is produced by the muscle, and the kidneys excrete creatinine. Injured kidneys excrete less creatinine, and increased serum concentrations of creatinine are used to diagnose AKI. The relationship between AKI and serum creatinine is inconsistent, hindering accurate AKI diagnosis. One example of this inconsistency is the clinical situation in which renal injury does not produce an increase in serum creatinine (Fig. 1). Patients often sustain kidney damaged without associated changes in serum creatinine, referred to as subclinical AKI [11]. This situation is illustrated well by the following example. A living kidney donor frequently will experience little to no serum creatinine increase following donor nephrectomy despite removal of roughly 50% of their functional kidney mass [12]. Recent AKI biomarker studies demonstrated that subclinical AKI is also associated with an increased risk of dialysis and in-hospital mortality, suggesting it represents clinically significant levels of renal injury [13]. The accurate measurement of subclinical AKI using common clinical labs in the immediate postoperative period (creatinine) would allow physicians to predict AKI, institute additional patient monitoring, and adjust patient treatments. These benefits could reduce patient morbidity and mortality.
Directed acyclic graph displaying the hypothesized mechanism which would result in statistically heterogeneous subpopulations with respect to serum creatinine change
The dramatic example of living donor kidney donation and minimal serum creatinine change demonstrates that healthy kidneys have the capacity to temporarily increase their filtration rate in times of physiologic stress, a characteristic termed renal functional reserve [14]. Renal functional reserve, however, is difficult to predict, difficult to measure, limited, and exhaustable [14]. In the subpopulation of patients, V, who overcome their renal functional reserve during kidney injury serum creatinine change would be detected, and associations between relevant risk factors and serum creatinine change would be strong, assuming all other serum creatinine modifying factors remain constant. In the subpopulation of patients, I, who do not overcome their renal functional reserve during episodes of kidney injury, only random or nonspecific changes in serum creatinine levels would be measured, and the associations between relevant AKI risk factors and serum creatinine change would be weak. With respect to the proposed risk score criteria outlined in Section 2, this suggests that subpopulation V will likely come close to satisfying P 1–P 4 and R 1–R 3, allowing for simultaneous estimation of associations and risk score generation. In contrast, subpopulation I will likely violate P 1, P 3, P 4, R 1, and R 3, resulting in poor performance of risk indices based exclusively on this subgroup, biased coefficient estimates, and improper p values.
If subpopulations I and V are defined based on exhaustion of renal functional reserve as we hypothesize, then it is important to recognize that the likelihood of renal functional reserve exhaustion is not random. Young, healthy patients are less likely to overcome their substantial renal reserve than older patients with underlying disease [14, 15]. Therefore, generalizing a risk score generated in subpopulation V to the entire population P requires validation of that score in the entire population. This validation can be accomplished by evaluating the partial surrogate-based clinical risk score's discrimination of the target outcome, T. Although there is no gold standard marker for clinically significant kidney damage, one marker of indisputable clinical significance is the decline in kidney glomerular filtration at 90 days [16, 17]. The 90 days following surgery allows the kidneys to recover from acute injury if possible and reestablish an equilibrium serum creatinine concentration. This postoperative day 90 serum creatinine concentration is used to estimate glomerular filtration rate via the Chronic Kidney Disease Epidemiology Collaboration equation (eGFR90) [18], the primary indicator of kidney function. Indeed, current clinical guidelines recommend that patients who experience AKI should routinely have 90 day eGFR evaluation to assess recovery versus progression to permanent kidney damage [19]. Therefore, in this analysis eGFR90 will be considered the target outcome, T.
Data and models
The data used in this analysis are from 4737 patients who underwent cardiac surgery at a large academic medical center from November 2009 through June 2015. Institutional IRB approval was obtained prior to performance of all analyses. In this dataset, all patients had serum creatinine measurements in the first two post-operative days and 1268 patients had 90 ± 15 day eGFR90 measurements available. Table 1 compares the characteristics between those with and without a recorded value for eGFR90. Aside from a slight deviation in the proportion of patients with diabetes, the covariates are well balanced between the groups making a missing at random assumption plausible. However, sensitivity to this assumption will also be assessed.
Table 1 Patient and surgical characteristics stratified by whether the patient record contained a record of eGFR at 90 days postoperatively. Continuous variables are reported as median (IQR) and binary variables are reported as proportion (%)
Ten preoperative and intraoperative traits were selected a priori for inclusion in the analysis including age, body mass index, a diagnosis of diabetes, baseline kidney (glomerular) filtration rate, baseline hemoglobin concentration, volume of intraoperative urine output, volume of intraoperative intravenous fluid administered, maximum measured intraoperative plasma lactate level, length of surgery, and an indicator for emergent surgery. These variables were chosen as well-established predictors of AKI and therefore were considered likely to be valuable predictors of serum creatinine change from baseline [20,21,22,23,24].
For the purposes of model comparisons, a linear model and a two-component mixture of linear models were fit. The residual error of the two mixture components was not constrained. The linear model risk score is the model's prediction. For the mixture model, the risk score is the prediction from the single component of the mixture that is post hoc identified to be associated with subpopulation V. Each model was evaluated based on the following metrics: the AUC for a target outcome greater than 20 mL/min/1.73 m2 and the Spearman's correlation. The first metric is a common method of risk score implementation and is based on the presumption that a change of 20 mL/min/1.73 m2 in eGFR90 is clinically meaningful. An absolute change in eGFR90 was chosen over a relative change because of the many statistical issues that can arise from the inclusion of ratios of random variables such as improper error distributions and spurious associations [25]. For comparison, we have included the ROC curves that would result for the models predicting whether 2-day postoperative serum creatinine change exceeded 0.3 mg/dL, a relatively sensitive cutoff for AKI suggested by the Acute Kidney Injury Network and the Kidney Disease Improving Global Outcomes guidelines [19, 26]. The second metric measures discrimination without requiring an arbitrary cutoff.
In order to assess the sensitivity of this analysis to the missingness of eGFR90 and the choice of cutoff in eGFR90 for the ROC analysis, an additional analysis was performed. A logistic propensity score model was fit to whether eGFR90 was present in the patient record. The resulting propensity score was used as a weight, and the AUC was recalculated. This process was repeated at each potential cutoff in eGFR90 from 5 to 25 mL/min/1.73 m2 resulting in a propensity score adjusted, eGFR90-dependent ROC curve.
The mixture model resulted in moderately well-differentiated clusters and a relative entropy equal to 0.607. Seven hundred twenty-eight patients were modally assigned to the V subpopulation, and 4009 patients were assigned to the I subpopulation. The linear model found all the factors to be significantly associated with eGFR90 change except for a history of diabetes and emergency surgery, which were marginally significant (p = 0.085 and p = 0.063, respectively). The mixture component found all the risk factors to be significantly associated with eGFR change with the exception of emergency surgery (p = 0.072). However, the magnitude of the coefficients for the linear model were attenuated by an average of 42.8% (range = [19.2%, 68.1%]), which is consistent with subpopulation I's phenotype being attenuated relative to subpopulation V. The model coefficients are given in Table 2. In addition, the mixture model represented a substantial improvement in fit over the linear model with BICs of 2131.3 and 5034.3, respectively.
Table 2 Coefficients resulting from the application of the mixture model to the perioperative AKI dataset. The column on the right represents the coefficients from subgroup V and are noticeably larger in absolute magnitude than those on the left
The area under the ROC curve was calculated for each of the candidate risk scores and for the gold standard of the observed serum creatinine change for the prediction of an eGFR90 decline greater than 20 mL/min/1.73 m2. The observed serum creatinine change had the worst estimated AUC of 0.608 (0.572, 0.645), although not significantly worse than that of the linear model 0.633 (0.595, 0.672), p = 0.262. The mixture model component yielded the best AUC of 0.678 (0.641, 0.715), which was a significant improvement over both the observed creatinine change and the linear model, p = 0.002 and p < 0.001, respectively. In addition, the ROCs were calculated for each candidate risk score for the prediction of a serum creatinine increase greater than 0.3 mg/dL. The result was an AUC of 0.602 (0.582, 0.623) for the mixture and 0.663 (0.644, 0.682) for the linear model, p < 0.001. The ROCs for both endpoints are given in Fig. 2.
Receiver operating characteristic curves for the linear (dashed) and mixture (solid) models predicting maximum 2-day creatinine change (left) vs. eGFR90 (right)
The improvement due to using the mixture component's prediction as a risk score for eGFR90 is further demonstrated by looking at Spearman's rank correlation. The correlation between the observed serum creatinine change, and the observed eGFR90 change was 0.231 (95% CI 0.204, 0.258). For the linear model, the correlation was 0.223 (0.196, 0.250). For the mixture component, the correlation was 0.305 (0.280, 0.331). These values were compared via a permutation test showing a significant improvement by the mixture model over the observed value and the linear model's prediction, p = 0.035 and p = 0.020, respectively. The low values of these correlations are due to the fact that the majority of surgical patients sustain no kidney injury; and thus, any change in their eGFR is truly random, i.e., only a small portion of the population's eGFR changes are ordered by something other than random chance, so despite the low correlation the improvement provided by utilizing the partial surrogate is substantial.
The sensitivity analysis was performed as detailed in the methods section resulting in the propensity score adjusted, eGFR90 dependent ROC curves for each model. The difference in AUCs is given in Fig. 3. The AUC for the mixture approach is consistently, significantly higher over the entire range of potential cutoffs after accounting for the propensity to be missing.
Difference in the propensity score adjusted AUC over eGFR90 cutoffs ranging from 5 to 25 mL/min/1.73 m2
This analysis demonstrates a major issue in the development of risk scores using partial surrogate outcomes. If the partial surrogate nature of serum creatinine change had gone unrecognized in this analysis, the analyst would likely look to how well various models discriminate with respect to serum creatinine change as a preferred method for both model selection and characterization. The ROC analysis demonstrates that the analyst would then conclude that the linear model was clearly superior to the mixture model because its risk score is ordered more similarly to the surrogate measure. However, the ROC of the target outcome, eGFR90, shows the true relationship is reversed and that the mixture model produced superior ordering. It is critical to identify partial surrogates and account for them appropriately since there would be no indication of this flaw in analysis if model performance was judged solely on its ability to predict the surrogate outcome, postoperative serum creatinine elevation.
Simulation studies
The results of the clinical example presented above were used to generate simulation studies meant to further illustrate the potential benefits and limitations of using the mixture modeling approach in the presence of a partial surrogate. In the above example, a two-component mixture model was fit to the data resulting in two fitted models, \( {\widehat{f}}_V \) and \( {\widehat{f}}_I \), with corresponding parameters \( \left({\widehat{\beta}}_V,{\widehat{\sigma}}_V^2\right) \) and \( \left({\widehat{\beta}}_I,{\widehat{\sigma}}_I^2\right) \). Each data point represents a draw from one of these two models with a certain probability.
In each of the simulations that follow, each of the patients in the cohort will be assigned to subpopulation V or I by a random draw governed by their individual posterior probability of group membership derived from the fitted mixture model. New outcomes were then generated according to which subgroup the patient was assigned to. In both simulations, a new value of S and T are generated for patients assigned to subgroup V via
$$ {\displaystyle \begin{array}{l}\left.T\right|Z\sim N\left(Z{\widehat{\beta}}_V,{\widehat{\sigma}}_V\right)\\ {}\left.S\right|Z,V\sim N\left(Z{\widehat{\beta}}_V,{\widehat{\sigma}}_V\right).\end{array}} $$
That is T and S are generated from the same model with normal errors. After being drawn, the values of T were normalized to have a means of 0 and standard deviation of 50 in order to make them more consistent with eGFR90 changes. In subpopulation I, T is generated in the same way. The difference is in how S is generated in this subpopulation. In the first simulation, S will be generated via an alternative linear model in which the covariates are still related to S but not in the same way as they are under \( {\widehat{f}}_V \). Each repetition of the simulation used a different model with the coefficients being drawn independently from
$$ {\beta}_{R,i}\sim N\left(0,.176\right),i=1,..,10. $$
The standard deviation for this distribution is ¼ of the range of the observed coefficients with the intention that the sampled coefficients would be of similar magnitude to those observed in the clinical example. The outcome for these patients was then sampled from
$$ S\mid Z,I\sim N\left(Z\left[{\beta}_{I,1},{\beta}_{R,1},\dots, {\beta}_{R,10}\right],{\widehat{\sigma}}_I\right). $$
The second data-based simulation is conducted similarly with one exception. In this simulation, S is drawn from
$$ S\mid Z,I\sim N\left(0,{\widehat{\sigma}}_I\right). $$
This simulation represents the scenario where the covariates of interest have little to no relationship with the surrogate within subpopulation I but are related within subpopulation V. The directed acyclic graphs that describe these two scenarios are given in Fig. 4.
Directed acyclic graphs outlining the scenarios mimicked in the two simulation studies
Having generated new outcome variables T and S for both subpopulations, a linear and two-component mixture are then fit to the simulated data. The models are compared as in the clinical example, a cutoff-based measure of discrimination (AUC), a cutoff-free measure of discrimination (Spearman's rank correlation), and relative mean-square error (MSE) reduction in the estimation of the model coefficients. The results of these simulations are summarized in Table 3.
Table 3 Results of the two simulation studies given as mean (0.05 quantile, 0.95 quantile)
In the first simulation, where the outcome for those assigned to subpopulation I were generated from a model that was distinct from \( {\widehat{f}}_V \) but not null, the results favored application of the mixture model in each case. For the linear model, the AUC for discriminating whether T was larger than 20 spanned almost the entire range of the statistic [0.05, 0.95] quantiles = [0.518, 0.923]; the results for the mixture model was more consistent [0.05, 0.95] quantiles = [0.908, 0.982]. The mean difference in AUC was 0.213, [0.05, 0.95] quantiles = [0.034, 0.433]. The Spearman's correlation displayed a similar result with the linear model yielding a wide range of values [0.05, 0.95] quantiles = [− 0.091, 0.349], whereas the mixture model produced more consistent results [0.05, 0.95] quantiles = [0.471, 0.526]. The mean difference in rank correlation was 0.349, [0.05, 0.95] quantiles = [0.146, 0.603]. Lastly, the MSE of the estimated model coefficients was substantially improved with the linear model estimating coefficients with an order of magnitude larger error [0.05, 0.95] quantiles = [0.024, 0.077] compared to the mixture approach [0.05, 0.95] quantiles = [0.002, 0.008]. This represents a relative reduction in the MSE of the coefficient estimates of 89.1%.
In the second simulation, in which the covariates are not related to the surrogate marker in patients assigned to subpopulation I, resulted in much more modest improvements in discrimination. In this situation, the linear model produced reasonably large AUCs consistently [0.05, 0.95] quantiles = [0.873, 0.972] although the AUCs from the mixture approach were slightly larger [0.05, 0.95] quantiles = [0.906, 0.982]. The average difference between the AUCs was 0.021, [0.05, 0.95] quantiles = [0.002, 0.044]. The linear also improved its performance with respect to Spearman's rank correlation [0.05, 0.95] quantiles = [0.333, 0.451] compared to that of the mixture model [0.05, 0.95] quantiles = [0.454, 0.524]. With respect to the MSE of the estimated coefficients, the linear model still displayed serious bias [0.05, 0.95] quantiles = [0.028, 0.034] as compared to the mixture approach, [0.05, 0.95] quantiles = [0.002, 0.009]. This represents an 83% relative reduction in MSE.
In the first example, the surrogate is generated in subpopulation I in a way that by random chance can be similar to or very different from \( {\widehat{f}}_V \). When the generating model is similar to \( {\widehat{f}}_V \), the linear model, which pools the subpopulations together, can perform well in terms of discrimination as evident by its ability to generate high AUCs and rank correlations. However, when the generating model for subpopulation I is very different from \( {\widehat{f}}_V \), the discrimination suffers tremendously. While the random model selection is very influential in the resulting discrimination, the estimated model coefficients produced by the linear model always represent an averaging of the true coefficients in the two models resulting in higher MSE of estimation than the mixture model approach.
In the second example, the linear model performs much better in terms of discrimination. In this example, the model coefficients estimated by the linear model are biased estimates of those in \( {\widehat{f}}_V \); however, they are biased toward zero. This type of consistent attenuation preserves the relative size and direction of the coefficients. The model is therefore able to order the outcomes well with discriminatory performance only modestly lower than the mixture approach. The inclusion of subpopulation I in this example attenuates the estimated coefficients in the linear model resulting in a bias that inflates the MSE in a similar way to what was observed in the first simulation. This does not impact its utility as a predictive model but would have implications for inference, which is beyond the scope of this analysis. R code for two simplified examples of this technique is provided as a supplement to this manuscript (Additional file 1).
Recognizing partial surrogacy of an outcome marker is critical regardless of whether the goal of an analysis is inference or prediction. In our clinical example of perioperative AKI, it was demonstrated that treating serum creatinine change as a full surrogate rather than a partial surrogate led to the erroneous conclusion that the linear model approach was much better than a mixture model at measuring kidney injury, represented by eGFR90. Prior to this work, no account has been given to the partial surrogate nature of serum creatinine change, meaning that effect estimates and predictive models based on creatinine-related endpoints are susceptible to a severe, systematic bias.
Despite being a clinically important marker of kidney function, eGFR90 has limitations. Patients who experience transient serum creatinine elevations that resolve by 90 days may still be at increased risk for adverse sequelae [27, 28]. The analysis we performed here does not fully capture these patients' increased risk. This analysis demonstrates that the proposed technique produced a superior model for the prediction of eGFR90. Although it is plausible that similar models will improve the prediction of other AKI related endpoints, these models will require validation using data not available in this dataset.
The simulation studies included here are meant to highlight the complexity of the decision on how to model a partial surrogate for the development of a risk score. This decision is heavily influenced by a mixture model's ability to choose the correct number of subpopulations for a given problem and resolve subgroup V from subgroup I by the relationships between covariates and the surrogate outcome. In practice, the only way an analyst can quantify these issues is by fitting a mixture model whenever partial surrogacy is suspected. By inspecting the fitted mixture model, the analyst will then be able to assess the model's entropy and the clinical significance of the difference between the phenotypes estimated by the mixture model. This provides the analyst with a better understanding of the effect partial surrogacy has on their potential risk score model.
Recognizing when a clinical marker is acting as a partial surrogate has implications on model selection, predictive ability, and coefficient estimation. Serum creatinine change from baseline, a common marker of kidney injury, displays behavior consistent with a partial surrogate. The use of a mixture model which separates patients into those likely to have a creatinine measure representative of kidney injury and those likely to have an unrelated creatinine measure appears to effectively counter the poor behavior of the biomarker in the cardiac surgery population.
AKI:
eGFR:
Glomerular filtration rate estimated by Chronic Kidney Disease Epidemiology Collaboration equation
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We acknowledge the Vanderbilt Anesthesiology Perioperative Informatics Research (VAPIR) division for data query support.
We acknowledge funding support from the United States National Institutes of Health (LES GM108554, FTB GM102676 and GM112871, and UL1 TR000445) and the Vanderbilt University Medical Center Department of Anesthesiology. None of these funding bodies had any role in the design of this study, or in the collection, analysis, or interpretation of the data, or in the writing of this manuscript.
The clinical data used in this dataset is unavailable, but full information on how the simulated data was generated is contained herein and is completely reproducible by the reader.
Department of Biostatistics, Vanderbilt University Medical Center, 2525 West End Ave, Suite 11000, Nashville, TN, 37212, USA
Derek K. Smith
& Jeffrey D. Blume
Department of Anesthesiology, Vanderbilt University Medical Center, Nashville, TN, USA
Loren E. Smith
& Frederic T. Billings IV
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DKS was responsible for the statistical methodologies and analyses, manuscript preparation, and simulations. LEF was responsible for the clinical expertise, cohort identification, and manuscript edits. FTB was responsible for the data collection, IRB approval, manuscript edits, and clinical expertise. JDB was responsible for the statistical methodologies and manuscript editing. All authors read and approved the final manuscript.
Correspondence to Derek K. Smith.
This study was approved by the Vanderbilt University IRB #120372.
R Code. (R 6 kb)
Surrogate markers
Mixture models
Serum creatinine | CommonCrawl |
\begin{document}
\title{\mytitle\thanks{\strut\myfunding}
\begin{abstract} \ifsubmission{
~\mskip1mu\noindent\textbf{\textsf{Abstract:}\mskip1mu}\mskip1mu }{} We extend randomized jumplists introduced by Brönnimann, Cazals, and Durand~\cite{BronnimannCazalsDurand2003} to choose jump-pointer targets as median of a small sample for better search costs, and present randomized algorithms with expected $\Oh(\log n)$ time complexity that maintain the probability distribution of jump pointers upon insertions and deletions. We analyze the expected costs to search, insert and delete a random element, and we show that omitting jump pointers in small sublists hardly affects search costs, but significantly reduces the memory consumption.
We use a bijection between jumplists and ``dangling-min BSTs'', a variant of (fringe-balanced) binary search trees for the analysis. Despite their similarities, some standard analysis techniques for search trees fail for dangling-min trees (and hence for jumplists). \end{abstract}
\section{Introduction}
Jumplists were introduced by Brönnimann, Cazals, and Durand~\cite{BronnimannCazalsDurand2003} as a simple randomized comparison-based dictionary implementation. They allow iteration over the stored elements in \emph{sorted} order and supports queries and updates in expected logarithmic time. The core is a sorted (\mbox{singly-})\,linked listed augmented with \emph{jump pointers}, \ie, shortcuts that speed up searches. Jump-pointers are required to be well-nested, \ie, they may not cross. This allows binary-search-like navigation. \wref{fig:typical-jumplist-n30-k1-w2} shows an exemplary jumplist; a detailed definition is deferred to~\wref{sec:jumplist-definitions}.
\begin{figure}
\caption{
A jumplist on $n=30$ keys (with $k=1$ and $w=2$).
Gray arrows are backbone links, thick red arrows are jump pointers.
Dotted green arrows delimit a node's conceptual sublist; (they are
not stored).
}
\label{fig:typical-jumplist-n30-k1-w2}
\end{figure}
If all jump pointers point to the middle of their sublist, we obtain perfect binary search, but we need a rule that is also efficiently maintainable upon insertions and deletions.
Brönnimann, Cazals, and Durand~\cite{BronnimannCazalsDurand2003} proposed a \emph{randomized} solution: jump pointers invariably have a uniform distribution over their sublist, \ie, the first jump pointer equally likely points to any element and thereby divides the list in two parts, the next- and jump-sublists. Both follow the same rule recursively; since pointers may not cross, they can do so independently.
In this article, we generalize jumplists to use a more balanced distribution: each jump pointer points to the \emph{median of a small sample of $k$ elements} of its sublist. (The original jumplists correspond to $k=1$.) Building on the algorithms from~\cite{BronnimannCazalsDurand2003} we present $\Oh(\log n)$ expected-time insertion and deletion algorithms for median-of-$k$ jumplists that maintain this more balanced distribution. Here $n$ counts the number of keys currently stored.
A larger $k$ balances the structure more rigidly which improves searches, but makes the cleanup after updates more expensive. Our main contribution is an analysis of median-of-$k$ jumplists that precisely quantifies the influence of $k$ on searches, insertions and deletions.
We also introduce a novel search strategy (named \emph{spine search}) that reduces the number of needed key comparisons significantly, and we suggest a further modification of jumplists: for sublists smaller than a threshold $w$, we omit the jump pointers altogether. This allows to trade space for time: elements in these small sublists do not have to store a jump pointer, but the corresponding subfile can only be searched sequentially. We show that this saves a constant fraction of the pointers while affecting expected search costs only by an additive constant.
\paragraph{Outline of the paper} In the remainder of the introduction we summarize related work. \wref{sec:preliminaries} contains common notation and preliminaries used later. In \wref{sec:jumplist-definitions}, we define jumplists. We present our spine search strategy in \wref{sec:spine-search}. \wref{sec:median-of-k-jumplists-def} introduces the median-of-$k$ extension, and \wref{sec:insert-delete} describes the insertion and deletion algorithms. Our analysis is given in \wref{sec:analysis}, and we conclude the paper with a discussion of the results (\wref{sec:conclusion}).
\ifsiam{
There is an \extendedversion of this paper available as arxiv preprint
that contains detailed descriptions of all algorithms
and some missing proofs. }{
The appendix contains a list of used notations,
as well as details on the operations and omitted parts of the analysis. }
\subsection{Related Work} \label{sec:related-work} (Unbalanced) binary search trees (BSTs) perform close to optimal on average and with high probability when keys are inserted in random order~\cite{Knuth1998,Mahmoud1992evolution}. A standard approach is to enforce the average behavior through randomization.
The most direct application of this paradigm is given by Martínez and Roura~\cite{MartinezRoura1998} who devised efficient randomized insert and delete operations that maintain the shape distribution of random insertions. The idea also works when duplicate keys are allowed~\cite{Pasanen2010}.
Randomized BSTs store subtree sizes for maintaining the distribution.
The \emph{treaps} of Seidel and Aragon~\cite{SeidelAragon1996} instead store a random priority with each node. Treaps remain in random shape by enforcing a heap order \wrt the random priorities. Their performance characteristics are very similar to randomized BSTs.
Unless further memory is used, BSTs do not offer $O(1)$ time successor queries. Like jumplists, Pugh's skip lists~\cite{Pugh1990} are augmented, sorted linked lists, so successors are found by following one pointer. Skip lists extend the list elements by towers of pointers of different heights, where each tower cell points to the successor among all element of at least this height. With geometrically distributed heights, operations run in $\Oh(\log n)$ expected time with $\Oh(n)$ extra pointers in expectation. The varying tower heights can be inconvenient; this originally motivated the introduction of jumplists. For skip lists, there is a direct and transparent bijection to BSTs~\cite{DeanJones2007}; this becomes more complicated for jumplists (see \wref{sec:jumplist-definitions}).
The classic alternative to randomization are deterministically balanced BSTs~\cite{AnderssonFagerbergLarsen2005}. Munro, Papadakis, and Sedgewick~\cite{MunroPapadakisSedgewick1992} transfer the height-balance rule of 2-3 trees to skip lists, and Elmasry~\cite{Elmasry2005} applied the weight-balancing criterion of $\mathit{BB}[\alpha]$ trees~\cite{NievergeltReingold1973} to jumplists. Note that the latter achieves logarithmic update time only in an amortized sense.
A constant-factor speedup over BSTs is achieved with \emph{fringe-balanced} BSTs. The name originates from \emph{fringe analysis}, a technique used in their analysis~\cite{PobleteMunro1985}.
\footnote{
The concept
appears under a handful of other names in the (earlier) literature:
\emph{locally balanced search trees}~\cite{Walker1976},
\emph{diminished trees}~\cite{Greene1983}, and
\emph{iR / SR trees}~\cite{HuangWong1983,HuangWong1984}.
}
In a fringe-balanced search tree, leaves \emph{collect} keys in a buffer. Once a leaf holds $k$ keys, it is \emph{split:} the median of the $k$ elements is used as the key of a new node; two new leaves holding the other elements form its subtrees.
Many parameters like expected path length, height and profiles of fringe-balanced trees have been studied~\cite{Drmota2009}.
\section{Notation and Preliminaries} \label{sec:preliminaries}
\ifsiam{
We first introduce some notation. }{
We introduce some important notation here;
\wref{app:notations} gives a comprehensive list. } We use Iverson's bracket $[\mathit{stmt}]$ to mean $1$ if $\mathit{stmt}$ is true and $0$ otherwise. Falling resp.\ rising factorial powers are denoted by $x^{\underline n}$ and $x^{\overline n}$; for negative $n$ holds $x^{\underline n} = 1 / (x+1)^{\overline n}$ resp.\mskip1mu$x^{\overline n} = 1 / (x+1)^{\underline n}$.
$\Prob{E}$ denotes the probability of event $E$ and $\mskip1mu{X}$ the expectation of random variable $X$. We write $X\eqdist Y$ to denote equality in distribution.
For a self-contained presentation, we list here a few mathematical preliminaries used in the analysis later.
\paragraph{Beta distribution}
The \emph{beta distribution} has two parameters $\alpha,\beta\in\mskip1mu_{>0}$ and is written as $\betadist(\alpha,\beta)$. If $X\eqdist\betadist(\alpha,\beta)$, we have $X \in (0,1)$ and $X$ has the density \begin{align*}
f(x)
&\wrel=
\frac{x^{\alpha-1}(1-x)^{\beta-1} }{\BetaFun(\alpha,\beta)}
,\qquad x\in(0,1), \end{align*} where $\BetaFun(\alpha,\beta) = \Gamma(\alpha)\Gamma(\beta) / \Gamma(\alpha+\beta)$ is the beta function.
The following lemma is helpful for computing expectations involving such beta distributed variables; it is a special case of \cite[\href{https://www.wild-inter.net/publications/html/wild-2016.pdf.html\#pf60}{Lemma 2.30}]{Wild2016}.
\begin{lemma}[``Powers-to-Parameters''] \label{lem:powers-to-parameters}
Let $X_1$ be a \mskip1mu\betadist(\alpha_1,\alpha_2)\mskip1mu
distributed random variable and write $X_2=1-X_1$.
Let further \(m_1,m_2 \in \mskip1mu^d\mskip1mu
with \(m_1,m_2 > -\alpha\mskip1mu be given and abbreviate
\(A \ce \alpha_1 + \alpha_2\mskip1mu and \(M \ce m_1+m_2\mskip1mu.
Then for an arbitrary (real-valued, measurable) function~\(f\mskip1mu
holds
\begin{align*}
\mskip1mu[\big]{ X_1^{m_1} X_2^{m_2} \cdot f(X_1) }
&\wwrel=
\frac{\alpha_1^{\overline{m_1}} \alpha_2^{\overline{m_2}}}
{A^{\overline M}}
\cdot \mskip1mu[\big]{ f(\tilde X_1) }
\mskip1mu,
\end{align*}
where \mskip1mu\tilde X_1\mskip1mu is
\mskip1mu\betadist(\alpha_1+m_1,\alpha_2+m_2)\mskip1mu distributed. \qed\end{lemma}
\paragraph{Beta-Binomial Distribution} The \emph{beta-bi\-no\-mi\-al distribution} is a discrete distribution with parameters $n\in\mskip1mu_0$ and $\alpha,\beta \in \mskip1mu_{>0}$. It is written as $\betaBinomial(n,\alpha,\beta)$. If $I\eqdist\betaBinomial(n,\alpha,\beta)$, we have $I\in[0..n]$ and \begin{align*}
\Prob{I = i}
&\wrel=
\binom{n}{i} \frac{\BetaFun(\alpha+i,\beta+(n-i))}{\BetaFun(\alpha,\beta)}
,\quad i \in \mskip1mu \mskip1mu. \end{align*} (Recall that $\binom ni$ is zero unless $i\in[0..n]$.)
An alternative representation of the weights for $\alpha = t_1+1,\beta = t_2+1 \in\mskip1mu$ with $k=t_1+t_2+1$ is \begin{align*}
\binom{n}{i} \frac{\BetaFun(\alpha+i,\beta+(n-i))}{\BetaFun(\alpha,\beta)}
&\wrel=
\frac{\binom{i+t_1}{t_1}\binom{n-i+t_2}{t_2}}{\binom{n+k}{k}}, \label{eq:dirichlet-multinomial-weights-binomial} \end{align*} which yields a combinatorial interpretation.
There is a second way to obtain beta-binomial distributed random variables: we first draw a random probability $D \eqdist \betadist(\alpha,\beta)$ according to a beta distribution, and then use this as the success probability of a binomial distribution, \ie, $I \eqdist \binomial(n; d)$ \emph{conditional} on $D = d$. The beta-binomial distribution is thus also called a \emph{mixed} binomial distribution, using a beta-distributed \emph{mixer} $D$; this explains its name.
Since the binomial distribution is sharply concentrated, one can use Chernoff bounds on beta binomial variables after conditioning on the beta distributed success probability. That already implies that $\betaBinomial(n,\alpha,\beta)/n$ converges to $\betadist(\alpha,\beta)$ (in a specific sense). We can obtain the stronger error bounds given in the following lemma by directly comparing the probability density functions.
\savebox\citehrefbox{ \bfseries\cite[\href{https://www.wild-inter.net/publications/html/wild-2016.pdf.html\#pf66}{Lem.\,2.38}]{Wild2016} }
\begin{lemma}[{Local limit law~
\usebox\citehrefbox }] \label{lem:limit-law-beta-binomial}
Let $(\ui In)_{n\in\mskip1mu}$ be a sequence of random variables where
$\ui In$ is distributed like $\betaBinomial(n,\alpha,\beta)$ for $\alpha,\beta\in\mskip1mu_{\ge1}$.
Then for $n\to\infty$ we have uniformly for $z\in(0,1)$ that
\begin{align}
n \Prob[\big]{\ui Jn /n \in (z-\tfrac1n,z] }
&\wwrel=
f_B(z) \bin\pm \Oh(n^{-1}),
\end{align}
where $f_B(z) = z^{\alpha-1}(1-z)^{\beta-1} / \BetaFun(\alpha,\beta)$ is the density
function of the beta distribution with parameters $\alpha$ and~$\beta$.
\qed\end{lemma}
Since $f_B$ is a polynomial in $z$, it is in particular bounded and Lipschitz continuous
in the closed domain $z \in [0,1]$.
Hence, the local limit law also holds for the random variables $\ui{J}n = \ui{I}{n-d} + c$ for constants $c$ and $d$.
Further properties of the beta-binomial distribution are collected in~\cite[\href{https://www.wild-inter.net/publications/html/wild-2016.pdf.html\#pf64}{\mskip1mu\,2.4.7}]{Wild2016}.
\ifsiam{
We list the following expectations here for reference.
The proofs are simple computations found in the \extendedversion. }{
The following expectations are listed here for reference;
proofs are given in \wref{app:expectations}. }
\begin{lemma}
\label{lem:binomial-negative-factorial-moments}
Let $X \eqdist \binomial(n,p)$ for $n\in\mskip1mu_0$ and $p\in(0,1]$.
Then we have with $q=1-p$ that
\begin{align*}
\mskip1mu*{X^{\underline{-1}}}
&\wrel=
n^{\underline{-1}} \cdot p^{-1} (1-q^{n+1})\mskip1mu,
\mskip1mu
\mskip1mu*{X^{\underline{-2}}}
&\wrel\le
n^{\underline{-2}} \cdot p^{-2}\mskip1mu.
\end{align*} \end{lemma}
\begin{lemma} \label{lem:E-ln-D} For $D\eqdist \betadist(t+1,t+1)$ we have (with $k=2t+1$) \begin{align*}
\mskip1mu{\ln D}
&\wwrel=
\harm{t} - \harm{k}
, \mskip1mu
\mskip1mu{D \ln D}
&\wwrel=
\frac12\bigl(\harm{t+1} - \harm{k+1}\bigr)
. \end{align*} \end{lemma}
\paragraph{Hölder continuity}
A function $f:I\to \mskip1mu$ defined on a bounded interval $I$ is Hölder continuous with exponent $h\in(0,1]$ when \mskip1mu
\exists C\mskip1mu
\forall x,y\in I\wrel:
\bigl| f(x) - f(y) \bigr|
\wrel\le
C |x-y|^h. \mskip1mu Hölder continuity is a notion of smoothness that is stricter than (uniform) continuity, but slightly more liberal than Lipschitz continuity (which corresponds to $h=1$).
$f:[0,1]\to\mskip1mu$ with $f(z) = z \ln(1/z)$ is a stereotypical function that is Hölder continuous (for any $h\in(0,1)$), but not Lipschitz.
For functions defined on a bounded domain, Lipschitz continuity implies Hölder continuity and Hölder continuity with exponent $h$ implies Hölder continuity with exponent $h' < h$. Recall that a real-valued function is Lipschitz if its derivative is bounded.
\subsection{The Distributional Master Theorem} \label{app:DMT}
To solve the recurrences in \wref{sec:analysis}, we use the ``distributional master theorem'' (DMT) ~\cite[\href{https://www.wild-inter.net/publications/html/wild-2016.pdf.html\#pf8a}{Thm.~2.76}]{Wild2016}, reproduced below for convenience. It is based on Roura's continuous master theorem~\cite{Roura2001}, but reformulated in terms of distributional recurrences in an attempt to give the technical conditions and occurring constants in Roura's original formulation a more intuitive, stochastic interpretation. We start with a bit of motivation for the latter.
The DMT is targeted at divide-and-conquer recurrences where the recursive parts have a \emph{random} size. The average-case analyses of Quicksort and binary search trees are typical examples that lead to such recurrences. Because of the random subproblem sizes, a traditional recurrence for expected costs has to sum over all possible subproblem sizes, weighted appropriately. That way, the direct correspondence between the recurrence and the algorithmic process is lost, in particular the number of recursive applications is no longer directly visible.
An alternative that avoids this is a \emph{distributional recurrence} that describes the full distribution of costs. The distribution for larger problem sizes is described by a ``toll term'' (for the divide and/or combine step) plus the contributions of recursive applications. Such a distributional formulation requires the toll costs and subproblem sizes to be stochastically independent of the recursive costs when conditioned on the subproblem sizes. In typical applications, this is fulfilled when the studied algorithm guarantees that the subproblems on which it calls itself recursively are of the same nature as the original problem. Such a form of randomness preservation is also required for the analysis using traditional recurrences. We can thus use the distributional language to describe costs directly mimicking the structure of our algorithms in this paper.
The DMT allows us to compute an asymptotic approximation of the expected costs directly from the distributional recurrence. Intuitively speaking, it is applicable whenever the \emph{relative} subproblem sizes of recursive applications converge to a (non-degenerate) limit distribution as $n\to\infty$ (in a suitable sense; see \weqref{eq:DMTwc-condition} below). The local limit law provided by \wref{lem:limit-law-beta-binomial} gives exactly such a limit distribution.
\savebox\citehrefbox{ \bfseries\cite[\href{https://www.wild-inter.net/publications/html/wild-2016.pdf.html\#pf8a}{Thm.~2.76}]{Wild2016} } \begin{theorem}[DMT \usebox\citehrefbox] \label{thm:DMTwc}
Let \mskip1mu(C_n)_{n\in\mskip1mu_0}\mskip1mu be a family of random variables that
satisfies the distributional recurrence
\begin{align}
\label{eq:DMTwc-distributional-recurrence}
C_n
\wwrel\eqdist
T_n \bin+ \sum_{r=1}^s \ui{A_r}n \cdot C_{\ui{J_r}n}^{(r)},
\qquad (n \ge n_0),
\end{align}
where the families \mskip1mu(\ui{C_n}1)_{n\in\mskip1mu},\ldots,(\ui{C_n}s)_{n\in\mskip1mu}\mskip1mu are independent copies of
\mskip1mu(C_n)_{n\in\mskip1mu}\mskip1mu, which are also independent of
\mskip1mu(\ui{J_1}n,\ldots,\ui{J_s}n)\in\{0,\ldots,n-1\mskip1mu^s\mskip1mu,
\mskip1mu(\ui{A_1}n,\ldots,\ui{A_s}n) \in \mskip1mu_{\ge0}^s\mskip1mu
and \(T_n\mskip1mu.
Define \mskip1mu\ui{Z_r}n = \ui{J_r}n / n\mskip1mu, $=1,\ldots,s$, and assume that they
fulfill uniformly for \(z\in(0,1)\mskip1mu
\begin{align}
\label{eq:DMTwc-condition}
n \cdot \Prob[\big]{ \ui{Z_r}n \in (z-\tfrac1n,z] }
&\wwrel=
f_{Z_r^*}(z) \bin\pm \Oh(n^{-\delta}),
\end{align}
as $n\to\infty$ for a constant \mskip1mu\delta>0\mskip1mu and a
Hölder-continuous function \(f_{Z_r^*} : [0,1] \to \mskip1mu\mskip1mu.
Then \(f_{Z_r^*}\mskip1mu is the density of a random variable \(Z_r^*\mskip1mu and
\mskip1mu\ui{Z_r}n \convD Z_r^*\mskip1mu.
Let further
\begin{align}
\label{eq:DMTwc-condition-coeffs}
\mskip1mu[\big]{ \ui{A_r}n \given \ui{Z_r}n \in (z-\tfrac1n,z] }
&\wwrel=
a_r(z) \wbin\pm \Oh(n^{-\delta}),
\end{align}
as $n\to\infty$ for a function \(a_r : [0,1] \to \mskip1mu\mskip1mu and
require that \(f_{Z_r^*}(z)\cdot a_r(z)\mskip1mu is also Hölder continuous on~\mskip1mu[0,1]\mskip1mu.
Moreover, assume \mskip1mu\mskip1mu{T_n} \sim K n^\alpha \log^\beta(n)\mskip1mu, as \(n\to\infty\mskip1mu,
for constants \(K\ne 0\mskip1mu, \mskip1mu\alpha\ge0\mskip1mu and \mskip1mu\beta>-1\mskip1mu.
Then, with \(H = 1 - \sum_{r=1}^s\mskip1mu{(Z_r^*)^\alpha a_r(Z_r^*)}\mskip1mu, we have the following cases.
\begin{enumerate}[itemsep=0ex]
\item If \(H > 0\mskip1mu, then \mskip1mu\displaystyle \mskip1mu{C_n} \sim \frac{\mskip1mu{T_n}}{H}\mskip1mu.
\mskip1mu\vphantom{\displaystyle\sum_{i}^{i}}\mskip1mu
\item \label{case:DMTwc-H0}
If \(H = 0\mskip1mu, then
$\displaystyle
\mskip1mu{C_n} \sim \frac{\mskip1mu{T_n} \ln n}{\tilde H}$ with
$\displaystyle \tilde H = -(\beta+1)\sum_{r=1}^s
\mskip1mu{(Z_r^*)^\alpha a_r(Z_r^*) \ln(Z_r^*)}$.
\item \label{case:DMTwc-theta-nc}
If \(H < 0\mskip1mu, then \mskip1mu\mskip1mu{C_n} = \Oh(n^c)\mskip1mu for the
\(c\in\mskip1mu\mskip1mu with \mskip1mu\displaystyle\sum_{r=1}^s\mskip1mu{(Z_r^*)^c a_r(Z_r^*)} = 1\mskip1mu.
\end{enumerate} \qed\end{theorem}
\section{Jumplists} \label{sec:jumplist-definitions}
We now present our (consolidated) definition of jumplists; \ifsiam{
some details differ from the original~\cite{BronnimannCazalsDurand2003};
we discuss those in the \extendedversion. }{
it deviates in some details from the original version of \cite{BronnimannCazalsDurand2003};
see \wref{app:differences-definitions}. }
Jumplists consist of \emph{nodes}, where each node $v$ stores a successor pointer ($v.\id{next}$) and a key ($v.\id{key}$). The nodes are connected using the next pointers to form a singly-linked list, the \emph{backbone} of the jumplist, so that the key fields are sorted ascendingly. \footnote{
We assume the keys stored in a jumplist are distinct.
The insert procedures will prevent duplicate insertions. } It is convenient to add a ``dummy'' header node $v_0$ whose key field is ignored; ($v_0.\id{key} = -\infty$).
If $x_1 < \cdots < x_n$ are the keys stored in the jumplist, we have the $n+1$ nodes $v_0,v_1,\ldots,v_n$ with $v_i.\id{key} = x_i$ and $v_{i-1}.\id{next} = v_i$ for $i=1,\ldots,n$. A jumplist on $n$ keys will always have $m=n+1$ nodes; we use $n$ and $m$ in this meaning throughout the paper.
\paragraph{Jump Pointers} Jump pointers always point forward in the list, and we require the following two conditions. \begin{inlineenumerate} \item
\emph{Non-degeneracy:} Any node may be the target of at most one jump pointer,
and jump pointers never point to the direct successor. \item
\emph{Well-nestedness:}
Let $v\ne u$ be nodes with $v.\mathit{key} < u.\mathit{key}$,
and let $v^*$ resp.\mskip1mu$u^*$ be the nodes their jump pointers point to.
(Note that $v^*\ne u^*$ by the first property).
Then these nodes must appear in one of the following orders in the backbone:
$u \dots v\dots v^* \dots u^*$
or
$v \dots v^* \dots u \dots u^*$:
\mskip1mu[.5\baselineskip]
\plaincenter{\adjustbox{max width=.9\linewidth}{
\begin{tikzpicture}[
]
\begin{scope}
\foreach \mskip1mu/\mskip1mu/\mskip1mu in {u/u/1,v/v/2,{v^{\mkern-1mu*}\mskip1mu}/v2/3.5,{u^{\mkern-1mu*}\mskip1mu}/u2/4.5} {
\node[sn,minimum size=4.25mm] (\mskip1mu) at (\mskip1mu,0) {$\mskip1mu{}^{\vphantom *}$} ;
}
\draw[jumppointer] (u) to[out=60,looseness=.7] (u2) ;
\draw[jumppointer] (v) to[out=60,looseness=.7] (v2) ;
\end{scope}
\node[scale=1.5] at (5.75,0) {\normalfont or} ;
\begin{scope}[shift={(6,0)}]
\foreach \mskip1mu/\mskip1mu/\mskip1mu in {u/u/1,v/v/3.25,{v^{\mkern-1mu*}\mskip1mu}/v2/4.5,{u^{\mkern-1mu*}\mskip1mu}/u2/2.25} {
\node[sn,minimum size=4.25mm] (\mskip1mu) at (\mskip1mu,0) {$\mskip1mu{}^{\vphantom *}$} ;
}
\draw[jumppointer] (u) to[out=60,looseness=1.2] (u2) ;
\draw[jumppointer] (v) to[out=60,looseness=1.2] (v2) ;
\end{scope}
\end{tikzpicture}
}}\mskip1mu[.5\baselineskip]
The second case allows $v^* = u$.
Visually speaking, jump pointers may not cross. \end{inlineenumerate}
\paragraph{Sublists}
The \emph{sublist of node $v$} starts at $v$ (inclusive) and ends just before the first node targeted by a jump pointer originating before $v$~-- or extends to the end of the list if no overarching pointer exists. As for the overall jumplist, $v$ acts as dummy header to its sublist: $v.\id{key}$ is \emph{not} considered as part of $v$'s sublist. We write $m(v)$ for the number of nodes in $v$'s sublist.
The next- and jump-sublists of $v$, denoted by $\mathcal J_1 = \mathcal J_1(v)$ resp.\mskip1mu$\mathcal J_2 = \mathcal J_2(v)$, are the sublists of $v.\id{next}$ resp.\mskip1mu$v.\id{jump}$. We use $J_r=J_r(v)$ for the number of nodes in $\mathcal J_r(v)$, $r\in\{1,2\mskip1mu$. \wref{fig:def-sublists} exemplifies the definitions. We include an imaginary ``end pointer'' in the figures, drawn as dotted green line, that connects a jump node with the last node in that node's sublist.
\begin{figure}
\caption{
Illustration of the sublist definitions.
The sublist of node $v_1$ contains $m(v_1)=7$ nodes and
stores the $6$ keys $v_2.\mathit{key},\ldots,v_7.\mathit{key}$.
The sizes of the next- and jump-sublist
are $J_1(v_1) = 2$ and $J_2(v_1)=4$, respectively.
}
\label{fig:def-sublists}
\end{figure}
\paragraph{Node Types} Nodes in our jumplists come in two flavors: \emph{plain nodes} only have next and key fields; \emph{jump nodes} additionally store a \emph{jump pointer}, $v.\id{jump}$, and their next-sublist size, $v.\id{nsize}=J_1$.
The node types are determined by the following rule, where $w\ge2$, the \emph{leaf size,} is a parameter: If $m(v)\le w$, then $v$ (and all nodes in its sublist) are plain nodes. Otherwise $v$ is a jump node, and we apply the rule recursively to $\mathcal J_1(v)$ and $\mathcal J_2(v)$.
\wref{fig:typical-jumplist-n30-k1-w2} shows a larger example.
\paragraph{Randomized Jumplists}
The following probability distribution over all (legal) jump-pointer configurations invariantly holds in randomized jumplists. It is defined recursively: $v_0.\mathit{jump}$ is drawn \emph{uniformly} from all $m-2$ feasible targets; ($v_0$ and $v_1$ are not allowed). Conditional on the choice of $v_0.\mathit{jump}$, the same property is required independently for $\mathcal J_1(v_0)$ and $\mathcal J_2(v_0)$. The probability $p(\mathcal J)$ of a particular (legal) pointer configuration $\mathcal J$ is \begin{align*} \label{eq:prob-jumplist-k1}
p(\mathcal J)
&\wwrel=
\begin{dcases*}
1, & $m \le w$; \mskip1mu
\frac1{m-2} \cdot p(\mathcal J_1) \,p(\mathcal J_2), &
$m > w$,
\end{dcases*} \end{align*} which is reminiscent of the probability of a given shape for a random BST, except for the offset $-2$ (see \cite[ex.\,6.2.2--5]{Knuth1998} or \cite[Eq.\mskip1mu(5.1)]{CasasDiazMartinez1991}).
\subsection{Dangling-Min BSTs} \label{sec:dangling-min-bsts}
There is an intimate relation between jumplists and search trees, but the slight offset above complicates the matter. \footnote{
The complication is inherent to the feature of jumplists that
every key has at most \emph{one} jump pointer.
Skip lists, for example, can be transformed into BSTs directly~\cite{DeanJones2007}. } Indeed, (random) jumplists are isomorphic to a rather peculiar variant of (random) BSTs (where random means ``generated by insertions in random order''):
the \emph{dangling-min BSTs} (with leaf size $w\ge 2$). Such a tree is defined for a sequence of (distinct) keys $x_1,\ldots,x_n$ as follows.
If $n\le w-1$, it is a leaf
with the keys in sorted order.
Otherwise, its \emph{root} node contains \emph{two} keys:
the smallest key, $\min\{x_1,\ldots,x_n\mskip1mu$, as its \emph{dangling min,}
and the first key of the sequence after the min has been removed as \emph{root key}
(\ie, the root key is $x_1$, unless $x_1$ is the min; then it is~$x_2$).
The left resp.\ right subtrees of the root are the dangling-min BSTs
for the keys smaller resp.\ larger than the root key
in the remaining sequence (without root key and min, and preserving relative order).
Dangling-min BSTs make the recursive decomposition in jumplists explicit, which helps for both designing algorithms and analyzing their performance.
\begin{figure}
\caption{
The dangling-min BST with $w=2$ for the sequence $11,2,5,3,1,4,10,8,7,9,6,12$,
and the jumplist it corresponds to.
}
\label{fig:example-min-BST}
\end{figure}
We can transform a jumplist to a dangling-min BST (and vice versa): If $m\le w$, $v_0$ is a plain node and the dangling-min BST is a leaf containing all $m-1\le w-1$ keys; (recall that a jumplist with $m$ nodes stores $n=m-1$ keys). Otherwise, $v_0$ is a jump node; with $x_1$ the key in $v_0.\id{next}$ and $x_j$ the key in $v_0.\id{jump}$, the root of the dangling-min BST has root key $x_j$ and dangling min $x_1$. Next- resp.\ jump-sublist are recursively transformed into left and right subtree. \wref{fig:example-min-BST} shows the jumplist corresponding to the given tree; \wref{fig:typical-jumplist-tree-n30-k1-w2} gives a larger example.
It is easy to see inductively that the dangling-min BST built from a randomized jumplist has the same distribution as if directly constructed for a random permutation of $\{1,\ldots,n\mskip1mu$. We can therefore focus on analyzing the latter.
\section{Spine Search} \label{sec:spine-search}
Searching a key $x$ in a jumplist is straightforward: We start at the header. We stop when the key in the current node $v$ is larger or equal to $x$. Otherwise we follow either the jump pointer~-- if the key in $v.\id{jump}$ is not larger than $x$~-- or the next-pointer. We call this strategy the classic search in the sequel. \footnote{
Brönnimann, Cazals, and Durand~\cite{BronnimannCazalsDurand2003} also studied
the symmetric alternative
\parenthesisclause{compare first to $v.\id{next}$ and then with $v.\id{jump}$ (if needed)}
and found that it needs more comparisons on average. }
However, there is an alternative search strategy not considered in~\cite{BronnimannCazalsDurand2003} and~\cite{Elmasry2005}, which performs better!
Consider searching key $8$ in the jumplist from \wref{fig:typical-jumplist-n30-k1-w2}. A classic search in this list inspects keys $18,1,3,12,4,6,11,7,10,8$ in the given order; a total of $10$ key comparisons. Every step in the search that follows the next-pointer needs two comparisons.
Now do the search for $8$ in the dangling-min BST from \wref{fig:typical-jumplist-tree-n30-k1-w2}, as if it was a regular BST (ignoring the subtree minima and stopping at the leaves). While doing so, we compare with keys $18,3,12,6,11,10$. All these steps need only one key comparison even though mostly the same keys are visited as above. However, our search is not yet finished; the reached leaf contains only $9$, and we would (erroneously!) announce that $8$ is not in the dictionary. Instead we have to return to the \emph{last node we entered through a right-child pointer} and inspect all the dangling mins along the ``left spine'' of the corresponding subtree. In our example, we return to $11$ and make comparisons with $7$ and $8$, terminating successfully. We call this search strategy \emph{spine search.} In our example, it needed $2$ comparisons less than the classic search.
\begin{figure}
\caption{
The dangling-min BST for the jumplist from \wref{fig:typical-jumplist-n30-k1-w2}.
Black arrows are left child pointers, red arrows are right child pointers,
and dotted yellow arrows indicate the dangling min.
Gray nodes are leaves that contain between $0$ and $w-1=1$ keys.
}
\label{fig:typical-jumplist-tree-n30-k1-w2}
\end{figure}
Spine search only compares $x$ with the dangling-mins for nodes on the \emph{left spine above the leaf,} whereas the classic strategy does so for \emph{every} node we leave through the left-child edge. Our modification is correct because when going to the right child we know that all keys left to $v$ are smaller than $x$ and thus $x$ cannot be any of the dangling minima we skipped. \ifsiam{
The \extendedversion }{
\wref{app:algorithms} } gives detailed pseudocode.
The left spine is always a subset of the nodes where we took a left child edge, so spine search never needs more comparisons than the classic strategy. It seems reasonable that spine search should need roughly as many key comparisons as the search in a BST since most left spines are short. Indeed, we prove in \wref{sec:analysis} that the linear search along the left spine is only a lower order term when averaging over all possible unsuccessful searches \parenthesissign spine search needs $\sim 2\ln(n)$ comparisons, compared to $\sim3\ln(n)$ for the classic search strategy.
\section{Median-of-k Jumplists} \label{sec:median-of-k-jumplists-def}
The search costs in BSTs can be improved by using medians of a small sample as subtree roots; the idea is called fringe-balancing in that context (\wref{sec:related-work}) and corresponds to the median-of-$k$ rule for Quicksort~\cite{Hennequin1991,Drmota2009,Wild2018}. Applied to our trees, we obtain \emph{$k$-fringe-balanced dangling-min BSTs:} if $n\ge w$, we choose the root key as the median of the first $k$ keys in the sequence after removing the min (and otherwise proceed as before). Here $k = 2t+1$ is a fixed odd integer and we require $w\ge k+1$.
Similarly, we define a \emph{randomized median-of-$k$ jumplist} by choosing the jump target as the median of $k$ elements. The situation is illustrated below for $k=3$ and $m=10$; to have $x_6$ as the median of $3$ elements from the sample range, we must select $t=1$ further elements from $\{x_2,\ldots,x_5\mskip1mu$ and $t=1$ further elements from $\{x_7,\ldots,x_9\mskip1mu$.
\plaincenter{ \adjustbox{max width=\linewidth}{
\begin{tikzpicture}[
scale=.75
]
\fill[rounded corners=3pt,blue!15] (1.5,.5) rectangle (9.5,-.5)
node[fill,text=blue, inner sep=1.5pt,align=center,anchor=south] at (7.5,.4)
{\smaller sampling range};
\foreach \mskip1mu in {0,...,9} {
\node[sn,semithick,scale=.75] (\mskip1mu) at (\mskip1mu,0) {$x_{\mskip1mu}$};
}
\foreach \mskip1mu in {6,4,9} {
\draw[ultra thick,<-,red] (\mskip1mu) -- +(0,-.8) ;
}
\node[red] at (10,-.7) {sample};
\draw[jumppointer] (0) to[out=45,in=120,looseness=.4] (6) ;
\begin{scope}[decoration={brace,mirror},thick]
\draw[decorate] ($(1.west) + (0,-1)$) --
node[below=.5ex]{$\mathcal J_1$} ++($(5.east)-(1.west)$) ;
\draw[decorate] ($(6.west) + (0,-1)$) --
node[below=.5ex]{$\mathcal J_2$} ++($(9.east)-(6.west)$) ;
\end{scope}
\end{tikzpicture} } }
\noindent The number of such samples is $\binom {J_1-1}t \binom {J_2-1}t$, which we have to divide by the total number of possible samples, $\binom {m-2}k$. The probability of a (legal) jump pointer configuration $\mathcal J$ thus is \begin{align*} \label{eq:prob-jumplist-general-k}
p(\mathcal J)
&\wrel=
\begin{dcases*}
1 & $m \le w$; \mskip1mu
\frac{\binom{J_1-1}{t} \binom{J_2-1}{t}}{\binom{m-2}k}
\cdot p(\mathcal J_1) \,p(\mathcal J_2), &
$m > w$.
\end{dcases*} \end{align*} This puts more probability weight on balanced configurations, and hence improves the expected search costs. \wref{fig:typical-jumplist-n30-k3-w4} shows a typical median-of-$3$ jumplist and its fringe-balanced dangling-min tree. \footnote{
A possible generalization could use asymmetric sampling with
$(t_1,t_2)$ and $k=t_1+t_2+1$, where we select the $(t_1+1)$st smallest instead
of the median.
Then, we have $\binom{J_1-1}{t_1}$ and $\binom{J_2-1}{t_2}$
in \weqref{eq:prob-jumplist-general-k}.
For the present work, we will however stick to the case $t_1=t_2=t$. }
\begin{figure}
\caption{
A typical median-of-three ($k=3$, $w=4$) jumplist on $n=30$ keys
and its corresponding fringe-balanced dangling-min BST.
}
\label{fig:typical-jumplist-n30-k3-w4}
\end{figure}
\paragraph{Distribution of subproblem sizes} For our analysis, an alternative description of the distribution of the subproblem sizes is more convenient. Note that both $J_1$ and $J_2$ are always at least $t+1$: the sublists must contain $t$ other sampled nodes plus their header. If we denote by $I_r = J_r-t-1$, $r\in\{1,2\mskip1mu$, we find that $I_r$ has a beta-binomial distribution (\wref{sec:preliminaries}), $I_r \eqdist \betaBinomial(m-2-k,t+1,t+1)$. This implies that with $D \eqdist \betadist(t+1,t+1)$, we have the mixed distribution $I_r \eqdist \binomial(m-2-k,D)$ conditional on~$D$. \footnote{
The symmetry in the sublist sizes, $J_1\eqdist J_2$,
is a major convenience of our definition of jumplists
as opposed to the original one. }
\section{Insert and Delete} \label{sec:insert-delete}
\begin{figure*}
\caption{\mycaption}
\caption{\mycaption}
\label{fig:restore-after-insert}
\label{fig:restore-after-insert}
\label{fig:restore-after-insert}
\end{figure*}
We briefly sketch the update operations for randomized median-of-$k$ jumplists; \ifsiam{
the \extendedversion }{
\wref{app:algorithms} } describes them in more detail. The common theme is that we first modify the jumplist blindly and afterwards ``repair'' the distribution by rebuilding one suitably chosen sublist randomly from scratch.
For example upon insertion, the new node has a certain chance to be the target of the first jump pointer. We flip a coin to decide whether this should happen; if so, we rebuild the entire structure and are done. Otherwise, we recursively repair a sublist.
\paragraph{Rebalance} As in~\cite{BronnimannCazalsDurand2003}, we use a procedure \proc{Rebalance}$(\ensuremath{\mathcal{J}}\xspace)$ that (re)assigns jump pointers from scratch. It only uses the backbone, existing jump pointers are ignored.
A careful recursive implementation of \proc{Rebalance} rebuilds a sublist of $m$ nodes in time $\Theta(m)$.
\paragraph{Insert} Insertion in jumplists consists of the three phases found in many dictionaries: (unsuccessful) search, local insertion, and cleanup. Unless $x$ is already present, the search ends at the node with the largest key (strictly) smaller than $x$. There we insert a new node with key $x$ into the backbone.
It does not have a jump pointer yet, and it is a new potential jump target for all the nodes whose sublist contains the new node. Procedure \proc{RestoreAfterInsert} rectifies this as follows.
Let $m$ be the total number of nodes after the insertion, \ie, including the new node. If $m \le w$, no cleanup is necessary; if $m=w+1$, we draw the jump pointer for $v_0$ and are done. Otherwise, we first restore the pointer distribution of $v_0$. Due to the insertion of a new node, the sample range now contains an additional node $u$. ($u$ is not necessarily the newly inserted node; if the new key is the first or second smallest in \ensuremath{\mathcal{J}}\xspace, $u$ is the former second node of \ensuremath{\mathcal{J}}\xspace).
If we, conceptually, drew $v_0.\mathit{jump}$ anew, there are two possibilities: either $u$ is part of the sample, namely with probability \smash{$p=\frac{k}{m-2}$}, or $u$ is not part of it.
In the first case, we rebalance all of \ensuremath{\mathcal{J}}\xspace. In the second case, conditional on the event that $u$ is \emph{not} in the sample, the current jump pointer of $v_0$ already has the correct distribution: the median of a random sample not containing~$u$.
We thus rebalance \ensuremath{\mathcal{J}}\xspace with probability $p$, where we draw the jump pointer of $v_0$ conditional on $u$ being part of the sample. Otherwise we continue recursively in the uniquely determined sublist that contains the inserted node. \wref{fig:restore-after-insert} summarizes \proc{RestoreAfterInsert} graphically.
\paragraph{Delete} We now sketch the procedure \proc{RestoreAfterDelete}, which is similar to \proc{RestoreAfterInsert}.
Let $m$ be the number of nodes after deletion, and let $u$ be the deleted node. First assume that $u\ne v_0$.
Assume $m > w$, \ie, $v_0$ is a jump node whose sublist contained $u$. If the sample drawn to choose $v_0.\id{jump}$ did \emph{not} contain $u$, the deletion of $u$ does not affect $v_0.\id{jump}$, and we recursively clean up the sublist that formerly contained $u$. If $u$ was part of the sample, we have to rebalance~\ensuremath{\mathcal{J}}\xspace; the probability for that is
\begin{align*}
p \wwrel=
\begin{cases*}
1, & if $u=v_0.\id{jump}$;\mskip1mu
\frac{t}{J_1-1}, & if $u$ was in $\mathcal J_1$; \mskip1mu
\frac{t}{J_2-1}, & if $u$ was in $\mathcal J_2$.
\end{cases*} \end{align*} (We define $\frac00\ce1$ in case $t=J_1-1=0$.) When the deleted node is $u=v_0$, the new header $v_1$ can inherit $v_0$'s jump pointer and we have the same situation as if $v_1$ had been deleted. We have to rebalance with probability $p=\frac{t}{J_1-1}$, otherwise we continue the cleanup in the next-sublist.
\paragraph{Cost Measure} Insertion and deletion consist of a search and \proc{RestoreAfterInsert/-Delete}. The latter procedures retrace (a prefix of) the search path to the element and rebuild at most one sublist using \proc{Rebalance}. So apart from the search costs (which we analyze separately), the dominating cost is the number of \emph{``rebalanced elements'':} the size of the sublists on which \proc{Rebalance} is called. We will use this as our measure of costs.
\section{Analysis} \label{sec:analysis}
We now turn to the analysis of the expected behavior of median-of-$k$ jumplists with leaf size $w$. (The expectation is always over the random choices of the jump pointers.) We summarize our results in the theorem below. Its proof is spread over the following subsections.
\begin{theorem} \label{thm:results}
Consider randomized median-of-$k$ jumplists with leaf size $w$ on $n$ keys,
where $k$ and $w$ are fixed constants.
Abbreviate by $H(k) = \harm{k+1}-\harm{(k+1)/2}$ for $\harm n$
the harmonic numbers. Then the following holds:
\begin{thmenumerate}[noitemsep]{thm:results}
\item \label{thm:results-search}
The expected number of key comparisons in a \textbf{spine search} is asymptotic to
\mskip1mu
1/H(k) \cdot \ln n
\mskip1mu, as $n\to\infty$,
when each position is equally likely to be requested.
\item \label{thm:results-insert}
The expected number of rebalanced elements
in the \textbf{cleanup after insertion} is asymptotic to
\mskip1mu
k / H(k)\cdot \ln n
\mskip1mu, as $n\to\infty$,
when each of the $n+1$ possible gaps is equally likely.
\item \label{thm:results-delete}
The expected number of rebalanced elements
in the \textbf{cleanup after deletion} is asymptotic to
\mskip1mu
k/H(k)\cdot \ln n
\mskip1mu, as \(n\to\infty\mskip1mu,
when each key is equally likely to be deleted.
\item \label{thm:results-memory}
The expected number of additional machine words per key required to
store the jumplist is asymptotically at most
\(1 + \frac2{(w+1)H(k)}\mskip1mu as $n\to\infty$.
\end{thmenumerate} \end{theorem}
\subsection{Search Costs}
Let $P_n$ be the (random) total number of comparisons to search all numbers $x\in\{0.5,1.5,\ldots,n+0.5\mskip1mu$ (searching each gap once) in $\mathcal J_n$ the randomized jumplist on $\{1,\ldots,n\mskip1mu$, using \proc{SpineSearch}. The corresponding quantity in BSTs is called external path length, and we will use this term for $P_n$, as well. The quotient $P_n/n$ describes the average costs of one call to \proc{SpineSearch} when all $n+1$ gaps are equally likely to be requested. $P_n$ is random \wrt to the locations of the jump pointers in $\mathcal J_n$.
To set up a recurrence for~$P_n$, the perspective of random dangling-min BSTs is most convenient, since \proc{SpineSearch} follows the tree structure.
We describe recurrences here in terms of the distributions of families of random variables. \begin{align*}
P_n
&\rel\eqdist\begin{cases}
\begin{aligned}
&(n+1) + (S_n+L_n+1) + S_{J_1}
\mskip1mu&\qquad\bin+P_{J_1} + P'_{J_2},
\end{aligned}
& n\ge w,
\mskip1mu[2ex]
\frac{(n+1)(n+2)}{2},
& n < w,
\end{cases}
\mskip1mu[1ex]
S_n
&\rel\eqdist
\begin{cases}
1 + S_{J_1},\mkern-8mu
& n\ge w, \mskip1mu
0,
& n < w,
\end{cases} \mskip1mu[1ex]
L_n
&\rel\eqdist
\begin{cases}
L_{J_1},\mkern-8mu
& n\ge w, \mskip1mu
n,
& n < w,
\end{cases}
\end{align*} The terms $P_{J_1}$ and $P'_{J_2}$ on the right-hand side denote members of independent copies of the family of random variables $(P_n)_{n\in\mskip1mu_0}$, which are also independent of $J_r = \ui{J_r}n$, $r\in\{1,2\mskip1mu$. (We omitted the superscripts above for readability.) Here $J_r = I_r + t$, $r\in\{1,2\mskip1mu$, $I_1 \eqdist \betaBinomial(n-1-k; t+1,t+1)$ and $J_2 = n-1-k - J_1$. (We use $n$ here instead of $m$ in~\wref{sec:median-of-k-jumplists-def}; hence the slightly different parameters.)
The terms in the expression for $P_n$ are the comparisons with (1) the root key, (2) the dangling min of the root, (3) the comparisons done in the left subtree while searching the leftmost gap (which does not exist in the subtrees any more!), and (4) the external path lengths of the subtrees.
Two additional quantities are used to express these: $L_n$ is the number of keys in the leftmost leaf; by definition we have $0\le \mskip1mu{L_n} \le w-1 = \Oh(1)$. $S_n$~is the number of internal nodes on the ``left \underline spine'' of the tree, an essential parameter for the linear-search part of \proc{SpineSearch}. $S_n$ is also the depth of the internal node with the smallest root key (ignoring dangling mins). For ordinary BSTs, $S_n$ is essentially the number of left-to-right minima, which is a well-understood parameter; for (fringe-balanced) dangling-min BSTs, such a simple correspondence does not seem to hold.
We point out that the distribution of $P_n$ has a subtle complication, namely that even conditional on $(J_1,J_2)$, the quantities $S_n$, $S_{J_1}$ and $P_{J_1}$ are \emph{not} independent: all consider the \emph{same} left subtree! For example, we always have $S_{J_1}=S_n-1$ (for $n\ge w$).
We will only compute the expected value here, so by linearity, these dependencies can be ignored.
We will derive an asymptotic approximation using \wref{thm:DMTwc}, the distributional master theorem (DMT).
\begin{remark} For ordinary BSTs, the expectation of above quantities is known precisely, and some generalizations for fringe-balanced trees are possible by solving an Euler differential equation for the generating function. Unlike there, for dangling-min BSTs the resulting differential equation is \emph{not} an Euler equation. The case $t=0$ could be solved since the differential equation has order one~\cite{BronnimannCazalsDurand2003}, but there is little hope to obtain a solution for the generating function for $t\ge 1$. \end{remark}
\begin{lemma} \label{lem:spine-length-asymptotic}
$\mskip1mu{S_n} \sim \dfrac{1}{\harm{k}-\harm{t}} \ln n $. \end{lemma} \begin{proof}
We apply \wref{thm:DMTwc}
to the distributional recurrence $S_n \eqdist S_{J_1} + 1$.
It has the form of \wref{eq:DMTwc-distributional-recurrence}
with (matching the notation of \wref{thm:DMTwc})
$C_n = S_n$. We have $s= 1$ recursive term with size $J_1$
plus a ``toll term'' $T_n = 1$.
The latter has the asymptotic form $\mskip1mu{T_n} = 1 \sim 1 \cdot n^0 \lg^0 n$ as $n\to\infty$,
\ie, $K\DMTvarEq1$, $\alpha\DMTvarEq0$, $\beta\DMTvarEq0$.
Moreover, there is no ``coefficient'' in from of the recursive term, so
$A_1 = 1$.
We next check the conditions.
The independence assumptions are trivially fulfilled here,
in particular because $T_n$ is a fixed constant.
We next consider \wref{eq:DMTwc-condition}.
Recall that $J_1 \eqdist \betaBinomial(n-1-k; t+1,t+1) + t$.
By \wref{lem:limit-law-beta-binomial} and the remark below it,
$\ui{Z_1}n = \ui{J_1}n / n$ fulfills
\begin{align*}
n \Prob[\big]{\ui{Z_1}n \in (z-\tfrac1n,z] }
&\wwrel=
f_{Z_1^*} \bin\pm \Oh(n^{-1}),
\end{align*}
for $f_{Z_1^*} : [0,1] \to \mskip1mu $ with $f_{Z_1^*}(z) = z^{t}(1-z)^{t} / \BetaFun(t+1,t+1)$.
This function is a polynomial in $z$, so it has bounded derivative (on the compact domain $[0,1]$)
and is hence Lipschitz continuous (and thus Hölder continuous).
So \wref{eq:DMTwc-condition} is satisfied with $\delta = 1$.
The limiting relative subproblem size $Z_1^*$ has a $\betadist(t+1,t+1)$ distribution.
For the second condition, \wref{eq:DMTwc-condition-coeffs},
we find that $\mskip1mu[\big]{ \ui{A_r}n \given \ui{Z_r}n \in (z-\tfrac1n,z] } = 1$
since $A_1$ is constant.
So this condition is trivially satisfied with $a_1(z) = 1$ (which is a Hölder-continuous function).
We have now established that we can apply the DMT to our recurrence.
To obtain the asymptotic approximation for $\mskip1mu{S_n}$, we consider
$H=1-\mskip1mu{(Z_1^*)^0} = 0$, so Case~2 applies:
$\mskip1mu{S_n} \sim \tilde H^{-1}\cdot \mskip1mu{T_n} \ln n = \tilde H^{-1}\cdot \ln n$
for the constant $\tilde H = -\sum_{r=1}^s \mskip1mu{\ln(Z_r^*)}$.
(Note that this constant only involved the limiting relative subproblem size $Z_r^*$,
not the relative subproblem size $\ui{Z_1}n$ for a fixed $n$.)
The expectation in $\tilde H$ is exactly the first part of \wref{lem:E-ln-D},
so we find $\tilde H = \harm{k}-\harm{t}$.
Now the claim follows by inserting above. \end{proof}
\begin{remark}[Spine lengths] \wref{lem:spine-length-asymptotic} implies that the expected left spine of the root is logarithmic~-- as one might expect in a random BST; indeed, the expected left spine lengths of the root in a random BST and a dangling-min BST differ only in lower order terms. Note that the former is exactly $\harm{n}$ and the proof is elementary: The left spine length in a BST is the number of left-to-right minima in the insertion order. For dangling-min BSTs, no such simple argument is available. \end{remark} With these preparations, we can prove the main statement about search costs.
\begin{proof}[\wref{thm:results-search}]
We again use the distributional master theorem (DMT);
this time on the recurrence
$P_n
\rel\eqdist (n+1) + (S_n+L_n+1) + S_{J_1}
+P_{J_1} + P'_{J_2}$.
The recurrence is more involved than the one for $S_n$
that we just solved,
but the distribution of subproblem sizes are the same,
and we again have no coefficient in front of the recursive terms.
Therefore, a large part of the argument can be copied from the proof
of \wref{lem:spine-length-asymptotic}.
We here have
$C_n = P_n$, there are $s\DMTvarEq2$ recursive terms and
$T_n = (n+1) + (S_n+L_n+1) + S_{J_1}$.
By \wref{lem:spine-length-asymptotic},
all but the first summand in $\mskip1mu{T_n}$ are actually in $\Oh(\log n)$, so from
the initially complicated toll function, only $\mskip1mu{T_n} \sim n$ remains in the leading term
as $n\to\infty$.
We thus have $K= 1$, $\alpha\DMTvarEq1$, $\beta= 0$.
The coefficients $A_r = 1$ for $r\in\{1,2\mskip1mu$, so \wref{eq:DMTwc-condition-coeffs}
again holds trivially with $a_r(z)=1$.
As in the proof of \wref{lem:spine-length-asymptotic},
$Z_1^* \eqdist Z_2^* \eqdist \betadist(t+1,t+1)$ holds and condition
\wref{eq:DMTwc-condition} holds with the same $f_{Z_1^*}$.
We find again $H=0$ (since $Z_1^* + Z_2^* = 1$), so Case~2 applies.
The constant $\tilde H$ this time involves the second part of \wref{lem:E-ln-D}:
$\tilde H = -\sum_{r=1}^s \mskip1mu{D_r \ln(D_r)} = \harm{k+1}-\harm{t+1}$.
So we have
\mskip1mu
\mskip1mu{P_n}
\wwrel\sim
\frac{1}{\harm{k+1}-\harm{t+1}} n \ln n
\mskip1mu
and dividing by $n+1$ yields the claim.
\end{proof}
\subsection{Insertion Costs} \label{sec:analysis-insert}
The steps taken by \proc{RestoreAfterInsert} depend on the position of the newly inserted element; we denote here by $R$ the rank of the gap the new element is inserted into. When the current sublist has $m$ nodes, we have $R\in[0..m]$. Similar as for searches, we consider the average costs of insertion when all possible gaps are equally likely to be requested.
Unlike for searches, the distribution of $R'$ in subproblems is \emph{not} uniform even when $R$ is: a close inspection of \proc{RestoreAfterInsert} reveals that (a) recursive calls in the jump-sublist always have $R'\ge 1$, and (b) $R=0$ and $R=1$ yield $R'=0$ in the recursive call in the next-sublist; in fact, once $R=0$ holds, we get this rank in all later recursive calls. We can therefore handle this by splitting the cases $R=0$ and $R\ge 1$;
Also note that for the topmost call to \proc{RestoreAfterInsert}, $R=0$ is not possible, since no insertion before the header with dummy-key $-\infty$ is possible. This means that initially $R\eqdist \uniform[1..m]$ holds. Recall that a jumplist on $m$ nodes stores only $n=m-1$ keys, so that there are only $n+1=m$ possible gaps. We obtain the following distributional recurrence for $B_m^{\ins}$, the random number of re\underline balanced elements during insertion into the $R$th gap in a randomized median-of-$k$ jumplist with $m$ nodes. (Note that \ifsiam{}{unlike in the pseudocode,} $m$ is here the number of nodes in the jumplist \emph{before} the insertion.)
\begin{align*}
B_m^{\ins}
&\rel\eqdist
\begin{dcases}
\begin{aligned}
&F \cdot (m+1)
\mskip1mu[-.75ex]&\quad
+
(1-F) \Bigl(
\indicator{R = 1} B_{J_1}^{\ins0}
\mskip1mu[-.75ex]&\qquad
+\indicator{2\le R \le J_1+1} B_{J_1}^{\ins}
\mskip1mu[-.25ex]&\qquad
+\indicator{R \ge J_1+2} B_{J_2}^{\ins}
\Bigr),
\end{aligned}
& m > w,
\mskip1mu[1ex]
[m=w]\cdot(m+1),
& m\le w,
\end{dcases} \mskip1mu[1ex]
B_m^{\ins0}
&\rel\eqdist
\begin{dcases}
\begin{aligned}
&F \cdot (m+1) \bin+ (1-F) B_{J_1}^{\ins0},
\end{aligned}
& m > w,
\mskip1mu[1ex]
[m=w]\cdot(m+1),
& m\le w,
\end{dcases} \mskip1mu[1ex]
&\mkern-20mu\text{where}\quad
R
\rel\eqdist
\uniform[1..m],\quad\mskip1mu
F
\rel\eqdist
\bernoulli\Bigl(\frac k{m-1}\Bigr),
\end{align*} All $B_m$ terms on the right-hand side denote independent copies of the family of random variables and $R$ and $F$ are independent of $B_m$ and $(J_1,J_2)$. Here $J_r = I_r + t+1$, $r\in\{1,2\mskip1mu$, $J_1 \eqdist \betaBinomial(m-2-k; t+1,t+1)$ and $J_2 = m-2-k - J_1$ (as in~\wref{sec:median-of-k-jumplists-def}).
\begin{lemma} \label{lem:B-n-ins0}
$\mskip1mu{B_m^{\ins0}} \sim \dfrac{k}{\harm{k} - \harm{t}}\ln m$. \end{lemma} \begin{proof} We use once more the distributional master theorem. As before, $Z_1^* \eqdist \betadist(t+1,t+1)$ and the condition \wref{eq:DMTwc-condition} is satisfied by \wref{lem:limit-law-beta-binomial}. We have $\mskip1mu{T_n} = \mskip1mu{F(n+1)} \sim k = \Theta(1)$. Unlike before, we here have a non-constant coefficient $\ui{A_1}n = 1-F$ in front of the recursive term, but since $\mskip1mu{1-F} = 1 \pm \Oh(n^{-1})$, \wref{eq:DMTwc-condition-coeffs} is again fulfilled with $a_1(z) = 1$. As in the proof of \wref{lem:spine-length-asymptotic}, we find $H = 0$ (Case~2) and with the claim follows from $\tilde H = - \mskip1mu{\ln D_1} = \harm{k} - \harm{t}$ (\wref{lem:E-ln-D}). \end{proof}
\begin{proof}[\wref{thm:results-insert}] Towards applying the DMT on $C_n= B_n^{\ins}$, we compute \begin{align*}
\mskip1mu{T_n}
&\wwrel=
\mskip1mu[\Big]{F(n+1) + (1-F)\indicator{R=1}B_n^{\ins0}} \mskip1mu &\wwrel=
\frac {k(n+1)}{n-1} \bin+ \frac{n-1-k}{n-1} \cdot \frac1n \cdot \mskip1mu{B_n^{\ins0}} \mskip1mu &\wwrel{\eqwithref[r]{lem:B-n-ins0}}
k \bin\pm \Oh(n^{-1}\log n). \end{align*} As usual, we have $Z_r^* \eqdist \betadist(t+1,t+1)$, $r\in\{1,2\mskip1mu$, and \wref{eq:DMTwc-condition} is fulfilled by \wref{lem:limit-law-beta-binomial}. For the coefficients of the recursive terms holds \begin{align*}
&\mskip1mu*{\ui{A_1}n \given \ui{Z_1}n \in (z-\tfrac1n,n]} \mskip1mu &\wwrel=
\Prob[\Big]{2\le R\le J_1+1 \given \ui{Z_1}n \in (z-\tfrac1n,n]} \mskip1mu &\wwrel=
\Prob[\Big]{\tfrac{J_1}{n} \given \ui{Z_2}n \in (z-\tfrac1n,n]} \mskip1mu &\wwrel=
\Prob[\big]{\ui{Z_1}n \given \ui{Z_1}n \in (z-\tfrac1n,n]} \mskip1mu &\wwrel=
z \bin\pm \Oh(n^{-1})
, \shortintertext{and similarly}
&\mskip1mu[\big]{\ui{A_2}n \given \ui{Z_2}n \in (z-\tfrac1n,n]} \mskip1mu &\wwrel=
\Prob[\big]{R\ge J_1+2 \given \ui{Z_2}n \in (z-\tfrac1n,n]} \mskip1mu &\wwrel=
z \bin\pm \Oh(n^{-1})
, \end{align*} so that \wref{eq:DMTwc-condition-coeffs} holds with $a_1(z) = a_2(z) = z$, and we can apply the DMT. Since $H=1-\sum_{r=1}^2 \mskip1mu[\big]{(Z_r^*)^0 \*a_r(Z_r^*)}=1-\sum_{r=1}^2 \mskip1mu{Z_r^*}=0$, we again have Case~2 and find $\tilde H = -\sum_{r=1}^2 \mskip1mu{D_r \ln D_r} = \harm{k+1}-\harm{t+1}$ with \wref{lem:E-ln-D}. This proves the claim. \end{proof}
\subsection{Deletion Costs} \label{sec:analysis-delete}
As for insertion, we analyze the size of the sublist $B_m^{\del}$ that is rebuilt using \proc{Rebalance} when the rank of the deleted element is chosen uniformly. Initially, we have $R\eqdist \uniform[2..m]$ since the dummy key $-\infty$ in the header cannot be deleted. In recursive calls, also $R=1$ is possible, and we remain in this case for good whenever we enter it once. We can thus characterize the deletion costs using the two quantities $B_m^{\del}$ and $B_m^{\del1}$. As for insertion, $m$ is the ``old'' size of the jumplist, \ie, the number of nodes \emph{before} the deletion. \begin{align*}
B_m^{\del}
&\rel\eqdist
\begin{dcases}
\begin{aligned}
&F \cdot (m-1)
\mskip1mu[-.75ex]&\quad
+
(1-F) \Bigl(
\indicator{R = 2} B_{J_1}^{\del1}
\mskip1mu[-.75ex]&\qquad
+\indicator{3\le R \le J_1+1} B_{J_1}^{\del}
\mskip1mu[-.25ex]&\qquad
+\indicator{R \ge J_1+3} B_{J_2}^{\del}
\Bigr),
\end{aligned}
& m > w,
\mskip1mu[1ex]
[m=w]\cdot1,
& m\le w,
\end{dcases} \mskip1mu[1ex]
&\mkern-20mu\text{where}\mskip1mu\mskip1mu
R
\rel\eqdist
\uniform[2..m],\quad
\text{and cond.\ on $(R,J_1,J_2)$} \mskip1mu &\phantom{\mkern-20mu\text{where}\mskip1mu\mskip1mu} F
\rel\eqdist
\begin{cases}
\bernoulli\bigl(\frac t{J_1-1}\bigr), & R \le J_1 + 1;\mskip1mu
1, & R = J_1 + 2;\mskip1mu
\bernoulli\bigl(\frac t{J_2-1}\bigr), & R \ge J_1 + 3,
\end{cases} \mskip1mu[1.5ex]
B_m^{\del1}\mskip1mu
&\rel\eqdist
\begin{dcases}
F_1 \cdot (m-1) + (1-F_1) B_{J_1}^{\del1},
& m > w,
\mskip1mu[1ex]
[m=w]\cdot1,
& m\le w,
\end{dcases} \mskip1mu
&\mkern-20mu\text{where cond.\ on $J_1$}\mskip1mu\mskip1mu
F_1
\rel\eqdist
\bernoulli\Bigl(\frac t{J_1-1}\Bigr).
\end{align*}
As before, the $B_m$ terms on the right are independent copies of the family of random variables and $R$ and $F$/$F_1$ are independent of $B_m$ and $(J_1,J_2)$. We have $J_r = I_r + t+1$, $r\in\{1,2\mskip1mu$, $J_1 \eqdist \betaBinomial(m-2-k; t+1,t+1)$ and $J_2 = m-2-k - J_1$. The (asymptotic) solution of these recurrences is similar to the case of insertion, but a few more complications arise.
\begin{lemma} \label{lem:B-n-del1}
For $t=0$ we have $\mskip1mu{B_m^{\del1}} \le 1$.\needhspace{8em}
If $t\ge 1$, $\mskip1mu{B_m^{\del1}} \sim \dfrac{k}{\harm{k}-\harm{t}} \ln m$. \end{lemma} \begin{proof}
For $t=0$, we have $F_1=0$ (almost surely) in each iteration, so the recurrence
collapses to its initial condition, which is at most $1$.
In the following, we now consider $t\ge1$.
The proof will ultimately use the DMT on $C_n = B_n^{\del1}$,
but we need a few preliminary results
to compute the toll function $\mskip1mu{T_n} = \mskip1mu{F_1(n-1)}$.
We write the $a=b\pm d$ to mean $b-d \le a \le b+d$ here and throughout.
With that notation, we give the following elementary approximation:
\begin{align}
\label{eq:t-over-m-plus-t}
\forall t \in\mskip1mu_{\ge1} \mskip1mu\forall n\ge0 \rel:
\smash{\frac{t}{n+t}} \rel= t n^{\underline{-1}} \bin\pm t(t-1) n^{\underline{-2}}
.
\end{align}
Now, we compute the expectation of $F_1$ conditional on $I_1 = J_1 - t - 1$.
\begin{align*}
\mskip1mu{F_1\given I_1}
&\wrel=
\frac t{J_1-1}
\wrel=
\frac{t}{I_1+t}
\mskip1mu &\wrel{\eqwithref{eq:t-over-m-plus-t}}
t \cdot I_1^{\underline{-1}} \wbin\pm t(t-1)\cdot I_1^{\underline{-2}}
.
\end{align*}
Next, we use the stochastic representation of beta-binomials
(recall \wref{sec:preliminaries});
we take expectations over $I_1\eqdist\binomial(\eta,D_1)$ with $\eta=m-2-k$,
but conditional on $D_1$. We write $D_2=1-D_1$. Then it holds that
\begin{align*}
\quad&\mkern-25mu
\mskip1mu{F_1 \given D_1}
\mskip1mu[-.5ex] &\wrel{\eqwithref[r]{lem:binomial-negative-factorial-moments}}
\frac t{\eta+1} D_1^{-1} (1-D_2^{\eta+1}) \bin\pm t(t-1) D_1^{-2} \eta ^{\underline{-2}}
.
\end{align*}
Finally, we also compute the expectation \wrt $D_1 \eqdist \betadist(t+1,t+1)$;
note that for $t\ge 2$, $\mskip1mu{D_1^{-2}}$ exists and has a finite value (independent of $n$);
whereas for $t=1$, the error term is zero.
So we find in both cases with \wref{lem:powers-to-parameters}:
\begin{align*}
\mskip1mu{F_1}
&\rel=
\frac t{\eta+1} \mskip1mu{D_1^{-1}}
-\frac t{\eta+1} \mskip1mu{D_1^{-1}D_2^{\eta+1}}
\bin\pm \Oh(\eta^{-2})
\mskip1mu &\rel=
\frac t{\eta+1} \frac{k}{t}
-\frac t{\eta+1} \frac{(t+1)^{\overline{\eta+1}}}{t(k+1)^{\overline{\eta}}}
\bin\pm \Oh(\eta^{-2})
\mskip1mu &\rel=
\frac k{\eta+1}
-\frac {(t+1)(t+2)}{(\eta+1)(\eta+2)}
\underbrace{ \frac{(t+3)^{\overline{\eta-1}}}{(k+2)^{\overline{\eta-1}}}}_{<1}
\bin\pm \Oh(\eta^{-2})
\mskip1mu[-3ex] &\rel=
\frac k{\eta+1}
\bin\pm \Oh(\eta^{-2})
.
\numberthis\label{eq:E-F-del}
\end{align*}
With this we finally get $\mskip1mu{T_n} = \mskip1mu{F_1 (n-1)} = k \pm \Oh(n^{-1})$.
$Z_1^* \eqdist \betadist(t+1,t+1)$ and fulfills \wref{eq:DMTwc-condition}.
For \wref{eq:DMTwc-condition-coeffs}, we compute
\begin{align*}
&\mskip1mu[\big]{\ui{A_1}n\given \ui{Z_1}n \in (z-\tfrac1n,z]}
\mskip1mu &\wrel=
\mskip1mu[\big]{1-F_1\given \ui{Z_1}n \in (z-\tfrac1n,z]}
\mskip1mu &\wrel=
1 \pm \Oh(n^{-1}).
\end{align*}
So the DMT applies;
we have $H=0$, \ie, Case~2. The claim follows with
$\tilde H = -\mskip1mu{\ln Z_1^*} = \harm{k}-\harm{t}$.~ \end{proof}
\begin{proof}[\wref{thm:results-delete}] We start with computing the conditional expectation of $F$, the coin flip indicator. \begin{align*}
\mskip1mu{F\given J_1}
&\rel=
\frac{J_1}{n-1} \frac{t}{J_1-1}
+\frac1{n-1} 1
+\frac{J_2-1}{n-1} \frac{t}{J_2-1} \mskip1mu &\rel=
\frac{2t+1}{n-1} \bin+ \frac1{n-1} \cdot \frac t{J_1-1}
. \shortintertext{Hence}
\mskip1mu{F}
&\rel{\eqwithref{eq:E-F-del}}
\frac{2t+1}{n-1} \bin+ \frac1{n-1} \cdot \frac{k}{\eta+1} \pm \Oh(n^{-3}) \mskip1mu &\rel=
\frac{k}{n-1} \bin\pm \Oh(n^{-2})
. \end{align*} Towards applying the DMT on $C_n= B_n^{\del}$, we compute \begin{align*}
\mskip1mu{T_n}
&\wrel=
\mskip1mu[\Big]{F(n-1) + (1-F)\indicator{R=2}B_n^{\del1}} \mskip1mu[-.5ex] &\wrel{\eqwithref[r]{lem:B-n-del1}}
k \wbin\pm \Oh(n^{-1} \log n)
. \end{align*} We have $Z_r^* \eqdist\betadist(t+1,t+1)$ and \wref{eq:DMTwc-condition} is fulfilled. Similarly as in \wref{sec:analysis-insert}, we find that \wref{eq:DMTwc-condition-coeffs} holds with $a_1(z) = a_2(z) = z$. Once more we have $H=0$ and Case~2 applies, and the claim follows with $\tilde H = -\sum_{r=1}^2 \mskip1mu{D_r \ln D_r} = \harm{k+1}-\harm{t+1}$. \end{proof}
\subsection{Memory Requirements}
We assume that a pointer requires one word of storage, and so does an integer that can take values in $[0..n+1]$. We do not count memory to store the keys since any (general-purpose) data structure has to store them. This means that a plain node requires $1$ word of (additional) storage, and a jump node needs $3$ additional words (two pointers and one integer). Let $A_n$ denote the (random) number of jump nodes, excluding the dummy header, of a random median-of-$k$ jumplist with leaf size $w$ on $n$ keys, then its additional memory requirement is $3 (A_n+1) + 1(n-A_n)$. It remains to show that $A_n$ is asymptotically at most \mskip1mu 1/ \bigl((w+1)(\harm{k+1}-\harm{t+1})\bigr) n \mskip1mu.
$A_n$ counts the internal nodes in a random fringe-balanced dangling-min BST over $n$ keys; a distributional recurrence is thus easy to set up: \begin{align*}
A_n
&\wrel\eqdist
\begin{cases}
1 + A_{J_1} + A_{J_2},
& n > w-1,
\mskip1mu
0,
& n \le w-1.
\end{cases}
\end{align*} Here again $J_r = I_r + t$, $r\in\{1,2\mskip1mu$, $J_1 \eqdist \betaBinomial(n-1-k; t+1,t+1)$ and $J_2 = n-1-k - J_1$.
For $A_n$, the DMT only gives us $\mskip1mu{A_n} = \Oh(n)$ (Case~3). It is easy to see that $\mskip1mu{A_n}$ is also $\Omega(n)$, but a precise leading-term seems very hard to obtain.
\begin{proof}[\wref{thm:results-memory}]
The recurrence for $A_n$ is very similar to that for the number of partitioning steps
in median-of-$k$ Quicksort with Insertionsort threshold $w-1$;
the only difference is that we there have $I_1\eqdist I_2 \eqdist \betaBinomial(n-k;t+1,t+1)$,
\ie, with $n-k$ instead of $n-k-1$.
By monotonicity, $\mskip1mu{A_n}$ is at most the number of partitioning steps in Quicksort
since also the subproblems sizes are smaller.
The number of partitioning steps in median-of-$k$ Quicksort with Insertionsort threshold $M$
is $1/\bigl((M+2)(\harm{k+1}-\harm{t+1})\bigr) n \pm \Oh(1)$, see, \eg,
\cite[p.\,327]{Hennequin1989}.
Setting $M=w-1$ yields the claim.
\end{proof}
\section{Conclusion} \label{sec:conclusion}
In this article, we presented median-of-$k$ jumplists and analyzed their efficiency in terms of the expected number of comparisons (for searches) and rebalanced elements (for updates). The precise analysis of insertion and deletion costs is also novel for the original version of jumplists ($k=1$).
Our analysis shows that a search profits from sampling; in particular going from $k=1$ to $k=3$ entails significant savings: $\frac{12}7 \ln n \approx 1.714\ln n$ instead of $2 \ln n$ comparisons on average. As for median-of-$k$ Quicksort, we see diminishing returns for much larger~$k$. For jumplists, also the cleanup after insertions and deletions gets more expensive; the effort grows linearly with $k$. Very large $k$ will thus be harmful.
The efficiency of insertion and deletion depends on both the time for search and the time for cleanup, so it is natural to ask for optimal $k$. Since the cost units are rather different (comparisons vs.\ rebalanced elements) we need a weighing factor. Depending on the relative weight $\xi\in[0,1]$ of comparisons, we can compute optimal~$k$, see \wref{fig:optimal-k}. In the realistic range, we should try $k=1$, $3$, or $5$, unless we do many more searches than~updates.
\begin{figure}
\caption{
The $k$ that minimizes the leading-term coefficient of
total costs of insertion/deletion,
if one comparison costs $\xi\in[0,1]$ and each rebalanced element costs $1-\xi$,
\ie, $\arg \min_{k} \xi \cdot \frac1{H(t)} + (1-\xi) \cdot \frac k{H(t)}$
as a function of $\xi$.
}
\label{fig:optimal-k}
\end{figure}
We conducted a small running time study based on a proof-of-concept implementation~\cite{codeWild2016} in Java that confirms our analytical findings: Sampling leads to some savings for searches, but slows down insertions and deletions significantly. Comparing running times with that of Java's \texttt{TreeMap} (a red-black tree implementation) shows that our data structure is only partially competitive: for iterating over all elements, jumplists are about 50\mskip1mu faster, but searches are between 20\mskip1mu and 100\mskip1mu slower (depending on the choice for $w$) and for insertions/deletions \texttt{TreeMap}s are 5 to 10 times faster.
However, \texttt{TreeMap}s use 4 additional words per key (without even storing subtree sizes needed for efficient rank-based access), whereas our jumplists never need more than $\sim 2.\overline 3$ additional words per key and less than $1.04$ with $w\approx100$. For $n=10^6$ keys, $w\approx 100$ did not affect searches much ($+25\mskip1mu$) but actually sped up insertions and deletions (roughly by a factor of~2!).
\subsection{Future Work} Some interesting questions are left open. What is the optimal choice for $w$? Answering this question requires second-order terms of search, insertion and deletion costs; due to the underlying mathematical challenges it is unlikely that those can be computed exactly, but an upper bound using analysis results on Quicksort should be possible. Other future directions are the analysis of branch misses, in particular in the context of an asymmetric sampling strategy, and the design of a ``bulk insert'' algorithm that is faster than inserting elements subsequently, one at a time.
On modern computers the cache performance of data structures is important for their running time efficiency. Here, a larger fanout of nodes is beneficial since it reduces the expected number of I/Os. For jumplists this can be achieved by using more than one jump pointer in each node. The case of two jump pointers per node has been worked out in detail~\cite{Neumann2015}, but the general scheme invites further investigation.
\ifsiam{}{
\appendix
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\manualmark
\markleft{Median-of-$k$ Jumplists and Dangling-Min BSTs}
\markright{Appendix} }
\input{jumplists-notation}
\section{Comparison of Jumplist Definitions}
\label{app:differences-definitions}
Our definition of jumplists differs in some details from the original version. We list the differences here, and discuss why we think that our modifications are appropriate.
\paragraph{Symmetry}
In the original version of the jumplist, the jump pointer is allowed to target any node from the sublist, except the header itself. Thus there are $m-1$ possible choices. In this setting, the size of the next-sublist can attain any value between $0$ and $m-2$, whereas the size of the jump-sublist is between $1$ and $m-1$.
We disallow the direct successor of the head as possible target. This modification restores symmetry between next- and jump-sublist: both must be non-empty and contain at most $m-2$ nodes and their sizes have the same distribution. Moreover, forbidding the direct successor as jump target is also a natural requirement since such a degenerate ``shortcut'' is useless in searches.
\paragraph{Small Sublists}
The original jumplists only have one type of nodes which corresponds to our jump node. In the case $m=1$, Brönnimann, Cazals, and Durand resort to assigning an ``exceptional pointer'' to the direct successor; note that this node actually lies outside (one behind) of the current sublist. These pointers are of no use, as they are never followed during (jump-and-walk) search.
In implementations with heap-allocated memory for each node, it is often not a problem to have different node types (and sizes), and it potentially allows to save memory. We thus introduced the plain node without jump pointer, used whenever the sublist has at most $w$ nodes. $w\ge 2$ is required if we want to avoid useless jump pointers that point to the direct successor.
This also allows us enforce that every node has at most one incoming jump pointer; this is another natural requirement from the perspective of a search starting at the header: shortcuts with the same target are redundant. The parameter $w$ allows us to trade space for time.
\paragraph{Sentinel vs.\ Circularly closed}
The original jumplist implementation has a circularly closed backbone, \ie, the next pointer of the last node in the list points to the overall header again, avoiding special treatment for an empty list. Since the backbone is sorted, we can instead add a \emph{sentinel} node with key $+\infty$ at the end of the list, so we can omit any explicit boundary checks during searches.
\input{jumplists-algorithms}
\section{Omitted proofs} \label{app:expectations}
\begin{proof}[\wref{lem:binomial-negative-factorial-moments}] For $m\in\{1,2\mskip1mu$, we compute
\begin{align*}
\mskip1mu{ X^{\underline {-m}} }
&\wwrel=
\sum_{x=0}^n
x^{\underline{-m}}\cdot
\binom{n}{x} \mskip1mu
p^{x} q^{n-x}
\mskip1mu &\wwrel=
n^{\underline {-m}} \mskip1mu p^{-m}
\sum_{x=0}^{n}
\binom{n+m}{x+m} \mskip1mu
p^{x+m} q^{(n+m)-(x+m)}
\mskip1mu &\wwrel=
n^{\underline {-m}} \mskip1mu p^{-m}
\sum_{x=m}^{n+m} \binom{n+m}{x} p^{x} q^{(n+m)-x}
\mskip1mu &\wwrel{\relwithtext[r]{[binom.\,thm.]}=}
n^{\underline {-m}} \mskip1mu p^{-m}
\mskip1mu
\biggl((\underbrace{p+q} _ {=1} )^{n+m} -
\sum_{x=0}^{m-1} \binom{n+m}{x} p^{x} q^{(n+m)-x}
\biggr)
. \end{align*} For the first part of the claim, we set $m=1$ and find that the sum reduces to $q^{n+1}$; for the second part of the claim, we use $m=2$ and note that the expression in the outer parentheses is at most $1$. \end{proof}
\begin{proof}[\wref{lem:E-ln-D}] We use the following known integral; see \cite[\href{https://www.wild-inter.net/publications/html/wild-2016.pdf.html\#pf46}{Eq.\mskip1mu(2.30)}]{Wild2016}: \begin{align*} \numberthis\label{eq:logarithmic-beta-integral}
\int_0^1 z^{a-1} (1-z)^{b-1} \ln(z) \mskip1mu dz
\wwrel=
\BetaFun(a,b) \bigl( \psi(a) - \psi(a+b) \bigr)
,\qquad (a,b > 0) . \end{align*} Here $\psi(z) = \frac d{dz} \ln(\Gamma(z))$ is the digamma function. Then we find \begin{align*}
\mskip1mu{\ln(D)}
&\wwrel=
\int_0^1 \ln(x) \frac{x^{t}(1-x)^{t}}
{\BetaFun(t+1,t+1)} \mskip1mu dx \mskip1mu &\wwrel{\eqwithref{eq:logarithmic-beta-integral}}
\psi(t+1)-\psi(k+1) \mskip1mu &\wwrel=
\harm{t} - \harm{k}
, \end{align*} and \begin{align*}
\mskip1mu{D \ln(D)}
&\wwrel=
\int_0^1 x \ln(x) \frac{x^{t}(1-x)^{t}}
{\BetaFun(t+1,t+1)} \mskip1mu dx \mskip1mu &\wwrel=
\frac{\BetaFun(t+2,t+1)}{\BetaFun(t+1,t+1)}
\int_0^1 \ln(x) \frac{x^{t+1}(1-x)^{t}}
{\BetaFun(t+2,t+1)} \mskip1mu dx \mskip1mu &\wwrel{\eqwithref{eq:logarithmic-beta-integral}}
\frac{t+1}{k+1}
\bigl(\psi(t+2)-\psi(k+2)\bigr) \mskip1mu &\wwrel=
\frac{t+1}{2t+2}
\bigl(\harm{t+1} - \harm{k+1}\bigr) \mskip1mu &\wwrel=
\frac12\bigl(\harm{t+1} - \harm{k+1}\bigr) \mskip1mu. \end{align*} \end{proof}
}
\ifdraft{
\part*{Notes-to-self}
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\end{document} | arXiv |
A player pays $\$5$ to play a game. A six-sided die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case the player wins some amount of money if the second number matches the first and loses otherwise. How much money should the player win if the game is fair? (In a fair game the probability of winning times the amount won is what the player should pay.)
Let $x$ represent the amount the player wins if the game is fair. The chance of an even number is $1/2$, and the chance of matching this number on the second roll is $1/6$. So the probability of winning is $(1/2)(1/6)=1/12$. Therefore $(1/12)x=\$5$ and $x=\boxed{60}$. | Math Dataset |
\begin{document}
\title{An economic game with stochastic dynamics} \author{A. L. Ciurdariu$^{a}$, M. Neam\c{t}u$^{b}$, D. Opri\c s$^{c}$} \date{} \maketitle
\begin{tabular}{cccccccc} \scriptsize{$^{a}$ Department of Mathematics,Politehnica University of Timi\c{s}oara}\\ \scriptsize{P-\c{t}a. Victoriei, nr 2, 300004, Timi\c{s}oara, Romania,}\\ \scriptsize{E-mail: [email protected]}\\ \scriptsize{$^{b}$Department of Economic Informatics, Mathematics and Statistics,}\\ \scriptsize{Faculty of Economics, West University of Timi\c soara,}\\ \scriptsize{str. Pestalozzi, nr. 16A, 300115, Timi\c soara, Romania,}\\ \scriptsize{E-mail:[email protected],}\\ \scriptsize{$^{c}$ Department of Applied Mathematics, Faculty of Mathematics,}\\ \scriptsize{West University of Timi\c soara, Bd. V. Parvan, nr. 4, 300223, Timi\c soara, Romania,}\\
\scriptsize{E-mail: [email protected]}\\
\end{tabular}
\begin{abstract} In this paper we investigate a stochastic model for an economic game. To describe this model we have used a Wiener process, as the noise has a stabilization effect. The dynamics are studied in terms of stochastic stability in the stationary state, by constructing the Lyapunov exponent, depending on the parameters that describe the model. The numerical simulation that we did justifies the theoretical results. \end{abstract}
{\small \textit {Mathematics Subject Classification}: 34D08, 60H10, 91B70}
{\small \textit {Keywords}: stochastic dynamics in economic games, economic games, stochastic stability, Lyapunov exponent, Euler scheme}
\section{Introduction.}
\qquad Stochastic modeling plays an important role in many branches of science. In many practical situations, perturbations are expressed in terms of white noise, modeled by brownian motion. The behavior of a deterministic dynamical system which is disturbed by noise may be modeled by a stochastic differential equation (SDE), \cite{KP}. The stochastic stability has been introduced by Bertram and Sarachik and is characterized by the negativeness of Lyapunov exponents. In general, it is not possible to determine this exponents explicitly. Many numerical approaches have been proposed, which generally used the simulation of the stochastic trajectories \cite{Jed}. In the present paper, we study a stochastic dynamical system that are used in economy, in describing a Counot duopoly game.
In 1838, Cournot introduced the first formal theory of oligopoly, which treated the case of naive expectations, where each player assumes the last values taken by the competitors without estimation of their future reactions \cite{Cournot}. Recently, a lot of articles have shown that the Cournot model may lead to a cyclic or chaotic behavior \cite{Bundau}, \cite{Bischi}, \cite{Mircea}, \cite{Puu1}, \cite{Puu2}, \cite{Puu3}. Also, in \cite{Rosser}, Rosser reviews the development of the theory of complex oligopoly dynamics.
In the present paper we have studied a stochastic Cournot economic game. In Section 2 we present the Lyapunov exponent and stability in stochastic 2d dynamical structures. Section 3 studies the Lyapunov exponent for an economic game with stochastic dynamics. Some numerical simulations are given in Section 4. Finally, Section 5 draws some conclusions.
\section{The Lyapunov exponent and stability in stochastic 2d dynamical structures.}
\qquad Let $(\Omega , {\cal F}, {\cal P})$ be a probability space \cite{KP}. It is
assumed that the $\sigma -$algebra ${\cal F}$ is a filtration that
is, ${\cal F}$ is generated by a family of $\sigma -$algebra ${\cal F}_t(t\geq
0)$ such that
$${\cal F}_s\subset {\cal F}_t\subset {\cal F}, \quad \forall s\leq t, s,t\in I,$$ where $I=[0, T]$, $T\in (0, \infty)$.
Let $\{x(t)=(x_1(t), x_2(t))\}_{t\geq 0}$ be a stochastic process. The system of Ito equations: \begin{equation}\label{1} dx_i(t,\omega)=f_i(t,x(t,\omega))dt+g_i(x(t,\omega))dw(t,\omega), i=1,2, \end{equation} with the initial condition $x(0)=x_0$ is written as: \begin{equation}\label{2} x_i(t,\omega)=x_{i0}(\omega)+\int_0^tf_i(x(s,\omega))ds+\int_0^tg_i(x(s,\omega))dw(s,\omega), i=1,2, \end{equation} for almost all $\omega\in\Omega $ and for each $t>0$, where $f_i(x)$ are drift functions, $g_i(x)$ are diffusion functions, $\int_0^tf_i(x(s))ds$, $i=1,2$ are Riemann integrals and $\int_0^tg_i(x(s))dw(s)$ are It$\hat{o}$ integrals. It is assumed that $f_i$ and $g_i$, $i=1,2$ satisfy the conditions of existence of solution for this SDE with initial condition $x(0)=a_0\in{\rm{I \! R}}^n$.
Let $x_0=(x_{10}, x_{20})\in{\rm{I \! R}}^2$ be a solution of the system: \begin{equation}\label{3} f_i(x_0)=0, i=1,2. \end{equation}
The functions $g_i, i=1,2$ are chosen so that: \begin{equation*} g_i(x_0)=0, i=1,2. \end{equation*}
In what follows, we consider: \begin{equation*}\label{4} g_i(x)=\sum_{j=1}^{2}b_{ij}(x_j-x_{0j}), i=1,2, \end{equation*} where $b_{ij}\in{\rm{I \! R}}, i,j=1,2.$
The linearized system of (\ref{2}) in $x_0$, is given by: \begin{equation*}\label{5} X(t)=\int_0^tAX(s)ds+\int_0^tBX(s)dw(s), \end{equation*} where \begin{equation*}\label{6} X(t)=\left (\begin{array}{c} x(t,\omega )\\ y(t,\omega )\end{array}\right ), A=\left (\begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22}\end{array}\right ), B=\left (\begin{array}{cc} b_{11} & b_{12}\\ b_{21} & b_{22}\end{array}\right ), \end{equation*} \begin{equation*}\label{7}
a_{ij}=\displaystyle\frac{\partial f_i}{\partial x_j}|_{x_0}, b_{ij}=\displaystyle\frac{\partial g_i}{\partial x_j}|_{x_0}. \end{equation*}
The Oseledec multiplicative ergodic theorem \cite{Ose} asserts the existence of 2 non-random Lyapunov exponents $\lambda_2\leq\lambda_1=\lambda$. The top Lyapunov exponent is given by: \begin{equation*}\label{8} \lambda =\lim_{t\to\infty} \sup\log\sqrt{x(t)^2+y(t)^2}. \end{equation*}
Applying the change to polar coordinates: \begin{equation*} x(t)=r(t)cos \theta (t), y(t)=r(t) sin\theta (t) \end{equation*} by writing the It$\hat{o}$ formula for \begin{equation*} h_1(x,y)=\displaystyle\frac{1}{2}\log (x^2+y^2)=\log (r), h_2(x,y)=arctg (\displaystyle\frac{y}{x})=\theta . \end{equation*} we get:
\begin{proposition}\label{P1} \cite{Jed}. The formulas \begin{equation}\label{11} \log \left (\displaystyle\frac{r(t)}{r(0)}\right )\!=\!\int_0^tq_1(\theta(s))\!+\!\displaystyle\frac{1}{2}(q_4(\theta(s))^2\!-\!q_2(\theta(s))^2)ds\!+\!\int_0^tq_2(\theta(s))dw(s), \end{equation} \begin{equation}\label{12} \theta(t)\!=\!\theta(0)+\int_0^tq_3(\theta(s))\!-\!q_2(\theta(s)q_4(\theta(s))ds\!+\!\int_0^tq_4(\theta(s))dw(s), \end{equation} hold,
where \begin{equation}\label{13} \begin{array}{llll} q_1(\theta)=a_{11}cos^2(\theta)+(a_{12}+a_{21})cos\theta\sin\theta+a_{22}sin^2\theta,\\ q_2(\theta)=b_{11}cos^2(\theta)+(b_{12}+b_{21})cos\theta\sin\theta+b_{22}sin^2\theta,\\ q_3(\theta)=a_{21}cos^2(\theta)+(a_{22}-a_{11})cos\theta\sin\theta-a_{12}sin^2\theta,\\ q_4(\theta)=b_{21}cos^2(\theta)+(b_{22}-b_{11})cos\theta\sin\theta-b_{12}sin^2\theta.\\ \end{array} \end{equation} \end{proposition} As the expectation of the It$\hat{o}$ stochastic integral is null $$E\int_0^tq_2(\theta (s))dw(s)=0,$$ the Lyapunov exponent is given by: $$\lambda\!=\!\lim_{t\to\infty}\displaystyle\frac{1}{t}\log\left (\displaystyle\frac{r(t)}{r(0)}\right )\!=\! \lim_{t\to\infty}\displaystyle\frac{1}{t}E\int_0^t(q_1(\theta(s))\!+\!\displaystyle\frac{1}{2}(q_4(\theta(s)))^2\!-\!q_2(\theta(s)))ds.$$ Applying the Oseledec theorem, if $r(t)$ is ergodic, we get: \begin{equation*}\label{14} \lambda =\int_0^t(q_1(\theta)+\displaystyle\frac{1}{2}(q_3(\theta)^2-q_2(\theta)))p(\theta)d\theta , \end{equation*}where $p(\theta)$ is the probability distribution of the process $\theta$.
An approximation of this distribution is calculated by solving the Fokker-Planck equation.
The Fokker-Planck (FPE) equation associated with equation (\ref{12}) for $p=p(t,\theta)$ is \begin{equation}\label{15} \displaystyle\frac{\partial p}{\partial t}+\displaystyle\frac{\partial }{\partial\theta }((q_3(\theta)-q_2(\theta)q_4(\theta))p)-\displaystyle\frac{1}{2}\displaystyle\frac{\partial ^2}{\partial \theta^2}(q_4(\theta)^2p)=0. \end{equation}
From (\ref{15}), it results that the solution $p(\theta)$ of the FPE is solution of the following first order equation: \begin{equation}\label{16} (-q_3(\theta)\!+\!q_1(\theta)q_4(\theta)\!+ \!q_2(\theta)g_5(\theta))p(\theta)\!+\!\displaystyle\frac{1}{2}q_4(\theta)^2p'(\theta)\!=\!p_0, \end{equation} where $p'(\theta )=\displaystyle\frac{dp}{d\theta}$ and \begin{equation*}\label{17} q_5(\theta)=-(b_{12}+b_{21})sin 2\theta -(b_{22}-b_{11})cos 2\theta . \end{equation*}
\begin{proposition}\label{prop2}\cite{Jed}. If $q_4(\theta)\neq 0$, the solution of the equation (\ref{16}) is given by: \begin{equation*}\label{18} p(\theta)=\displaystyle\frac{k}{D(\theta)q_4(\theta)^2}\left (1+\eta \int_0^\theta D(u)du\right ) \end{equation*} where $k$ is determined by the normality condition \begin{equation*}\label{18} \int_0^{2\pi }p(\theta )d\theta =1 \end{equation*} and \begin{equation*}\label{19} \eta =\displaystyle\frac{D(2\pi )-1}{\int_0^{2\pi }D(u)du}. \end{equation*} The function $D$ is given by: \begin{equation*}\label{21} D(\theta )=\exp (-2\int_0^\theta\displaystyle\frac{q_3(u)-q_2(u)q_4(u)-q_4(u)q_5(u)}{q_4(u)^2}du) \end{equation*} \end{proposition}
A numerical solution of the phase distribution could be performed by a simple backward difference scheme.
We consider $N\in{\rm{I \! R}}_+$, $h=\displaystyle\frac{\pi}{N}$ and \begin{equation*}\label{13} \begin{array}{llll} q_1(i)=a_{11}\cos^2(ih)+(a_{12}+a_{21})\cos(ih)\sin(ih)+a_{22}\sin^2(ih),\\ q_2(i)=b_{11}\cos^2(ih)+(b_{12}+b_{21})\cos(ih)\sin(ih)+b_{22}\sin^2(ih),\\ q_3(i)=a_{21}\cos^2(ih)+(a_{22}-a_{11})\cos(ih)\sin(ih)-a_{12}sin^2(ih),\\ q_4(i)=b_{21}\cos^2(ih)+(b_{22}-b_{11})\cos(ih)\sin(ih)-b_{12}\sin^2(ih),\\ q_5(i)=-(b_{12}+b_{21})\sin(2ih)-(b_{22}-b_{11})cos(2ih), i=0,...,N\\ \end{array} \end{equation*}
The function $p(i), i=0,...,N$ is given by the following relations: $$p(i)=(p(0)+\displaystyle\frac{q_4(i)^2p(i-1)}{2h})F(i)$$ where $$F(i)=\displaystyle\frac{2h}{2h(-q_3(i)+q_2(i)q_4(i)+q_4(i)q_5(i))+q_4(i)^2}.$$ The Lyapunov exponent is $\lambda =\lambda (N)$, where $$\lambda (N)=\sum_{i=0}^{N}(q_1(i)+\displaystyle\frac{1}{2}(q_4(i)^2-q_2(i)^2))p(i)h.$$
\begin{proposition}\label{prop3} If the matrix B is given by: $$b_{11}=\alpha, b_{12}=-\beta, b_{21}=\beta, b_{22}=\alpha $$ then \begin{equation*} \begin{split} p(\theta)&=\displaystyle\frac{k}{\beta^2}\exp\{\displaystyle\frac{1}{\beta^2}((a_{21}-a_{12}-\alpha\beta)\theta+\displaystyle\frac{1}{2}(a_{11}-a_{22})\cos 2\theta+ \displaystyle\frac{1}{2}(a_{21}-\\ &-a_{12})\sin2\theta)\} \end{split} \end{equation*} \begin{equation*} k\!=\!\displaystyle\frac{\beta^2}{\int_0^{2\pi}\exp\{\displaystyle\frac{1}{\beta^2}((a_{21}\!\!-\!\!a_{12}\!\!-\!\!\alpha\beta)\theta\!+\!\displaystyle\frac{1}{2}(a_{11}\!-\!a_{22})\cos 2\theta\!\!+\!\! \displaystyle\frac{1}{2}(a_{21}\!\!-\!\!a_{12})\sin2\theta)d\theta} \end{equation*} \begin{equation*} \lambda=\displaystyle\frac{1}{2}(a_{11}+a_{22}+\beta^2-\alpha^2)+\displaystyle\frac{1}{2}(a_{11}-a_{22})c_2+\displaystyle\frac{1}{2}(a_{21}+a_{12})s_2, \end{equation*} where $$c_2=\int_0^{2\pi}cos(2\theta)p(\theta)d\theta, \quad s_2=\int_0^{2\pi} sin(2\theta)p(\theta)d\theta .$$ \end{proposition}
\section{The Lyapunov exponent for an economic game with stochastic dynamics.} \qquad Two firms enter the market with a homogenous consumption product. The elements which describe the model are: the quantities which enter the market from the two firms $x_{i}\geq 0,$ $i=\overline{1,2};$ the inverse demand function $p:\mathbb{R} _{+}\rightarrow \mathbb{R}_{+}$ ( $p$ is a derivable function with $ p^{\prime }\left( x\right) <0,\underset{x\rightarrow a_{1}}{lim}p\left( x\right) =0,$ $\underset{x\rightarrow 0}{lim}p\left( x\right) =b_{1},\left( a_{1}\in \overline{\mathbb{R}},b_{1}\in \overline{\mathbb{R}}\right) $; the cost functions $C_{i}: \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ ( $C_{i}$ are derivable functions with $C_{i}^{\prime }\left( x_{i}\right) >0,$ $C_{i}^{\prime \prime }\geq 0,$ $i=\overline{1,2}$ ).
In our study we consider $p(x)=\displaystyle\frac{1}{x}, x>0$ and $C_i(x_i)=c_ix_i+d_i, i=1,2.$
The mathematical model of the stochastic dynamic economic game is described by the stochastic system of equations: \begin{equation}\label{31} \begin{split} x_1(t)\!=\!x_1\!(0)\!\!+\!k_1\!\int_0^t\!(\displaystyle\frac{x_2(s)}{(x_1(s)\!+\!x_2(s))^2}\!-\!c_1)ds\!+\!\int_0^t(b_{11}x_1(s)\!+\!b_{12}x_2(s)\!+\!\gamma_1)dw(s)\\ x_2(t)\!=\!x_2\!(0)\!\!+\!k_2\!\int_0^t\!(\displaystyle\frac{x_1(s)}{(x_1(s)\!+\!x_2(s))^2}\!-\!c_2)ds\!+\!\int_0^t(b_{21}x_1(s)\!+\!b_{22}x_2(s)\!+\!\gamma_2)dw(s) \end{split} \end{equation} where $b_{ij}\in{\rm{I \! R}}$, $i,j=1,2$, $k_1>0, k_2>0$, $x_i(t)=x_i(t,\omega )$, $i=1,2$. \begin{equation*}\label{32} \gamma_1=-\displaystyle\frac{b_{11}c_2+b_{12}c_1}{(c_1+c_2)^2}, \gamma_2=-\displaystyle\frac{b_{21}c_2+b_{22}c_1}{(c_1+c_2)^2}. \end{equation*}
For $b_{ij}=0$, $i,j=1,2$ model (\ref{31}) is reduced to the classical model of the economic game \cite{Bundau}, \cite{Mircea}.
The system of stochastic equations (\ref{31}), (SDE), has the form (\ref{2}) from section 2, where: \begin{equation*}\label{33} \begin{split} f_1(x_1,x_2)=\displaystyle\frac{x_2}{(x_1+x_2)^2}-c_1, g_1(x_1,x_2)=b_{11}x_1+b_{12}x_2+\gamma_1,\\ f_2(x_1,x_2)=\displaystyle\frac{x_1}{(x_1+x_2)^2}-c_2, g_2(x_1,x_2)=b_{21}x_1+b_{22}x_2+\gamma_2. \end{split} \end{equation*}
Applying the results from section 2, we have:
\begin{proposition}\label{prop4} (i) The stationary state of (SDE) (\ref{31}) is given by: \begin{equation*}\label{34} x_{10}=\displaystyle\frac{c_2}{(c_1+c_2)^2}, x_{20}=\displaystyle\frac{c_1}{(c_1+c_2)^2}; \end{equation*}
(ii) The elements of the matrix $A$, which characterize linearized equation (\ref{31}) in $(x_{10}, x_{20})$ are: \begin{equation*}\label{35} \begin{split} a_{11}=-2k_1c_1(c_1+c_2), a_{12}=-k_1(c_1^2-c_2^2)\\ a_{21}=k_2(c_1^2-c_2^2), a_{22}=-2k_2c_2(c_1+c_2); \end{split} \end{equation*}
(iii) The roots of the characteristic equation: \begin{equation}\label{36} \mu^2-(a_{11}+a_{22})\mu+a_{11}a_{22}-a_{12}a_{21}=0 \end{equation} have the real part: \begin{equation*}\label{37} Re(\mu_{1,2})=-(k_1c_1+k_2c_2)(c_1+c_2); \end{equation*}
(iv) If $b_{11}=\alpha$, $b_{12}=-\beta$, $b_{21}=\beta$, $b_{22}=\alpha$, $\beta\neq 0$, then the Lyapunov coefficient of (SDE) (\ref{3}) is: \begin{equation}\label{38} \begin{split} \lambda\! &=\!-(k_1c_1+k_2c_2)(c_1+c_2)+\displaystyle\frac{1}{2}(\beta^2-\alpha^2)- (k_1c_1-k_2c_2)(c_1+c_2)D_2+\\ &+\displaystyle\frac{1}{2}(k_2-k_1)(c_1^2-c_2^2)E_2 \end{split} \end{equation} where \begin{equation*}\label{39} D_2=\int_0^{2\pi}cos(2\theta)p(\theta)d\theta, E_2=\int_0^{2\pi}sin(2\theta)p(\theta)d\theta \end{equation*} and \begin{equation*}\label{40} \begin{split} & p(\theta)=kg(\theta), k=\displaystyle\frac{1}{\int_0^{2\pi}g(\theta)d\theta},\\ & g(\theta)=\displaystyle\frac{1}{\beta^2}\exp\{\displaystyle\frac{1}{\beta^2}((k_1+k_2)(c_1^2-c_2^2)+\alpha\beta)\theta- (k_1c_1-k_2c_2)(c_1+c_2)\cos(2\theta)+\\ &+\displaystyle\frac{1}{2}(k_1+k_2)(c_1^2-c_2^2)\sin(2\theta)\}. \end{split} \end{equation*}
\end{proposition}
\section{Numerical Simulations.}
\qquad We have done the numerical simulations using a program in Maple 12. For $c_1=0.2$, $c_2=2$, $k_1=0.2$, $k_2=0.4$, $\beta=2$, in figure 1 is displayed $(\alpha, \lambda (\alpha))$, where $\lambda(\alpha)$ is given by (\ref{38}). For $\alpha\in (-\infty, -1.2) \cup (1.1, \infty)$, the Lyapunov exponent is negative, then (SDE) has an asymptotically stable stationary state. For $\alpha\in (-1.2, 1.1)$, the Lyapunov exponent is positive and (SDE) has an asymptotically unstable stationary state.
\begin{center}\begin{tabular}{ccc} \\ Fig 1. $(\alpha, \lambda(\alpha))$\\ \includegraphics[width=5cm]{Fig1.eps}
\end{tabular} \end{center}
If $\beta$ is a real parameter and $\alpha=2$, the figure 2 shows the behavior of the top Lyapunov exponent as a function of $\beta$: $(\beta, \lambda(\beta))$.
\begin{center}\begin{tabular}{ccc} \\ Fig 2. $(\beta, \lambda(\beta))$\\ \includegraphics[width=5cm]{Fig2.eps}
\end{tabular} \end{center}
For $\beta \in (-\infty, -2.6)\cup (2.6,\infty)$ the Lyapunov exponent is positive and (SDE) has an asymptotically unstable stationary state. For $\beta (-2.6, 2.6)$ the Lyapunov exponent is negative and (SDE) has an asymptotically stable stationary state.
The Euler second order scheme for (SDE) (2) is given by: \begin{equation*} \begin{split} &x_1(n+1)=x_1(n)+h\left (\displaystyle\frac{x_2(n)}{(x_1(n)+x_2(n))^2}-c_1\right )+(b_{11}x_1(n)+b_{12}x_2(n)+\gamma_1)\\ &\cdot G(n)+b_{11}(b_{11}x_1(n)+b_{12}x_2(n)+\gamma_1)\displaystyle\frac{G(n)^2-h}{2}+ (-\displaystyle\frac{2x_1(n)x_2(n)}{(x_1(n)+x_2(n))^3}\\ &\cdot\left (\displaystyle\frac{x_2(n)}{(x_1(n)+x_2(n))^2}-c_1\right )+(b_{11}x_1(n)+b_{12}x_2(n)+\gamma_1)\displaystyle\frac{x_1(n)x_2(n)}{(x_1(n)+x_2(n))^3} )\displaystyle\frac{h^2}{2}+\\ &(b_{11}-\displaystyle\frac{2x_2(n)}{(x_1(n)+x_2(n))^3})(b_{11}x_1(n)+b_{12}x_2(n)+\gamma_1)\displaystyle\frac{hG(n)}{2}, \end{split} \end{equation*}
\begin{equation*} \begin{split} &x_2(n+1)=x_2(n)+h\left (\displaystyle\frac{x_1(n)}{(x_1(n)+x_2(n))^2}-c_2\right )+(b_{21}x_1(n)+b_{22}x_2(n)+\gamma_2)\\ &\cdot G(n)+b_{22}(b_{21}x_1(n)+b_{22}x_2(n)+\gamma_2)\displaystyle\frac{G(n)^2-h}{2}+ (-\displaystyle\frac{2x_1(n)x_2(n)}{(x_1(n)+x_2(n))^3}\\ &\cdot\left (\displaystyle\frac{x_1(n)}{(x_1(n)+x_2(n))^2}-c_2\right )+(b_{21}x_1(n)+b_{22}x_2(n)+\gamma_2)\displaystyle\frac{x_1(n)x_2(n)}{(x_1(n)+x_2(n))^3} )\displaystyle\frac{h^2}{2}+\\ &(b_{21}-\displaystyle\frac{2x_1(n)}{(x_1(n)+x_2(n))^3})(b_{21}x_1(n)+b_{22}x_2(n)+\gamma_2)\displaystyle\frac{hG(n)}{2}, \end{split} \end{equation*} where $G(n)=w((n+1)h)-w(nh)$, $n=1,2,...$, and $x_i(n)=x_i(nh,\omega)$,
$i=1,2$.
In figures 3 and 4 are displayed the orbits: $(n, x_1(n, \omega))$ for (SDE) and $(n, x_1(n))$ for (ODE):
\begin{center}\begin{tabular}{ccc} \\ Fig 3. $(n, x_1(n, \omega))$ & Fig 4. $(n, x_1(n))$\\ \includegraphics[width=5cm]{Fig3.eps} & \includegraphics[width=5cm]{Fig4.eps}
\end{tabular} \end{center}
In figures 5 and 6 are displayed the orbits: $(n, x_2(n, \omega))$ for (SDE) and $(n, x_2(n))$ for (ODE):
\begin{center}\begin{tabular}{ccc} \\ Fig 5. $(n, x_2(n, \omega))$ & Fig 6. $(n, x_2(n))$\\ \includegraphics[width=5cm]{Fig5.eps} & \includegraphics[width=5cm]{Fig6.eps}
\end{tabular} \end{center}
In figures 7 and 8 are displayed the orbits: $(x_1(n,\omega), x_2(n, \omega))$ for (SDE) and $(x_1(n), x_2(n))$ for (ODE):
\begin{center}\begin{tabular}{ccc} \\ Fig 7. $(x_1(n,\omega), x_2(n, \omega))$ & Fig 8. $(x_1(n), x_2(n))$\\ \includegraphics[width=5cm]{Fig7.eps} & \includegraphics[width=5cm]{Fig8.eps}
\end{tabular} \end{center}
\section{Conclusions.}
\qquad In the present paper we investigate an economic game with stochastic dynamics. We focus on a particular game and determine the Lyapunov exponent for the stochastic system of equations that describes the mathematical model. The calculation of the top Lyapunov exponent enables us to decide whether a stochastic system is stable or not. Using a program in Maple 12, we display the Lyapunov exponent and the system orbits.
\end{document} | arXiv |
# Data preprocessing for time series
Before diving into the Chirp Z-transform, it's important to understand the basics of data preprocessing for time series analysis. Data preprocessing involves cleaning and transforming raw data into a format that can be easily analyzed and understood.
In the context of time series analysis, data preprocessing typically involves:
- Removing outliers or anomalies: These are data points that are significantly different from the rest of the data.
- Handling missing values: If there are any missing values in the data, you'll need to decide whether to fill them in with interpolated values or remove the corresponding time series data.
- Scaling and normalization: Scaling and normalization help to ensure that all features in the data have the same scale, which can improve the performance of machine learning algorithms.
- Time series decomposition: This involves breaking down the time series data into its constituent components, such as trend, seasonality, and noise.
## Exercise
Instructions:
1. Load a time series dataset from a CSV file.
2. Remove any outliers or anomalies from the dataset.
3. Fill in any missing values using interpolation.
4. Scale and normalize the data.
5. Perform time series decomposition on the data.
Correct answer:
```python
import pandas as pd
from sklearn.preprocessing import StandardScaler
from statsmodels.tsa.seasonal import seasonal_decompose
# Load the time series data
data = pd.read_csv('time_series_data.csv')
# Remove outliers or anomalies
# (assuming the data is in a column named 'value')
Q1 = data['value'].quantile(0.25)
Q3 = data['value'].quantile(0.75)
IQR = Q3 - Q1
data = data[~((data['value'] < (Q1 - 1.5 * IQR)) | (data['value'] > (Q3 + 1.5 * IQR)))]
# Fill in missing values using interpolation
data['value'].interpolate(inplace=True)
# Scale and normalize the data
scaler = StandardScaler()
data['value'] = scaler.fit_transform(data[['value']])
# Perform time series decomposition
decomposition = seasonal_decompose(data['value'], model='multiplicative')
```
# The Chirp Z-transform: definition and properties
The Chirp Z-transform (CZT) is a powerful tool for analyzing time series data. It is a generalization of the Fourier transform that allows for more flexibility in the analysis of time series data.
The CZT is defined as follows:
$$y_n = \frac{1}{2\pi}\int_{-\infty}^{\infty} x(t) e^{-j\omega_n t} dt$$
where:
- $y_n$ is the $n$-th point of the transformed sequence
- $x(t)$ is the original time series data
- $\omega_n$ is the angular frequency of the $n$-th point
The CZT has several important properties:
- It is a linear transformation, which means that the sum or product of two transformed sequences is equal to the sum or product of their original sequences.
- It is a time-invariant transformation, which means that the transformed sequence does not change if the original sequence is shifted in time.
- It is a time-reversible transformation, which means that the transformed sequence can be easily inverted to obtain the original sequence.
## Exercise
Instructions:
1. Implement the Chirp Z-transform in Python.
2. Use the CZT to transform a given time series sequence.
Correct answer:
```python
import numpy as np
def chirp_z_transform(x, N):
y = np.zeros(N)
for n in range(N):
y[n] = np.sum(x * np.exp(-1j * np.pi * n * np.arange(len(x)) / N)) / (2 * np.pi)
return y
# Example usage:
x = np.array([1, 2, 3, 4, 5])
N = len(x)
y = chirp_z_transform(x, N)
```
# Applications of the Chirp Z-transform in time series analysis
The Chirp Z-transform has a wide range of applications in time series analysis, including:
- Frequency analysis: The CZT can be used to determine the dominant frequencies in a time series.
- Spectral analysis: It can be used to analyze the spectral characteristics of a time series, such as the power spectral density.
- Time-frequency analysis: The CZT can be used to visualize the time-frequency relationships in a time series.
- Anomaly detection: The CZT can be used to identify anomalies or outliers in a time series.
- Signal reconstruction: The CZT can be used to reconstruct a time series from its transformed sequence.
These applications can be particularly useful in fields such as signal processing, control systems, and data science.
# Implementing the Chirp Z-transform in Python
To implement the Chirp Z-transform in Python, you can use the following code:
```python
import numpy as np
def chirp_z_transform(x, N):
y = np.zeros(N)
for n in range(N):
y[n] = np.sum(x * np.exp(-1j * np.pi * n * np.arange(len(x)) / N)) / (2 * np.pi)
return y
```
This function takes as input a time series sequence `x` and the desired number of points in the transformed sequence `N`. It returns the transformed sequence `y`.
## Exercise
Instructions:
1. Implement the Chirp Z-transform in Python.
2. Use the CZT to transform a given time series sequence.
Correct answer:
```python
import numpy as np
def chirp_z_transform(x, N):
y = np.zeros(N)
for n in range(N):
y[n] = np.sum(x * np.exp(-1j * np.pi * n * np.arange(len(x)) / N)) / (2 * np.pi)
return y
# Example usage:
x = np.array([1, 2, 3, 4, 5])
N = len(x)
y = chirp_z_transform(x, N)
```
# Creating Python functions for time series analysis
To perform time series analysis using the Chirp Z-transform, you can create Python functions that perform specific tasks, such as:
- Loading time series data from a file or database
- Preprocessing the time series data
- Applying the Chirp Z-transform to the time series data
- Analyzing the transformed data
- Visualizing the results
Here's an example of a function that loads time series data from a CSV file and applies the Chirp Z-transform:
```python
import pandas as pd
def load_and_transform_data(file_path, N):
# Load the time series data
data = pd.read_csv(file_path)
# Preprocess the time series data
# (assuming the data is in a column named 'value')
data['value'] = data['value'].interpolate()
# Apply the Chirp Z-transform
transformed_data = chirp_z_transform(data['value'], N)
return transformed_data
```
# Working with time series data: importing and exporting
When working with time series data, you'll often need to import data from files or databases and export the results of your analysis.
To import time series data from a CSV file, you can use the `pandas` library:
```python
import pandas as pd
data = pd.read_csv('time_series_data.csv')
```
To export the results of your analysis, you can use the `pandas` library to write the data to a new CSV file:
```python
data.to_csv('transformed_time_series_data.csv', index=False)
```
# Analyzing time series data using the Chirp Z-transform
After applying the Chirp Z-transform to your time series data, you can analyze the transformed data to gain insights into the underlying patterns and trends.
Some common analysis techniques include:
- Identifying dominant frequencies or frequency bands
- Calculating the power spectral density
- Visualizing the time-frequency relationships
- Detecting anomalies or outliers
## Exercise
Instructions:
1. Load time series data from a CSV file.
2. Preprocess the data.
3. Apply the Chirp Z-transform to the data.
4. Analyze the transformed data to identify dominant frequencies or frequency bands.
Correct answer:
```python
import pandas as pd
from sklearn.preprocessing import StandardScaler
# Load the time series data
data = pd.read_csv('time_series_data.csv')
# Preprocess the time series data
# (assuming the data is in a column named 'value')
data['value'] = data['value'].interpolate()
# Apply the Chirp Z-transform
N = len(data['value'])
transformed_data = chirp_z_transform(data['value'], N)
# Analyze the transformed data
# (e.g., identify dominant frequencies or frequency bands)
# ...
```
# Visualizing time series data with Python
Visualizing time series data is an essential part of analyzing the data. You can use libraries like `matplotlib` and `seaborn` to create various types of plots, such as line plots, scatter plots, and histograms.
Here's an example of how to create a line plot of the original time series data and its transformed sequence:
```python
import matplotlib.pyplot as plt
# Plot the original time series data
plt.plot(data['value'], label='Original time series')
# Plot the transformed sequence
plt.plot(transformed_data, label='Transformed sequence')
# Add labels and a title
plt.xlabel('Time')
plt.ylabel('Value')
plt.title('Time series data and its transformed sequence')
# Add a legend
plt.legend()
# Show the plot
plt.show()
```
# Advanced topics: multidimensional time series and frequency analysis
In some cases, you may need to work with multidimensional time series data, such as time series data with multiple dimensions or time series data with complex frequency patterns.
To analyze multidimensional time series data, you can use techniques such as:
- Principal component analysis (PCA) to reduce the dimensionality of the data
- Time-frequency analysis to visualize the time-frequency relationships in the data
- Spectral clustering to identify clusters or segments in the data
## Exercise
Instructions:
1. Perform principal component analysis (PCA) on a multidimensional time series dataset.
2. Visualize the time-frequency relationships in the data.
3. Identify clusters or segments in the data.
Correct answer:
```python
from sklearn.decomposition import PCA
from statsmodels.tsa.seasonal import seasonal_decompose
# Perform PCA on the multidimensional time series dataset
pca = PCA(n_components=2)
reduced_data = pca.fit_transform(data)
# Visualize the time-frequency relationships in the data
# (e.g., using the Chirp Z-transform or other techniques)
# ...
# Identify clusters or segments in the data
# (e.g., using spectral clustering or other techniques)
# ...
```
# Case study: real-world applications of the Chirp Z-transform
The Chirp Z-transform has been used in various real-world applications, such as:
- Analyzing the vibration patterns of mechanical systems
- Detecting anomalies in financial time series data
- Studying the frequency content of audio signals
- Investigating the time-frequency relationships in seismic data
By studying these case studies, you can gain a deeper understanding of the practical applications of the Chirp Z-transform and how it can be applied to solve real-world problems. | Textbooks |
This article is about
structured content
LaTeX example
NPR entity diagram
Media embedding in Drupal 8.8
On simplicity & maintainability: CDN module for Drupal 8
Eaton & Urbina: structured, intelligent and adaptive content
Drupal 8: best authoring experience for structured content?
Drupal 8 will ship with big authoring experience improvements: WYSIWYG editing & in-place editing, thanks to the Spark distribution that Acquia — my employer — is sponsoring.
But how well does it fare with the growing importance of structured content? Do Drupal 8's WYSIWYG & in-place editing enable it or prevent it?
The new web world order: many form factors
The Big Thing of the last few years: the advent of mobile. Inherent to that: websites that are optimized for mobile devices and act as data providers for apps.
A new form factor — mobile devices — changed web development forever. Before mobile, the life of web developers and authors (content creators) was relatively simple: make sure websites work well on a few typical screen sizes (let's deny the existence of Internet Explorer 6 and all the misery it caused).
But … we cannot predict what's next. We cannot predict new content consumption form factors. That's where content strategy becomes vitally important:
content strategy is to copywriting as information architecture is to design
We have to make sure that our content is structured and has enough metadata to successfully reuse the same (structured) content for different content consumption form factors. Without having to edit each piece of content again.
Structured content: successfully dealing with form factors
NPR's Create Once, Publish Everywhere is the most often cited example of a content strategy that successfully provides content for many form factors. They create content once, then publish it to >10 different platforms. With a small team, they do more than some other companies, because of their excellent content strategy. It took them years to evolve their systems in this direction, and it paid off.
Another example is TV Guide. They decided back in the 1980s to capture all semantic metadata, to build a database and extracting a magazine from that, rather than just creating a nicely formatted magazine every time. Thanks to that, they're still relevant today.
It appears that the reuse of content is something every website should strive towards. There's nothing inherently bad about it. However, there are downsides.
TV Guide editors used a mainframe application (and maybe still do?). NPR editors use this UI:
NPR editors are encouraged to only think about content, not presentation — hence a very basic data entry UI is all they get 1. This UI looks more like a web front-end to a database than a CMS (anybody else who's reminded of PHPMyAdmin?)…
So, while this may be true:
The goal of any CMS should be to gather enough information to present the content on any platform, in any presentation, at any time.
No CMS really aims to have a poor authoring experience, of course.
Drupal & structured content
Drupal is already well prepared for structured content.
All of the principles that are being used when reviewing code that is being proposed for Drupal core inclusion, are a superset of the principles applied to structured content. Drupal demands full separation of concerns at every level. Everything must be overridable/alterable. Separation of concerns for CSS files, to ensure clean overriding of styling without having to duplicate all CSS. Content may never contain CSS nor depend on CSS. And so on.
Five features in particular stand out with regards to structured content and content reuse:
Structured content: Field API.
It allows content to be modeled as granularly as desired.
Clean content: Filter system.
Ensures fancy mark-up is only added on output, and the stored content is as clean as possible. e.g. the fancy typographic features in this very piece of text is automatically added by Typogrify.
Different presentations of the same content: view modes.
A view mode defines the order of the fields and the field formatter & label of each field. 2
Internal reuse of content (within the website): Views module.
To create lists, grids, tables, galleries etc. of content, while showing related content. A listing can be configured to use a specific view mode.
External reuse of content (outside the website): REST module.
To provide JSON, XML, HAL, JSON-LD, YourCustomMarkupLanguage output.
Drupal authoring experience
Drupal's authoring experience used to be remarkably similar to that of NPR's COPE. We've gone through a lot of effort in Drupal 6, 7 and 8 to improve usability in general. In Drupal 8, the Spark distribution on which I work has specifically targeted the improving of the authoring experience.
Some of the authoring experience improvements in Drupal 8 (in part) thanks to Spark:
two-column backend content editing (with publishing options/meta configuration in a sidebar)
in-place editing for fields
CKEditor-powered WYSIWYG editing
The first is noncontroversial when looking at it from a structured content perspective. It's the second and third that appear to be counter to the premise of structured content — to quote Karen McGrane about WYSIWYG editing:
[…] we allow content creators to embed layout and styling information directly into their content. Unfortunately, the code added by content creators can be at odds with the style sheet, and it's difficult for developers to parse what's style and what's substance. When it comes time to put that content on other platforms, we wind up with a muddled mess.
or Jeff Eaton about in-place editing:
The editing interfaces we offer to users send them important messages, whether we intend it or not. They are affordances, like knobs on doors and buttons on telephones. If the primary editing interface we present is also the visual design seen by site visitors, we are saying: "This page is what you manage! The things you see on it are the true form of your content."
First, let me state that I in fact do not disagree with either of them. We've actually taken that into account while adding WYSIWYG editing and in-place editing to Drupal core. Let me explain how.
WYSIWYG in Drupal 8: enforces clean markup
By default (in the Standard install profile), Drupal 8 will not ship with formatting/layout tools enabled in its WYSIWYG editor (CKEditor).
We make sure in Drupal 8 to prevent crappy markup and format/layout markup (style, font attributes). It's not only impossible to set these kinds of "bad attributes" in the WYSIWYG editor using the toolbar, it's also impossible to paste them in and to use the "source mode" (where you can type HTML directly) to insert them — you can type them in the latter case, but they will be stripped upon going back to WYSIWYG mode from source mode, or upon save if you try to save it without going back to WYSIWYG mode.
This is powered by the new "Advanced Content Filter" feature in CKEditor 4.1, which was added specifically on our request to make this possible.
Furthermore, we made it very easy to configure CKEditor in Drupal 8, yet at the same time very hard to break the above strictness. Only HTML tags and attributes allowed by a specific CKEditor toolbar button will be allowed, even if you add more buttons. So the above "guaranteed clean HTML" will not only be true for the default WYSIWYG configuration, but for any configuration. Drupal 8 will even automatically sync WYSIWYG configuration with filter system configuration:
In the past, configuring WYSIWYG editors was a pain, and in part because of that, the configuration of the WYSIWYG editor and corresponding filter system settings were too permissive.
Finally, we're currently working on making sure that when you insert an image into a piece of text (with or without a WYSIWYG editor), that won't result in the final HTML like <img src="/files/styles/thumbnail/llama.jpg" width="100" height="100" alt="Awesome llama!" />, but instead in a placeholder that the filter system will transform into the final HTML upon output: <img data-file-uuid="aa657593-0da9-42c0-9a05-5d63d27ad27d" data-image-style="thumbnail" />.
In other words: the text should only contain text and programmatic references to other content; the filter system should then handle "upcasting" these into their final form. This will make it much, much easier to upgrade existing content to new image styles, to modify referenced media, to migrate to a new CDN, and whatnot.
WYSIWYG in Drupal 8: from brochureware to newspapers
Drupal needs to cater to both the extreme of very structured content for maximal reuse and to the extreme of unstructured content (where pretty much all data is in a single "blob" called the "body" field, besides maybe a "title" and a "tags" field). It also needs to deal with everything in between.
Drupal may be used for news sites, but also for brochureware sites. By having the WYSIWYG editor be configurable, and hence letting the site builder choose whether formatting/layout tools are available or not, we empower the user to choose.
WYSIWYG in Drupal 8: previews are evil? WYSIWYM to the rescue?
A WYSIWYG editor by definition provides a preview — a best effort preview, that is not guaranteed to be accurate. Providing a preview is not a problem in and of itself, as long as the author knows and understands that the content will be used in multiple contexts, where it will look different.
Of course, reality is that not every author will be sufficiently educated, so we have to take potential abuse into account. Drupal's filter system and very strict WYSIWYG editing in Drupal 8 do precisely that.
What might be even better though, is if we were to make it explicitly visually obvious that the WYSIWYG editor is indeed providing a best-effort preview: visualize the building blocks of the content that the author is using, to make him very aware of the structure of the content that he's creating.
This is what is some people have called WYSIWYM: "What You See Is What You Mean". 3 Wikipedia defines it as follows:
WYSIWYM (an acronym for "what you see is what you mean") is a paradigm for editing a structured document. It is an adjunct to the better-known WYSIWYG (what you see is what you get) paradigm, which displays a formatted document on screen as it will appear in only one mode of presentation.
The main advantage of this system is the total separation of presentation and content: users can structure and write the document once, rather than repeatedly altering it for each mode of presentation, which is left to the export system.
A HTML text editor specifically built for to be a WYSIWYM HTML editor exists: WYMeditor.
WYMeditor's main concept is to leave details of the document's visual layout, and to concentrate on its structure and meaning, while trying to give the user as much comfort as possible (at least as WYSIWYG editors).
You may have tried a full-featured WYSIWYG editor, but you apprehend that your clients use it inappropriately, with the risk it degenerates visually and on the code quality.
You may also have tried the BBcode syntax, Markdown or the wiki-style syntax, but you don't want to force your clients to solutions that are too technical/complex for them, even if it tends to generate good quality code.
The downside of WYMeditor (besides its utilitarian UI and absence of keyboard accessibility) is that it doesn't support the whole range of websites that Drupal needs to support: some people want to do everything in a WYSIWYG editor, and for the simplest websites, that's acceptable. Drupal tries to impose as few choices as possible.
So, ideally, we'd use CKEditor, with a way to turn on a "WYSIWYM mode". The great news: this already exists to a certain extent in the form of its "Show Blocks" plugin! (Which we're already shipping with Drupal core specifically to accomodate this.)
If we find this an acceptable solution, then all we need to do is improve CKEditor's "Show Blocks" plugin!
Of course, this line of reasoning might come across as a superficial solution that isn't a real solution. But let me demonstrate that the core a this pattern has been used for almost 20 years: in the LaTeX world.
WYSIWYM & LaTeX: LyX
I'm sure many of you know LaTeX. It's a "document markup language and document preparation system". It's typically used for writing papers, but also books. 4
LaTeX is based on the philosophy that authors should be able to focus on the content of what they are writing without being distracted by its visual presentation. In preparing a LaTeX document, the author specifies the logical structure using familiar concepts such as chapter, section, table, figure, etc., and lets the LaTeX system worry about the presentation of these structures. It therefore encourages the separation of layout from content while still allowing manual typesetting adjustments where needed.
That really captures the gist of it: authors focus on content, don't think about visual presentation. That's up to "the system" to figure out. Now, here too, it is the domain markup, and complete knowledge of it, that is problematic: the plethora of LaTex commands.
That's why tools like LyX exist. LyX is essentially an easier to use interface to generate LaTeX. It shields the user (mostly) from the rather complex LaTeX markup. It provides a preview of sorts, but one that clearly looks completely different from the end result that LaTeX's typesetting will generate: LyX encourages writing based on structure (WYSIWYM) rather than appearance (WYSIWYG).
If all of the above sounded rather abstract, let's look at an example:
Writing LaTeX: here's a tiny subset of the LaTeX code — see the attached file for more:
In inline formulas it looks like this:
\begin_inset Formula $\lim_{x\rightarrow\infty}f(x)$
\end_inset
Writing LaTeX in Lyx:
The output for both:
LyX' initial release was in 1995. It's still actively being used. Many, many papers have been written it as well as many books.
But … WYSIWYG editors suck!
Sure, WYSIWYG editors sucked… because they allowed for formatting & layout, which Drupal 8's WYSIWYG editing doesn't allow.
We still have work to do to stress the importance of content structure over content presentation — see the WYSIWYM section above. But that can be bolted on top of the solid foundations that we already have.
So, these wonderfully colorful quotes used to be painfully true, but they're not applicable to Drupal 8's WYSIWYG:
WYSIWYG Editors suck because they promote thinking about style rather than content. While content editors are busy changing headings to Comic Sans, pondering the use of a grimacing smiley on their about us page or getting creative with colour, they are not considering the actual copy they are adding to the site.
WYSIWYG Editors suck because as a designer you lose control over big chunks of the design. Anywhere that allows people to enter HTML via an editor allows them to get as creative as they like, using any mark-up that they like. Unless you carefully go through and remove all the creativity that stuff is going to stay there. For developers, even if you switch off most of the buttons, just allowing the administrator to enter simple formatting and links, you still have a situation where a user is entering HTML which you then display on the website. This can enable all kinds of stuff to get into your content, which is then very hard to remove and fundamentally tied to the current design of the site.
In-place editing
In-place editing does not inherently conflict with structured content. In fact, for most things, Drupal's implementation of in-place editing stresses the fact that the content is structured: most structured data is impossible to edit in the same way as it is presented. Only for textual fields, we offer the überfancy "true WYSIWYG in-place editing" capability, where Jeff Eaton's quote from above is most relevant. Even there though, abuse is prevented by the very restrictively configured WYSIWYG editor. For other fields, like taxonomy terms, image fields, boolean fields and so on, we still offer a form-based editing UI while editing in-place, and the danger of letting content presentation prevail is extremely limited.
To a degree, in-place editing can even be useful in increasing awareness of the need for structured content. If the content isn't structured (i.e. one blob of data, for example a "body" field containing all content besides the title), then that becomes immediately and painfully obvious: no specialized, optimized in-place editors appear to edit the particular piece of content; instead you'd have to find your way to the particular thing you want to edit in the body field.
In-place editing in the way we've implemented it encourages structured content.
In our initial implementation of in-place editing, there was more potential for misunderstanding and abuse. But we've made two important changes:
in-place editing is no longer triggered on the page level, but at the entity level: the user must declare his intent to edit a specific entity in-place. So the user can no longer get the impression he's "editing the page": he's explicitly made aware of the type of content (entity type) he's editing (node, taxonomy term, custom block …) and of the field within that piece of content (entity) that he's currently editing (Title, Author, Body, Tag, Image …).
in-place editing is no longer saving each field individually, instead the modified fields for a specific entity are queued up and saved at once, this strengthens the communication to the user that he's editing a singular piece of content that just happens to be rendered on this particular page. (In progress.)
Finally, in-place editing is only designed to be used for quick edits (hence it being triggered by a "Quick edit" action in the contextual links of entities). It's intended to bring a level of "delightful interaction" to editing, instead of being forced to go back to the overwhelming back-end form every single time, even if you don't need to modify metadata.
Education, understanding, awareness of content reuse
It is absolutely essential that authors (content creators) understand the entire flow of the content: from creating it first, using each field for its proper purpose, to the different ways that content might end up in output.
Because in-place editing happens on the output, and output can happen in many ways, in-place editing never allows all the content to be edited: at the very least it is going to be impossible to edit metadata. From that last perspective, it's definitely possible for an author to abuse in-place editing.
We need to provide omnipresent, explicit awareness whenever an author is creating or editing content. Both when editing on the back-end and on the front-end. Low-fidelity, simultaneous previews of the different view modes and preferably on multiple form factors would be the ideal here.
Embedding this explicit awareness is something we still have to achieve for Drupal.5
Data storage in NPR's COPE
We saw NPR's UI earlier in this article. What we didn't see yet, are two fundamentally different ways of storing the data within what is presented as a single field to the end user:
Each paragraph of a single text field is stored as a distinct database record. This also implies that the position of the paragraph needs to be stored. (See the full diagram for details.)
When saving a paragraph, all HTML markup it contains is stored independently: it stores just the text in one database record, and then there is one database record per HTML tag used within that paragraph, which stores the type of tag, the start and end position of that tag within the text, and the attributes for that tag. They call this Markup Addressing:
In essence: extreme database normalization!
Drupal does not yet support this out of the box. The question is whether this is actually necessary? There's a lot of additional overhead to going so far in normalizing data. What is the use case for storing individual paragraphs in separate database records, when many paragraphs are meaningless without the surrounding paragraphs?
The use case for storing the markup separately from the text it was applied to is more clear: to easily facilitate those platforms that don't use HTML markup, and to support changes in markup more easily (e.g. <b> → <strong>). NPR decided against the alternative: storing the markup in the database and filter (strip/transform) it on the way out.
The main gripe Daniel Jacobson had with "filter on output" is based on how he'd seen that implemented before: hard-to-maintain scripts and most systems allowed all markup to be used. However, Drupal already has a mature system to deal with that: its filter system.
Both architectures have downsides. Neither is clearly superior6. Time will tell whether Drupal's data storage approach needs to evolve.
WYSIWYG and in-place editing can clearly be highly problematic when it's implemented like it has been for many websites for about a decade now. For many websites, they have been (ab)used to the extreme point of entire HTML pages being built by a WYSIWYG editor, which has caused consistent inconsistency and utter lack of reuse. Liked by authors at first, until things went bad — or until the next redesign.
The other extreme is a system like NPR's COPE, where it is guaranteed that content is consistent and reusable. At the cost of the authoring experience.
However, I believe that using WYSIWYG editing in a very disciplinary manner combined with a well-defined system for filtering on output and a data model similar to NPR's COPE, can yield equally successful results as NPR's COPE, but with a significantly better authoring experience.
Sources & related reading
http://en.wikipedia.org/wiki/Content_strategy
http://blog.programmableweb.com/2009/10/13/cope-create-once-publish-everywhere/
http://blog.programmableweb.com/2009/10/21/content-modularity-more-than-just-data-normalization/
http://blog.programmableweb.com/2009/11/11/content-portability-building-an-api-is-not-enough/
http://karenmcgrane.com/2013/05/23/drupalcon-keynote-video-and-talk-notes/
https://www.lullabot.com/blog/articles/inline-editing-and-cost-leaky-abstractions
http://alistapart.com/column/wysiwtf
http://www.rachelandrew.co.uk/archives/2011/07/27/your-wysiwyg-editor-sucks/
Both examples are content businesses. The efficient managing and reusing of that content is the whole reason they exist and survive. Hence it is acceptable for them to have a very poor authoring experience. Also: the data model has to be right from the beginning; if something was missing or wrong, it may be impossible to transform old content to the updated data model. Hence there is also an intentional lack of flexibility. ↩︎
Use the Entity View Modes module to create new view modes. ↩︎
Not in the sense that it was discussed at the WYSIWYM BoF at DrupalCon Portland, where it was really about semantic annotation. ↩︎
The whole reason it exists is because somebody got fed up with messing with WYSIWYG editors to get everything just right: the typography, the whitespace, the layout, and so on. Instead, that person wanted to just write the content and have software automatically calculate optimal whitespace, optimal typesetting. ↩︎
The Spark team has already been working on this to a certain extend: the responsive previews patch. However, it is not tightly integrated with editing; neither on back-end nor front-end. ↩︎
Ideally, there would a domain-specific markup (as in, a markup with annotations for the specific knowledge domain of your site) that has more expressive semantics and would then be transformed to HTML when the content gets rendered for web purposes, and to something else than HTML for other purposes. We should explore this.
But at the same time, the threshold would become rather high: which sites, besides those whose primary business is the longevity of their content, the long-term relevance and reusability of their content, will want to invest to build their domain-specific language?
It requires a lot of discipline and research, to come up with a sufficiently expressive domain-specific markup. Precisely because once you've begun expressing content using your domain-specific markup, there is no way back. You cannot automatically enrich existing content with newly added domain-specific markup. The domain-specific markup must be complete before you begin using it.
Not to mention that either the author will need a complete understanding of the complete domain-specific markup as well, because otherwise it will all have been a measure for nothing. Once you enter this realm, it's also very realistic (and human) for authors to forget about a few elements of the domain-specific markup. So then something like a WYSIWYG editor, but with buttons that generate the domain-specific markup could be a great help. This is once again WYSIWYM. ↩︎
Jeff Eaton ∞
Well, I've been called out so I'd best weigh in. ;-)
I won't bang my drum about inline editing beyond what I've already written, but the "body field" WYSIWYG issue is a related one that I've been digging into a lot lately.
First off, I think the approach you're describing is a huge improvement over the "kitchen sink" approach that is often used when exposing WYSIWYG editing functionality. A lot of the pain and suffering inherent in WYSIWYG can be reduced by stripping out the egregiously presentation-oriented markup features like colors and fonts, and taming the dreaded "paste from Word" feature.
We've found (consistently) that the "standard" markup elements like em, strong, blockquote, h1-h6, img, ul, li, and even table are pretty straightforward. As long as people aren't abusing the tables for layout purposes inside the body, and Drupal fields are being used to manage appropriately "chunked" data, we're in pretty solid shape. Sufficiently creative output filtering and CSS can adapt that markup to responsive sites and alternative output channels quite effectively. I like to think of this aspect of the issue as the "formatting" problem.
The challenges come when richer semantic concepts enter the picture: captioned figures, document transclusion, interactive elements like header-oriented collapsible text, inline charts and graphs, etc. Those things almost never correspond to a single simple HTML element, and the underlying semantic meaning may need to be represented with different markup depending on the output channel. I've seen other writers refer to this as the "upstream meaning, downstream markup" divide.
The formatting problem is (IMO) completely solvable using the kinds of techniques you're discussing. The meaning/complex structure problem requires some different approaches, and in many cases the toughest aspects will have to be site-specific. The problem isn't just in visual vs. markup representation, it's the mismatch between HTML's vocabulary and the concepts that need to be expressed.
We're actually in the process of generalizing a couple of the tools we've used on previous projects, and working on ways to smooth the implementation curve for the site/business specific pieces that inevitably arise.
I'll reiterate Karen's comments from the Drupalcon keynote – I have no objections to assistive editors, even ones called 'WYSIWYG!' ;-) A toolbar, visual cues inside the text area instead of "ugly markup," buttons and tools that make the editing process easier… all of these are really important parts of improving the editorial experience. The challenge is figuring out how to do this without recreating the long-term markup reuse problems that have plagued other systems.
Thanks for the hard work in articulating this stuff, and the great UX and development work that's been going into the D8 editing interface!
To clarify: I'm basically agreeing with the domain-specific markup perspective you discuss in the footnotes. ;-) Figuring out how to crack that nut in an editor-friendly fashion is the big issue that we're chewing on at the moment.
By now, we've talked about this in real life, so I think we're indeed on the same page :)
It's interesting and very insightful of you to split the problem in formatting (which is solvable and arguably solved in Drupal 8) and meaning/complex structure. The latter is indeed much, much harder. And would be elegantly solvable using domain-specific markup, for which the technical/financial/educational setup cost would be too high for many sites, unfortunately.
I also like the specific challenges you call out:
The challenges come when richer semantic concepts enter the picture: captioned figures, document transclusion, interactive elements like header-oriented collapsible text, inline charts and graphs, etc. Those things almost never correspond to a single simple HTML element, and the underlying semantic meaning may need to be represented with different markup depending on the output channel.
I believe we have solved captioned figures by shipping the Caption filter with Drupal 8 (which equates to "custom markup": data-caption and data-align attributes) and made using that to caption images — the most common use case — usable for all thanks to a CKEditor Widgets-powered UX. And AFAICT you agree with those claims — let me know if I'm mistaken there :)
I believe that in the case of document transclusion, custom markup (e.g. <drupal:entity type="node" id="345" />) + filter + assistive "WYSIWYG" editing UX is once again the solution. Drupal core could and should provide a built-in solution for that.
The others are less clear cut and different sites may want to use different approaches, but I think that in general the "just write a filter to deal with your custom markup" approach is solid, and as long as you only have to implement a handful of them, it should also be manageable.
I think the big challenge there is to come up with a system of no longer requiring custom filters to be written for each use case plus accompanying custom assistive "WYSIWYG" editor plugins to be written to make the UX nice. It is my hope that Drupal 8 contrib will experiment a lot in that area, so that we hopefully will learn enough by the time we work on Drupal 9 to make that a reality :)
Matthew Oliveira ∞
Great article, really enjoyed it.
I second Eaton's comment about the limitations of WYSIWYG. When you have a simple mapping from what the semantic meaning you as an editor are trying to express to a simple HTML element(s), you're golden. As soon as you want something like an image caption, which doesn't map to single representation in HTML, suddenly the WYSIWYG editor falls apart.
I was encouraged by Nate's talk here: https://portland2013.drupal.org/node/2878 that talked about solving this by hijacking CKEditor's default dialogs with something custom for Drupal, e.g. an image insert dialog that has an option for a caption. Not sure how this is implemented, but it would be good to have some intermediate representation of that image caption, something like Wordpress does with it's short tags API:
[caption id="attachment_120" align="alignleft" width="300"]<a href="http://local.alro.com/wp-content/uploads/2013/05/5953291314_74d8e8b37e_o.jpg"><img src="image.jpg" width="300" height="190" /></a> This is a caption[/caption]
On output, it's filtered into the HTML markup needed, which can change without the content needing to change.
Yep, that shortcode-style approach was the starting point of the mechanism we're leaning on now. My only concern is that it's essentially inventing a parallel markup format inside of the custom markup format.
The approach taken by for image captioning in Drupal 8 – overloading the standard HTML element with data-* attributes – feels like a much more flexible system that could be used in other, similar situations.
Glad you enjoyed it, Matthew! :)
I share your concerns, but this is in fact a solved problem by now — when I wrote the article, that was still a work in progress, but by now it has landed. From my reply above to Jeff Eaton:
I believe we have solved captioned figures by shipping the Caption filter with Drupal 8 (which equates to "custom markup": data-caption and data-align attributes) and made using that to caption images — the most common use case — usable for all thanks to a CKEditor Widgets-powered UX.
That implements the spirit of what you were suggesting, but using a different method, for a reason that Jeff Eaton already pointed out in his reply to your comment:
My only concern is that it's essentially inventing a parallel markup format inside of the custom markup format.
Exactly! That's highly problematic. It makes it unnecessarily different to manage, maintain, massage, transform that content.
The advantages of data-* attributes in comparison are numerous:
simply HTML: parsing & transforming can be implemented much more robustly
much more extensible1
graceful degradation when the output filters are missing
i.e. add not only a data-caption attribute, but also data-source and data-license attributes to an <img />, which would translate into a crazy nested syntax in the [caption …]<img />[/caption] example. ↩︎
Joseph ∞
First of all, I found this post encouraging for the future of Drupal.
Second of all, how do you do footnotes on this site? I like it. Is there a module you're using or something?
Thanks, -Joseph
tenken ∞
? https://drupal.org/project/footnotes
I'm using the Markdown filter :)
Footnotes are a standard Markdown feature.
Larry Garfield ∞
As others have noted, having clean and tidy and semantic HTML is only part of the picture. That assumes an HTML output. "The Web" contains more than HTML now (weird as that sounds). The REST API project I'm on now has both browser-based and non-browser-based clients. For that reason, we're not allowing HTML anywhere but instead planning to use Markdown and ship that straight to the client applications to render to the appropriate format locally. Effectively we're using Markdown in place of a DSL.
Both Wim and Jeff are right that a DSL is the ultimate optimal solution, but hardest. The trick, though, is that a DSL is simply an inline form of chunked data. Fields are (as Wim correctly points out) awesome for chunked data.
That is, in the ideal case… Drupal is your DSL. :-)
Oh, the siren song of custom XML schemas. If we keep talking, someone's inevitably going to say that we should use DITA. ;-)
This is really the heart of the problem, though. Semantically structured HTML, managed with care, can be transformed into other forms but we have to plan for it rather than slathering that on after the fact. I'd also argue that certain techniques (like core tags with data-* attributes to layer additional meaning, or custom HTML5 element types) can get us some of the advantages without going whole-hog XML.
While Drupally field chunking is often a good solution, I don't think we're ever going to overcome the need for some rich content in text fields. Fields capture the fact that a piece of data is associated with an entity, but not where that piece of data lives in the narrative flow of a larger body of text. When that aspect is actually important, we enter the world of semantic editors. ;-)
Bojhan Somers ∞
Whoa, this is quite an interesting read. I am surprised to learn how many tools and considerations we already took to better support structured data. Ever since CCK, I think Drupal has tried to provide structured content. The fact that it can do this, even somewhat from the UI has been a big contributor to its succes.
I personally don't think WYSIWYM, is an answer to this need — it's a way to expose the structure that is applied. It highly depends on education whether content creators can add meaning to that structure. I think it's in the same realms as markup, since it makes the relationship between content and structure more explicit. I think LaTeX has been so successful, because content creators wish to publish their content within a certain system — the presentation is less important to them than conforming to this system. Although this is true for Drupal, the system is much more free in how you express the content.
I think the best solution would be a mix of #6 and more advanced previewing. Currently previewing is largely a "best attempt" because our technology doesn't come close, but there are many ways it can get a lot closer. What we really want is content creators to be able to see their content in different contexts, it should be part of their workflow to preview and adjust/optimise. However there is currently still a disconnect between the places where "chunks" live, and your ability to see that through the creation/editing interfaces. I think Drupal's job would be to keep track of those connections, and provide the ability to see different contexts, devices is really just one of them (as you note, there are view modes, Views and even REST).
IPE, in many ways, brings editing and these different contexts a lot closer — in a way contextual links did this too, but I feel like IPE adds another dimension to it.
The normalisation is quite an interesting approach. I always wondered if the truly future approach isn't more in the realms of machine learning, where (search) tools have a better understanding of meaning in sentences. Currently this requires a lot of data attributes, e.g. Wolfram Alpha to work. But the holy grail is from my point of view in being able to extract meaning not just from phrases/words, but to divide a paragraph and sentence into meaningful parts (objects, prepositions, modifiers etc.) that can be used as chunks elsewhere.
Just philosophising here :)
@eaton It's good to see we are still missing essential parts, I hope plugins are able to capture the more advanced elements. The question is how it maps to the user experience, it's often that these advanced elements come with a heavy set of configuration.
I think LaTeX has been so successful, because content creators wish to publish their content within a certain system — the presentation is less important to them than conforming to this system.
I don't think that's true. It is possible to provide additional metadata or even specific instructions in LaTeX markup for the LaTeX processor, to respectively direct the presentation or specifically control the presentation.
It's a simple fact that most of us don't have the necessary skills to perfectly align every single symbol to yield an optimal reading experience. I know I don't. LaTeX takes those worries away and lets you worry about the content.
Just like you could — in Drupal — add a custom <drupal:entity type="node" id="345" /> HTML tag and write a filter to transform that into something useful, you can write custom commands to accommodate your semantical needs in LaTeX:
\newcommand{name}[num]{definition}
The differences are that LaTeX is oriented towards page output (just like HTML 4) and Drupal's stored HTML is oriented towards stand-alone pieces of content intended for reuse inside and outside of the website. But in theory, I think it's perfectly plausible to transform every single piece of "filtered text field" content in Drupal into LaTeX or vice versa.
Anyway, enough about LaTeX.
I personally don't think WYIWYM, is an answer to this need — it's a way to expose the structure that is applied. It highly depends on education whether content creators can add meaning to that structure. I think it's in the same realms as markup, since it makes the relationship between content and structure more explicit.
I'm not sure what you mean by adding meaning to structure. In the context of what we're talking about, they're the same? You see the structure of your text. You mean to apply a blockquote structure to a piece of text, so that is what you see in WYSIWYM.
You're right that it's in the same realm as markup (and having to know mark-up): you have to know the different concepts. But a big difference is that you don't have to know the syntax anymore. WYSIWYM to me is just about making writing markup a lot easier.
The point is that authors should not think about what the blockquote or heading or paragraph or code sample looks like, but that the thing they're writing is in fact a blockquote, heading, paragraph or code sample: WYSIWYM — meaning over a pretending preview.
The cool thing is that we can offer three ways of content creation in Drupal 8:
WYSIWYG (as is implemented today in Drupal 8)
WYSIWYG + WYSIWYM (by simply enabling the "Show Blocks" plugin that ships with Drupal 8) — you can see this in the screenshot in the article.
WYSIWYM (can be implemented in Drupal 8 by overriding the CKEditor stylesheet to something that styles all content in a monospaced font etc.)
However, using WYSIWYM does not mean that we should abandon previews altogether. It's merely inappropriate (except for brochureware sites) to be editing inside a preview (i.e. WYSIWYG). Previewing in different contexts is indeed very useful.
It's the WYSIWYG expectation — "what you see while editing is precisely what you'll get when viewing" — that is problematic. Hence my proposal to make it visually obvious that it's a best-effort preview… of a single channel/context.
znerol ∞
Thanks for that very interesting writeup. I especially appreciate the look behind the NPR scenes.
On a Drupal 7 newspaper site we allow our editors to put literally anything into articles. Interactive maps, tables, figures (including captions), embedded YouTube movies, etc. However the body field is restricted to a very tight set of HTML tags allowing not much more than structured text. We do not even allow there.
In order to protect the body from text-unrelated markup and still make it possible to insert fancy stuff, we developed a scheme largely based on the Field Collection and Field Injector modules. Instead of inserting the tag directly into the body field (or having that inserted by some plugin), our editors upload the image into a field collection item along with a caption (of course the form is embedded into the node editing form). A simple integer field allows them to choose the paragraph number where the image should be displayed — if the default value is not good enough.
We use the same mechanism to support non-restricted HTML content for special cases. The body text remains the same, even when editors insist on an embedded YouTube movie in the middle of the text.
Because field collection items are entities, we can have bulk operations-enabled administrative Views for them. And when that's not enough, there is still EFQ Entity API.
This mix works out pretty well for us.
Very interesting — thanks for sharing! :)
What you describe is indeed another way to achieve this. However, it seems more restrictive and more brittle to me at first sight: what if there is no paragraph, but only a <blockquote> and a <ul>, for example? Sure, you can accommodate those cases, but it's easy to think of such edge cases.
Furthermore, that does not solve the case of wanting to "inject" things inline (e.g. a link to a node whose title is automatically updated when the node title changes). For such cases, you still need filtering on output.
I do see a broad range of use cases where this will work just fine though :)
Irakli Nadareishvili ∞
Hey Wim, really nice post and thanks for all your efforts in improving Drupal's content authoring experience.
Some thoughts:
Can you make your comments Markdown enabled? :)
What you are doing with images is very similar in its intent with how we ended-up handling rich media assets (audio, video, images etc.) in Public Media Platform. I should be able to share some of that thinking with you soon, in case it's useful. So: stay tuned.
I think that in-place editing, for a content-management system, is evil always and under any circumstances. Any way you try to cover it, it's still extremely tight coupling of presentation with the content, promotes editor's thinking that what they see is what everybody else will see – a perception that is increasingly and completely wrong.
Tight coupling of content with presentation is a large topic and an extremely important one. It's not enough to avoid it at a single content-item level but it should be a cross-cutting concern.
Here's the thing: in this day and age our main concern is not just that content destination is diversified (which is what COPE was addressing years ago) but that content's sources are also highly diversified. For all but the simplest use-cases, it's smart to embrace the notion that: There is no single CMS anymore!
This notion is so important that we had to revise COPE into CAPE to facilitate it: http://bit.ly/capeapi (slides have minimal text in it, but there's full narrative in slide notes).
Bottom-line is: assuming that all content on your website comes from a single CMS is wrong, dead wrong. The reality is that content comes from many sources and it's because of the "traditional" tight coupling of content editing and content presentation that those sources need to unnecessarily go through the "main CMS".
I believe that the future is not in a monolithic CMS. Not even one as customizable as Drupal. I believe that the future of content management lies in an elegant collaboration of loosely coupled Content Tools, each one of which exposes some vertical of content via a web API. Each tool can be written in a completely different language/framework and deployed on separate servers.
Content through those APIs can then be run through a web rendering layer to produce HTML. It's very important to assume that the rendering layer is built in different technology than content tools and deployed separately.
Security: your website must be public, but your content management may need to be behind VPN, deployed separately.
We need to stop thinking of website as something that is allowed to access database systems directly. Web is just one of the many target platforms which we push our content to. Don't make it special! You would never dream about letting an iOS native app access your databases directly: you'd route them through an API. Do the same for your website as well.
These comments are Markdown-enabled! It even says so in Drupal's helpful (yet terribly crappy UX-wise) filter tips right below the commenting <textarea> ;) :)
Glad to hear we're apparently applying the same reasoning — that's usually a good sign :) Any news about
I should be able to share some of that thinking with you soon, in case it's useful. So: stay tuned.
This is very interesting:
Bottom-line is: assuming that all content on your website comes from a single CMS is wrong, dead wrong.
And I think it's indeed increasingly true. But I don't think it applies to everybody. I'm sure it applies to large organizations such as NPR.
I believe that the future of content management lies in an elegant collaboration of loosely coupled Content Tools, each one of which exposes some vertical of content via a web API. Each tool can be written in a completely different language/framework and deployed on separate servers.
I believe you're right — but only for big sites. The part I emphasized is what indicates to me that this no longer applies to smaller websites. For smaller websites, what you describe is simply too technically advanced (at least in the foreseeable future). Each "vertical of content" (I assume that means articles versus videos versus…) gets its own tools, you say — but that means many UIs to learn, many systems to connect, manage and scale.
Everything in my article benefits small sites as well as large ones.
I think Drupal will indeed need to become better at becoming "just" a Content Repository, so that it can become a viable component in the architecture you describe. Maybe Drupal should even be split into two parts — then it could very well meet your two requirements/reasons.
But there will always be a need for an integrated system.
Jeff Noyes ∞
Picking up from our Twitter conversation… A great content authoring experience should have the ability to control how text flows around image. WordPress has this ability, and as you say, maybe their tool creates bad markup, to which I say "oh well". Often there is not a silver bullet in software design, so we have to pick the lesser of two evils. In this case, I think the content author's experience should win over the themer's experience because if the author's experience isn't great, then there won't be anything to theme. Also, there are MANY MANY more content authors than themers.
The author doesn't need font colors, or underline. The theme can take care of that, but they MUST be able to control the flow of content. Authors live in the world of Word, PPT, or similar. Most pick themes to guide their work and won't change that because they know they're not designers. But imagine picking a PPT or Keynote theme, and being forced to place images on the left, or right. Like it or not, content authors will compare Drupal's editing experience to it. You need to get as close as possible.
it's not the author's experience vs. themer's experience. The only "experience" that matters is that of of the reader!
Unfortunately, we see this confusion way too often: developers think they work for authors (mostly because "authors" pay for CMSes… at least when they pay). The reality is: developers, designers, themers and authors all work for the end-user: the reader, listener, watcher. It's either that or they all fail.
When put in that perspective, priorities do change. Readers couldn't care less if author has freedom to place images on the right or left, they care about pleasant and intuitive reading experience.
And: there may be more authors than themers, but there're millions of times more readers than authors :)
A great content authoring experience should have the ability to control how text flows around image. Wordpress has this ability, and as you say, maybe their tool creates bad markup, to which I say oh well. Often there is not a silver bullet in software design, so we have to pick the lessor of two evils.
To be clear, I think a lot of this discussion gets tangled up in the idea of "bad markup," with old-timers fighting off flashbacks of Adobe Pagemill and MS Word HTML output. The concern isn't simple cases like ordered lists, emphasis, or even inline images – the markup for those things has always been quite straightforward. Rather, it's the more complex scenarios like fully captioned images, embedded slideshows, inlined media elements, and structurally-intensive "house styles" like citation and design treatment.
The assertion that "We need to get as close as we can to Word and Powerpoint and Keynote" is an assumption that needs to be examined, IMO. It's no different than someone from 1995 saying that we need an HTML editor that feels like Photoshop, with pixel-perfect drag and drop alignment of every image and text block. Quite a few of those tools were built, in fact, and we learned a lot of painful lessons from them…
Indeed! The idea of "give the user control" vs "not" is, IMO, a false dichotomy. Even if we just consider variable sized screens for the moment (and there are plenty more issues besides that), the question is better phrased:
Do we make the user have to think about the layout anywhere from 2-8 different layout considerations when entering content, or do we automate it so they *don't* have to be graphic designers just to make a page not look like ass on the latest phone?
Most users really don't want to have to think about that, I wager. They may think they want all the controls!, but they change their mind very quickly. And that's still just dealing with the visual layout, before we get into questions of content strategy, reuse, Views, multi-channel publishing, and other complications.
"With great power comes great responsibility", and most content editors, I wager, don't actually want that responsibility. If they did, they'd be designers, not content editors. :-)
The reader's experience is created by the author and the themer. That is to say, the reader's experience will be poor if the author writes poorly or the content flow is choppy. It will also be poor if themer fails at vertical rhythm, contrast, uses hard-to-read fonts, etc. Drupal can only control reader experience by giving the author and themer the right tools, so I think "the reader's experience being the most important" is of little relevance here.
In providing the right tools… A themer may have to hold his nose while working with bad WYSIWYG markup, but they know how to maneuver the cruft, they're motivated by getting paid, and it's set-it-and-forget-it task. In contrast, an author uses this tool to send his reader messages, again, and again, and again. They will only hold their nose if made too. And they do not get paid unless what they're authoring has an impact. To have an impact, the author needs:
writing skills (we can't help here)
a content authoring tools that helps them deliver their message
a theme for ensuring their content looks nice / on brand.
I think you guys are over complicating my point. I'm not saying we need a WYSIWYG that's comparable to having Photoshop spit out HTML, or by making sure the authors experience factors in N number of layouts across different devices. I'm also not saying we need to recreate the Word or PPT experience in Drupal. All I'm saying is that you have to consider the author's point of reference — which often will be Word, PPT or similar. And with that, there is a certain bar that will be expected, and that bar has to include the ability to wrap text around images.
FWIW, I just went through this with a client last week. He didn't know what WYSIWYG meant. He didn't know that he was authoring markup underneath. All he knew was that he wanted his page to look like X — which was a single column layout, had headers, some bold text, and some images with text wrapping around.
Ryan Aslett ∞
All he knew was that he wanted his page to look like X
How did he want his content to look on a mobile device? How did he want his content to look when somebody shared it on Facebook? How did he want his SERP to appear in Google? How did he want his content to perform on a screen reader?
If your client doesn't know what a WYSIWYG is, he probably also doesnt know the importance of COPE/CAPE.
This isn't really a question of a themer simply working with crufty WYSIWYG markup to make it look right to the author in one singular context. The author in a modern publishing world is creating content that has the potential to be presented in myriad ways. Creating an editing environment that encourages the author to think in those terms helps to decouple the idea that they are creating a page that looks like X. And instead fosters the idea that they are creating content that has value in multiple contexts, and sometimes it will look like X, and sometimes Y.
@Jeff,
I hear you more than I am able to express using this tiny comment text-box. I've been in your shoes, working for editors. And then I had a chance to work alongside the editors, designers and product managers for the benefit of users. These things are not the same, not by a long shot.
You are absolutely correct that many editors use MS Word as a point of reference. And to that if they are paying the bill and that pretty much explains the horrible experience most websites deliver. I agree with you 100% in the description of the status quo, but that doesn't make status quo right.
Those editors aren't trained user interface or user experience designers. It's none of their business to make calls whether text is laid out as one column or two columns. They simply aren't qualified for it. Add to that everything Ryan said about mobile/tablet/Facebook/Google Glass view of the content and it becomes abundantly obvious that using Word as the point of reference is the absolute worst thing the editors can do. To be fair, though, not all editors/writers are equal. Many have no designer to pretend being a designer.
The success of writing platforms such as Medium is also very important. How far is Medium from WYSIWYG/MS Word? As far at it gets. And it is intentional. And plenty people love it. There's an important lesson somewhere there.
If you haven't already, it might be worth thinking about whether all of those services are defaults. Should a vast number of users get an editing experience that doesn't map to their mental model because some users also want their content tool look good on Facebook, Google, etc.? Does it make sense to break advanced features into modular components?
For displays, can you specify wrapping conditions for just the two extremes, phone and desktop, and make others extensible by modules? For multi-channel distributions, can you port to one or more modules?
I'm not sure what the answer is, but it feels to me that you're falling into the typical Drupal trap and designing for the community, or thinking so hard about how to scale to the power users that you're possibly neglecting others. Many users want responsive content, multi-distribution channels, e.g. Facebook, Google, Twitter, etc. But many more don't even know how to think like that. Are you designing for the minority and neglecting the majority? What can be done to do both?
The reason many people want or realize they should want multi-channel digital publishing is because there's overwhelmingly abundant data clearly showing desktop-browsing is steadily decreasing in overall web browsing.
I absolutely agree that many don't know how do to it correctly, but if Drupal is to stay competitive it's exactly the reason why Drupal should enable the new way for everybody (current majority as well as current minority) rather than encourage and cater to the dying trend. At least that is what sounds like a reasonable approach to me.
I agree with both of you :) Like I said in the article:
Not everybody cares about every potential way a website could be displayed. For most websites, it's impossible to optimize for every possible channel, especially because channels change and new ones get added over time — to do this well, you almost by definition need a full-time team.
That being said, I think there's a big distinction between optimizing for every channel (which is very expensive) and ensuring your content is well-structured, so that it is future-proof (which comes at only a fraction of the cost). If you do the latter, you're still keeping the former open as an option.
Drupal 8 core does not cover every possible use case — and it can't do that, because there will always be site-specific needs — but it does make a few very common things much more affordable! Plus, by setting a precedent, Drupal developers now have a blueprint of how to solve such challenges.
From Jeff's comments, I think that all he wanted to point out is that Drupal 8's WYSIWYG editor makes it easy to align a display: block image, but not a display: inline image, and that the latter is necessary for text flow control, and a great reader experience.
I think he got much more reactions than he expected :D
That's a fair point, but at least it is now trivial to support that: just add a few lines of CSS.
Funny enough, he's the first to have made a remark about this in all that time :)
harmless ∞
I would be happiest with Markdown in my database instead of HTML. Anyone know how uphill/against the grain that would be? Is that a thing that can happen while retaining most of Drupal 8's content authoring toolkit?
I know CKEditor has some form of Markdown functionality but am unsure if that is compatible with everything D8 is doing with it?
You will of course still be able to use Markdown in D8. But that would indeed mean you lose some of the nice authoring improvements, such as CKEditor, and specifically CKEditor widgets.
That is normal, since Markdown is designed to be written by hand, not to be generated using UI tools. | CommonCrawl |
\begin{document}
\begin{center}
\begin{Huge} \textbf{Polylogarithms for \textsl{GL}\textsubscript{2}\\ over totally real fields} \end{Huge}
Philipp Graf
\textbf{Abstract} \end{center}
We give a new, purely topological construction of Eisenstein cohomology classes for Hilbert-Blumenthal varieties using the polylogarithm for families of topological tori and a decomposition with respect to the units in the center of $GL_2$. These classes are explicitly calculated in de Rham chomology and compared with Harder's Eisenstein classes. For non-trivial coefficient systems the whole Eisenstein cohomology in positive degrees is generated by these topological Eisenstein classes. This gives an alternative proof for the rationality of Harder's Eisenstein operator without using any multiplicity one arguments. The text constitutes my 2016 Regensburg PhD thesis.
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\tableofcontents
\chapter{Introduction and overview}
\section{Eisenstein series and cohomology}\label{G_Eis}
Let $G$ be a reductive linear algebraic group over ${\mathbb Q}$ and ${\mathbb A}$ the adeles over ${\mathbb Q}$. Let us consider the double quotient space \begin{equation*} X_{G,K}:=K\backslash G({\mathbb A})/G({\mathbb Q})A_G({\mathbb R})^0. \end{equation*} Here $A_G$ is the maximal split torus in the center of $G$, $A_G({\mathbb R})^0\subset A_G({\mathbb R})$ is the connected component of the identity, $K=K_\infty K_f$ with $K_f\subset G({\mathbb A}_f)$ compact open and $K_\infty \subset G({\mathbb R})$ maximal compact. The space $X_{G,K}$ has the Borel-Serre compactification $\overline {X_{G,K}}$. It is a manifold with corners, which has a stratification with respect to classes $\left\{P\right\}$ of associate parabolic ${\mathbb Q}$-subgroups of $G$.
A representation $(E,\rho)$ of $G$ in a finite dimensional ${\mathbb Q}$-vector space defines a local system $\tilde E$ on $X_{G,K}$ and one can consider the cohomology groups $H^\bullet(X_{G,K},\tilde E)$ and $H^\bullet(X_{G},\widetilde E):=\varinjlim_{K_f} H^\bullet(X_{G,K},\widetilde E)$. The last cohomology group can be considered as the colimit of the group cohomologies of all arithmetic subgroups $\Gamma\subset G({\mathbb Q})$ with values in $E$. From this point of view it is clear that these cohomology groups are of fundamental arithmetic interest.\\ When coefficients are extended to ${\mathbb C}$, \cite{Fr} showed that the cohomology classes in these groups may be represented by automorphic forms and that one has a direct sum decomposition of the cohomology with respect to classes $\left\{P\right\}$ of associate parabolic ${\mathbb Q}$-subgroups of $G$ as $G({\mathbb A}_f)$-module, whenever $A_G$ acts by a central character on $E$. The summand corresponding to the parabolic $G$ itself is denoted by $H^\bullet _{cusp}(X_G,\widetilde{E\otimes{\mathbb C}})$ and is called the cuspidal cohomology. It is build up by cusp forms, in other words, automorphic forms whose constant terms at parabolic subgroups different from $G$ are zero. The cuspidal cohomology does not contribute to the cohomology of the boundary. The complement to the cuspidal cohomology in $H^\bullet(X_{G},\widetilde {E\otimes {\mathbb C}})$ is build up by Eisenstein series and residues of such.
As this decomposition of the cohomology is obtained by the theory of Eisenstein series and therefore by an analytic summation process, it is not clear whether it respects the natural ${\mathbb Q}$-structure of the cohomology. The ${\mathbb Q}$-rationality of the decomposition has been proven in the case $G=Res_{K/{\mathbb Q}}GL_n$ with $K/{\mathbb Q}$ a number field, see \cite{Fr} Theorem 20 and $\cite{FrSch}$ 4.3. Theorem, using a classification theorem of \cite{J-Sh} of cuspidal automorphic representations of $GL_n({\mathbb A}_K)$.
The ${\mathbb Q}$-rationality of the decomposition above is of great arithmetic interest, as it can be used to derive rationality results for special values of $L$-functions, which may occur as constant terms of Eisenstein series or as integrals of Eisenstein series over cycles, see \cite{Ha3} 3.1 or \cite{Ha1} (4.2.2) and V.
\section{Eisenstein cohomology for Hilbert-Blumenthal varieties}
As the main goal of this thesis is the construction of so called Eisenstein cohomology classes for $G=Res_{F/{\mathbb Q}}GL_2$ with $F$ a totally real number field, we recall Harder's definition of Eisenstein cohomology in this situation in more detail. \cite{Ha1} considers the space $X^\prime _{G,K} :=K\backslash G({\mathbb A})/G({\mathbb Q})Z({\mathbb R})^0$, where $Z\subset G$ is the center and $Z({\mathbb R})^0$ the connected component of the identity, and local systems $\widetilde E$ on $X^\prime _{G,K}$ associated to $G$-representations $(E,\rho)$. The space $X^\prime _{G,K}$ is a disjoint union of so called Hilbert-Blumenthal varieties and also has the Borel-Serre compactification whose boundary is homotopy equivalent to $\partial X^\prime_{G,K}:= K\backslash G({\mathbb A})/B({\mathbb Q})Z({\mathbb R})^0$ with $B\subset G$ the standard Borel subgroup of upper triangular matrices. The natural map $\partial X^\prime_{G,K}\to X^\prime_{G,K}$ induces by pullback a restriction map $\operatorname{res}_K:H^\bullet(X^\prime_{G,K},\widetilde E)\to H^\bullet(\partial X^\prime_{G,K},\widetilde E)$ on cohomology and by passing to the colimit a $G({\mathbb A}_f)$-equivariant map $\operatorname{res}:H^\bullet(X^\prime_{G},\widetilde E)\to H^\bullet(\partial X^\prime_{G},\widetilde E)$. The subspace $H^\bullet _{\operatorname{Eis}}(X^\prime_{G},\widetilde E)\subset H^\bullet(X^\prime_{G},\widetilde E)$, such that $\operatorname{res}:H^\bullet_{\operatorname{Eis}}(X^\prime_{G},\widetilde E)\to \operatorname{im}(\operatorname{res})$ is an isomorphism, is called the Eisenstein cohomology.
Harder determines the Eisenstein cohomology in two steps. First he describes the cohomology of the boundary. It may be understood as a sum of induced modules $\operatorname{Ind}^{G({\mathbb A}_f)}_{B({\mathbb A}_f)}{\mathbb C}\phi_f$, where the $\phi_f$ are the finite components of algebraic Hecke characters $\phi:T({\mathbb A})/T({\mathbb Q})\to {\mathbb C}^*$ and $T$ is the maximal torus in $G$, see \cite{Ha1} Theorem 1. More precisely, if a cohomology class on $\partial X^\prime_{G}$ is represented by a $B({\mathbb Q})$-invariant differential form $\omega$, then $\omega$ is in particular $U({\mathbb Q})$- invariant, where $U\subset B$ is the unipotent radical, and therefore $\omega$ may be developed into a Fourier series with respect to the group $U({\mathbb Q})$. The class $\omega$ is then already determined by the constant term of the differential form $\omega$, this means by its zeroth Fourier coefficient.
As a second step Harder constructs a $G({\mathbb A}_f)$-equivariant operator \begin{equation*} \operatorname{Eis}: \operatorname{im}(\operatorname{res})\otimes {\mathbb C}\to H^\bullet_{\operatorname{Eis}}(X^\prime_{G},\widetilde E\otimes {\mathbb C}), \end{equation*}
which is a section for $\operatorname{res}$. Explicitly it may be described as follows: If $\omega \in\operatorname{im}(\operatorname{res})\otimes {\mathbb C}$ is represented by a $B({\mathbb Q})$-invariant differential form, then $\operatorname{Eis}(\omega):=\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q})}\gamma^*\omega$ where the sum has to be defined by analytic continuation in general. If we have a trivialization $\tilde E\cong {\mathbb C}$, it turns out that $\operatorname{res}:H^{2\xi-1}(X^\prime_{G},{\mathbb C})\to H^{2\xi-1}(\partial X^\prime_{G},{\mathbb C})$ is not surjective, where we set $\xi:=[F:{\mathbb Q}]$.
\section{The topological polylogarithm and associated Eisenstein classes}\label{pol_intro}
The connection between Eisenstein classes and special values of $L$-functions may be seen as motivation to construct ${\mathbb Q}$-rational Eisenstein classes geometrically. One way to do so is to specialize polylogarithms. \cite{Be-L} constructed polylogarithms for relative elliptic curves. In the case of the universal elliptic curve with level-$N$-structure $\mathcal E\to X_{G,K}$, $G=SL_{2}$, $K_f=\ker(SL_2(\hat{\mathbb Z})\to SL_2({\mathbb Z}/N{\mathbb Z}))$, Beilinson proved that the polylogarithm specialized along non-zero $N$-torsion sections actually yields Eisenstein classes for $SL_2$.\\ Following the ideas of Beilinson and \cite{No} we can adapt this construction easily to our topological situation. Given a group $G$ as before with a finite dimensional representation $\psi:G\to Aut(V)$ we may consider the space \begin{equation*} \pi:T_{G,K}:=V(\hat {\mathbb Z})\rtimes KA_G({\mathbb R})^0\backslash V({\mathbb A})\rtimes G({\mathbb A})/V({\mathbb Q})\rtimes G({\mathbb Q})\to X_{G,K} \end{equation*} for $K_f\subset \ker(Aut(V(\hat{\mathbb Z}))\to Aut(V({\mathbb Z}/N{\mathbb Z})))$. It is a group object over $X_{G,K}$ and its fibers are topological tori, in other words, isomorphic to products of $S^1$. One has the pro local system $Log$ associated to the $V({\mathbb Q})\rtimes G({\mathbb Q})$-module $\prod_{k\geq 0}\operatorname{Sym}^kV({\mathbb Q})$, where $V({\mathbb Q})$ acts by multiplication with the exponential series and $G({\mathbb Q})$ via $\psi$. Furthermore, we consider $D\subset T_{G,K}$, which is the union of the images of non-zero $N$-torsion sections with open complement $U$, and the relative orientation bundle $\mu:=\widetilde{\det(V) }$. From the cohomological vanishing properties of $Log$ one easily derives short exact localization sequences \begin{equation*}
0\to H^{dim(V)-1}(U,Log\otimes \pi^{-1}\mu^{n+1})\to H^0(D, \pi_{|D}^{-1}\prod_{k\geq0 }\operatorname{Sym}^k\widetilde{V}\otimes \mu^n)\to H^{0}(X_{G,K},\mu^n)\to 0 \end{equation*}
for all $n\in {\mathbb Z}$. Given a section $f\in H^0(D, \pi_{|D}^{-1}\prod_{k\geq0 }\operatorname{Sym}^k\widetilde{V}\otimes \mu^n)$ mapping to zero on the right-hand side we get a unique cohomology class called the polylogarithm associated to $f$ \begin{equation*} \operatorname{pol}(f)\in H^{dim(V)-1}(U,Log\otimes \pi^{-1}\mu^{n+1}). \end{equation*} This class may be specialized along the zero section to obtain polylogarithmic Eisenstein classes \begin{equation*} (\operatorname{Eis}^k(f))_{k\geq 0}:=0^*pol(f)\in \prod_{k\geq 0}H^{dim(V)-1}(X_{G,K},\operatorname{Sym}^k\widetilde{V}\otimes\mu^{n+1}). \end{equation*} If we have $n=0$, typical examples for such $f$ may be given by functions $f:V({\mathbb Z}/N{\mathbb Z})\to {\mathbb Q}$ with $f(0)=\sum_{v\in V({\mathbb Z}/N{\mathbb Z})}f(v)=0$. By the naturality properties of the polylogarithm these polylogarithmic Eisenstein classes glue to $G({\mathbb A}_f)$-equivariant operators \begin{equation*} \operatorname{Eis}^k:\mathcal S(V({\mathbb A}_f),{\mathbb Q})^0\otimes H^0(X_G,\mu^{n})\to H^{dim(V)-1}(X_{G},\operatorname{Sym}^k\widetilde{V}\otimes\mu^{n+1}), \end{equation*} where $\mathcal S(V({\mathbb A}_f),{\mathbb Q})^0$ are the ${\mathbb Q}$-valued Schwartz-Bruhat functions with $\int_{V({\mathbb A}_f)}f(v)dv=f(0)=0$. \cite{L} and \cite{No} represented the polylogarithm cohomology classes by explicit currents in the cohomology with ${\mathbb C}$-coefficients. The polylogarithmic Eisenstein classes are then represented by Eisenstein-Kronecker series. With this explicit description \cite{Bl2} and \cite{Ki1} proved in the situation $G=Res_{F/{\mathbb Q}}SL_2$ with $F/{\mathbb Q}$ a totally real number field, that again the polylogarithmic Eisenstein classes are non-trivial Eisenstein classes, as their constant terms turned out to be special values of partial $L$-functions associated to the field $F$. In this way they also proved that these special $L$-values have to be rational numbers, a result which already goes back to Siegel.
Here we only addressed the topological realization of the polylogarithm. Indeed, by \cite{Ki2} the polylogarithm for abelian schemes is of motivic origin and a powerful tool to tackle deep arithmetic problems and conjectures. See for example \cite{Ki3}, where the \'{e}tale elliptic polylogarithm is a decisive instrument for the proof of the Tamagawa number conjecture for CM elliptic curves.
\section{Decomposition of Eisenstein classes}\label{Eis_intro}
Even though the polylogarithmic Eisenstein classes are a mighty tool in arithmetic, there is one obvious flaw: They are all stuck in cohomological degree $dim(V)-1$.
In this work we present a decomposition principle for polylogarithmic Eisenstein classes in the case $G=Res_{F/{\mathbb Q}}GL_2$, $V=Res_{F/K}\mathbb{G}_a^2$ and $F/{\mathbb Q}$ a totally real number field. \\ The idea is very easy. We may see $X_{G,K}$ as a fiber bundle \begin{equation*} \varphi:X_{G,K}\to X^\prime _{G,K}= K\backslash G({\mathbb A})/G({\mathbb Q})Z({\mathbb R})^0, \end{equation*} where again $X^\prime_{G,K}$ is a space considered by \cite{Ha1} in his landmark paper. The fiber is $\varphi^{-1}(1)=(A_G({\mathbb R})^0K_\infty \cap Z({\mathbb R})) \backslash Z({\mathbb R})/Z_K$ and $Z_K:=Z({\mathbb Q})\cap K_f\subset \mathcal O_F ^{\times}$ is a subgroup of finite index of the units of $\mathcal O_F\subset F$ the ring of integers. By Dirichlet's unit theorem we know that the fiber is compact and that we have for the cohomology \begin{equation*} H^\bullet(\varphi^{-1}(1),{\mathbb Q})=H^\bullet(Z_K,{\mathbb Q})=H^\bullet(\mathcal O_F ^{\times},{\mathbb Q})=\bigwedge^\bullet Hom(\mathcal O_F ^{\times},{\mathbb Q}). \end{equation*} Using the coordinate on $G({\mathbb R})$ coming from the determinant we see that $\varphi$ is up to a finite covering a trivial bundle. So one has cohomology classes in $H^\bullet (X_{G,K},{\mathbb Q})$ restricting to a basis of the cohomology of all fibers of $\varphi$ and this gives a Leray-Hirsch isomorphism \begin{equation*} H^\bullet(\varphi^{-1}(1),{\mathbb Q})\otimes H^\bullet(X^\prime _{G,K},\varphi_*\operatorname{Sym}^k\widetilde{V}\otimes \mu^{n+1}) \to H^\bullet(X_{G,K},\operatorname{Sym}^k\widetilde{V}\otimes \mu^{n+1}). \end{equation*} The polylogarithmic Eisenstein classes may then be decomposed with respect to this isomorphism. We get by evaluation \begin{equation*} \mathcal S(V({\mathbb A}_f),{\mathbb Q})^0\otimes H^0(X_G,\mu^{n})\otimes H^q(\varphi^{-1}(1),{\mathbb Q})^{*}\stackrel{\operatorname{Eis}^k_q}{\rightarrow} H^{dim(V)-1-q}(X_{G}^\prime,\varphi_*\operatorname{Sym}^k\widetilde{V}\otimes\mu^{n+1}) \end{equation*} for $q=0,...,\xi-1=dim(\varphi^{-1}(1))$.
We want to show that this decomposition of polylogarithmic Eisenstein classes is as non-trivial as possible. In other words, we want to get as many of Harder's Eisenstein cohomology classes as possible. To do so we follow \cite{L} and \cite{No} to represent the polylogarithm by a current and hence the polylogarithmic Eisenstein classes by differential forms in de Rham cohomology. The decomposition isomorphism will be made explicit by fiber integration on the level of de Rham cohomology. This gives the decomposed polylogarithmic Eisenstein classes as Mellin transforms of theta series as considered by \cite{Wi}.
To determine the image of our polylogarithmic Eisenstein classes we calculate the constant terms, in other words, the restriction to the cohomology of the boundary. We already have mentioned that we have thanks to \cite{Ha1} a complete understanding of the cohomology of the boundary of $\overline{X^\prime_{G,K}}$ as a sum of induced modules $\operatorname{Ind}^{G({\mathbb A}_f)}_{B({\mathbb A}_f)}{\mathbb C}\phi_f$. The precise relation between $f$ and the constant term of the polylogarithmic Eisenstein class $\operatorname{Eis}^k(f)$ is then controlled by the horospherical map \begin{equation*} \rho:\mathcal S(V({\mathbb A}_f),{\mathbb Q})^0\otimes H^0(X_G,\mu^{n})\to \bigoplus \operatorname{Ind}^{G({\mathbb A}_f)}_{B({\mathbb A}_f)}{\mathbb C}\phi_f. \end{equation*} The horospherical map gives also the relation to special values of $L$-functions.
To understand the image of our polylogarithmic Eisenstein classes we have to determine which of the induced functions above may be realized by the horospherical map.
Our main result is as follows \begin{thm} The operators $\operatorname{Eis}^k_q$ factor through the Eisenstein cohomology. The image of $\operatorname{Eis}^k_q$ is the whole Eisenstein cohomology in degree $2\xi-1-q$, if $k>0$. If $k=0$, $\operatorname{Eis}^0_0$ generates the Eisenstein cohomology in degree $2\xi-1$, for $q>0$ we get all Eisenstein classes but not those associated to spherical functions (see \cref{spherical}). \end{thm} We want to remark a few things. \begin{enumerate} \item The spherical functions are exactly those functions which are not in the image of the restriction map to the cohomology of the boundary in degree $2\xi-1$. As the $\operatorname{Eis}^k_q(f)$ have the same constant term for all $q$, we cannot obtain the Eisenstein classes associated to spherical functions in cohomological degrees $\xi,...,2\xi-1$.
\item The Eisenstein cohomology of $X^\prime _{G,K}$ is supported in cohomological degrees $0$ and $\xi,...,2\xi-1$, so that our Eisenstein operator actually generates most of the relevant part.
\item Even though Harder started with more general coefficient systems than we do, see \cite{Ha1} 1.4, we get all Eisenstein cohomology classes for non-trivial representations. The reason is that after extending coefficients to $\overline {\mathbb Q}$ his representations are direct summands in $\operatorname{Sym}^k V\otimes \det(V)^n$, $k\geq1$, and all weights $\phi_{|T({\mathbb R})^0}$ occurring in the cohomology of the boundary have to factor through the norm character, see \cite{Ha1} 2.8.
\item We even get a much finer result. The sheaf $Log$ has an integral structure which allows us to define our polylogarithmic Eisenstein classes over the Ring ${\mathbb Z}[\frac{1}{N}]$, when we consider $X_{G,K}$ and $K_f$ defines the level-$N$-structure. From this we also deduce integrality results for special values of partial $L$-functions for totally real fields.
\item However, it seems as if one cannot expect that the polylogarithm always generates much of the Eisenstein cohomology. To demonstrate this problem look at $F={\mathbb Q}(\sqrt{-1})$ and $G=Res_{F/{\mathbb Q}}GL_2$. One would naively choose $V=Res_{F/{\mathbb Q}}\mathbb{G}_a^2$. We have $dim(V)=4$ and this gives polylogarithmic Eisenstein classes in cohomological degree $3$. Now we have $X_{G,K}=X^\prime_{G,K}$, since $A_{G}({\mathbb R})^0K_\infty={\mathbb R}_{>0}U(2)={\mathbb C}^{\times}U(2)=Z({\mathbb R})^0K_\infty$ and the part of $Z({\mathbb R})$, which does not split over ${\mathbb R}$, already lies in the maximal compact subgroup. So there is no decomposition and $H^3(X_{G,K},\widetilde{E})=0$ by Poincar\'{e} duality. In this case the polylogarithm gives nothing. Nevertheless, we hope for applications of the polylogarithm to other groups. Especially the case of $G$ different from $GL_{n,F}$ with $F$ a number field would be interesting, as one does not know the ${\mathbb Q}$-rationality of the decomposition of the cohomology in this case, see \cref{G_Eis}.
\end{enumerate} \section{Outline of the thesis}
This thesis has three main parts. We want to discuss them here in more detail.
\subsection{The topological polylogarithm}
In the first part we define the general geometric setup. We begin with the definition of a family of topological tori over a manifold $S$ as a proper submersion $\pi:T\to S$ with connected fibers, which is a commutative group object over $S$. If $S$ is connected one identifies the category of families of topological tori with the category of finitely generated free abelian groups $L$ with $\pi_1(S)$-action (\Cref{tori}). This is done by associating to $L$ the quotient-torus $(L\otimes {\mathbb R}/L)\times_{\pi_1(S)}\widetilde S $. Here $\widetilde S$ is the universal cover of $S$ where $\pi_1(S)$ acts on by deck transformations and $\pi_1(S)$ acts on the left factor by the module structure. So the torus $T$ is determined by a representation $\pi_1(S)\to Aut(L)$, which we call the representation associated to $T$.
This allows us to define locally constant sheaves on $T$ by using $L\rtimes \pi_1(S)$-modules. Let $A$ be any noetherian ring. $L\rtimes \pi_1(S)$ acts on $L$ by affine transformations. This makes the group ring $A[L]$ a $L\rtimes \pi_1(S)$-module. The augmentation $A[L]\stackrel{\operatorname{aug}}{\rightarrow} A$ is equivariant making the augmentation Ideal $\mathfrak a:=\ker(\operatorname{aug})$ a $L\rtimes \pi_1(S)$-module. Consequently, $A[[L]]:=\varprojlim_{n\in {\mathbb N}_0}A[L]/\mathfrak a ^{n+1}$ is a $L\rtimes \pi_1(S)$-module and the associated locally constant sheaf on $T$ is called the logarithm sheaf $Log$.
Next we deduce a purity result for projective systems of local systems indexed over the natural numbers. If $i:D\to T$ is a closed submanifold of codimension $c$ of our torus we get $Ri^!Log=i^{-1}Log\otimes or_{D/T}[-c]$ (\Cref{lim_purity}), where $or_{D/T}$ is the relative orientation bundle. This gives a natural localization triangle \begin{equation*} i_*i^{-1}Log\otimes or_{D/T}[-c]\to Log\to j_*j^{-1}Log\stackrel{+1}{\rightarrow}, \end{equation*} where $j:U\to T$ is the inclusion of the open complement of $D$ (\Cref{localization_triangle}). We calculate the well known higher right derived images of $Log$ \begin{equation*} R^p\pi_*(Log)=0, p\neq d,\ R^d\pi_*(Log)\stackrel{\operatorname{aug} \cong}{\rightarrow}R^d\pi_*(A), \end{equation*} where $d$ is the fiber dimension of $\pi$. From this we deduce the localization sequence \begin{equation*} 0\to H^{d-1}(U,Log\otimes or_{T/S}^{n+1})\to H^0(D,i^{-1}Log\otimes or_{T/S}^{n})\stackrel{\operatorname{aug}}{\rightarrow} H^0(S,or_{T/S}^{n}), \end{equation*} when $\pi:D\to S$ is a finite cover (\Cref{global_Log_localization}).
To profit from the localization sequence, this means to get non-trivial polylogarithmic classes, we need a good understanding of $H^0(D,i^{-1}Log\otimes or_{T/S}^{n})$. Following \cite{BKL} we show that there is a section $A\to i^{-1}Log$ of $\operatorname{aug}$, if $D$ is the union of the images of torsion sections whose order is invertible in $A$ (\Cref{trivializationLog}). This gives us the definition of $\operatorname{pol}(f)\in H^{d-1}(U,Log\otimes or_{T/S}) $ for locally constant functions $f:D\to A$ with $\operatorname{aug}(f)=0$, as we have $H^0(D,A)\subset H^0(D,i^{-1}Log)$ (\Cref{pol}).
For computational purposes we also need a trivialization of the pro vector bundle associated to $Log$. It comes from the isomorphism \begin{equation*} {\mathbb Q}[[L]]\to \prod_{k\geq0}\operatorname{Sym}^k(L\otimes {\mathbb Q}),l\mapsto \exp(l), \end{equation*} which gives the nowhere vanishing $\mathcal C^\infty$-section $v\mapsto \exp(-v)$, $v\in L\otimes{\mathbb R}$, and the relation of $Log$ to the symmetric powers of the representation $\pi_1(S)\to Aut(L)$ associated to $T$: $0^{-1}Log=\prod_{k\geq0}\operatorname{Sym}^k\widetilde{(L\otimes {\mathbb Q})}$ (\Cref{cont_triv}), where $0:S\to T$ is the zero section of the group object $T$.
Moreover, we show that the localization sequence of $Log$ may be calculated by resolving the logarithm sheaf by non-continuous functionals. This gives a characterization of the polylogarithms by differential equations (\Cref{{differential_equ}}).
\subsection{Polylogarithmic Eisenstein classes for Hilbert-Blumenthal varieties}
First we recall the geometric situation $\varphi:\mathcal M_K:=X_{G,K}\to \mathcal S_K:=X^\prime_{G,K}$, when we have $G=Res_{F/{\mathbb Q}}GL_2$ for a totally real field $F$. Let us keep the notation already fixed in \Cref{pol_intro} and \Cref{Eis_intro}. We construct a torus over $\mathcal M_K$ using the standard representation $G\to Aut(V)$, $V=Res_{F/{\mathbb Q}}\mathbb{G}_a^2$. This allows us to define polylogarithms and polylogarithmic Eisenstein classes for $\mathcal M_K$. We use the naturality properties of $Log$ and the localization sequence to show that we can glue our Eisenstein classes to $G({\mathbb A}_f)$-equivariant operators as described in \Cref{pol_intro} (\Cref{Eis^k}).
As we have the polylogarithmic Eisenstein classes on $\mathcal M_K$, we want to decompose the cohomology. The proof is divided up into several parts.
First we calculate the higher direct images of $\operatorname{Sym}^k\widetilde V\otimes \mu^{n+1}$. As $\varphi$ is a fiber bundle it suffices to understand the cohomology of the fibers, which we calculate using group cohomology $H^\bullet(Z_K,\operatorname{Sym}^k V\otimes \det(V)^{n+1})$. As the action of $Z_K$ is semi-simple we easily calculate these groups as $H^0(Z_K,\operatorname{Sym}^k V\otimes \det(V)^{n+1})\otimes H^\bullet(Z_K,{\mathbb Q})$ (\Cref{{cohomology_abelian_group}}).
This gives $R^\bullet\varphi_*(\operatorname{Sym}^k\widetilde V\otimes \mu^{n+1})=\varphi_*\operatorname{Sym}^k\widetilde V\otimes \mu^{n+1}\otimes R^\bullet\varphi_*({\mathbb Q})$ (\Cref{fiber_cohom}) and we trivialize $R^\bullet\varphi_*({\mathbb Q})$ by forms coming from global classes $\mathfrak H^\bullet \subset H^\bullet (\mathcal M_K,{\mathbb Q})$ using the fact that $\varphi$ is up to a finite cover a trivial fiber bundle (\Cref{global_classes}).
These global classes define then by cup-product the decomposition isomorphism (\Cref{decomp}) $H^\bullet(\mathcal S_K,\varphi_*\operatorname{Sym}^k\widetilde V\otimes \mu^{n+1})\otimes \mathfrak H^\bullet\stackrel{\cup}{\rightarrow} H^\bullet(\mathcal M_K,\operatorname{Sym}^k\widetilde V\otimes \mu^{n+1})$, which we finally discuss in the setting of de Rham cohomology with the theory of fiber integration (\Cref{trace_integration}).
\subsection{Comparison with Harder's Eisenstein classes}
In the last part we represent the polylogarithmic Eisenstein classes by differential forms and compare them with those of Harder.
Using the ideas of \cite{No} we represent the polylogarithms explicitly by currents describing the Eisenstein classes as differential forms. We give the description in adelic coordinates and calculate the decomposition of the polylogarithmic Eisenstein classes by fiber integration (\Cref{decomp_Eis}). We reinterpret this as a Mellin transform of a theta series as considered by \cite{Wi} (\Cref{Wi-series}).
As Harder's Eisenstein classes are determined by their restriction to the cohomology of the boundary, we recall Harder's calculation of the cohomology of the boundary of $\mathcal S_K$ (\Cref{boundary}). We restrict the polylogarithmic Eisenstein classes to the boundary where they are determined by their constant terms (\Cref{boundary_residueI}) and derive rationality and integrality results for these constant terms (\Cref{integral_L_value}). Then we define the horospherical map controlling the relation between $f$ and the constant term of $\operatorname{Eis}^k(f)$ and explain how the horospherical map determines the image of our Eisenstein operators (\Cref{hor}).
Next we translate our cohomology classes to $(\mathfrak g,K)$-cohomology and compare them there with Harder's Eisenstein operator. We see that the polylogarithmic Eisenstein classes are in the image of Harder's Eisenstein operator and therefore our operators $\operatorname{Eis}^k_q$ actually factor through the Eisenstein cohomology (\Cref{comparison}).
Finally, we determine the image of our operators $\operatorname{Eis}^k_q$ by studying the horospherical map. The main ingredient is to show that the horospherical map is surjective, when one allows general Schwartz-Bruhat functions, this means functions $f$ where we do not necessarily have $\int_{V({\mathbb A}_f)}f(v)dv=f(0)=0$ (\Cref{hor_surjective}).
\section{Acknowledgments}
I would like to thank my advisor Guido Kings for giving me the opportunity to work on his beautiful idea to decompose the polylogarithmic Eisenstein classes for $Res_{F/{\mathbb Q}}GL_2$ by using the units in the center. His encouragement was decisive for the success of this work and I want to thank him for everything I have learned during my studies.
\chapter{The topological polylogarithm}
\section{The logarithm sheaf on families of topological tori}
We want to start with the definition and construction of the logarithm sheaf on families of topological tori. These will be our main geometric objects. As we want to calculate cohomology classes on them explicitly, we need practical and explicit descriptions for locally constant sheaves on them. This is achieved by the theory of representations of the fundamental group and equivariant sheaves on the universal cover. \subsection{Families of topological tori}
For any site $C$ we denote by $Sh(C)$ the category of abelian sheaves on $C$. For a $\mathcal C^\infty$-manifold $\mathcal S$ we denote by $Mfd/\mathcal S$ the category of manifolds over $\mathcal S$. \begin{definition} Let $\pi: \mathcal T\rightarrow \mathcal S$ be a proper submersion of $\mathcal C^\infty$-manifolds. We call $\pi: \mathcal T\rightarrow \mathcal S$ a \textit{family of topological tori} or simply a \textit{torus over $\mathcal S$}, if it is a commutative group object in $Mfd/\mathcal S$ with connected fibers. Families of topological tori together with group homomorphisms form a subcategory of $Mfd/\mathcal S$, which we denote by $Tori/\mathcal S$. \end{definition} \begin{remark}
As the fibers of $\pi$ are compact commutative Lie groups, they are topological tori, in other words products of $S^1:=\left\{z\in {\mathbb C}:|z|=1\right\}$. \end{remark} \begin{remark} We will always consider $Mfd/\mathcal S$ with the usual topology of open covers. It follows that we have a Yoneda embedding \begin{equation*} Tori/\mathcal S\rightarrow Sh(Mfd/\mathcal S),\ \mathcal T \mapsto \left\{X\mapsto \mathcal T(X)=Hom_{Mfd/\mathcal S}(X,\mathcal T)\right\} \end{equation*} \end{remark} Let us fix a base point $s_0\in \mathcal S$ and let us take $t_0:=0(s_0)\in \mathcal T$ as base point for $\mathcal T$, where $0:\mathcal S\rightarrow \mathcal T$ is the zero section. If $\mathcal S$ is not connected, we choose a base point for any connected component and do all our constructions for each connected component separately. Any proper submersion of $\mathcal C^\infty$-manifolds is a fiber bundle, see \cite{E}. From this we deduce a short exact sequence \begin{equation*} 0\rightarrow \pi_1(\mathcal T_{s_0},t_0)\stackrel{i_*}{\rightarrow} \pi_1(\mathcal T,t_0)\stackrel{\pi_*}{\rightarrow} \pi_1(\mathcal S,s_0)\rightarrow 0 , \end{equation*} where as usual $\mathcal T_{s_0}:=\pi^{-1}(s_0)$ denotes the fiber over $s_0$. This sequence is split exact, since $\pi_*$ has the section $0_*$, and we have a natural left $\pi_1(\mathcal S,s_0)$-action on $\pi_1(\mathcal T_{s_0},t_0)=H_1(\pi^{-1}(s_0),{\mathbb Z})$ given by conjugation, as the latter group is commutative. In other words, there is an isomorphism \begin{equation*}
\pi_1(\mathcal T_{s_0},t_0)\rtimes \pi_1(\mathcal S,s_0)\cong \pi_1(\mathcal T,t_0),\ (l,g)\mapsto i_*(l)0_*(g),
\end{equation*} where the structure of the semidirect product is determined by $0$ and the by the action above. \\ We are going to construct locally constant sheaves on $\mathcal T$, which are induced by $\pi_1(\mathcal T,t_0)$-modules. The functor $\mathcal F\mapsto \mathcal F_{t_0}$, which assigns to a locally constant sheaf $\mathcal F$ its stalk at the base point, establishes an equivalence between the category of locally constant sheaves on a connected topological manifold $\mathcal T$ and the category of left-$\pi_1(\mathcal T,t_0)$-modules, see \cite{Iv} IV Theorem 9.7. Note that we consider the universal cover $\widetilde{\mathcal T}$ of $\mathcal T$ equipped with a right $\pi_1(\mathcal T,t_0)$-action. \begin{example} If $\pi:\mathcal T\to \mathcal S$ is a torus, we have the locally constant sheaf $\mathcal H:=\underline{Hom}_{\mathbb Z}(R^1\pi_*({\mathbb Z}),{\mathbb Z})$ and $\mathcal H_{s_0}=H_1(\mathcal T_{s_0},{\mathbb Z})$ by Poincar\'{e} duality. \end{example} \begin{definition} A \textit{lattice $L$} is a free ${\mathbb Z}$-module of finite rank. Let $A\to A^\prime $ be a ring homomorphism and $M$ a $A$-module. We set $M_{A^\prime}:=M\otimes_A A^\prime$. \end{definition}
\begin{proposition}\label{tori} Let $\mathcal S$ be a connected $\mathcal C^\infty$-manifold, say with base point $s_0\in \mathcal S$. Then we have an equivalence of categories \begin{equation*} Tori/\mathcal S\rightarrow \left\{\text{lattices $L$ with left-$\pi_1(\mathcal S,s_0)$-action and equivariant maps}\right\} \end{equation*} \begin{equation*} \mathcal T\mapsto H_1(\mathcal T_{s_0},{\mathbb Z}) \end{equation*} \begin{proof} Given a $\pi_1(\mathcal S,s_0)$-module $L$ we have the torus $T:=L_{\mathbb R}/L$. As $\pi_1(\mathcal S,s_0)$ acts on $L$, we get an induced action on $T$. We denote by $\tilde{\mathcal S}$ the universal cover of $\mathcal S$ and set $\mathcal T_{L}:=\tilde{\mathcal S}\times_{\pi_1(\mathcal S,s_0)}T$, where $\tilde{\mathcal S}\times_{\pi_1(\mathcal S,s_0)}T:= \tilde{\mathcal S}\times T/\pi_1(\mathcal S,s_0)$ and $\gamma\in \pi_1(\mathcal S,s_0)$ acts on $(s,t)\in\tilde{\mathcal S}\times T$ by $(s,t)\gamma=(s\gamma,\gamma^{-1}t)$. We have a structure map $\pi:\mathcal T_{L}\rightarrow \mathcal S$ coming from the first projection. The action of $\pi_1(\mathcal S,s_0)$ on $\tilde{\mathcal S}$ is properly discontinuous and fixpoint free. So $\tilde{\mathcal S}$ may be covered by $\tilde{U}\subset \tilde{\mathcal S}$ open where $\tilde{U}\gamma\cap \tilde{U}\neq \emptyset$ implies $\gamma=\operatorname{id}$. If we denote by $U$ the image of $\tilde{U}$ in $\mathcal S$ under the canonical map, we get $\pi^{-1}(U)\cong U\times T$. So our quotient space is a fiber bundle with typical fiber $T$ and $\pi$ is proper. Since the action is by diffeomorphisms, $\mathcal T_{L}$ is even a manifold.\\ We endow $pr_1:U\times T\rightarrow U$ with the constant group structure as a group object in manifolds over $U$. By construction all transition maps of our fiber bundle come from the $\pi_1(\mathcal S,s_0)$-action on $T$. So all transition maps are group homomorphisms and this means that we may glue all group structures $pr_1:U\times T\rightarrow U$ to obtain a global group structure on $\mathcal T_{L}$. Obviously, equivariant maps $f:L\rightarrow L^\prime$ induce maps on the corresponding spaces. This gives the desired quasi-inverse functor \begin{equation*} \left\{\text{lattices $L$ with left-$\pi_1(\mathcal S,s_0)$-action and equivariant maps}\right\}\to Tori/\mathcal S \end{equation*} \begin{equation*} L\mapsto \tilde\mathcal S\times_{\pi_1(\mathcal S,s_0)} L_{\mathbb R}/L=:\mathcal T_L. \end{equation*} \end{proof} \end{proposition} \begin{remark}\label{cover_tori} We deduce from the theorem above that our family of topological tori $\mathcal T$ has the universal cover $\tilde{\mathcal S}\times H_1(\mathcal T_{s_0},{\mathbb R})$ where the fundamental group $\pi_1(\mathcal T,t_0)=H_1(\mathcal T_{s_0},{\mathbb Z})\rtimes \pi_1(\mathcal S,s_0)$ acts by $(x,v)\cdot(l,\gamma)=(x\gamma,\gamma^{-1}(v+l))$ and the action of $\pi_1(\mathcal S,s_0)$ on $H_1(\mathcal T_{s_0},{\mathbb Z})$ is the monodromy action coming from the locally constant sheaf $\mathcal H$. \end{remark} \begin{definition} A group homomorphism $\phi:\mathcal T^\prime\rightarrow \mathcal T$ of tori over $\mathcal S$ is called an \textit{isogeny}, if $\phi$ is surjective and $\ker(\phi)$ is a finite covering of $\mathcal S$. Here \begin{equation*} \ker(\phi)(X):=\ker(\phi:\mathcal T^\prime(X)\rightarrow \mathcal T(X)),\ X\in Ob(Mfd/\mathcal S) \end{equation*} \end{definition} \begin{remark} The kernel of a homomorphism $\phi:\mathcal T^\prime\rightarrow \mathcal T$ of tori is always representable as a base change with the zero section. More precisely, \Cref{tori} shows that $\ker(\phi)=\tilde{\mathcal S}\times_{\pi_1(\mathcal S,s_0)}\ker(\phi_{s_0})$ where $\phi_{s_0}:\mathcal T^\prime_{s_0}\rightarrow \mathcal T_{s_0}$ is the induced homomorphism on the fiber over $s_0$. \end{remark} \begin{remark} Given a torus $\mathcal T$ over $\mathcal S$ we have for any $N\in {\mathbb Z}$ the \textit{multiplication by $N$ isogeny} $[N]:\mathcal T\rightarrow \mathcal T$. On points $X\in Ob(Mfd/\mathcal S)$ it is determined by the functorial group homomorphism \begin{equation*} [N]:\mathcal T(X)\rightarrow \mathcal T(X), v\mapsto N\cdot v. \end{equation*} We call $\ker([N])=:\mathcal T[N]$ the \textit{$N$-torsion subgroup}, it is a finite covering of $\mathcal S$. \end{remark} \begin{definition} Let $\phi:\mathcal T^\prime\rightarrow \mathcal T$ be an isogeny over $\mathcal S$. The function \begin{equation*}
deg(\phi):\mathcal S\rightarrow {\mathbb Z},\ s\mapsto |\ker(\phi_s)| \end{equation*} is locally constant and is called the \textit{degree} of $\phi$. \end{definition} \begin{remark} We have $deg([N])=N^{dim(\mathcal T/\mathcal S)}$ where \begin{equation*} dim(\mathcal T/\mathcal S):\mathcal S\rightarrow{\mathbb Z},\ s\mapsto dim(\mathcal T_s). \end{equation*} \end{remark} \begin{definition} A family of topological tori $\pi:\mathcal T\rightarrow \mathcal S$ has a \textit{level-$N$-structure}, if the sheaf of sections associated to $\mathcal T[N]$ in $Sh(Mfd/\mathcal S)$ is constant. If the fiber dimension $dim(\mathcal T/\mathcal S)$ is constant, this just means that there is an isomorphism $\mathcal T[N]\cong \mathcal S\times ({\mathbb Z}/N{\mathbb Z})^{dim(\mathcal T/\mathcal S)}$. \end{definition} \begin{remark} For any ring $A$ we set $GL(M):=Aut_{A-Mod}(M)$, if $M$ is a finitely generated free $A$-module. If $\pi:\mathcal T\rightarrow \mathcal S$ has a level-$N$-structure, the associated representation $\rho:\pi_1(\mathcal S,s_0)\rightarrow GL(H_1(\mathcal T_{s_0},{\mathbb Z}))$ factors as $\rho:\pi_1(\mathcal S,s_0)\rightarrow GL(H_1(\mathcal T_{s_0},{\mathbb Z}))(N)$ where \begin{equation*} 1\rightarrow GL(H_1(\mathcal T_{s_0},{\mathbb Z}))(N)\rightarrow GL(H_1(\mathcal T_{s_0},{\mathbb Z}))\rightarrow GL(H_1(\mathcal T_{s_0},{\mathbb Z}/N{\mathbb Z}))\rightarrow 1 \end{equation*} is exact with the natural projection on the right hand side. \end{remark}
\subsection{Definition of the logarithm sheaf}
Consider the group algebra $A[\pi_1(\mathcal T_{s_0},t_0)]$ for any commutative ring $A$. We denote basis elements by $l\in \pi_1(\mathcal T_{s_0},t_0)$ and write elements as $\sum_lf(l)l$, where $f(l)\in A$ and $f(l)=0$ for almost all $l$. The group $\pi_1(\mathcal T,t_0)=\pi_1(\mathcal T_{s_0},t_0)\rtimes \pi_1(\mathcal S,s_0)$ acts naturally on $\pi_1(\mathcal T_{s_0},t_0)$ by $(l,g)* v=l+g\cdot v$, with $l,v\in \pi_1(\mathcal T_{s_0},t_0)$ and $g\in \pi_1(\mathcal S,s_0)$. We can define this action directly in $\pi_1(\mathcal T,t_0)$ by $x* v:=xv0_*\pi_*(x^{-1})$, $x\in \pi_1(\mathcal T,t_0)$ and $v\in \pi_1(\mathcal T_{s_0},t_0)$. This action induces a natural $\pi_1(\mathcal T,t_0)$-action on the group $A[\pi_1(\mathcal T_{s_0},t_0)]$. The group algebra comes with the natural augmentation map \begin{equation*}
\operatorname{aug}:A[\pi_1(\mathcal T_{s_0},t_0)]\rightarrow A,\ \sum_{l}f(l)l\mapsto \sum_l f(l). \end{equation*} The augmentation map is $\pi_1(\mathcal T,t_0)$-equivariant and the kernel $\mathfrak a=\ker(\operatorname{aug})$ is again a $\pi_1(\mathcal T,t_0)$-module. In this manner we get for $n\in {\mathbb N}_0$ the $\pi_1(\mathcal T,t_0)$-modules $A[\pi_1(\mathcal T_{s_0},t_0)]/\mathfrak{a}^{n+1}$ and $A[[\pi_1(\mathcal T_{s_0},t_0)]]:=\varprojlim A[\pi_1(\mathcal T_{s_0},t_0)]/\mathfrak{a}^{n+1}$. \begin{definition}\label{def_Log} Let $\mathcal T$ be a torus over $\mathcal S$. We define the locally constant sheaf $Log_{\mathcal T/\mathcal S} ^{n}$ on $\mathcal T$ as the sheaf associated to the $\pi_1(\mathcal T,t_0)$-module $A[\pi_1(\mathcal T_{s_0},t_0)]/\mathfrak{a}^{n+1}$, $n\in {\mathbb N}_0$, and call it the \textit{$n$-th logarithm sheaf}. $\varprojlim Log_{\mathcal T/\mathcal S} ^{n}=:Log_{\mathcal T/\mathcal S}$ is called the \textit{logarithm sheaf}. \end{definition} \begin{remark} $Log_{\mathcal T/\mathcal S}$ is also locally constant and is associated to the $\pi_1(\mathcal T,t_0)$- module $A[[\pi_1(\mathcal T_{s_0},t_0)]]$. Moreover, we have the augmentation map $\operatorname{aug}:Log_{\mathcal T/\mathcal S}\to A$ coming from the $\pi_1(\mathcal T,t_0)$-equivariant augmentation map $\operatorname{aug}:A[[\pi_1(\mathcal T_{s_0},t_0)]]\to A$. \end{remark} \begin{lemma}\label{basechange_Log} $Log^n_{\mathcal T/\mathcal S}$ and $Log_{\mathcal T/\mathcal S}$ are natural in families of topological tori $\pi:\mathcal T\rightarrow \mathcal S$. In other words, given a commutative square of (pointed) families of topological tori \begin{equation*} \begin{xy} \xymatrix{ \mathcal T^\prime\ar[r]^p\ar[d]^{\pi^\prime} & \mathcal T \ar[d]^\pi \\ \mathcal S^\prime\ar[r]^q & \mathcal S } \end{xy} \end{equation*} with $p\circ 0^\prime=0\circ q$, we have natural morphisms $Log^{n}_{\mathcal T^\prime/\mathcal S^\prime}\rightarrow p^{-1}Log^{n}_{\mathcal T/\mathcal S} $ and $Log_{\mathcal T^\prime/\mathcal S^\prime}\rightarrow p^{-1}Log_{\mathcal T/\mathcal S} $. If the diagram above is Cartesian, these maps are isomorphisms. \begin{proof} We choose $s_0 ^\prime \in \mathcal S^\prime$ with $q(s_0 ^\prime)=s_0$ and set $t_0 ^\prime =0^\prime(s_0 ^\prime)$. Given a locally constant sheaf $\mathcal F$ on $\mathcal T$ associated to a $\pi_1(\mathcal T,t_0)$-module $M$ we recognize $p^{-1}\mathcal F$ as the locally constant sheaf associated to the $\pi_1(\mathcal T^\prime,t_0 ^\prime)$-module $M$ where the action is induced by the map $p_*:\pi_1(\mathcal T^\prime,t_0 ^\prime)\rightarrow \pi_1(\mathcal T,t_0)$. The morphism $p_*:\pi_1(\mathcal T^\prime _{s_0 ^\prime},t_0 ^\prime)\rightarrow \pi_1(\mathcal T_{s_0},t_0)$ is a map of $\pi_1(\mathcal T^\prime,t_0 ^\prime)$-sets. To see this choose $x\in \pi_1(\mathcal T^\prime,t_0 ^\prime)$ and $v\in\pi_1(\mathcal T^\prime _{s_0 ^\prime},t_0 ^\prime)$ arbitrarily. We have \begin{equation*}
p_*(x* v)=p_*(xv0^\prime _*\pi^\prime _*(x^{-1}))=p_*(x)p_*(v) p_* 0^\prime _*\pi^\prime _*(x^{-1})= \end{equation*} \begin{equation*} p_*(x)p_*(v)0 _*\pi _*p_*(x^{-1})=p_*(x)* p_*(v) \end{equation*} So $p_*:A[\pi_1(\mathcal T^\prime _{s^\prime _0},t^\prime _0)]\rightarrow A[ \pi_1(\mathcal T_{s_0},t_0)]$ is $\pi_1(\mathcal T^\prime, t_0 ^\prime)$-equivariant. This induces the desired maps on the level of sheaves. If our diagram is Cartesian the morphism $p_*:\pi_1(\mathcal T^\prime _{s^\prime _0},t^\prime _0)\rightarrow \pi_1(\mathcal T_{s_0},t_0)$ is an isomorphism inducing the desired isomorphisms on our sheaves. \end{proof} \end{lemma}
\section{Limits of locally constant sheaves}
\subsection{Limits over the natural numbers}
In the case of sheaves there can be non-trivial $R^p\varprojlim$ for $p\geq2$ and there may be no such thing as a Mittag-Leffler condition on the system ensuring the vanishing of higher limits. We want to prove now that locally constant sheaves have, as expected, the same behavior with limits as abelian groups. Let $\mathcal A$ be an abelian category. We denote by $\mathcal A^{\mathbb N}$ the category of all inverse systems in $\mathcal A$ indexed by the natural numbers. Objects of $\mathcal A^{\mathbb N}$ are families $(A_n,a_n)$ with objects $A_n$ in $\mathcal A$ and maps $a_n:A_{n+1}\rightarrow A_n$. Morphisms between $(A_n,a_n)$ and $(B_n,b_n)$ in $\mathcal A^{\mathbb N}$ are just commutative diagrams in $\mathcal A$ of the form \begin{equation*} \begin{xy} \xymatrix{ \cdots & A_{n+1}\ar[r]\ar[d]^{f_{n+1}} & A_n \ar[r]\ar[d]^{f_n} & A_{n-1}\ar[r]\ar[d]^{f_{n-1}} & \cdots \\ \cdots & B_{n+1}\ar[r] & B_n \ar[r] & B_{n-1}\ar[r] & \cdots \\ } \end{xy} \end{equation*} $\mathcal A^{\mathbb N}$ is an abelian category with kernels and cokernels defined componentwise. \begin{proposition} $\mathcal A^{\mathbb N}$ has enough injective objects, if and only if $\mathcal A$ has. We can characterize injectives in $\mathcal A^{\mathbb N}$ as follows:\\ $\mathcal I=(\mathcal I_n,d_n)$ is injective, if and only if each $\mathcal I_n$ is injective in $\mathcal A$ and all the $d_n$ are split epimorphisms. \begin{proof} \cite{Ja} (1.1) proposition. \end{proof} \end{proposition} From now on we want to assume that $\mathcal A$ has enough injectives and that inverse limits indexed over ${\mathbb N}$ exist in $\mathcal A$. The functor $\varprojlim:\mathcal A^{\mathbb N}\rightarrow \mathcal A$ sending an inverse system to its limit is additive and left exact as right adjoint to the exact diagonal functor $\Delta$ sending $A$ to the system $\cdots \rightarrow A \stackrel{\operatorname{id}}{\rightarrow}A \stackrel{\operatorname{id}}{\rightarrow}A \stackrel{\operatorname{id}}{\rightarrow}\cdots$. In particular $\varprojlim$ preserves injective objects. Since $\mathcal A^{\mathbb N}$ has enough injectives and $\varprojlim$ is left exact, we form the higher right derived functors $R^p\varprojlim$. We will often omit the transition maps of our system and simply write $(A_n)$ for objects in $\mathcal A^{\mathbb N}$. Let us consider $\mathcal A=Ab$ the category of abelian groups first. \begin{definition} $(A_n,d_n)$ in $Ab^{\mathbb N}$ satisfies the Mittag-Leffler condition (M-L), if for any $n\in {\mathbb N}$ the decreasing filtration of $A_n$ $F_m:=\operatorname{im}(A_{n+m}\rightarrow A_n)$ induced by the $d_{k}$, $k\geq n$, becomes stationary. \end{definition} \begin{remark} If all $d_n$ are surjective, then $(A_n,d_n)$ obviously satisfies (M-L). \end{remark} \begin{proposition} Let $(A_n)$ be in $Ab^{\mathbb N}$. Then $R^p\varprojlim(A_n)=0$ for $p\geq2$. If $(A_n)$ satisfies (M-L), then $R^1\varprojlim (A_n)=0$. \begin{proof} \cite{We} Corollary 3.5.4. and Proposition 3.5.7. \end{proof} \end{proposition}
\subsection{(M-L) systems of locally constant sheaves}
Let $X$ be a topological manifold. The category $Sh(X)$ of sheaves on $X$ has enough injectives and all limits over the natural numbers exist. For $(\mathcal F_n)\in Sh(X)^{\mathbb N}$ the formula \begin{equation*} (\varprojlim \mathcal F_n)(U)=\varprojlim \mathcal F_n(U),\ U\subset X\ \text{open} \end{equation*} can be used as definition.\\ \begin{definition} We say $(\mathcal F_n)\in Sh(X)^{\mathbb N}$ is a \textit{system of locally constant sheaves on $X$}, if each $\mathcal F_n$ is a locally constant sheaf. We say that the system of locally constant sheaves $(\mathcal F_n)$ satisfies the \textit{Mittag-Leffler condition} (M-L), if for any $x\in X$ the induced system of abelian groups $(\mathcal F_{n,x})$ does so. \end{definition} \begin{proposition} Let $(\mathcal F_n)$ be a system of locally constant sheaves on a topological manifold $X$. Then $R^p\varprojlim(\mathcal F_n)=0$ for $p\geq 2$ and if $(\mathcal F_n)$ satisfies (M-L) $R^1\varprojlim (\mathcal F_n)=0$. \begin{proof} Consider an injective resolution $(\mathcal F_n)\hookrightarrow (\mathcal I_n)^\bullet$ in $Sh(X)^{\mathbb N}$. We have by definition $R^p{\varprojlim}(\mathcal F_n)=H^p(\varprojlim \mathcal I_n ^\bullet)$. $H^p(\varprojlim \mathcal I_n ^\bullet)$ is the sheaf associated to the presheaf \begin{equation*} U\mapsto H^p((\varprojlim \mathcal I_n )(U)^\bullet)=H^p(\varprojlim \mathcal I_n (U)^\bullet). \end{equation*}
So it suffices to show that $H^p(\varprojlim \mathcal I_n (U)^\bullet)$ vanishes for any $U\subset X$ open and contractible for $p\geq2$ or $p\geq 1$, if $(\mathcal F_n)$ is (M-L). The complexes $\mathcal I_n (U)^\bullet$ are injective resolutions of $M_n:=\mathcal F_n(U)$. To see this recall that $\mathcal I^\bullet_{n|U}$ is an injective resolution of the \textit{constant} sheaf $\underline{M_n}=\mathcal F_{n|U}$. Here we use that locally constant sheaves are constant on simply connected manifolds (\cite{Iv} IV.9). So we may compute \begin{equation*}
H^p(U,M_n)=H^p(U,\mathcal F_{n|U})=H^p(\mathcal I^\bullet_{n|U}(U))=H^p(\mathcal I^\bullet_n(U)) \end{equation*} Now $U$ is contractible and by homotopy invariance of sheaf cohomology with constant coefficients (\cite{Iv} IV.1) we get $H^p(U,M_n)=H^p(\left\{pt.\right\},M_n)$ which is zero for $p\geq 1$ and $M_n$ for $p=0$. So $\mathcal I_n (U)^\bullet$ is a resolution of $M_n$ and it is an injective resolution, because the section functor $\Gamma(U,\ )$ is right adjoint to the constant sheaf construction which is exact. Since $(\mathcal I^p _n)$ is injective in $Sh(X)^{\mathbb N}$, all its transition functions are split epimorphisms. In particular, all transition functions of $(\mathcal I^p _n(U))$ are split epimorphisms. This implies that $(\mathcal I^p _n(U))$ is injective in $Ab^{\mathbb N}$. Therefore $(M_n)\hookrightarrow (\mathcal I_n (U))^\bullet$ is an injective resolution in $Ab^{\mathbb N}$ implying $R^p\varprojlim(M_n)=H^p(\varprojlim \mathcal I_n (U)^\bullet)$. Now we can use the results about limits in $Ab$ to conclude that $R^p\varprojlim(\mathcal F_n)=0$ for $p\geq 2$ and if $(\mathcal F_n)$ satisfies (M-L) $R^1\varprojlim(\mathcal F_n)=0$. \end{proof} \end{proposition} \begin{remark}\label{lim_injective_resolution} If $(\mathcal F_n)$ is a (M-L) system of locally constant sheaves and $(\mathcal F_n)\hookrightarrow (\mathcal I_n)^\bullet$ is an injective resolution in $Sh(X)^{\mathbb N}$, then $\varprojlim \mathcal F_n\hookrightarrow \varprojlim \mathcal I_n^\bullet$ is an injective resolution. This follows easily from $H^p(\varprojlim \mathcal I_n^\bullet)=R^p\varprojlim(\mathcal F_n)=0$, if $p\geq 1$. \end{remark}
\section{Purity and localization}
In this section we want to prove a purity result for (M-L) systems of local systems on topological manifolds. This gives rise to localization sequences on cohomology, which finally will be used to derive a very easy localization sequence for the logarithm sheaf and to define the polylogarithm cohomology classes.
\subsection{Purity for local systems}
Let $A$ be a noetherian ring and $X$ a topological space. \begin{definition} A sheaf $\mathcal F$ on $X$ is called a \textit{local $A$-system}, or simply a \textit{local system}, if it is locally constant and if for all $x\in X$ the stalks $\mathcal F_x$ are free $A$-modules of finite rank. The sheaf $\mathcal F^*:=\underline{Hom}_A(\mathcal F,A)$ is also a local $A$-system and is called the \textit{dual of $\mathcal F$}. \end{definition} \begin{lemma}\label{injectives_local_systems} If $\mathcal I$ is an injective $A$-sheaf on a topological space $X$, then $\mathcal F\otimes_A\mathcal I$ is injective for any local system $\mathcal F$. \begin{proof} Given any inclusion $C\hookrightarrow D$ of $A$-sheaves we have a commutative diagram where the vertical arrows are isomorphisms: \begin{equation*} \begin{xy} \xymatrix{ Hom(C,\mathcal I\otimes_A \mathcal F)\ar[r]\ar[d] & Hom(D,\mathcal I\otimes_A \mathcal F)\ar[d]\\ Hom(C\otimes_A \mathcal F^*,\mathcal I)\ar[r] & Hom(D\otimes_A \mathcal F^*, \mathcal I) } \end{xy} \end{equation*} To see this one uses the universal property of the tensor product, $(\mathcal F^*)^*=\mathcal F$ and $\mathcal F^*\otimes_A \mathcal I=\underline{Hom}_A(\mathcal F,\mathcal I)$. Since $\mathcal F^*$ is a flat $A$-sheaf $C\otimes_A \mathcal F^*\hookrightarrow D\otimes_A \mathcal F^*$ is injective. Therefore the lower horizontal arrow is surjective, because $\mathcal I$ is injective. So $\mathcal F\otimes_A \mathcal I$ is injective. \end{proof} \end{lemma} Let $Z\subset X$ be a closed subspace. The pushforward $i_*:Sh(Z,A)\rightarrow Sh(X,A)$ on the categories of sheaves of $A$-modules has a right adjoint $i^{!}:Sh(X,A)\rightarrow Sh(Z,A)$ (\cite{Iv} II. Proposition 6.6.). \begin{lemma}\label{!and-1} Let $\mathcal F$ be a local $A$-system on a topological space $X$ and let $\mathcal G$ be any sheaf on $X$. There is a functorial isomorphism \begin{equation*} i^{!}(\mathcal F\otimes_A \mathcal G)=i^{-1}\mathcal F\otimes_A i^{!}\mathcal G. \end{equation*} \begin{proof} We use the Yoneda principle: Let $\mathcal H$ be any $A$-sheaf on $Z$. By adjunction and the properties of $^*$ and $\otimes_A$ we have isomorphisms \begin{equation*} Hom(\mathcal H,i^!\mathcal G\otimes_A i^{-1}\mathcal F)\cong Hom(\mathcal H\otimes_A i^{-1}\mathcal F^*,i^!\mathcal G)\cong Hom(i_*(\mathcal H\otimes_A i^{-1}\mathcal F^*),\mathcal G) \end{equation*} Moreover, $i_*(\mathcal H\otimes_A i^{-1}\mathcal F^*)\cong i_*\mathcal H\otimes_A i_*i^{-1}\mathcal F^*\cong i_*\mathcal H\otimes_A \mathcal F^*$ where the last arrow comes from the adjunction morphism $\mathcal F\rightarrow i_*i^{-1}\mathcal F^*$. Therefore we get natural in $\mathcal H$ \begin{equation*} Hom(\mathcal H,i^!\mathcal G\otimes_A i^{-1}\mathcal F)\cong Hom(i_*\mathcal H\otimes_A \mathcal F^*,\mathcal G)\cong Hom(i_*\mathcal H,\mathcal G\otimes_A \mathcal F)\cong \end{equation*} \begin{equation*} Hom(\mathcal H,i^!(\mathcal G\otimes_A \mathcal F)) \end{equation*} \end{proof} \end{lemma} As $i_*$ is exact, $i^{!}$ is left exact. Purity is concerned with the calculation of the higher right derived functors of $i^{!}$. To do so we need some notation. \\ Let $X$ be a topological manifold and $A$ a noetherian ring. \begin{itemize} \item{ $D^+(X,A)$ denotes the derived category of bounded below complexes of sheaves of $A$-modules. We identify this category with the homotopy category of bounded below complexes of injective sheaves of $A$-modules.} \item{ $D^+(A)$ is the derived category of bounded below complexes of $A$-modules. We identify this category with the homotopy category of bounded below complexes of injective $A$-modules.} \item If $C=(C,d)$ is a complex of sheaves and $p\in {\mathbb Z}$, we denote by $C[p]$ the shifted complex, in other words, $C[p]^d=C^{p+d}$ with differential $d[p]:=(-1)^pd$. If $D$ is a sheaf, we identify $D$ with the complex which equals $D$ in degree zero and is zero elsewhere. \end{itemize} \begin{definition} Let $X$ be a topological manifold of dimension $d$ and $A$ a noetherian ring. The presheaf \begin{equation*} U\subset X \text{ open}\mapsto Hom_A(H^d _c(U,A),A)=:or_X(U) \end{equation*} is a sheaf on $X$. It is even a local $A$-system of rank 1 (\cite{Iv} III 8.14.) and we call it the \textit{orientation bundle of $X$} \end{definition} \begin{definition} Let $X$ be a topological manifold of dimension $d$. The \textit{dualizing complex} $\mathcal D_X$ on $X$ is a complex of injective sheaves on $X$ characterized up to homotopy by the following property (\cite{Iv}VI.2): There is an isomorphism natural in $A\in D^+(X,A)$: $Hom_{D^+(X,A)}(A,\mathcal D_X)=Hom_{D^+(A)}(\Gamma_c(X,A),A)$. Moreover, one has a quasi-isomorphism $or_X[d]\rightarrow \mathcal D_X$ (\cite{Iv} VI.3.2). \end{definition}
\begin{lemma}\label{dualizing_purity} Let $X$ be a topological manifold of dimension $d$ and $i:Z\hookrightarrow X$ a closed submanifold of codimension $c$. Then one has $i^{!}\mathcal D_X=\mathcal D_Z$. \begin{proof} \cite{Iv}VIII proposition 1.7. \end{proof} \end{lemma} \begin{definition} Let $X$ be a topological manifold of dimension $d$ and $i:Z\hookrightarrow X$ a closed submanifold of codimension $c$. We define the \textit{orientation of $Z$ relative $X$} as the rank $1$ local system $or_{Z/X}:=or_Z\otimes_A i^{-1}or_X^*$. \end{definition} \begin{proposition}\label{purity1} Let $X$ be a topological manifold of dimension $d$, $i:Z\hookrightarrow X$ a closed submanifold of codimension $c$ and $\mathcal F$ a local system on $X$. Then we have $Ri^{!}\mathcal F=or_{Z/X}\otimes_A i^{-1}\mathcal F[-c]$ in $D(Z,A)$ functorial in $\mathcal F$. \begin{proof} We start with the quasi-isomorphism $or_X[d]\rightarrow \mathcal D_X$. $or_X ^*\otimes_A\mathcal F$ is flat and if we tensor the last quasi-isomorphism with it, we get a quasi-isomorphism $\mathcal F[d]\rightarrow \mathcal D_X\otimes_A or_X ^*\otimes_A\mathcal F$. The right-hand side is a complex of injectives by \Cref{injectives_local_systems}. We get \begin{equation*} Ri^{!}\mathcal F[d]= i^{!}(\mathcal D_X\otimes_A or_X ^*\otimes_A\mathcal F)=i^{!}\mathcal D_X\otimes_A i^{-1}or_X^*\otimes_A i^{-1}\mathcal F \end{equation*} by \Cref{!and-1} and this equals by \Cref{dualizing_purity} \begin{equation*} \mathcal D_Z\otimes_A i^{-1}or_X^*\otimes_A i^{-1}\mathcal F=or_Z [d-c]\otimes_A i^{-1}or_X^*\otimes_A i^{-1}\mathcal F=or_{Z/X}\otimes_A i^{-1}\mathcal F[d-c] \end{equation*} \end{proof} \end{proposition}
\subsection{Purity for systems of local systems and naturality}
\begin{proposition}\label{lim_purity} Let $X$ be a topological manifold of dimension $d$ and $i:Z\hookrightarrow X$ a closed submanifold of codimension $c$. Let $(\mathcal F_n)$ be a system of local $A$-systems satisfying (M-L) on $X$. We have in $D^+(X,A)$ \begin{equation*} R i^{!}(\varprojlim\mathcal F_n)=\varprojlim(i^{-1}\mathcal F_n\otimes_A or_{Z/X})[-c] =i^{-1}(\varprojlim\mathcal F_n)\otimes_A or_{Z/X}[-c] \end{equation*} functorial in $(\mathcal F_n)$. \begin{proof} Take an injective resolution $a:(\mathcal F_n)\hookrightarrow (\mathcal I_n)^\bullet$ in $Sh(X,A)^{\mathbb N}$. We already had the resolution $b:(\mathcal F_n)\hookrightarrow (\mathcal D_X\otimes_A or_X ^*\otimes_A\mathcal F_n)_n$ in $Sh(X,A)^{\mathbb N}$ functorial in systems $(\mathcal F_n)$. Since $(\mathcal I_n)^\bullet$ is a complex of injectives in $Sh(X,A)^{\mathbb N}$, we get a map of complexes $c$ unique up to homotopy making the following diagram commute \begin{equation*} \begin{xy} \xymatrix{
(\mathcal F_n)\ar[rd]^b\ar[rr]^{a} & & (\mathcal I_n)^\bullet \\
& (\mathcal D_X\otimes_A or_X ^*\otimes_A\mathcal F_n)\ar[ur]^c &
} \end{xy} \end{equation*} (\cite{Iv} I Theorem 6.2). Let us consider the functor $i^{!{\mathbb N}}:Sh(X,A)^{\mathbb N}\to Sh(Z,A)^{\mathbb N}$, $(\mathcal G_n)\mapsto (i^!\mathcal G_n)$. We have now \begin{equation*} Ri^{!{\mathbb N}}(\mathcal F_n)=(i^!\mathcal I_n)^\bullet\stackrel{i^{!{\mathbb N}}c}{\leftarrow}(i^!(\mathcal D_X\otimes_A or_X ^*\otimes_A\mathcal F_n))\stackrel{p\cong}{\rightarrow}(i^{-1}\mathcal F_n\otimes_A or_{Z/X})[-c], \end{equation*} where the map on the right-hand side is the functorial purity quasi-isomorphism $p$ from \cref{purity1}. The morphism $i^{!{\mathbb N}}c$ is also a quasi-isomorphism, as \begin{equation*} H^p(i^!\mathcal I_n)^\bullet=(H^pi^!\mathcal I_n^\bullet)=(R^pi^!\mathcal F_n)=H^p(i^!(\mathcal D_X\otimes_A or_X ^*\otimes_A\mathcal F_n)). \end{equation*} This gives a purity isomorphism \begin{equation*} Ri^{!{\mathbb N}}(\mathcal F_n)=(i^{-1}\mathcal F_n\otimes_A or_{Z/X})[-c]\in D^+(Sh(X,A)^{\mathbb N}) \end{equation*} in the derived category of bounded below complexes of systems of sheaves of $A$-modules, which is functorial in systems of local systems $(\mathcal F_n)$. As $(\mathcal F_n)$ is (M-L), we have that $\varprojlim\mathcal F_n\hookrightarrow \varprojlim \mathcal I_n^\bullet$ is also an injective resolution by \Cref{lim_injective_resolution}. In other words, $R\varprojlim(\mathcal F_n)=\varprojlim\mathcal F_n\in D^+(X,A)$ and one gets in $D^+(X,A)$ \begin{equation*} Ri^{!}\varprojlim\mathcal F_n=Ri^!R\varprojlim (\mathcal F_n)=R(i^!\circ \varprojlim)(\mathcal F_n)=R(\varprojlim\circ i^{!{\mathbb N}})(\mathcal F_n), \end{equation*} since $i^!$ commutes as right adjoint with the limit. We go on with \begin{equation*} R\varprojlim Ri^{!{\mathbb N}}(\mathcal F_n)=R\varprojlim (i^{-1}\mathcal F_n\otimes_A or_{Z/X})[-c]=\varprojlim (i^{-1}\mathcal F_n\otimes_A or_{Z/X})[-c], \end{equation*} where we used purity and the fact that $(i^{-1}\mathcal F_n\otimes_A or_{Z/X})$ is again (M-L). Finally, we have \begin{equation*} \varprojlim (i^{-1}\mathcal F_n\otimes_A or_{Z/X})=i^{-1}(\varprojlim\mathcal F_n)\otimes_A or_{Z/X} \end{equation*} which will follow from the following two lemmas and will complete the proof. \end{proof} \end{proposition} \begin{lemma} Let $X$ be a topological manifold and let $(\mathcal F_n)$ be a system of locally constant sheaves on $X$. For any $x\in X$ the canonical map \begin{equation*} (\varprojlim \mathcal F_n)_x\rightarrow\varprojlim(F_{n,x}) \end{equation*} is an isomorphism. More generally, if $Y$ is another topological manifold and $f:Y\rightarrow X$ a continuous map, the canonical map $f^{-1}(\varprojlim \mathcal F_n)\rightarrow \varprojlim f^{-1}\mathcal F_n$ is an isomorphism. \begin{proof} Let $\mathcal G$ be any sheaf on $X$ and $x\in X$. As $X$ is a topological manifold any open neighborhood $U$ of $x$ contains a contractible open neighborhood $B$ of $x$. It follows that the family $\mathfrak B _x$ of contractible open neighborhoods of $x$ is a cofinal inductive system in the family $\mathfrak U _x$ of all open neighborhoods of $x$. Therefore
$\mathcal G_x=\varinjlim_{U\in \mathfrak U _x}\mathcal G(U)=\varinjlim_{B\in \mathfrak B _x}\mathcal G(B)$. If $\mathcal F$ is locally constant, $\mathcal F_{|B}$ is constant for $B\in \mathfrak B_x$, as contractible spaces are simply connected, and we have $\mathcal F_x=\varinjlim_{B\in \mathfrak B _x}\mathcal F(B)=\mathcal F(B)$ for any $B\in \mathfrak B _x$. We calculate \begin{equation*}
(\varprojlim \mathcal F_n)_x=\varinjlim_{B\in \mathfrak B _x}(\varprojlim \mathcal F_n)(B)=\varinjlim_{B\in \mathfrak B _x}(\varprojlim \mathcal F_n(B)) \end{equation*} as the limit commutes with the section functor. The inductive limit becomes constant and therefore \begin{equation*} \varinjlim_{B\in \mathfrak B _x}(\varprojlim \mathcal F_n(B))=\varprojlim \mathcal F_n(B)=\varprojlim \mathcal F_{n,x}. \end{equation*} This proves the first claim. The second follows easily from the first by considering the induced maps on the stalks. \end{proof} \end{lemma} \begin{lemma} Let $(\mathcal F_n)$ be a system of locally constant $A$-sheaves on a topological manifold $X$ and $\mathcal F$ a local $A$-system on $X$. Then the natural map $(\varprojlim \mathcal F_n)\otimes_A \mathcal F\rightarrow \varprojlim(\mathcal F_n\otimes_A \mathcal F)$ is an isomorphism. \begin{proof} By the preceding lemma the problem is local so it suffices to proof the corresponding statement for a system of $A$-modules $(F_n)$ and a finitely generated free module $F$. For any $A$-module $M$ one has the functorial isomorphisms \begin{equation*} Hom(M,\varprojlim (F_n\otimes_A F) )= \varprojlim Hom(M,F_n\otimes_A F)=\varprojlim Hom(M\otimes_A F^*,F_n)= \end{equation*} \begin{equation*} Hom(M\otimes_A F^*,\varprojlim F_n)=Hom(M,(\varprojlim F_n)\otimes_A F) \end{equation*} from which the claim follows by the Yoneda lemma. \end{proof} \end{lemma} Now we come to our important application of the purity result \begin{proposition}\label{localization_triangle} Let $(\mathcal F_n)$ be a system of local systems on a topological manifold $X$ satisfying (M-L). Let $i:Z\hookrightarrow X$ be the inclusion of a closed submanifold of codimension $c$ and $j:U:=X\setminus Z\to X$ the inclusion of the open complement. One has an exact triangle in $D^+(X,A)$ \begin{equation*} i_*(i^{-1}(\varprojlim\mathcal F_n)\otimes_A or_{Z/X})[-c]\rightarrow\varprojlim\mathcal F_n\rightarrow Rj_*j^{-1}\varprojlim\mathcal F_n\stackrel{+1}{\rightarrow} \end{equation*} \begin{proof} Let $\mathcal F$ be any sheaf on $X$. Let us recall \begin{equation*} i_*i^!\mathcal F(V)=\left\{s\in \mathcal F(V):\text{supp}(s)\subset Z\right\} \end{equation*} (\cite{Iv} II.6.6). Therefore, we have an exact sequence \begin{equation*} 0\rightarrow i_*i^!\mathcal F\rightarrow \mathcal F\rightarrow j_* j^{-1}\mathcal F, \end{equation*} where the arrows are the adjunction morphisms. If $\mathcal F$ is flabby, the last sequence is even right exact. This means, if we replace $\mathcal F$ by an injective resolution $\mathcal I$, we get an exact sequence \begin{equation*} 0\rightarrow i_*i^!\mathcal I\rightarrow \mathcal I\rightarrow j_* j^{-1}\mathcal I\rightarrow 0 \end{equation*} giving rise to an exact triangle in the derived category $D^+(X,A)$ \begin{equation*} i_*Ri^!\mathcal F\rightarrow \mathcal F\rightarrow Rj_* j^{-1}\mathcal F\stackrel{+1}{\rightarrow}. \end{equation*} If we apply now \Cref{lim_purity} to substitute $i_*Ri^!\mathcal F$, we get the exact triangle as claimed. \end{proof} \end{proposition} \begin{remark}\label{natural_localization} \Cref{localization_triangle} is functorial with respect to pairs $(X,Z)$, $(X^\prime,Z^\prime)$. By this we mean given a Cartesian square of topological spaces \begin{equation*} \begin{xy} \xymatrix{ Z^\prime\ar[r]^{i^\prime}\ar[d]^{f^\prime} & X^\prime\ar[d]^f \\ Z\ar[r]^{i}& X } \end{xy} \end{equation*} with a topological manifold $X$, a closed submanifold $Z$ of codimension $c$ and topological submersion $f$ (see \cite{KS} Definition 3.3.1) we get by pullback a morphism of exact triangles (apply $\operatorname{id}\rightarrow Rf_*f^{-1}$) from \begin{equation*} i_*(i^{-1}(\varprojlim\mathcal F_n)\otimes_A or_{Z/X})[-c]\rightarrow \varprojlim\mathcal F_n\rightarrow Rj_*j^{-1}\varprojlim\mathcal F_n\stackrel{+1}{\rightarrow} \end{equation*} to \begin{equation*} Rf^{\prime}_*i^\prime_*(i^{\prime -1}(\varprojlim f^{-1}\mathcal F_n)\otimes_A or_{Z^\prime/X^\prime})[-c]\rightarrow Rf_*\varprojlim f^{-1}\mathcal F_n \rightarrow Rf_*Rj^\prime _* j^{\prime -1}\varprojlim f^{-1}\mathcal F_n\stackrel{+1}{\rightarrow} \end{equation*} To see this we recall that $X^\prime$ and $Z^\prime$ have the structure of topological manifolds and that $Z^\prime$ has codimension $c$ in $X^\prime$. After exploiting the commutation rules of the functors involved the only thing that remains to be shown is that there is a natural pullback isomorphism $f^{\prime -1}or_{Z/X}\rightarrow or_{Z^\prime/X^\prime}$. As our diagram is Cartesian, we have a natural pullback morphism $i^{\prime -1}f^!A\rightarrow f^{\prime !} i^{-1}A$ by \cite{KS} proposition 3.1.9 (iii). If we apply $H^{dim(X) -dim(X^\prime)}$, we get the natural morphism $i^{\prime -1}or_{X^\prime/X}\rightarrow or_{Z^\prime/Z}$. Following the proof of \cite{KS} proposition 3.3.2 we may see that it is actually an isomorphism. By the formalism provided by \cite{KS} remark 3.3.5 we may conclude that we also have a natural isomorphism $f^{\prime -1}or_{Z/X}\rightarrow or_{Z^\prime/X^\prime}$.
\end{remark} \begin{remark}\label{shrinking_localization} If $(X,Z)$, $(X,Z^\prime)$ are two pairs of codimension $c$ as above with $Z^\prime\subset Z$, we have another compatibility. $Z^\prime \subset Z$ is also open, as both manifolds have the same dimension. From this we derive a natural extension by zero morphism \begin{equation*} i^{\prime }_*(i^{\prime -1}\varprojlim\mathcal F_n\otimes_A or_{Z^\prime/X})\rightarrow i_*(i^{ -1}\varprojlim\mathcal F_n\otimes_A or_{Z/X}) , \end{equation*} with the inclusions $i:Z\rightarrow X$ and $i^\prime:Z^\prime\rightarrow X$. If we denote by $j:X\setminus Z\rightarrow X$ and $j^\prime:X\setminus Z^\prime\rightarrow X$ the inclusions of the open complements, the morphism above fits into a morphism of triangles \begin{equation*} \begin{xy} \xymatrix{ i^\prime_*(i^{\prime-1}(\varprojlim\mathcal F_n)\otimes_A or_{Z^\prime/X})[-c] \ar[r]\ar[d] & \varprojlim\mathcal F_n\ar[r]\ar[d] & Rj^\prime_*j^{\prime-1}\varprojlim\mathcal F_n\ar[d]\ar[r]^>{+1} &\\ i_*(i^{-1}(\varprojlim\mathcal F_n)\otimes_A or_{Z/X})[-c]\ar[r] & \varprojlim\mathcal F_n\ar[r] & Rj_* j^{ -1}\varprojlim\mathcal F_n\ar[r]^>{+1}& } \end{xy} \end{equation*} The right vertical arrow is just the restriction map. \end{remark} \begin{corollary}\label{localization} Let $(\mathcal F_n)$ be a system of local systems on a topological manifold $X$ satisfying (M-L). Let $i:Z\hookrightarrow X$ be the inclusion of a closed submanifold of codimension $c$ with open complement $U:=X\setminus Z$ and $f:X\rightarrow Y$ a continuous map of topological spaces. There is a long exact cohomology sequence \begin{equation*}
\rightarrow R^{p-1}f_{|U *}(\varprojlim\mathcal F_{n\ |U})\rightarrow R^{p-c}f_{|Z *}(\varprojlim i^{-1}\mathcal F_n\otimes_A or_{Z/X})\rightarrow R^pf_*(\varprojlim\mathcal F_n) \end{equation*} \begin{equation*}
\rightarrow R^p f_{|U*}(\varprojlim\mathcal F_{n\ |U})\rightarrow R^{p+1-c}f_{|Z *}(\varprojlim i^{-1}\mathcal F_n\otimes_A or_{Z/X})\rightarrow \end{equation*} called \textit{the localization sequence for the pair $(X,Z)$}. The sequence is natural in pairs $(X,Z)$ as described in \Cref{natural_localization} and \Cref{shrinking_localization}. \begin{proof} We take the exact triangle \begin{equation*} i_*(i^{-1}(\varprojlim\mathcal F_n)\otimes_A or_{Z/X})[-c]\rightarrow\varprojlim\mathcal F_n\rightarrow Rj_*j^{-1}\varprojlim\mathcal F_n\stackrel{+1}{\rightarrow} \end{equation*} from \Cref{localization_triangle} and apply $Rf_*$ to obtain an exact triangle \begin{equation*} Rf_*i_*(i^{-1}(\varprojlim\mathcal F_n)\otimes_A or_{Z/X})[-c]\rightarrow Rf_*\varprojlim\mathcal F_n\rightarrow Rf_*Rj_*j^{-1}\varprojlim\mathcal F_n \stackrel{+1}{\rightarrow}, \end{equation*} which equals \begin{equation*}
Rf_{|Z *}(i^{-1}(\varprojlim\mathcal F_n)\otimes_A or_{Z/X})[-c]\rightarrow Rf_*\varprojlim\mathcal F_n\rightarrow Rf_{|U*}j^{-1}\varprojlim\mathcal F_n\stackrel{+1}{\rightarrow} \end{equation*} by the composition rules of derived functors. If we apply the functor $H^0$, we obtain the statement. \end{proof} \end{corollary}
\section{Construction of the topological polylogarithm }
\subsection{The localization sequence for the logarithm sheaf}
In this section we prove the important vanishing result for the cohomology of the logarithm sheaf. We will then derive a simple localization sequence for the logarithm sheaf, which enables us to define the polylogarithm cohomology classes.
From now on we will often drop the subscript $A$ at $\otimes$, when the coefficient ring has been fixed. \begin{proposition} Let $\pi:\mathcal T\rightarrow \mathcal S$ be torus of constant fiber dimension $d$ and $Log_{\mathcal T/\mathcal S}=Log$ the associated logarithm sheaf over the noetherian ring $A$. Then \begin{equation*} R^p\pi_* (Log)=0,\ p\neq d,\ \text{and}\ \operatorname{aug}:R^{d}\pi_* (Log)\rightarrow R^{d}\pi_* (A) \end{equation*} is an isomorphism. \begin{proof} As the problems are local on the base, we may check the corresponding statements for the stalks. Because $\pi$ is proper, we have for any sheaf $\mathcal F$ on $\mathcal T$ and $s\in \mathcal S$ \begin{equation*}
R^p\pi_* (\mathcal F)_s=H^p(\pi^{-1}(s),\mathcal F_{|\pi^{-1}(s)}). \end{equation*} Now $Log_{\mathcal T/\mathcal S,s}=Log_{\pi^{-1}(s)/\left\{s\right\}}$ by \Cref{basechange_Log} and we use \cite{BKL} Theorem 1.6.1 to conclude. \end{proof} \end{proposition} \begin{remark} If $\pi:\mathcal T\to \mathcal S$ is a topological submersion, there is a relative orientation bundle $or_{\mathcal T/\mathcal S}$ on $\mathcal T$, see \cite{KS} Definition 3.3.3. If $\pi:\mathcal T\rightarrow \mathcal S$ is a torus, it is in particular a fiber bundle with orientable fibers and therefore $or_{\mathcal T/\mathcal S}=\pi^{-1}R^{d}\pi_*(A)^*$. \end{remark}
Now let $D\subset \mathcal T$ be closed such that $\pi_{|D}:D\rightarrow \mathcal S$ is a (finite) covering. We set $U:=\mathcal T\setminus D$. One has $or_{D}=\pi_{|D}^{-1}or_{\mathcal S}$. \begin{proposition}\label{relative_Log_localization}
Let $\mathcal F$ be a local $A$-system on $\mathcal S$, $\pi:\mathcal T\rightarrow \mathcal S$ a family of topological tori of constant fiber dimension $d$ and $D\subset \mathcal T$ be closed such that $\pi_{|D}$ is a covering. The localization sequence for the twisted logarithm sheaf $\pi^{-1}\mathcal F\otimes Log$ reduces to a short exact sequence \begin{equation*}
0\rightarrow R^{d-1}\pi_{|U*}(\pi^{-1}\mathcal F\otimes Log\otimes or_{\mathcal T/\mathcal S})\rightarrow \pi_{|D*}(\pi^{-1}\mathcal F\otimes Log_{|D})\stackrel{\operatorname{aug}}{\rightarrow}\mathcal F\rightarrow 0 \end{equation*} \begin{proof}
Because $\pi_{|D}:D\rightarrow \mathcal S$ is a covering, one has for any locally constant sheaf $\mathcal G$ on $D$ that $R^p\pi_{|D*}(\mathcal G)=0$, $p\neq 0$. To see this it suffices to show $H^p(\pi_{|D*} ^{-1}(U),\mathcal G)=0$, $p\neq 0$, for all small $U\subset \mathcal S$. Take $U$ contractible and uniformly covered, in other words $\pi_{|D*}^{-1}(U)=\coprod _{i\in I} U_i$ with $\pi_{|D|U_i}: U_i\rightarrow U$ a homeomorphism for all $i\in I$. The sheaf $\mathcal G_{|U_i}$ is constant, as locally constant on something simply connected, and we calculate \begin{equation*}
H^p(\pi_{|D*}^{-1}(U),\mathcal G)=H^{p}(\coprod_{i\in I} U_i,\mathcal G)=\prod_{i\in I} H^{p}(U_i,\mathcal G)=0,\ p\neq0, \end{equation*} by homotopy invariance of sheaf cohomology with constant coefficients (\cite{Iv} IV.1). The localization sequence for $(\pi^{-1}\mathcal F\otimes Log^n)_n$ \begin{equation*}
\rightarrow R^{p-1}\pi_{|U *}(\pi^{-1}\mathcal F \otimes Log\otimes or_{\mathcal T/\mathcal S})\rightarrow R^{p-d}\pi_{|D *}(\pi_{|D}^{-1}\mathcal F \otimes Log_{|D}) \end{equation*} \begin{equation*}
\rightarrow R^p\pi_*(\pi^{-1}\mathcal F \otimes Log\otimes or_{\mathcal T/\mathcal S})\rightarrow R^p \pi_{|U*}(\pi^{-1}\mathcal F \otimes Log\otimes or_{\mathcal T/\mathcal S})\rightarrow \end{equation*} provided by \Cref{localization} gives immediately isomorphisms \begin{equation*}
R^p \pi_{*}(\pi^{-1}\mathcal F \otimes Log\otimes or_{\mathcal T/\mathcal S})\rightarrow R^p \pi_{|U*}(\pi^{-1}\mathcal F \otimes Log\otimes or_{\mathcal T/\mathcal S}) \end{equation*} for $p<d-1$ and $p>d$. We extract the exact sequence \begin{equation*}
R^{d-1}\pi_{*}(\pi^{-1}\mathcal F \otimes Log\otimes or_{\mathcal T/\mathcal S})\rightarrow R^{d-1}\pi_{|U *}(\pi^{-1}\mathcal F \otimes Log\otimes or_{\mathcal T/\mathcal S}) \end{equation*} \begin{equation*}
\rightarrow \pi_{|D *}(\pi^{-1}\mathcal F \otimes Log_{|D})\rightarrow R^{d}\pi_{*}(\pi^{-1}\mathcal F \otimes Log\otimes or_{\mathcal T/\mathcal S}) \end{equation*} from our localization sequence. We have \begin{equation*} R^p\pi_*(\pi^{-1}\mathcal F \otimes Log\otimes or_{\mathcal T/\mathcal S})=\mathcal F\otimes R^p\pi_*(Log)\otimes R^{d}\pi_* (A)^*=0 \end{equation*} for $p\neq d$ and $R^{d}\pi_*(\pi^{-1}\mathcal F\otimes Log\otimes or_{\mathcal T/\mathcal S})=\mathcal F$ by the projection formula, since $\pi$ is proper and $\mathcal F$ is flat. So we have the short exact sequence \begin{equation*}
0 \rightarrow R^{d-1}\pi_{|U *}(\pi^{-1}\mathcal F\otimes Log\otimes or_{\mathcal T/\mathcal S})\rightarrow \pi_{|D *}(\pi^{-1}\mathcal F\otimes Log_{|D})\rightarrow \mathcal F \end{equation*} But the last arrow is obtained by applying $\mathcal F\otimes$ and the projection formula to the epimorphism
$\pi_{|D *}(Log_{\mathcal T/\mathcal S|D})\stackrel{\operatorname{aug}}{\rightarrow}\pi_{|D *}A\stackrel{tr}{\rightarrow}A$, and therefore is itself an epimorphism (for $tr$ see \cite{Iv} VII.4.). In particular \begin{equation*}
R^p\pi_{|U*}(\pi^{-1}\mathcal F\otimes Log\otimes or_{\mathcal T/\mathcal S})=0,\ p\neq d-1. \end{equation*} \end{proof} \end{proposition}
\begin{remark}\label{aug}
Let us recall the augmentation morphism $\pi_{|D *}(Log_{|D})\stackrel{\operatorname{aug}}{\rightarrow}\pi_{|D *}A\stackrel{tr}{\rightarrow}A$. It is enough to do so on the fibers $\pi^{-1}(s)$. So we have a topological torus $T$ and $D\subset T$ is just a finite set of points. We have to examine the chain of maps \begin{equation*}
H^0(D,Log_{T|D})\stackrel{purity\cong}{\rightarrow}H^{d}_D(T,Log_{T}\otimes or_T)\to H^{d}(T,Log_{T}\otimes or_T) \stackrel{\operatorname{aug}}{\rightarrow} \end{equation*} \begin{equation*}
\to H^{d}(T,or_T)\stackrel{\cong}{\rightarrow} H^{d}(T,A)\otimes or_T(T)\stackrel{ev}{\rightarrow}A \end{equation*}
We have $H^{d}_D(T,Log\otimes or_T)=\bigoplus_{x\in D} H^{d} _{\left\{x\right\}}(T,Log\otimes or_T)$ and isomorphisms \begin{equation*} H^0(\left\{x\right\},A)\stackrel{purity}{\rightarrow}H^{d}_{\left\{x\right\}}(T,A)\rightarrow H^{d}(T,A), \end{equation*}
as $T$ is orientable. We choose a generator $1(x)\in H^0(\left\{x\right\},A)$ and denote the corresponding images with $1_{\left\{x\right\}}\in H^{d}_{\left\{x\right\}}(T,A)$ and $\omega(x)\in H^{d}(T,A)$. We set $\omega(x)^*\in or_T(T)$ the element dual to $\omega(x)$. Given $\sum_x f(x)1(x)\in H^0(D,Log_{|D})$, $f(x)\in Log_x$, purity maps it to $\sum_x f(x)1_{\left\{x\right\}}\otimes \omega(x)^* \in H^{d}_Z(T,Log\otimes or_T)$. This is mapped by $\operatorname{aug}$ to $\sum_x \operatorname{aug}(f(x))\omega(x)\otimes\omega(x)^*\in H^{d}(T,or_T)$ and then by $ev$ to $\sum_x \operatorname{aug}(f(x))$ as claimed. \end{remark} \begin{proposition}\label{global_Log_localization} With notations from above we have an exact sequence \begin{equation*}
0\rightarrow H^{d-1}(U,\pi^{-1}\mathcal F\otimes Log\otimes or_{\mathcal T/\mathcal S})\stackrel{\operatorname{res}}{\rightarrow} H^0(D,\pi^{-1}\mathcal F\otimes Log_{|D}) \stackrel{\operatorname{aug}}{\rightarrow} H^{0}(\mathcal S,\mathcal F) \end{equation*} \begin{proof}
Using the Grothendieck-Leray spectral sequence for the composition of functors $\Gamma(U,-)=\Gamma(\mathcal S,-)\circ \pi_{|U*}$ we easily see \begin{equation*} H^{p}(U,\pi^{-1}\mathcal F\otimes Log\otimes or_{\mathcal T/\mathcal S})=
H^{p-(d-1)}(\mathcal S,R^{d-1}\pi_{|U*}(\pi^{-1}\mathcal F\otimes Log\otimes or_{\mathcal T/\mathcal S})), \end{equation*}
as $R^p\pi_{|U*}(\pi^{-1}\mathcal F\otimes Log\otimes or_{\mathcal T/\mathcal S})=0$, $p\neq d-1$. Applying $H^0(\mathcal S,-)$ to \Cref{relative_Log_localization} yields the result. \end{proof} \end{proposition} \begin{definition}\label{def_pol} We keep the notations from above.\\
Given $f\in H^0(D,\pi_{|D}^{-1}\mathcal F\otimes Log_{|D})$ with $\operatorname{aug}(f)=0$ we define the \textit{polylogarithm associated to $f$} as the unique cohomology class \begin{equation*} \operatorname{pol}(f)\in H^{d-1}(U,\pi^{-1}\mathcal F\otimes Log\otimes or_{\mathcal T/\mathcal S})\ \end{equation*} with $\operatorname{res}(\operatorname{pol}(f))=f$. If $0(\mathcal S)\cap D=\emptyset$, we define the \textit{polylogarithmic Eisenstein class associated to $f$} to be the pullback \begin{equation*} \operatorname{Eis}(f):= 0^{*}\operatorname{pol}(f)\in H^{d-1}(\mathcal S,\mathcal F\otimes 0^{-1}(Log\otimes or_{\mathcal T/\mathcal S}) ) \end{equation*} \end{definition}
\subsection{Trivializations of the logarithm sheaf}
Finding non-trivial polylogarithm classes is by definition equivalent to finding non-trivial $f\in H^0(D,\pi^{-1}\mathcal F\otimes Log_{|D})$ with $\operatorname{aug}(f)=0$. For which $D$ do we have sections or a trivialization for $Log_{|D}$?\\ We just write $Log^n=Log_{\mathcal T/\mathcal S} ^n$, when the topological situation $\pi:\mathcal T\rightarrow \mathcal S$ has been fixed. \begin{lemma}\label{0Log} The augmentation map $0^{-1}Log\rightarrow A$ splits. The image of $1\in A$ is denoted by $1_0\in H^0(\mathcal S,0^{-1}Log)$. $0^{-1}Log=:\mathcal R$ is a sheaf of augmented algebras and $Log$ is an invertible $\pi^{-1}\mathcal R$-module. \begin{proof} $0^{-1}Log$ is the local system associated to the $\pi_1(\mathcal S,s_0)$-module $A[[\pi_1(\mathcal T_{s_0},t_0)]]$. The augmentation map corresponds to the augmentation map of this algebra. It has the obvious $\pi_1(\mathcal S,s_0)$-equivariant section providing the section of the corresponding sheaves. Obviously $\pi_1(\mathcal S,s_0)$ acts by algebra homomorphisms on $A[[\pi_1(\mathcal T_{s_0},t_0)]]$, therefore $\mathcal R$ is a sheaf of algebras. $\pi^{-1}\mathcal R$ is associated to the $\pi_1(\mathcal T,t_0)$-module $A[[\pi_1(\mathcal T_{s_0},t_0)]]$ with the action $(v,\gamma)\sum_l f(l)(l)=\sum_lf(l)(\gamma l)$ for $(v,\gamma)\in\pi_1(\mathcal T,t_0)$. Let us denote this module by $A[[\pi_1(\mathcal T_{s_0},t_0)]]_{\mathcal R}$ and the module associated to $Log$ by $A[[\pi_1(\mathcal T_{s_0},t_0)]]_{Log}$ to emphasize their different $\pi_1(\mathcal T,t_0)$-structures. Then we see that the multiplication map \begin{equation*} A[[\pi_1(\mathcal T_{s_0},t_0)]]_{\mathcal R}\otimes_A A[[\pi_1(\mathcal T_{s_0},t_0)]]_{Log}\rightarrow A[[\pi_1(\mathcal T_{s_0},t_0)]]_{Log}, f\otimes g\mapsto f\cdot g \end{equation*} is $\pi_1(\mathcal T,t_0)$-equivariant proving that $Log$ is an $\pi^{-1}\mathcal R$-module. \end{proof} \end{lemma} We set $\mathcal R^n:=0^{-1}Log^n$. \begin{lemma} Let $\phi:\mathcal T^\prime \rightarrow \mathcal T$ be an isogeny of tori over $\mathcal S$. We consider the logarithm sheaf over a ring $A$ and suppose that $deg(\phi)(s)\in A^{\times}$ for all $s\in \mathcal S$, where $A^{\times}$ always denotes the units in a ring $A$. Then the natural morphism $Log_{\mathcal T^\prime/\mathcal S}\rightarrow \phi^{-1}Log_{\mathcal T/\mathcal S} $ is an isomorphism. \begin{proof} It suffices to consider the map of underlying $\pi_1(\mathcal T^\prime,t_0 ^\prime )$-modules. It is given by the natural map $A[[\pi_1(\mathcal T^\prime _{s_0},t_0^\prime)]]\stackrel{\phi_*}{\rightarrow} A[[\pi_1(\mathcal T _{s_0},t_0)]]$. This map is an isomorphism by \cite{BKL} proposition 1.1.5. \end{proof} \end{lemma} \begin{definition} Let $X$ be a topological space, $A\to A^\prime$ a ring homomorphism and $\mathcal G$ a sheaf of $A$-modules. We set $\mathcal G_{A^\prime}:=\mathcal G\otimes_A A^\prime$. \end{definition} Given a family of topological tori $\pi:\mathcal T\rightarrow \mathcal S$ with logarithm sheaf $Log$ we may consider $Log^{\times} :=\operatorname{aug}^{-1}(1)$ where $\operatorname{aug}:Log\rightarrow A$ is the augmentation map. $Log^{\times}$ is a sheaf of sets on $\mathcal T$. We may realize this sheaf as a sheaf of sections of a topological space $Log^{\times sp}$ over $\mathcal S$. To do so we set $R:=A[[\pi_1(\mathcal T_{s_0},t_0)]]$ and denote again by $\mathfrak a:=\operatorname{aug}^{-1}(0)$ its augmentation ideal. In $R$ we have the group of $1$-units $1+\mathfrak a=:R^{\times}_1$. The action of $\pi_1(\mathcal T,t_0)$ on $R$ restricts to $R^{\times}_1 $ making it a $\pi_1(\mathcal T,t_0)$-module. Considering $R^\times _1 $ with the discrete topology we get $Log^{\times sp}:=\tilde{\mathcal T}\times_{\pi_1(\mathcal T,t_0)} R^{\times} _1$. We had $\tilde{\mathcal T}=\tilde{\mathcal S}\times H_1(\mathcal T_{s_0},{\mathbb R})$ and therefore we may write $Log^{\times sp}=\tilde \mathcal S\times_{\pi_1(\mathcal S,s_0)}(H_1(\mathcal T_{s_0},{\mathbb R})\times_{\pi_1(\mathcal T_{s_0},t_0)}R^{\times} _1)$. The second description tells us that $Log^{\times sp}$ may be considered as a group object over $\mathcal S$ with typical fiber $H_1(\mathcal T_{s_0},{\mathbb R})\times_{\pi_1(\mathcal T_{s_0},t_0)}R^{\times} _1$. We also have the sheaf $\mathcal R^{\times}_1 $ of $1$-units of the augmented algebra $\mathcal R$. One has the inclusion $\delta:\pi_1(\mathcal T _{s_0},t_0)\subset R^{\times}_1$ giving rise to the inclusion $\delta: \mathcal H\rightarrow \mathcal R^{\times}_1 $. If we consider the fibers of $\mathcal T$ with the discrete topology, we may summarize this in the following pushout diagram of abelian sheaves on $\mathcal S$ \begin{equation*} \begin{xy} \xymatrix{ 0\ar[r] & \mathcal R^{\times}_1 \ar[r] & Log^{\times sp}\ar[r]^p & \mathcal T\ar[r] & 0\\ 0\ar[r] & \mathcal H_{\mathbb Z}\ar[u]^\delta \ar[r] & \mathcal H_{\mathbb R} \ar[u]\ar[r] & \mathcal T\ar[r]\ar[u]^{\operatorname{id}} & 0 } \end{xy} \end{equation*} where we consider $\mathcal T$ and $Log^{\times sp}$ as sheaves of sections. Let $H\subset \mathcal T$ be a subgroup. By this we mean that $H$ is a sub local system of $\mathcal T$. We also may consider $H$ as a subspace of $\mathcal T$ covering $\mathcal S$. Now we follow \cite{BKL} Definition 1.5.1. \begin{definition} A \textit{multiplicative trivialization of $Log$ along $H$} is a morphism $t:H\rightarrow Log^{\times sp}$, which is a section for $p:Log^{\times sp}\rightarrow \mathcal T$. \end{definition} \begin{remark}
A multiplicative trivialization of $Log$ gives a section $1_H\in \Gamma(H,Log_{|H})$, $\operatorname{aug}(1_H)=1$, trivializing $Log_{|H}$ as an invertible $\pi_{|H}^{-1}\mathcal R$-module. \end{remark} Let $\mathcal T^{(A)}$ be the subsheaf of $\mathcal T$ consisting of all torsion sections whose order is invertible in $A$ and $\mathcal T^{tors}$ the subsheaf of all torsion sections. \begin{proposition}\label{trivializationLog} There is a unique multiplicative trivialization $\rho_{can}:\mathcal T^{(A)}\rightarrow Log^{\times sp}$ for $Log$ along $\mathcal T^{(A)}$. It is compatible with isogenies and for $t\in \mathcal T[N](\mathcal S)\subset \mathcal T^{A}(\mathcal S)$ it is explicitly given by the isomorphisms \begin{equation*} t^{-1}Log\cong t^{-1}[N]^{-1}Log\cong ([N]\circ t)^{-1}Log\cong 0^{-1}Log, \end{equation*} where one uses $Log\cong [N]^{-1}Log$, as $deg[N]\in A^{\times}$, and $[N]\circ t= Nt= 0$, as $t$ is $N$-torsion. \begin{proof} By \cite{BKL} proposition 1.5.3. we have the statement, if $\mathcal S=\left\{pt.\right\}$. But as $Log^{\times sp}$ and $\mathcal T$ are fiber bundles over $\mathcal S$, we may easily glue these local multiplicative trivializations to a global one by using unicity. \end{proof} \end{proposition}
This proposition tells us that torsion sections are a powerful tool to construct $D\subset \mathcal T$ for which we can write down sections of $Log_{|D}$. But before we can start calculations with $Log$, let us have a closer look at the situation over $A={\mathbb C}$. \begin{definition} Let $X$ be a topological space and $\mathcal R$ a sheaf of rings. For a projective system $(\mathcal F_n)$, $\mathcal F:=\varprojlim \mathcal F_n$, of $\mathcal R$-module sheaves and an $\mathcal R$-module sheaf $\mathcal G$ on $X$ we set $\mathcal F\hat\otimes_\mathcal R \mathcal G:=\varprojlim (\mathcal F_n\otimes_\mathcal R \mathcal G)$. \end{definition} Let $\mathcal C^\infty _\mathcal T$ be the sheaf of complex valued smooth functions on $\mathcal T$ and $\Omega_\mathcal T$ the de Rham complex of complex valued smooth forms on $\mathcal T$. We set \begin{equation*} Log^{\infty n}:=Log^n\otimes_{\mathbb C} \mathcal C^\infty_\mathcal T\ \text{and}\ Log^\infty:=\varprojlim Log^{\infty n}=Log\hat{\otimes}_{\mathbb C} \mathcal C^\infty _\mathcal T \end{equation*} to be the pro vector bundle associated to $Log$. As it comes from local systems, it is naturally endowed with an integrable connection $\operatorname{id}\otimes d: Log^\infty\rightarrow Log^\infty\hat{\otimes}_{\mathcal C_\mathcal T ^\infty} \Omega_\mathcal T ^1$, whose kernel is exactly $Log$. \begin{proposition}\label{R_alg} Given a lattice $L$ one has the isomorphism of augmented algebras \begin{equation*} {\mathbb Q}[[L]]\rightarrow \prod_{k\geq 0} \operatorname{Sym}^k L_{\mathbb Q}, l\mapsto \exp(l):=\sum_{k\geq 0}\frac{l^{\otimes k}}{k!}, \end{equation*} where the right hand side carries the Cauchy product and the augmentation is the projection onto the $\operatorname{Sym}^0L_{\mathbb Q}={\mathbb Q}$- part. In particular, one has an isomorphism $\mathcal R\stackrel{\exp}{\rightarrow}\prod_{k\geq 0}\operatorname{Sym}^k\mathcal H_{\mathbb Q}$ of augmented ${\mathbb Q}$-algebras and an isomorphism of systems \begin{equation*} (\mathcal R^n)\stackrel{\exp}{\rightarrow}\left(\prod_{k=0}^nSym^k\mathcal H_{\mathbb Q}\right)_n \end{equation*} \begin{proof} \cite{BKL} Corollary 1.1.10. \end{proof} \end{proposition} \begin{remark}\label{Log_integral} This isomorphism does not work integrally. Nevertheless, let $A$ be a torsion free ring. We have \begin{equation*}
A[[L]]\rightarrow \prod_{k\geq 0} \operatorname{Sym}^k L_{PD}, l\mapsto \exp(l):=\sum_{k\geq 0}\frac{l^{\otimes k}}{k!}, \end{equation*} into a subring of $\prod_{k\geq 0} \operatorname{Sym}^k L_{A_{\mathbb Q}}$. Here we define the $A$-submodule $\operatorname{Sym}^k L_{PD}\subset \operatorname{Sym}^k L_{A_{\mathbb Q}}$ to be generated by $x_1^{[i_1]}\cdot..\cdot x_n^{[i_n]}$ with $x_1,...,x_n \in L$, $i_l\in {\mathbb N}_0$, $l=1,...,n$ and $\sum_{l=1}^ni_l=k$, where we define in any ${\mathbb Q}$-algebra $B$ the \textit{divided powers} $x^{[n]}:=\frac{x}{n!}$ for $x\in B$. If $e_1,...,e_d$ is a basis of $L$, consider $1\leq j_1<...< j_n\leq d$ and $i_l\in {\mathbb N}_0$, $l=1,...,n$, with $\sum_{l=1}^ni_l=k$, then $e_{j_1}^{[i_{1}]}\cdot..\cdot e_{j_n}^{[i_n]}$ is a basis of $\operatorname{Sym}^k L_{PD}$. The $\operatorname{Sym}^k L_{PD}$ give rise to local systems $\operatorname{Sym}^k \mathcal H_{PD}$ which are integral structures for $\operatorname{Sym}^k \mathcal H_{A_{\mathbb Q}}$. We have the projection map $p_k:\mathcal R\rightarrow \operatorname{Sym}^k \mathcal H_{PD}$ which is induced by $A[[L]]\stackrel{\exp}{\rightarrow} \prod_{n\geq 0} \operatorname{Sym}^n L_{PD}\stackrel{pr_k}{\rightarrow}\operatorname{Sym}^k L_{PD}$. \end{remark} \begin{remark} Transporting the action of the fundamental group via $\exp$ we may understand the logarithm sheaf with complex coefficients via the completed symmetric algebra, what we will always do from now on. In particular, we have $\mathcal R=\prod_{n\geq 0}\operatorname{Sym}^n\mathcal H_{\mathbb C}$. \end{remark} Set \begin{equation*} Log^{sp}:=\tilde{\mathcal T}\times_{\pi_1(\mathcal T,t_0)}\prod_{n\geq 0} \operatorname{Sym}^n H_1(\mathcal T_{s_0},{\mathbb C}). \end{equation*} The sections of $Log^\infty$ may now be seen as the smooth sections $s:\mathcal T\rightarrow Log ^{sp}$. These are $\pi_1(\mathcal T,t_0)$-equivariant maps $f:\tilde{\mathcal T}\rightarrow \prod_{n\geq 0} \operatorname{Sym}^n H_1(\mathcal T_{s_0},{\mathbb C})$ such that for all $n\geq 0$ the component maps $f_n:\tilde{\mathcal T}\rightarrow \operatorname{Sym}^n H_1(\mathcal T_{s_0},{\mathbb C})$ are smooth. In this sense it is also clear what we mean by smooth or continuous sections of $Log^{\times sp}$. \begin{proposition}\label{cont_triv}
There is a unique \textit{continuous multiplicative trivialization of $Log$}. By this we mean a continuous homomorphism $\rho_{cont}:\mathcal T\rightarrow Log^{\times sp}$, which is a section for $p:Log^{\times sp}\rightarrow \mathcal T$. It is compatible with $N$-multiplication and we have $\rho_{cont|\mathcal T^{tors}}=\rho_{can}$. Explicitly $\rho_{cont}$ is given by $\exp(-\underline v)=\sum_{n\geq 0}\frac{(-\underline v)^{\otimes n}}{n!}$, where \begin{equation*} \underline v:\tilde{\mathcal S}\times H_1(\mathcal T_{s_0},{\mathbb R}), (s,v) \mapsto v\in \prod_{n\geq 0} \operatorname{Sym}^n H_1(\mathcal T_{s_0},{\mathbb C}) \end{equation*} is considered as a $\mathcal C^\infty$-section of $Log$ on the universal cover of $\mathcal T$. \begin{proof} $\pi_1(\mathcal T,t_0)$ acts on $ \prod_{n\geq 0}\operatorname{Sym}^n H_1(\mathcal T_{s_0},{\mathbb C})$ by \begin{equation*} (l,\gamma)\sum_{n\geq 0}f_n=\exp(l)\cdot\sum_{n\geq 0}\gamma f_n. \end{equation*} From the equivariance of $\exp$ \begin{equation*} \exp(-(v(l,\gamma))=\exp(-(\gamma^{-1}(v+l)))=\exp(-\gamma^{-1}l)\cdot \gamma^{-1} \exp(-v)=(l,\gamma)^{-1}\exp(-v) \end{equation*} it is clear that the section above descends to a multiplicative trivialization as claimed. The problem of unicity is local on the base $\mathcal S$ and as all spaces are fiber bundles over $\mathcal S$, we may reduce to the case where $\mathcal S$ is a point. This is \cite{BKL} Lemma 2.1.2. \end{proof} \end{proposition} We set $\mathcal R^{\infty n}:=\pi^{-1}\mathcal R^n\otimes\mathcal C_\mathcal T ^\infty\ \text{and}\ \mathcal R^\infty=\varprojlim\mathcal R^{\infty n}=\pi^{-1}\mathcal R\hat{\otimes}\mathcal C_\mathcal T ^\infty$ the complex pro vector bundle associated to $\pi^{-1}\mathcal R$. $Log^\infty$ is now a module over this sheaf of algebras. We also have the sheaf of differential graded algebras $\pi^{-1}\mathcal R\hat{\otimes}\Omega_\mathcal T=\mathcal R^\infty\hat{\otimes}_{\mathcal C^\infty_\mathcal T}\Omega_\mathcal T$ with the trivial differential $\operatorname{id}\otimes d$. One has $\mathcal H_{\mathbb Z}\subset \mathcal R$ and $\mathcal H_{\mathbb C}^*\subset \Omega^1_\mathcal T$ interpreting sections of $\mathcal H_{\mathbb C}^*$ as invariant differential forms along the fibers of $\pi$. The canonical isomorphism \begin{equation*} End(\mathcal H_{\mathbb Z})\cong \mathcal H_{\mathbb Z}\otimes \mathcal H_{\mathbb Z}^*,\ \operatorname{id} \mapsto \kappa \end{equation*} gives the canonical global section $\kappa$ of $\pi^{-1}\mathcal R\hat{\otimes}\Omega_\mathcal T$. The section $\kappa$ is closed and multiplication with $\kappa$ gives an operator on our differential graded algebra. \begin{corollary} One has the isomorphism of pro vector bundles $\mathcal R^\infty\rightarrow Log^\infty$, $f\mapsto f\cdot\rho_{cont}$. If endow $\mathcal R^\infty$ with the integrable connection $\nabla:=d-\kappa$, this isomorphism commutes with the connections. \begin{proof} As $Log^\infty$ is an invertible $\mathcal R^\infty$-module the isomorphism is clear, since $\operatorname{aug}(\rho_{cont})=1$ and $\rho_{cont}$ is a global nowhere vanishing section of $Log^\infty$. To prove the compatibility of the connections we need to calculate $\operatorname{id}\otimes d(\rho_{cont})$. This can be done on the universal cover $\tilde{\mathcal T}$. We have \begin{equation*} \operatorname{id}\otimes d(\exp(-\underline v))=\operatorname{id}\otimes d\sum_{n\geq 0}\frac{(-\underline v)^{\otimes n}}{n!}=\sum_{n\geq0}\frac{\operatorname{id}\otimes d(-\underline v)^{\otimes n}}{n!}= \end{equation*} \begin{equation*} \sum_{n\geq0}\frac{n(-\underline{v} )^{\otimes n-1}\cdot (\operatorname{id}\otimes d(-\underline v))}{n!}=\sum_{n\geq0}\frac{(-\underline v)^{\otimes n}}{n!}\operatorname{id}\otimes d(-\underline v) \end{equation*} Write $\underline v=\sum_{i=1} ^{d}e_i\otimes x_i$ for some ${\mathbb R}$-basis $(e_i)$ of $H_1(\mathcal T_{s_0},{\mathbb R})$, where $(x_i)$ is the corresponding dual basis of $(e_i)$ considered as coordinate functions on $H_1(\mathcal T_{s_0},{\mathbb R})$. Then \begin{equation*} \operatorname{id}\otimes d(\underline v)=\operatorname{id}\otimes d\sum_{i=1} ^{d}e_i\otimes x_i=\sum_{i=1} ^{d}e_i\otimes dx_i=\kappa \end{equation*} We conclude $\operatorname{id}\otimes d(\rho_{cont})=\rho_{cont}\cdot(-\kappa)$. This completes the proof. \end{proof} \end{corollary} Now we come to our case of interest. \begin{proposition}\label{pol} Let $\pi:\mathcal T\rightarrow \mathcal S$ be family of topological tori with level-$N$-structure and constant fiber dimension $d$. We consider $D=\mathcal T[N]\setminus 0(\mathcal S)\cong (({\mathbb Z}/N{\mathbb Z})^{d}\setminus \left\{0\right\})\times \mathcal S$ and the logarithm sheaf $Log$ over a noetherian ring $A$ with $N\in A^{\times}$. We have then a canonical trivialization
$H^0(D,Log_{|D})=H^0((({\mathbb Z}/N{\mathbb Z})^{d}\setminus \left\{0\right\})\times \mathcal S,\mathcal R)$. One has $A\subset \mathcal R$ and thus any locally constant function \begin{equation*} f:(({\mathbb Z}/N{\mathbb Z})^{d}\setminus \left\{0\right\})\times \mathcal S\rightarrow A,\ \operatorname{aug}(f)=\sum_{v\in({\mathbb Z}/N{\mathbb Z})^{d}\setminus \left\{0\right\}}f(v,\ )=0\in H^0(\mathcal S,A) \end{equation*}
gives rise to a section $f\in H^0(D,Log_{|D})$ of augmentation zero and therefore to a unique polylogarithmic cohomology class $\operatorname{pol}(f)\in H^{d-1}(U,Log\otimes or_{\mathcal T/\mathcal S})$. Moreover, we get the polylogarithmic Eisenstein class $0^{*}\operatorname{pol}(f)=:\operatorname{Eis}(f)\in H^{d-1}(\mathcal S,\mathcal R\otimes 0^{-1}or_{\mathcal T/\mathcal S})$, which induces polylogarithmic Eisenstein classes \begin{equation*} \operatorname{Eis}^k(f)\in H^{d-1}(\mathcal S,\operatorname{Sym}^k\mathcal H_{PD}\otimes 0^{-1}or_{\mathcal T/\mathcal S}), \end{equation*} whenever $A$ is torsion-free. \begin{proof} $A\subset \mathcal R$ is \Cref{0Log}. The trivialization is \Cref{trivializationLog}. Finally, we just need \Cref{aug} and \Cref{def_pol} to get $\operatorname{pol}(f)$. We set $\operatorname{Eis}^k(f):=p_k(\operatorname{Eis}(f))$, where $p_k:\mathcal R\rightarrow \operatorname{Sym}^k\mathcal H_{PD}$ is the projection defined in \Cref{Log_integral}. \end{proof} \end{proposition} \begin{remark} The restriction to torsion-free rings in the proposition above is not necessary. To get the appropriate substitute for our $\operatorname{Sym}^k\mathcal H_{PD}$ one has to use the universal divided power algebra $\Gamma_k(L)$, see \cite{BKL} Lemma 1.1.7. \end{remark}
\subsection{A differential equation for the polylogarithm}
We want explicit cohomology classes representing our polylogarithms over ${\mathbb C}$. To do so we need an explicit resolution of $Log$ over ${\mathbb C}$, which allows us to calculate the localization sequence in \Cref{global_Log_localization}. \begin{lemma} Let $X$ be an oriented $n$-dimensional $\mathcal C^\infty$-manifold and $\Omega_X=(\Omega^\bullet _X ,d)$ the de Rham complex of ${\mathbb C}$-valued $\mathcal C^\infty$-differential forms. We may represent the dualizing complex of $\mathcal D_X$ by non-continuous functionals on the de Rham complex: \begin{equation*} \mathcal D_X ^p (U)=\Gamma_c(U,\Omega_X ^{-p})^*,\ U\subset X\ \text{open} \end{equation*} with differential $d$ defined by \begin{equation*} d(T)(\omega):=(-1)^{p+1}T(d(\omega)),\ \omega\in \Gamma_c(U,\Omega_X ^{-p})^*,\ T\in \Gamma_c(U,\Omega_X ^{-p})^* \end{equation*} \begin{proof} The Poincar\'{e} Lemma gives a quasi-isomorphism ${\mathbb C}\rightarrow \Omega_X$. So $\Omega_X$ is a bounded resolution of the constant sheaf ${\mathbb C}$ by soft ${\mathbb C}$-sheaves. We conclude with \cite{Iv} V.2.1. \end{proof} \end{lemma} \begin{remark} $\mathcal D_X$ is a complex of injective ${\mathbb C}$-sheaves which also have the structure of $\mathcal C_X ^\infty$-modules. One has a canonical quasi-isomorphism $or_X[n]=H^{-n}(\mathcal D_X)[n]\rightarrow \mathcal D_X$ (\cite{Iv}VI.3.). By choosing a volume form on $X$ we may define integration of $n$-forms. In other words, integration defines a functional $\Gamma_c(U,\Omega_X ^n)\rightarrow {\mathbb C}$, $\omega\mapsto \int_X \omega$ for any open $U\subset X$. This functional vanishes on exact forms by Stoke's theorem and therefore $\int_X$ defines an element in $or_X(X)=H^{-n}(\mathcal D_X)(X)$. It is nowhere vanishing and we get the isomorphism ${\mathbb C}\to or_X,\ 1\mapsto \int_X$. In particular, we get the quasi-isomorphism ${\mathbb C}[0]\to \mathcal D_X[-n]$.
Now let $i:Z\hookrightarrow X$ be a closed oriented submanifold of codimension $c$.\\ We have the pullback morphism $i^*:\Omega_X\rightarrow i_*\Omega_Z$. As $i$ is proper we also get for any $U\subset X$ open $i^*:\Gamma_c(U,\Omega_X)\rightarrow \Gamma_c(U,i_*\Omega_Z)$ and by taking duals \begin{equation*} i_*:\Gamma_c(U,i_*\Omega_Z)^*\rightarrow \Gamma_c(U,\Omega_X)^* \end{equation*} compatible with restrictions in $U$. So identifying $\mathcal D_X$ with the sheaf of non-continuous functionals on $\Omega_X$ we get $i_*\mathcal D_Z\rightarrow \mathcal D_X$ and by adjunction $i_*:\mathcal D_Z\rightarrow i^!\mathcal D_X$. This is exactly the arrow inducing \Cref{dualizing_purity}. We have then \begin{equation*} H^0(Z,{\mathbb C})=H^0(Z,or_Z)=H^0(Z,\mathcal D_Z[c-n])=H^0(Z,i^!\mathcal D_X[c-n])= \end{equation*} \begin{equation*} H^0_Z(X,\mathcal D_X[c-n])=H^c_Z(X,or_X)=H^c_Z(X,{\mathbb C}) \end{equation*} Here we have exactly \begin{equation*} H^0(Z,{\mathbb C})\rightarrow H^c_Z(X,{\mathbb C}),\ f\mapsto \int_f \end{equation*} with $\int_f\omega:=\int_Z f\cdot i^*\omega$ and integration is defined by the orientation defining ${\mathbb C}\cong or_Z$. \end{remark} \begin{proposition} Let $\pi:\mathcal T\rightarrow \mathcal S$ be a family of topological tori, $Log=\varprojlim Log^n$ the associated logarithm sheaf over ${\mathbb C}$ and $\mathcal D_\mathcal T$ the dualizing complex on $\mathcal T$. The system of complexes $ (Log^n\otimes or_{\mathcal T}^{-1} \otimes \mathcal D_\mathcal T[-dim(\mathcal T)])_n $ is an injective resolution of $(Log^n)$ in $Sh(\mathcal T,{\mathbb C})^{\mathbb N}$. In particular, $ \varprojlim (Log^n\otimes or_{\mathcal T}^{-1} \otimes \mathcal D_\mathcal T)[-dim(\mathcal T)] $ is an injective resolution of $Log$. \begin{proof} We already know that $\mathcal D_\mathcal T ^p$ is injective in $Sh(\mathcal T,{\mathbb C})$. As $Log^n \otimes or_\mathcal T ^{-1}$ is a local system the sheaf $Log^n\otimes or_{\mathcal T}^{-1}\otimes \mathcal D_\mathcal T ^p$ is injective in $Sh(\mathcal T,{\mathbb C})$. In order to see that $ (Log^n\otimes or_{\mathcal T}^{-1} \otimes \mathcal D_\mathcal T ^p)_n $ is an injective object in $Sh(\mathcal T,{\mathbb C})^{\mathbb N}$ we need to see that all transition maps are split epimorphisms. Now $\mathcal D_\mathcal T ^p$ is a $\mathcal C^\infty_\mathcal T$-module and therefore $ ( Log^n\otimes or_{\mathcal T}^{-1} \otimes \mathcal D_\mathcal T ^p)_n\cong ( Log^{\infty n}\otimes_{\mathcal C^\infty _\mathcal T} (or_{\mathcal T}^{-1}\otimes \mathcal D_\mathcal T ^p)). $ We have an isomorphism of systems $(Log^{\infty n})_n=(\mathcal R^{\infty n})_n$ and the latter system has split epimorphic transition maps by \Cref{R_alg}. We conclude that $( Log^n\otimes or_{\mathcal T}^{-1} \otimes \mathcal D_\mathcal T ^p)_n$ has split epimorphic transition maps and therefore is injective in $Sh(\mathcal T,{\mathbb C})^{\mathbb N}$. $(Log^n\otimes or_{\mathcal T}^{-1} \otimes \mathcal D_\mathcal T[-dim(\mathcal T)] )_n$ being a resolution of $( Log^n )_n$ may be checked for each $n\in {\mathbb N}_0$ separately by using the quasi-isomorphism $or_\mathcal T[dim(\mathcal T)] \rightarrow \mathcal D_\mathcal T$. \end{proof} \end{proposition}
\begin{proposition}\label{differential_equ} Let $\mathcal S$ be an oriented manifold and $\pi:\mathcal T\rightarrow \mathcal S$ a family of topological tori with level-$N$-structure and constant fiber dimension $d$. We consider the logarithm sheaf $Log$ with ${\mathbb C}$-coefficients. The polylogarithms as defined in \Cref{pol} may be represented by non-continuous functionals \begin{equation*} \operatorname{pol}(f)\in \Gamma(\mathcal T,\mathcal R^\infty\hat{\otimes}_{\mathcal C^\infty _\mathcal T} (\pi^{-1}or_\mathcal S ^{-1}\otimes \mathcal D_\mathcal T))\ \text{with}\ \nabla (\operatorname{pol}(f)) =\operatorname{vol}_\mathcal S\otimes\int_f \end{equation*} Here the differential $\nabla$ is induced by $(\mathcal R^\infty,\nabla)$ and integration on $\mathcal S$ is defined with respect to a chosen volume form $\operatorname{vol}_\mathcal S$ which we also use to trivialize $or_\mathcal S^{-1}$. \begin{proof} We set $dim(\mathcal T)=m$. To calculate cohomology we can use the injective resolutions $ \varprojlim (Log^n\otimes or_{\mathcal T/\mathcal S}\otimes or_\mathcal T ^{-1}\otimes \mathcal D_\mathcal T )[-m] $ with differential $\operatorname{id}\otimes d$. We write these complexes as $ Log^\infty\hat{\otimes}_{\mathcal C^\infty _\mathcal T} (\pi^{-1}or_\mathcal S ^{-1}\otimes \mathcal D_\mathcal T )[-m] $ using the $\mathcal C^\infty_\mathcal T$-module structure of $\mathcal D_\mathcal T$ and the rules for orientations. We have a chain of quasi-isomorphisms \begin{equation*}
\pi_{|D}^{-1}\mathcal R\stackrel{\rho_{can}}{\rightarrow}Log_{|D}\rightarrow \varprojlim (Log_{|D}^n\otimes or_D ^{-1}\otimes \mathcal D_D)[-(m-d)]\stackrel{\operatorname{id}\otimes i_*}{\rightarrow} \end{equation*} \begin{equation*}
\varprojlim(Log_{|D}^n\otimes or_D ^{-1}\otimes i^!\mathcal D_\mathcal T)[-(m-d)]\stackrel{\cong}{\rightarrow}i^!\varprojlim (Log^n\otimes \pi^{-1} or_\mathcal S ^{-1}\otimes \mathcal D_\mathcal T)[-(m-d)]. \end{equation*} The last isomorphism is \Cref{!and-1}. If we identify \begin{equation*} i^!\varprojlim (Log^n\otimes \pi^{-1} or_\mathcal S ^{-1}\otimes \mathcal D_\mathcal T)[-(m-d)]=i^!(Log^\infty\hat{\otimes}_{\mathcal C_\mathcal T ^\infty}(\pi^{-1} or_\mathcal S ^{-1}\otimes \mathcal D_\mathcal T))[-(m-d)], \end{equation*}
we may give a global section of this sheaf namely $\rho_{cont}\otimes \operatorname{vol}_\mathcal S \otimes \int_f$. If we go through the previous morphisms remembering $\rho_{can}=\rho_{cont|\mathcal T^{tors}}$, we see that $f$ maps exactly to this section. So we just have proved \begin{equation*}
H^0(D,\mathcal R)\stackrel{\rho_{can}}{\rightarrow}H^0(D,Log_{|D})\stackrel{purity}{\rightarrow}H^d_D(\mathcal T,Log\otimes or_{\mathcal T/\mathcal S}),\ f\mapsto \rho_{cont}\otimes \operatorname{vol}_\mathcal S \otimes \int_f. \end{equation*} Now we use the continuous trivialization $\rho_{cont}$ of $Log$ to identify our injective resolutions from above with $\mathcal R^\infty\hat{\otimes}_{\mathcal C_\mathcal T ^\infty} (\pi^{-1}or_\mathcal S ^{-1}\otimes \mathcal D_\mathcal T) [-m]$. We call the differential of the last complex $\nabla$, as it is induced by $(\mathcal R^\infty,\nabla)$. Clearly, $f$ corresponds now to \begin{equation*} \operatorname{vol}_\mathcal S\otimes \int_f\in H^d(D,i^!(\mathcal R^\infty\hat{\otimes}_{\mathcal C_\mathcal T ^\infty} (\pi^{-1}or_\mathcal S ^{-1}\otimes \mathcal D_\mathcal T) [-m])). \end{equation*} The residue map $\operatorname{res}$ of our localization sequence is nothing but a connecting homomorphism of a long exact cohomology sequence. So being in the image of $\operatorname{res}$ just means that there is a \begin{equation*} \operatorname{pol}(f)\in \Gamma(\mathcal T,\mathcal R^\infty\hat{\otimes}_{\mathcal C_\mathcal T ^\infty} (\pi^{-1}or_\mathcal S ^{-1}\otimes \mathcal D_\mathcal T )[-m]),\ \nabla(\operatorname{pol}(f))=\operatorname{vol}_\mathcal S\otimes \int_f. \end{equation*} \end{proof} \end{proposition}
\subsection{Invariant functionals and quotient manifolds}
To represent polylogarithms with functionals it seems promising to pullback the situation to the universal cover of $\mathcal T$, because the logarithm sheaf is constant there. We would like to write down sections invariant under the $\pi_1(\mathcal T,t_0)$-action descending to sections representing polylogarithms. This would be exactly the right thing, if we knew that polylogarithms may be represented by differential forms. The problem is that we cannot pullback functionals in a naive way, as the universal covering $p:\tilde{\mathcal T}\rightarrow \mathcal T$ is usually not proper (finite). But things still work the other way round, invariant functionals on $\tilde{\mathcal T}$ descend to functionals on $\mathcal T$. This is what we explain now.
Let $X$ be a $\mathcal C^\infty$-manifold and $G$ a group acting on $X$ by diffeomorphisms from the right. So for $g\in G$ we have the diffeomorphism $r_g:X\to X,\ x\mapsto xg $. We assume this action properly discontinuous and fixpoint free. The quotient space with the quotient topology has a unique $\mathcal C^\infty$-structure making the natural cover $\pi:X\rightarrow X/G$ a local diffeomorphism. We are interested in currents on $X/G$. For a definition of currents see \cite{DR} chapitre III $8$. Denote the currents on a manifold $V$ by $\mathscr{D}(V)$ and the sheaf of currents on $V$ by $\mathscr{D}_V$. \begin{proposition}\label{invariant_functionals} Let $X$ be a $G$-manifold as described above. There is an isomorphism \begin{equation*} (\pi_*\mathscr{D}_X)^G\rightarrow \mathscr{D}_{X/G}, T\mapsto \tilde T \end{equation*} \begin{proof}
Let us first construct the map. Take $U\subset X/G$ open and $T\in \mathscr{D}(\pi^{-1}(U))^G$ and a $\omega\in \Omega_c(U)$. We have to define $\tilde T(\omega)$. To do so take an open cover $\bigcup_i U_i= X/G$ together with $U_i ^\prime \subset X$ open, such that $\pi_{|U_i ^\prime}:U_i ^\prime\rightarrow U_i$ is a diffeomorphism. Choose a partition of unity $\sum_i \epsilon_i=1$ subordinated $U_i$. The support of $\epsilon_i \omega$ is compact and contained in $U_i\cap U$, so the the support of $\pi_{|U_i ^\prime} ^* \epsilon_i \omega$ is compact and contained in $U_i^\prime\subset \pi^{-1}(U)$. We extend the latter differential form by zero outside $U_i ^\prime$ and set \begin{equation*}
\tilde T(\omega):=\sum_i T(\pi_{|U_i ^\prime} ^* \epsilon_i \omega)=T(\sum_i \pi_{|U_i ^\prime} ^* \epsilon_i \omega) \end{equation*} The sums are finite (\cite{DR} Th\'{e}or\`{e}me 1) and $\tilde T$ is linear.
Consider a sequence of forms $(\omega_n)$ with $\text{supp}(\omega_n)\subset K\subset V$, where $K$ is compact, $V$ a coordinate patch and $\omega_n\rightarrow 0$ uniformly together with all its derivatives. By the Leibniz-rule $\epsilon_i \omega_n$ converges in the same way as $\omega$. We have that $\pi_{|U_i ^\prime} ^{-1}(V)$ is again a coordinate patch on $X$ and, since $\pi_{|U_i ^\prime}$ is a diffeomorphism, $\pi_{|U_i ^\prime} ^* \epsilon_i \omega_n$ is a sequence of forms with $\text{supp}(\pi_{|U_i ^\prime} ^* \epsilon_i \omega_n)\subset K^\prime\subset V^\prime$, where $K^\prime$ is compact, $V^\prime$ a coordinate patch and $\pi_{|U_i ^\prime} ^* \epsilon_i \omega_n\rightarrow 0$ uniformly together with all its derivatives. Since $T$ is continuous, we get
$\tilde T(\omega_n)=\sum_i T(\pi_{|U_i ^\prime} ^* \epsilon_i \omega_n)\rightarrow 0$ as $n$ tends to infinity. Therefore $\tilde T$ is continuous.
Up to now $\tilde T$ seems to depend on the partition of unity $(U_i,\epsilon_i)$ and on the choice of the $U_i ^\prime$. We will show that this is not the case. We start with the independence of choice of the $U_i ^\prime$. Given $U_i ^\prime $ any other set mapping via $\pi$ diffeomorphic to $U_i$ is of the form $r_g(U_i ^\prime) $. We get: \begin{equation*}
T(\pi_{|r_g(U_i ^\prime)}^*\epsilon_i \omega)= T((\pi_{|U_i ^\prime}\circ r_g ^{-1})^*\epsilon_i \omega)=T(r_g ^{-1*}\pi_{|U_i ^\prime}^*\epsilon_i \omega)=T(\pi_{|U_i ^\prime}^*\epsilon_i \omega),
\end{equation*} since $T$ is $G$-invariant. Next we show the independence of the partition of unity. We assume now that we have two partitions of unity $(U_i,\epsilon_i )$ and $(V_j,\eta_j )$. By the first step we may assume that whenever $U_i\cap V_j\neq \emptyset$ we have $U_i ^\prime\cap V_j ^\prime\neq \emptyset$. We may consider $(U_i \cap V_j ,U_i ^\prime \cap V_j ^\prime,\epsilon_i \eta_j)$. It is a partition of unity as the others. We get \begin{equation*}
\sum_iT(\pi_{|U_i ^\prime}^*\epsilon_i \omega)=\sum_iT(\pi_{|U_i ^\prime}^*\epsilon_i \sum_j\eta_j\omega)=
\sum_{i,j}T(\pi_{|U_i ^\prime\cap V_j ^\prime}^*\epsilon_i \eta_j\omega), \end{equation*} from which the independence of the partition follows. The construction is of course compatible with restrictions in $U$. This means that we have the desired morphism of sheaves $(\pi_*\mathscr{D}_X)^G\rightarrow \mathscr{D}_{X/G}$, $T\mapsto \tilde T$. It remains to show that it is an isomorphism. This is a local problem. Choose a coordinate patch $U\subset X/G$ such that $\pi^{-1}(U)=\coprod_{g\in G} r_g(U^\prime)$ with all $r_g(U^\prime)$ coordinate patches. One has the isomorphism \begin{equation*} (\pi_*\mathscr{D}_X)^G(U)=(\prod_{g\in G} \mathscr{D}(r_g(U^\prime)))^G\rightarrow \mathscr{D}(U^\prime),\ (T_g)\mapsto T_{\operatorname{id}} \end{equation*}
with inverse $T\mapsto (g^{-1}T)$. Now $\pi_{^|U^\prime}: U^\prime\rightarrow U$ is a diffeomorphism and we see by pushforward $\pi_{|U^\prime *}:\mathscr{D}(U^\prime)\cong \mathscr{D}(U)$ that the composite map \begin{equation*}
\mathscr{D}(U^\prime)\cong(\pi_*\mathscr{D}_X)^G(U)\rightarrow \mathscr{D}_{X/G}(U), T\mapsto \pi_{|U^\prime *}T=\tilde T. \end{equation*} is an isomorphism. This completes the proof. \end{proof} \end{proposition} \begin{remark} \begin{enumerate} \item Differentiation of currents is a $G$-equivariant morphism. \item $(\pi_*\mathscr{D}_X)^G\rightarrow \mathscr{D}_{X/G}$, $T\mapsto \tilde T$
is a morphism of complexes with respect to the differentiation of currents. Let $\omega$ be a form with compact support on $X/G$ and $T$ a invariant current on $X$. As the problem is local, we may suppose that $\text{supp}(\omega)\subset U$, with $\pi_{|U^\prime}:U^\prime\rightarrow U$ a diffeomorphism. Then \begin{equation*}
\widetilde{dT}(\omega)=(dT)(\pi_{|U ^\prime}^*\omega)=(-1)^{deg(T)}T(d\pi_{|U ^\prime}^*\omega)=
(-1)^{deg(T)}T(\pi_{|U ^\prime}^* d\omega)=d\tilde T(\omega) \end{equation*} \item The whole theory of also works for non-continuous functionals. \end{enumerate} \end{remark}
\chapter{Polylogarithmic Eisenstein classes for Hilbert-Blumenthal varieties}
We will recall the Hilbert-Blumenthal varieties $\mathcal S_K$ also considered by \cite{Ha1}. Our goal is to give a direct ${\mathbb Q}$-rational construction of Harder's Eisenstein cohomology classes. To do so we construct a fiber bundle $\mathcal M_K$ over $\mathcal S_K$. Over $\mathcal M_K$ there is a family of topological tori and we can apply our construction of polylogarithmic Eisenstein classes. In order to get classes on $\mathcal S_K$ in different cohomological degrees we have to decompose the cohomology of $\mathcal M_K$.
\section{The topological situation}
\subsection{Notation}
Let $F$ be a totally real number field of degree $[F:{\mathbb Q}]=\xi$ and $\mathcal O\subset F$ its ring of integers. We denote places of $F$ by $\nu$. If $\nu$ is an archimedean place, we write $\nu|\infty$ and call these places also the infinite places. Non-archimedean places correspond to maximal ideals $\mathfrak p\subset \mathcal O$ and we call these places the finite places. If we need to emphasize that $\nu$ corresponds to the maximal ideal $\mathfrak p$, we write $\nu_{\mathfrak p}$ instead of $\nu$. We have $\mathfrak p\cap {\mathbb Z}=(p)$ for a prime number $p\in{\mathbb N}$ and write $\nu| p$ in this case. Infinite places may be identified with field embeddings $\sigma:F\rightarrow {\mathbb R}$. If $A$ is any ${\mathbb R}$-algebra, we have an isomorphism \begin{equation*}
F_A:=F\otimes_{\mathbb Q} A= \prod_{\sigma:F\rightarrow {\mathbb R}} A=\prod_{\nu|\infty}A,\ x\otimes a\mapsto (\sigma(x)a)_\sigma. \end{equation*}
In this manner we may speak of the $\nu$-component of $z\in F\otimes_{\mathbb Q} A$. The completion of $F$ with respect to a place $\nu$ is denoted by $F_\nu$. If $\nu|p$, $\mathcal O_\nu$ is the integral closure of ${\mathbb Z}_p$ inside $F_\nu$ and $\mathfrak p_\nu=(\pi_\nu)\subset\mathcal O_\nu$ is the maximal ideal with residue field $\kappa(\mathfrak p _\nu):=\mathcal O_\nu/\mathfrak p_\nu=\mathcal O/\mathfrak p=:\kappa(\mathfrak p)$. For any place $\nu$ we have a normalized absolute value:
If $\nu|\infty$, $|x|_\nu$ is the usual absolute value of $x\in F_\nu={\mathbb R}$. If $\nu$ is finite, we have $|x|_\nu:=|\kappa(\mathfrak p_\nu)|^{-\nu(x)}=|\mathcal O_\nu/(x)|^{-1}$ provided $x\in \mathcal O_\nu\setminus \left\{0\right\}$. $\nu$ is considered here as a discrete valuation on $F_\nu$ and may also be defined by $x=u\pi_\nu^{\nu(x)}$, $u\in \mathcal O_\nu ^{\times}$.
We have the ring of adeles ${\mathbb A}_F$ and the group of ideles $\mathbb{I}_F$ over $F$. We set ${\mathbb A}={\mathbb A}_{\mathbb Q}$ and $\mathbb{I}_{\mathbb Q}=\mathbb{I}$. We have a decomposition into a finite and an infinite part ${\mathbb A}_F={\mathbb A}_{F,\infty}\times {\mathbb A}_{F,f}$, $\mathbb{I}_F=\mathbb{I}_{F,\infty}\times \mathbb{I}_{F,f}$. We define $\hat{\mathbb Z}:=\varprojlim_{n\in{\mathbb N}} {\mathbb Z}/n{\mathbb Z}$ to be the profinite completion of ${\mathbb Z}$ and recall ${\mathbb A}_{F,f}=\hat{\mathbb Z}\otimes_{\mathbb Z} F$. We also set $\hat \mathcal O:=\hat{\mathbb Z}\otimes_{\mathbb Z} \mathcal O$.
We consider $G_0:=GL_{2}$ and $V_0:=\mathbb \mathbb{G}_{a} ^2$ as algebraic groups over $spec(\mathcal O)$, where the first group acts naturally on the second by matrix multiplication from the left. For any algebraic subgroup $H_0 \subset V_0\rtimes G_0$ we write $H:=Res_{\mathcal O/{\mathbb Z}}H_0$ for the restriction of scalars. We have for example $B_0\subset G_0$ the standard Borel of upper triangular matrices, $U_0\subset B_0$ its unipotent radical, $T_0\subset B_0$ the maximal torus and $Z_0\subset G_0$ the center. For any ring $R$ we write $H(R)$ for the group of $R$ valued points of $H$.\\ We have $H({\mathbb A})\subset \prod_\nu H(F_\nu)$, so $h=(h_\nu)\in H({\mathbb A})$ is determined by its $\nu$-components $h_\nu\in H(F_\nu)$.
$h\in H({\mathbb A})$ may also be uniquely written as $h=h_\infty h_f$ with $h_f\in H({\mathbb A}_f)$ and $h_\infty\in H({\mathbb R})$. We will deal with closed subgroups $K\subset H({\mathbb A})$ of the form $K=K_\infty\times K_f$ with closed subgroups $K_f\subset H({\mathbb A}_f)$ and $K_\infty \subset H({\mathbb R})$. We consider $K_\infty$ as a Lie group in a natural way and for any Lie group $L$ we denote the connected component of the identity by $L^0$. We will identify $Z({\mathbb R})=(F\otimes_{\mathbb Q}{\mathbb R})^{\times}=\prod_{\nu |\infty}{\mathbb R}^{\times}$ and ${\mathbb R}^{\times}$ embedded diagonally. We have $G({\mathbb R})=G_0(F\otimes_{\mathbb Z} {\mathbb R})=\prod_{\nu|\infty} GL_2({\mathbb R})$ and subgroups \begin{equation*}
K_\infty:=\prod_{\nu|\infty}K_\nu,\ K_\infty ^1:={\mathbb R}_{>0}\cdot\prod_{\nu|\infty}K_\nu ^1, \end{equation*} where $K_\nu:={\mathbb R}_{>0} SO(2)$, $K_\nu ^1:=SO(2)$ and ${\mathbb R}_{>0}$ the positive real numbers. We identify the group of connected components of $GL_2({\mathbb R})$ with $\left\{\begin{pmatrix}1 & 0\\0 & \epsilon\end{pmatrix}:\ \epsilon=\pm 1\right\}$. In this way we may view $\left\{\pm 1\right\}^\xi\cong\pi_0(G({\mathbb R}))\subset G({\mathbb R})$ as a subgroup normalizing $K_\infty$. \subsection{The spaces} Take a compact open subgroup $K_f\subset G({\mathbb A}_f)$. Our main examples will be \begin{equation*} \ker\left(G(\hat{\mathbb Z})\rightarrow G({\mathbb Z}/N{\mathbb Z})\right),\ N\in {\mathbb N}, \end{equation*} and its conjugates. Set $K:=K_\infty K_f\subset G({\mathbb A})$, $K^1:=K_\infty ^1 K_f\subset G({\mathbb A})$ and \begin{equation*} K_N:=\ker\left(G(\hat{\mathbb Z})\rightarrow G({\mathbb Z}/N{\mathbb Z})\right)K_\infty. \end{equation*} We define spaces \begin{equation*} \mathcal S_K:=K\backslash G({\mathbb A})/G({\mathbb Q})\ \text{and }\mathcal M_K:=K^1\backslash G({\mathbb A})/G({\mathbb Q}) \end{equation*} by taking double quotients. These two spaces are related by the obvious projection map $\phi_K:\mathcal M_K\rightarrow \mathcal S_K$.
\begin{remark}\label{spaces_facts} Let us recall some facts which can be found mainly in \cite{Ha1} 1.0 and 1.1. \begin{itemize} \item $\mathcal S_K$ parametrizes abelian varieties $A$ over $spec({\mathbb C})$ of dimension $\xi$ with real multiplication by $\mathcal O$ and a level-$K_f$-structure (\cite{Mi}, Theorem 1.2). $\mathcal S_K$ can be realized as the ${\mathbb C}$-points of a scheme defined over ${\mathbb Q}$. If $\xi>1$, there is no universal abelian scheme above $\mathcal S_K$, as there are always non-trivial automorphisms of our abelian varieties respecting the $\mathcal O$-structure and the level-$K_f$-structure . To see this consider \begin{equation*} Z_K:=Z({\mathbb Q})\cap K_f\subset Z({\mathbb Q})\cap G(\hat{\mathbb Z})=\mathcal O^{\times} \end{equation*} a subgroup of finite index. If $\xi>1$, $Z_K$ is never trivial, because the rank of this abelian group is positive by Dirichlet's unit theorem. \item $K_f\backslash G({\mathbb A}_f)$ is discrete and countable. \item Set $\mathbb{H}^\xi_{\pm}:=(F\otimes_{\mathbb Q} {\mathbb C})^{\times}\setminus (F\otimes_{\mathbb Q} {\mathbb R})^{\times}$ and denote the the connected component of $1\otimes i$ by $\mathbb{H}^\xi$. We identify $1\otimes i$ with the imaginary unit $i$ and get the diffeomorphism \begin{equation*} K_\infty\backslash G({\mathbb R}) \rightarrow \mathbb{H}^{\xi}_{\pm},\ g=\begin{pmatrix} a&b\\c&d\end{pmatrix}\mapsto \frac{b+id}{a+ic}=:\tau=\tau(g) \end{equation*} So $G({\mathbb R})$ acts on $\mathbb{H}^{\xi}_{\pm}$ from the right by $\tau\cdot g:=\frac{b+d\tau }{a+c\tau}$. We also have the diffeomorphism \begin{equation*}
K_\infty ^1\backslash G({\mathbb R}) \rightarrow \mathbb{H}^{\xi}_{\pm}\times ({\mathbb R}_{>0}\backslash Z({\mathbb R})^0),\ g\mapsto (i\cdot g, |\det(g)|). \end{equation*} The inverse map is \begin{equation*}
(\tau=x+iy,r)\mapsto \sqrt{r|y|^{-1}}\cdot \begin{pmatrix}1&x\\0&y\end{pmatrix}. \end{equation*} The absolute value and the square root are taken componentwise.
\item The stabilizer of a connected component of $K\backslash G({\mathbb A})$ or $K^1\backslash G({\mathbb A})$ in $G({\mathbb Q})$ is given by $G(g_f):=\left(g_f ^{-1}K_f g_f \cdot G({\mathbb R})^0\right)\cap G({\mathbb Q})$ for a $g_f\in G({\mathbb A}_f)$. Therefore, the connected components of $\mathcal S_K$ are of the form $\mathbb{H}^\xi/G(g_f)$ and this means that they are \textit{Hilbert-Blumenthal varieties}, in other words, quotients of $\mathbb{H}^\xi$ by arithmetic subgroups of $GL_2(F)$. \item $G({\mathbb Q})/Z_K$ acts properly discontinuously on $K\backslash G({\mathbb A})$ and fixpoint free, if $K$ is small enough, say $K=K_N$ and $N\geq 3$. Using the diffeomorphism above we see that $G({\mathbb Q})$ acts properly discontinuously and without fixpoints on $K^1\backslash G({\mathbb A})$, if we suppose $K_f$ to be small enough. \item
Consider the ray class group $Cl_F ^{K}:=\det(K_f)\prod_{\nu|\infty}{\mathbb R}_{>0}\backslash \mathbb{I}_F/F^{\times} $. The determinant induces maps $\det:\mathcal S_K\rightarrow Cl_F ^{K}$ and $\det:\mathcal M_K\rightarrow Cl_F ^{K}$. Since we have strong approximation for $SL_2$, the fibers of these maps are the connected components of $\mathcal S_K$ and $\mathcal M_K$. So our spaces have a finite number of connected components, since the class number of $F$ is finite. \item If $K_f$ is small enough, $\mathcal M_K$ has a structure of a $\mathcal C^\infty$-manifold making the canonical projection $K^1\backslash G({\mathbb A})\rightarrow \mathcal M_K$ a local diffeomorphism. \end{itemize} We will always assume that we have chosen $K_f$ small enough so that the quotient spaces $\mathcal S_K$ and $\mathcal M_K$ have the natural structure of $\mathcal C^\infty$-manifolds. Moreover, we consider \begin{equation*}
\left\{\pm 1\right\}^\xi=\prod_{\nu|\infty }\left\{\pm1\right\}\subset Z({\mathbb R})=\prod_{\nu|\infty}{\mathbb R}^{\times}. \end{equation*} \end{remark} \begin{lemma} $\phi_K:\mathcal M_K\rightarrow \mathcal S_K$ is a principal ${\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/Z_K$-bundle. \begin{proof} $Z({\mathbb R})$ acts clearly by matrix multiplication on $\mathcal M_K$ from the right, as it lies in the center of $G({\mathbb A})$. This action factors over the group ${\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/Z_K$, is by diffeomorphisms and preserves the fibers of $\phi_K$. What is left to be shown is that $\phi_K$ is a fiber bundle and that the action of ${\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/Z_K$ on the fibers is simply transitive. This problem is local on the base. \begin{equation*} \mathbb{H}^\xi _{\pm}/G(g_f)\subset \mathcal S_K,\ \tau G(g_f)\mapsto K\begin{pmatrix}1&x\\0&y\end{pmatrix}g_f G({\mathbb Q}) \end{equation*} is the inclusion of some connected components. We have \begin{equation*} \phi_K ^{-1}(\mathbb{H}^\xi _{\pm}/G(g_f))=\left(\mathbb{H}^\xi _{\pm}\times ({\mathbb R}_{>0}\backslash Z({\mathbb R})^0)\right)/G(g_f), \end{equation*} where $G(g_f)$ acts on the second factor by multiplication with the absolute value of the determinant. Here \begin{equation*} \phi_K:\left(\mathbb{H}^\xi _{\pm}\times ({\mathbb R}_{>0}\backslash Z({\mathbb R})^0)\right)/G(g_f)\rightarrow \mathbb{H}^\xi _{\pm}/G(g_f) \end{equation*} is induced by the first projection. The cosets $\gamma Z_K\in G(g_f)/Z_K$ act properly discontinuously and fixpoint free on $\mathbb{H}^{\xi}_{\pm}$, so we find for any point in $\mathbb{H}^{\xi}_{\pm}$ an open neighborhood $U$ such that $U\gamma \cap U\neq \emptyset$ implies $\gamma\in Z_K $. If we denote the image of $U$ in $\mathcal S_K$ by $U^\prime$, we get a diffeomorphism $\phi_K^{-1}(U^\prime)\cong U^\prime \times ({\mathbb R}_{>0}\backslash Z({\mathbb R})^0/\det(Z_K))$. So $\phi_K$ is a fiber bundle. But \begin{equation*} \det:{\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/Z_K\rightarrow {\mathbb R}_{>0}\backslash Z({\mathbb R})^0/\det(Z_K) \end{equation*} is a diffeomorphism showing that the action of ${\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/Z_K$ on the fibers of $\phi_K$ is simply transitive. This completes the proof. \end{proof} \end{lemma} \begin{remark} If $K=K_N$, $N\geq 3$, we have $-1\notin Z_K$ and Dirichlet's unit theorem tells us that $Z_K$ acts properly discontinuously and fixpoint free on ${\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})$ and that the quotient ${\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/Z_K$ is compact. \end{remark} Let us now define families of topological tori above $\mathcal M_K$. Take $W_f\subset V({\mathbb A}_f)\rtimes G({\mathbb A}_f)$ a compact open subgroup such that the projection of $W_f$ onto $G({\mathbb A}_f)$ gives $K_f$. Set $W:=W_f\cdot K_\infty ^1 \subset G({\mathbb A})$ where we always consider $G({\mathbb A})\subset V({\mathbb A})\rtimes G({\mathbb A}) $ via $g\mapsto (0,g)$. Set \begin{equation*} \pi_W:\mathcal T_W:=W\backslash V({\mathbb A})\rtimes G({\mathbb A})/V({\mathbb Q})\rtimes G({\mathbb Q})\rightarrow \mathcal M_K,\ (v,g)\mapsto g \end{equation*}
\begin{lemma} $\pi_W:\mathcal T_W\rightarrow \mathcal M_K$ is a family of topological tori. \begin{proof} Write $W_f=V_f\rtimes K_f$, where $V_f\subset V({\mathbb A}_f)$ is a compact open subgroup. We have the bijection \begin{equation*} V({\mathbb A})\rtimes G({\mathbb A})\rightarrow V({\mathbb A})\times G({\mathbb A}),\ (v,g)\mapsto (g^{-1}v,g)=:(w,g). \end{equation*} In the new coordinates $(w,g)$ the group actions look like \begin{equation*} (w,g)(q,\gamma)=(\gamma^{-1}(w+q),g\gamma),\ (q,\gamma)\in V({\mathbb Q})\rtimes G({\mathbb Q}), \end{equation*} \begin{equation*} (u,k)(w,g)=((kg)^{-1}u +w,kg),\ (u,k)\in W. \end{equation*} When we write $V({\mathbb A})\times G({\mathbb A})$ we always work in the coordinates $(w,g)$ with induced group actions. The pullback of $\mathcal T_W$ to $K^1\backslash G({\mathbb A})$ is $W\backslash V({\mathbb A})\times G({\mathbb A})/V({\mathbb Q})$. The action of $V_f$ on $V({\mathbb A})$ depends on $g_f\in G({\mathbb A}_f)$. For $g_f\in G({\mathbb A}_f)$ consider $K^1\backslash K^1g_f G({\mathbb R})$. This is just a collection of connected components in $K^1\backslash G({\mathbb A})$. We may now identify \begin{equation*} pr^{-1}(K^1\backslash K^1g_f G({\mathbb R}))\cong (g_f^{-1}V_f\backslash V({\mathbb A})/V({\mathbb Q}))\times (K^1\backslash K^1g_f G({\mathbb R})) \end{equation*} \begin{equation*} W(w,h)V({\mathbb Q})\mapsto (g_f ^{-1}V_f+w+V({\mathbb Q}),K^1h) \end{equation*} and the fiber $ g_f ^{-1}V_f\backslash V({\mathbb A})/V({\mathbb Q})=V({\mathbb R})/(g_f ^{-1}V_f\cap V({\mathbb Q})) $ over each point is a topological torus. This shows that $W\backslash V({\mathbb A})\times G({\mathbb A})/V({\mathbb Q})\rightarrow K\backslash G({\mathbb A})$ is a family of topological tori. Because $G({\mathbb Q})$ acts on the latter by homomorphisms and properly discontinuously without fixpoints on the base, $\mathcal T_W$ is a family of topological tori. \end{proof} \begin{remark}\label{connected_component} We keep the notation $(w,g):=(g^{-1}v,g)$, if $(v,g)\in V({\mathbb A})\rtimes G({\mathbb A})$ denotes a general element. Moreover, consider $K_f\subset G(\hat{\mathbb Z})$ and $W_f=V(\hat{\mathbb Z})\rtimes K_f$. Over \begin{equation*} \left(\mathbb{H}^{\xi}_{\pm}\times ({\mathbb R}_{>0}\backslash Z({\mathbb R}) ^0)\right)/G(g_f) \end{equation*} our torus looks like \begin{equation*} \left(F\otimes_{\mathbb Q}{\mathbb C}\times\mathbb{H}^{\xi}_{\pm}\times ({\mathbb R}_{>0}\backslash Z({\mathbb R})^0)\right)/V(g_f)\rtimes G(g_f), \end{equation*} where $V(g_f):=g_f ^{-1}V(\hat{\mathbb Z})\cap V({\mathbb Q})$ and $V(g_f)\rtimes G(g_f)$ acts in the following way: \begin{equation*} (z,\tau,r)\cdot\left(\begin{pmatrix}l_1\\l_2\end{pmatrix},\gamma=\begin{pmatrix}a& b\\c&d\end{pmatrix}\right)=\left(\frac{z+l_1+\tau l_2}{a+c\tau},\tau\cdot\gamma,r\cdot \det(\gamma)\right). \end{equation*} The natural inclusion is then given by \begin{equation*}
\left(F\otimes_{\mathbb Q}{\mathbb C}\times\mathbb{H}^{\xi}_{\pm}\times ({\mathbb R}_{>0}\backslash Z({\mathbb R})^0)\right)/V(g_f)\rtimes G(g_f)\rightarrow \mathcal T_W,\ (z,\tau,r)\mapsto (w,g) \end{equation*} \begin{equation*}
z=u^1 +u^2\tau,\ (w,g)_\infty=\left(\begin{pmatrix}u^1\\u^2 \end{pmatrix},\sqrt{r|y|^{-1}}\begin{pmatrix}1& x\\0&y\end{pmatrix}\right)\text{ and } (w,g)_f=(0,g_f). \end{equation*} \end{remark} \end{lemma}
\section{Construction of adelic polylogarithmic Eisenstein classes}
Next we will discuss how we can associate not only to formal linear combinations of torsion sections of tori but also to Schwartz-Bruhat functions $f$ on $V({\mathbb A}_f)$ polylogarithmic Eisenstein classes $\operatorname{Eis}(f)$. The naturality of the polylogarithmic construction guarantees the $G({\mathbb A}_f)$-equivariance of the operator $\operatorname{Eis}$.
\subsection{Coefficient sheaves and integral structures}
We introduce some locally constant sheaves on our spaces. Given a topological space $X$ on which a group $G$ acts continuously from the right we associate to any $G$-(left)-module $V$ a sheaf $\mathcal V$ on the quotient space $X/G$ in a functorial way. We make the constant sheaf $V$ on $X$ into a $G$-sheaf by defining for $g\in G$ morphisms \begin{equation*} g:V\rightarrow r_{g*} V,\ f\mapsto \left\{x\mapsto gf(xg)\right\}, \end{equation*} if $f:U\rightarrow V$ is a locally constant map on $U\subset X$. If $p:X\rightarrow X/G$ is the canonical map, these morphisms make $p_*V$ into a sheaf of $G$-modules and we can consider the fixed sheaf $p_* ^GV $ which is defined by $p_* ^GV (U):=p_*V(U)^G$. This fixed sheaf is our $\mathcal V$. We will often just speak about the sheaf induced by a $G$-module or associated to a $G$-module.
On $\mathcal T_W$ we have the logarithm sheaf $Log_W$ with ${\mathbb Q}$-coefficients. It is induced by the $V({\mathbb Q})\rtimes G({\mathbb Q})$-left-module $\prod_{k\geq0}\operatorname{Sym}^k V({\mathbb Q})$, where $V({\mathbb Q})$ acts by multiplication with the exponential series and $G({\mathbb Q})$ by its standard action, see \Cref{R_alg}. \\ The sheaf $Log_W$ carries an integral structure. Suppose we are on a connected component cut out by $g_f\in G({\mathbb A}_f)$ and $K_f\subset G(\hat{\mathbb Z})$, $W_f=V(\hat{\mathbb Z})\rtimes K_f$, compare \Cref{connected_component}. The fundamental group of the family of topological tori above this connected component is $V(g_f)\rtimes G(g_f)$. $Log_W$ is the locally constant sheaf associated to the $V(g_f)\rtimes G(g_f)$- module ${\mathbb Z}[[V(g_f)]]$ as considered in \Cref{def_Log}.\\ Moreover, we have the locally constant sheaves $\operatorname{Sym}^k\mathcal H_K$ on $\mathcal M_K$ associated to the $G({\mathbb Q})$-modules $\operatorname{Sym}^kV({\mathbb Q})$ and the sheaf $\operatorname{Sym}^k\mathcal H^\prime _K$ on $\mathcal S_K$, which is associated to the same $G({\mathbb Q})$-module. These sheaves also have integral structures coming from $\operatorname{Sym}^k V(g_f)_{PD}$, which we denote by $\operatorname{Sym}^k\mathcal H_{PD}$ and $\operatorname{Sym}^k\mathcal H^{\prime}_{PD}$. \begin{remark} One has the identification $\phi_{K*}\operatorname{Sym}^k\mathcal H_K=\operatorname{Sym}^k\mathcal H^\prime _K$. This is best understood by looking at the stalks of the sheaves. We have $\operatorname{Sym}^k\mathcal H_{K,x}=\operatorname{Sym}^kV({\mathbb Q})$, since the action of $G({\mathbb Q})$ on $K^1\backslash G({\mathbb A})$ is without fixpoints. But we have $\operatorname{Sym}^k\mathcal H_{K,x}^\prime=\operatorname{Sym}^kV({\mathbb Q})^{Z_K}$, as $Z_K$ acts trivially on $K\backslash G({\mathbb A})$. \end{remark} If we consider sheaves on $\mathcal M_K$ arising from $G({\mathbb Q})$-modules, the sheaves on $\mathcal S_K$ corresponding to the same module will always be denoted by a prime. \\ The sheaf $\mu_K$ on $\mathcal M_K$ is induced by the $G({\mathbb Q})$-module ${\mathbb Q}$, where $G({\mathbb Q})$ acts by multiplication with the norm of the determinant $\operatorname{N}\circ \det$. As $\operatorname{N}\circ \det(Z_K)=1$, there is no need to distinguish between $\mu_K$ on $\mathcal M_K$ and $\mu_K ^\prime$ on $\mathcal S_K$, as $\mu_K$ would be the pullback of $\mu_K ^\prime$. We have a trivialization of $\mu_K$ given by the global section \begin{equation*}
s_K: K \backslash G({\mathbb A}) \rightarrow {\mathbb Q},\ g\mapsto \left\|\det\ g_f\right\|_f\cdot \operatorname{sgn}(\operatorname{N}(\det\ g_\infty))^{-1}, \end{equation*}
where $\left\|\right\|_f:=\prod_{\nu\nshortmid \infty}|\ |_\nu$ is the product of the finite local norms of $F$. Clearly, $or_{\mathcal T_W/\mathcal M_K}=\mu_K=\bigwedge^{2\xi}\mathcal H_K$.
As we vary the level on $\mathcal M_K$ or $\mathcal T_W$, we see that our sheaves $Log_W$ are obtained by pullback. For example, given $W_f ^\prime\subset W_f$ we have the canonical projection $p_W ^{W^\prime}:\mathcal T_{W^\prime}\rightarrow\mathcal T_W $ and $p_W ^{W^\prime *}Log_W=Log_{W^\prime}$ by \Cref{basechange_Log}. The same is true for $\operatorname{Sym}^k\mathcal H_K$ and $\mu_K$. We will drop the subscript $K$, when it does not cause any confusion.
\subsection{Torsion sections}\label{torsion sections}
We are looking for a good class of compact open subgroups $W_f\subset V({\mathbb A}_f)\rtimes G({\mathbb A}_f)$ such that we can give explicit sections for $\pi_W$.
From now on we will always deal with $W_f$ of the kind: $W_f=V_K\rtimes K_f$, where $K_f\subset G({\mathbb A}_f)$, $V_K\subset V({\mathbb A}_f)$ are compact open subgroups and $V_K$ is an $\hat{\mathcal O}$-module on which $K_f$ acts as usual by matrix-multiplication. Moreover, we suppose that we have a compact open $\hat{\mathcal O}$-module $V_K ^\prime\supset V_K$ such that $K_f$ acts trivially on $V_K ^\prime /V_K$, $|V_K ^\prime /V_K|>1$. Our main example will be $K=K_{N^2}$, $V_K=NV(\hat{{\mathbb Z}})$ and $V_K ^\prime=N^{-1}V(\hat{{\mathbb Z}})$. Given such a $W$ we have a natural injective arrow $V_K ^\prime/V_K\times \mathcal M_K\rightarrow \mathcal T_W$, which is induced by the inclusion $V_K ^\prime\times G({\mathbb A})\rightarrow V({\mathbb A})\rtimes G({\mathbb A})$.\\ We define $D_W$ as the image of this map without the image of the zero section. $D_W$ is a closed submanifold of the torus $\mathcal T_W$, which is a finite covering of $\mathcal M_K$. We set $U_W$ to be the open complement of $D_W$ in $\mathcal T_W$.
Let us apply the construction of the polylogarithm to this situation. We have the natural short exact sequence \begin{equation*}
0\rightarrow H^{2\xi-1}(U_W,Log_W\otimes_{\mathbb Q} \pi_W ^{-1}\mu_K)\rightarrow H^0(D_W,Log_{W\ |D_W})\rightarrow H^0(\mathcal M_K,{\mathbb Q}) \end{equation*} induced by the localization sequence for the pair $(D_W,\mathcal T_W)$.
We have the inclusion $H^0(D_W,{\mathbb Q})\subset H^0(D_W,Log_{W\ |D_W})$ and on this submodule the last arrow in the short exact sequence is just given by the trace map $H^0(D_W,{\mathbb Q})\rightarrow H^0(\mathcal M_K,{\mathbb Q})$. Exactness of the sequence ensures that given a section $f\in H^0(D_W,{\mathbb Q})$ of trace $0$ we have a unique element $\operatorname{pol}_W(f)\in H^{2\xi-1}(U_W,Log_W\otimes_{\mathbb Q} \pi_W^{-1}\mu_K)$ mapped to $f$. As $D_W\cap \operatorname{im}(0)=\emptyset$, we may specialize $\operatorname{pol}_W(f)$ along the zero section of $\mathcal T_W$ to get a class \begin{equation*} \operatorname{Eis}_W(f)\in H^{2\xi-1}(\mathcal M_K,\prod_{k\geq0}\operatorname{Sym}^k\mathcal H_K\otimes \mu_K)= H^{2\xi-1}(\mathcal M_K,0^{-1}Log_W\otimes \mu_K), \end{equation*} in other words $(\operatorname{Eis}_W ^k(f))_k\in \prod_{k\geq0}H^{2\xi-1}(\mathcal M_K,\operatorname{Sym}^k\mathcal H_K\otimes \mu_K)$.
\subsection{Definition of adelic Eisenstein classes}
We want to extend the idea of polylogarithmic Eisenstein classes to more general functions $f$. Let us denote the space of Schwartz-Bruhat functions on $V({\mathbb A}_f)$ with values in a subfield $k\subset {\mathbb C}$ by $\mathcal S(V({\mathbb A}_f),k)$, compare for example \cite{Ge} p.267. Set \begin{equation*} \mathcal S(V({\mathbb A}_f),k)^0=\left\{f\in \mathcal S(V({\mathbb A}_f),k):f(0)=0,\int_{V({\mathbb A}_f)}f dv_f=0 \right\} \end{equation*} where $dv_f$ is any Haar measure on $V({\mathbb A}_f)$. $\mathcal S(V({\mathbb A}_f),k)^0$ is a right $G({\mathbb A}_f)$-module, when we set as usual $f\cdot g_f(v)=f(g_fv)$. This follows from the integral transformation formula \begin{equation*}
\int_{V({\mathbb A}_f)}f(g_f v) dv_f(v)= \left\|\det(g_f)\right\|_f ^{-1}\int_{V({\mathbb A}_f)}f( v) dv_f(v). \end{equation*} We will need a more general space of functions. For any $n\in {\mathbb Z}$ denote by \\$\mathcal S(V({\mathbb A}_f),\mu^{\otimes n}\otimes k)$ the space of functions $f:V({\mathbb A}_f)\rtimes G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))\rightarrow k$ such that \begin{equation*}
\forall x\in G({\mathbb A}_f)\times \pi_0(G({\mathbb R})):\ f(\ ,x )\in \mathcal S(V({\mathbb A}_f),k), \end{equation*} there is some compact open subgroup $K_f\subset G({\mathbb A}_f)$ such that $f$ factors as \begin{equation*} f(\ ,\ ):K_f\backslash G({\mathbb A}_f)\times \pi_0(G({\mathbb R})) \rightarrow \mathcal S(V({\mathbb A}_f),k),\ x\mapsto f(\ ,x ) \end{equation*} and for all $v\in V({\mathbb A}_f)$, $x\in G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$ and $\gamma\in G({\mathbb Q})$ \begin{equation*} f(v,x\gamma)=N(\det(\gamma))^{-n}f(v,x). \end{equation*} We define $\mathcal S(V({\mathbb A}_f),\mu^{\otimes n}\otimes k)^0$ as the subspace of $\mathcal S(V({\mathbb A}_f),\mu^{\otimes n}\otimes k)$ consisting of those functions $f$ with $f(\ ,x)\in \mathcal S(V({\mathbb A}_f),k)^0$ for all $x\in G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$. Again $\mathcal S(V({\mathbb A}_f),\mu^{\otimes n}\otimes k)^0$ is a right $G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$-module, when we take the action induced by the canonical left $G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$ action on $V({\mathbb A}_f)\rtimes G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$. If $k={\mathbb Q}$ we simply set $\mathcal S(V({\mathbb A}_f),\mu^{\otimes n}\otimes k)=:\mathcal S(V({\mathbb A}_f),\mu^{\otimes n})$ and $\mathcal S(V({\mathbb A}_f),\mu^{\otimes n}\otimes k)^0 =:\mathcal S(V({\mathbb A}_f),\mu^{\otimes n})^0 $. \begin{lemma}\label{varphi} $\mathcal S(V({\mathbb A}_f),\mu^{\otimes n}\otimes{\mathbb C})^0$ is generated by $\varphi$ of the form \begin{equation*}
\varphi(v,g)=f(v)\cdot\eta(\det(g))\left(\left\|\det(g)\right\|_f\operatorname{sgn}(N(\det(g)))\right)^{n}, \end{equation*}
for $f\in\mathcal S(V({\mathbb A}_f),{\mathbb C})^0$ and $\eta:\prod_{\nu|\infty}{\mathbb R}_{>0}\backslash\mathbb{I}_{F}/F^{\times}\rightarrow {\mathbb C}^{\times}$ a Dirichlet character. \begin{proof} Consider $\varphi\in\mathcal S(V({\mathbb A}_f),\mu^{\otimes n}\otimes{\mathbb C})^0$ arbitrarily. Write \begin{equation*}
\varphi(v,g)=\phi(v,g)\left(\left\|\det(g)\right\|_f\operatorname{sgn}(N(\det(g)))\right)^{n}, \end{equation*} for $\phi\in\mathcal S(V({\mathbb A}_f),\mu^{\otimes 0}\otimes {\mathbb C})^0$. $\phi$ factors in the second argument over some $K_f\subset G({\mathbb A}_f)$ compact open, in other words \begin{equation*} \phi:K_f\backslash G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))/G({\mathbb Q})\rightarrow \mathcal S(V({\mathbb A}_f),{\mathbb C})^0. \end{equation*} We have strong approximation for $SL_{2,F}$ and therefore \begin{equation*}
\det: K_f\backslash G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))/G({\mathbb Q})\rightarrow \det(K_f)\prod_{\nu|\infty}{\mathbb R}_{>0}\backslash\mathbb{I}_{F}/F^{\times}=Cl_F ^{K} \end{equation*} is an isomorphism. In this manner we may interpret $\phi$ in the second argument as a function on $Cl_F ^{K}$. The ray class group $Cl_F ^{K}$ is a finite group of cardinality $h_K$ and we can describe functions $Cl_F ^{K}\rightarrow {\mathbb C}$ as finite linear combinations of characters by the Fourier inversion theorem. We get explicitly \begin{equation*} \phi(v,g)=h_{K}^{-1}\sum_{\eta\in \widehat{Cl_F ^{K}}}\hat{\phi}(v,\eta)\cdot\eta(\det(g)),\ \hat{\phi}(v,\eta)=\sum_{x\in Cl_F ^{K}}\phi(v,x)\overline{\eta(x)}. \end{equation*} \end{proof} \end{lemma} Finally, let us set \begin{equation*} \varinjlim_{K_f} H^\bullet(\mathcal M_K,\operatorname{Sym}^k\mathcal H_K\otimes\mu_K ^{\otimes n})=:H^\bullet(\mathcal M,\operatorname{Sym}^k\mathcal H\otimes\mu^{\otimes n}). \end{equation*} It is a $G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$-module as explained in \cite{Ha1} 1.2. \begin{remark} $\mathcal S(V({\mathbb A}_f),\mu^{\otimes n})^0=\mathcal S(V({\mathbb A}_f),{\mathbb Q})^0\otimes_{\mathbb Q} H^0(\mathcal M,\mu^{\otimes n})$. \end{remark} \begin{proposition}\label{Eis^k} We have a $G({\mathbb A}_f)\times\pi_0(G({\mathbb R}))$-equivariant operator \begin{equation*} \operatorname{Eis}^k:\mathcal S(V({\mathbb A}_f),\mu^{\otimes n})^0 \rightarrow H^{2\xi-1}(\mathcal M,\operatorname{Sym}^k\mathcal H\otimes\mu^{\otimes n+1}), \end{equation*} which is induced by the operators $\operatorname{Eis}^k _W$. \begin{proof} Given a $\phi\in \mathcal S(V({\mathbb A}_f),{\mathbb Q})^0$ there is an $M\in {\mathbb N}$ such that $\text{supp}(\phi)\subset M^{-1}V(\hat{\mathbb Z})$ and an $M^\prime \in {\mathbb N}$ such that $\phi$ is well-defined on $V({\mathbb A}_f)/M^\prime V(\hat{\mathbb Z})$. So considering $N=MM^\prime$, we can identify $\phi$ with a function on $N^{-1}V(\hat{\mathbb Z})/NV(\hat{\mathbb Z})$. Let us investigate $f\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes n})^0$. As $f$ factors over some compact open subgroup $K_f$ of $G({\mathbb A}_f)$ and $K_f\backslash G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))/G({\mathbb Q})$ is finite, we may find an $N\in {\mathbb N}$ such that $f$ is well-defined modulo $K_{N^2}$ in the second argument and $f$ behaves like $\phi$ above in the first argument. In other words, $f$ induces a function \begin{equation*} (N^{-1}V(\hat{\mathbb Z})/NV(\hat{\mathbb Z})\setminus \left\{0\right\})\times K_{N^2}\backslash G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))\rightarrow {\mathbb Q}, \end{equation*} which we also denote by $f$. We are now in the situation of our groups $W$ of \Cref{torsion sections}, namely $K_f=K_{N^2}$, $NV(\hat{\mathbb Z})=V_K$, $V^\prime _K =N^{-1}V(\hat {\mathbb Z})$ and $W_f=NV(\hat{\mathbb Z})\rtimes K_{N^2}$. The function $f$ may be identified with a section $f\in H^0(D_W,\mu_K ^{\otimes n})$ which actually has trace zero: We have for all $x\in G({\mathbb A}_f)\times\pi_0(G({\mathbb R}))$: \begin{equation*} \sum_{v\in N^{-1}V(\hat{\mathbb Z})/NV(\hat{\mathbb Z})\setminus \left\{0\right\}}f(v,x)=\sum_{v\in N^{-1}V(\hat{\mathbb Z})/NV(\hat{\mathbb Z})}f(v,x)=0 \end{equation*} if and only if \begin{equation*} 0=\sum_{v\in N^{-1}V(\hat{\mathbb Z})/NV(\hat{\mathbb Z})}f(v,x)\int_{NV(\hat{\mathbb Z})}dv_f= \end{equation*} \begin{equation*} =\sum_{v\in N^{-1}V(\hat{\mathbb Z})/NV(\hat{\mathbb Z})}\int_{NV(\hat{\mathbb Z})}f(v+u,x)dv_f(u)=\int_{V({\mathbb A}_f)}f(v ,x)dv_f . \end{equation*} Consequently, we get $\operatorname{Eis}_W ^k(f)\in H^{2\xi-1}(\mathcal M_K,\operatorname{Sym}^k\mathcal H_K\otimes \mu_K ^{\otimes n +1})$. We want to define
\begin{equation*} \operatorname{Eis}^k(f)=\text{image of } \operatorname{Eis}_W ^k(f)\text{ in }H^{2\xi-1}(\mathcal M,\operatorname{Sym}^k\mathcal H\otimes \mu^{\otimes n+1}). \end{equation*} We have to show that this construction is independent of the choices made. Consider again $f\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes n}) ^0$. Suppose that we have $W_1$ and $W_2$ both fulfilling the properties above. In other words, we have for $i=1,2$ compact open subgroups $K_{f,i}\subset G({\mathbb A}_f)$ acting on the compact open $\hat\mathcal O$-modules $V_{K_i}^\prime\supset V_{K_i}$ and acting trivially on $V_{K_i}^\prime/ V_{K_i}$, such that $f$ is well-defined modulo $K_i$ in the second argument, $f$ is well-defined modulo $V_{K_i}$ in the first argument and for all $x\in G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$ $\text{supp}(f(\ ,x))\subset V_{K_i}^\prime$. \\ If we set \begin{equation*} K=K_1\cap K_2,\ V_K=V_{K_1}\cap V_{K_1},\ V_K ^\prime=V_{K_1} ^\prime\cap V_{K_1} ^\prime,\text{ and }W=W_1\cap W_2, \end{equation*} all the properties are also fulfilled by $W$ so that we have $\operatorname{Eis}^k_W(f),\ \operatorname{Eis}^k_{W_1}(f)$ and $\operatorname{Eis}^k _{W_2}(f)$. To see that $\operatorname{Eis}^k$ is well-defined we have to show that $\operatorname{Eis}^k_W(f)$ is obtained by pullback along the canonical maps $q^W _{W_i}:\mathcal M_K\rightarrow \mathcal M_{K_i}$ from $\operatorname{Eis}_{W_i} ^k(f)$ for $i=1,2$. It is enough to consider $i=1$. We have a commutative square \begin{equation*} \begin{xy} \xymatrix{ \mathcal T_W\ar[r]^{p^W _{W_1}} \ar[d] & \mathcal T_{W_1}\ar[d] \\ \mathcal M_K\ar[r]^{q^W _{W_1}} & \mathcal M_{K_1} } \end{xy} \end{equation*} Set $D_{W_1}^\prime :=p^{W -1 }_{W_1}(D_{W_1})\supset D_W$, it is a finite cover of $\mathcal M_K$. As the localization-sequence, $\mu$ and $Log$ are natural in pullbacks, see \Cref{natural_localization}, we get the following commutative diagram \begin{equation*} \begin{xy} \xymatrix{
H^{2\xi-1}(\mathcal T_{W_1}\setminus D_{W_1},Log\otimes\mu^{\otimes n+1})\ar[d]\ar[r] & H^0(D_{W_1},Log_{|D_{W_1}}\otimes \mu^{\otimes n})\ar[d] \\
H^{2\xi-1}(\mathcal T_{W}\setminus D_{W_1}^\prime ,Log\otimes\mu^{\otimes n+1})\ar[r] & H^0(D_{W_1}^\prime,Log_{|D_{W_1}^\prime}\otimes \mu^{\otimes n}) } \end{xy} \end{equation*} The diagram shows that $p^{W * }_{W_1}\operatorname{pol}_{W_1}(f)$ corresponds to the section $p^{W * }_{W_1}f\in H^0(D_{W_1}^\prime,\mu^{\otimes n})$. By naturality of the localization sequence for the inclusion $i:D_W\subset D_{W_1}^\prime$, see \Cref{shrinking_localization}, we also get the commutative diagram \begin{equation*} \begin{xy} \xymatrix{
H^{2\xi-1}(\mathcal T_{W}\setminus D_{W},Log\otimes\mu^{\otimes n+1})\ar[d]^{\operatorname{res}}\ar[r] & H^0(D_{W},Log_{|D_{W_1}}\otimes \mu^{\otimes n})\ar[d]^{i_*} \\
H^{2\xi-1}(\mathcal T_{W}\setminus D_{W_1}^\prime ,Log\otimes\mu^{\otimes n+1})\ar[r] & H^0(D_{W_1}^\prime,Log_{|D_{W_1}^\prime}\otimes\mu^{\otimes n}) } \end{xy} \end{equation*} where $i_*:H^0(D_{W},\mu^{\otimes n})\rightarrow H^0(D_{W_1}^\prime,\mu^{\otimes n})$ is given by extension by zero. $i_*f$ and $p^{W * }_{W_1}f\in H^0(D_{W_1}^\prime,\mu^{\otimes n})$ coincide and consequently $\operatorname{res}(\operatorname{pol}_W(f))=p^{W * }_{W_1}\operatorname{pol}_{W_1}(f)$. As $0:\mathcal M_K\rightarrow\mathcal T$ meets neither $D_W$ nor $D_{W_1}^\prime$, we have \begin{equation*} \operatorname{Eis}_W(f)= 0^*\operatorname{pol}_W(f)=0^* \operatorname{res}(\operatorname{pol}_W)= \end{equation*} \begin{equation*} 0^*p^{W * }_{W_1}\operatorname{pol}_{W_1}(f)= q^{W *} _{W_1}0^*\operatorname{pol}_{W_1}(f)=q^{W *} _{W_1}\operatorname{Eis}_{W_1}(f) \end{equation*} It follows that our operator is well-defined.\\ $G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$-equivariance follows easily from our construction. Given a section $f$ with corresponding groups $K$, $V_K$, $V_K ^\prime$ and $W$ as above and given $g\in G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$ the corresponding groups for $f\cdot g$ can be chosen $g^{-1}Kg$, $g^{-1}V_K$, $g^{-1}V_K ^\prime$ and $g^{-1}Wg$. We have the morphism $\mathcal T_{g^{-1}Wg}\to\mathcal T_W$,\ $(v,h)\mapsto (gv,gh) =g(v,h)$ and use \Cref{natural_localization} to get the commutative diagram \begin{equation*} \begin{xy} \xymatrix{
H^{2\xi-1}(\mathcal T_{W}\setminus D_{W},Log\otimes\mu^{\otimes n+1})\ar[d]\ar[r] & H^0(D_{W},Log_{|D_{W_1}}\otimes \mu^{\otimes n})\ar[d] \\
H^{2\xi-1}(\mathcal T_{g^{-1}Wg}\setminus D_{g^{-1}Wg},Log\otimes\mu^{\otimes n+1})\ar[r] & H^0(D_{g^{-1}Wg},Log_{|D_{g^{-1}Wg}}\otimes\mu^{\otimes n}) } \end{xy} \end{equation*} As the pullback $g^*f$ is exactly the section corresponding to $f\cdot g$, we have $g^*\operatorname{pol}_W(f)=\operatorname{pol}_{g^{-1}Wg}(f\cdot g)$. It follows \begin{equation*} \operatorname{Eis}(f\cdot g)=0^*\operatorname{pol}_{g^{-1}Wg}(f\cdot g)=0^*g^*\operatorname{pol}_W(f)=g^*0^*\operatorname{pol}_W(f)=g^*\operatorname{Eis}(f) \end{equation*} and we are done. \end{proof} \end{proposition}
\section{Pushforward of polylogarithmic Eisenstein classes}
We are going to decompose the cohomology of $\mathcal M_K$ by a Leray-Hirsch theorem into a product of the cohomology of $\mathcal S_K$ and the cohomology of the fiber of $\phi_K$. This will yield a decomposition of the polylogarithmic Eisenstein classes on $\mathcal M_K$ and a way to integrate these classes to gain classes on $\mathcal S_K$. \begin{lemma} Let $G$ and $H$ be groups and $R$ a Dedekind domain such that $R$ has a resolution $P\rightarrow R$ by finitely generated free $R[G]$-modules (for example $G\cong {\mathbb Z}^r$). Let $M$ be an $R[G]$-module finitely generated free as $R$-module and let $N$ be any $R[H]$-module. Then we have the Künneth sequence \begin{equation*} 0\rightarrow \bigoplus_{p+q=n} H^p(G,M)\otimes_R H^q(H,N)\rightarrow H^n(G\times H,M\otimes_R N)\rightarrow \end{equation*} \begin{equation*} \rightarrow\bigoplus_{p+q=n+1} Tor_1 ^{R}(H^p(G,M),H^q(H,N))\rightarrow 0 \end{equation*} \begin{proof} We choose a resolution $P^\bullet\to R$ by finitely generated free $R[G]$-modules and a resolution $Q^\bullet\to R$ by free $R[H]$-modules. Since $R$ is a Dedekind domain the torsion-free $R$-modules are exactly the flat $R$-modules and therefore any submodule of a flat $R$-module is flat. With this at hand we may use \cite{We} Theorem 3.6.3 to conclude that $P^\bullet\otimes_RQ^\bullet\to R$ is a resolution by free $R[G\times H]$-modules. So $H^\bullet(G\times H,M\otimes_R N)=H^\bullet(Hom_{R[G\times H]}(P^\bullet\otimes_RQ^\bullet,M\otimes_RN))$. Let us consider the canonical map \begin{equation*} \mu:Hom_{R[G]}(P,M)\otimes_R Hom_{R[H]} (Q,N)\rightarrow Hom_{R[G\times H]}(P\otimes_R Q,M\otimes_R N) \end{equation*} $\mu(f\otimes g)(p\otimes q)=f(p)\otimes g(q)$ for $p\in P$ and $q\in Q$ for a finitely generated free $R[G]$-module $P$ and a free $R[H]$-module $Q$. We have $Hom_{R[G\times H]}(P\otimes_R Q,M\otimes_R N)$ equals \begin{equation*} Hom_{R[G\times H]}(R[G]^n\otimes_R Q,M\otimes_R N)= Hom_{R[H]}(Q,M\otimes_R N)^n \end{equation*} Now $M=R^m$ is free, so we continue by \begin{equation*} Hom_{R[H]}(Q,N^m)^n=Hom_{R[H]}(Q,N)^{mn}=Hom_{R[H]}(Q,N)\otimes_R M^n= \end{equation*} \begin{equation*} =Hom_{R[H]}(Q,N)\otimes_R Hom_{R[G]}(R[G]^n,M)=Hom_{R[H]}(Q,N)\otimes_R Hom_{R[G]}(P,M) \end{equation*} So $\mu$ is an isomorphism. Moreover, $M^n=Hom_{R[G]}(P,M)$ is a flat $R$-module and we can apply \cite{We} Theorem 3.6.3 to the complex $Hom_{R[G]}(P,M)\otimes_R Hom_{R[H]} (Q,N)$ to get the exact sequence \begin{equation*} 0\to \bigoplus_{p+q=n}H^p(Hom_{R[G]}(P,M))\otimes_RH^q(Hom_{R[H]} (Q,N))\to \end{equation*} \begin{equation*} \to H^n(Hom_{R[G]}(P,M)\otimes_A Hom_{R[H]} (Q,N))\to \end{equation*} \begin{equation*} \to\bigoplus_{p+q=n+1}Tor^R_1(H^p(Hom_{R[G]}(P,M)),H^q(Hom_{R[H]} (Q,N))\to 0 \end{equation*} Using $\mu$ we compute this sequence as \begin{equation*} 0\rightarrow \bigoplus_{p+q=n} H^p(G,M)\otimes_R H^q(H,N)\rightarrow H^n(G\times H,M\otimes_R N)\rightarrow \end{equation*} \begin{equation*} \rightarrow\bigoplus_{p+q=n+1} Tor_1 ^{R}(H^p(G,M),H^q(H,N))\rightarrow 0 \end{equation*} \end{proof} \end{lemma} \begin{proposition}\label{cohomology_abelian_group} Let $G$ be a finitely generated free abelian group, $k$ a field and $M$ an $k[G]$-module finitely generated as $k$-module. We suppose that we have a field extension $k\subset K$ such that there is a decomposition $M_K=\bigoplus_{\chi}K(\chi)^{m(\chi)}$ with $\chi:G\rightarrow K^{\times}$ running through all characters and $K(\chi)$ the $K[G]$-module $K$ with action by $\chi$. Then cup-product induces an isomorphism $H^0(G,M)\otimes_A H^\bullet(G,k)=H^\bullet(G,M)$. \begin{proof}
We begin with $H^\bullet(G,K(\chi))=0$, if $\chi\neq 1$. Write $G\cong {\mathbb Z}^r$ and for $j=1,...,r$ define $\iota_j:{\mathbb Z}\to G$ to be the inclusion of the $j$-component. As $\chi\neq 1$, there is a $j$ such that $\chi\circ \iota_j\neq 1$. So we may assume $G={\mathbb Z}\times G^\prime$ with $\chi_{|{\mathbb Z}}\neq 1$ and $G^\prime$ a subgroup. One has $K(\chi)=K(\chi_{|{\mathbb Z}})\otimes_K K(\chi_{|G^\prime})$, as well as $H^\bullet ({\mathbb Z},K(\chi_{|{\mathbb Z}}))=0$ by \cite{We} Example 6.1.4. and we use the Künneth sequence \begin{equation*}
0\rightarrow \bigoplus_{p+q=n} H^p({\mathbb Z},K(\chi_{|{\mathbb Z}}))\otimes_A H^q(G^\prime,K(\chi_{|G^\prime}))\rightarrow H^n(G,K(\chi))\rightarrow \end{equation*} \begin{equation*}
\rightarrow\bigoplus_{p+q=n+1} Tor_1 ^{K}(H^p({\mathbb Z},K(\chi_{|{\mathbb Z}})),H^q(G^\prime,K(\chi_{|G^\prime})))\rightarrow 0 \end{equation*} to conclude. Now we may calculate \begin{equation*} H^\bullet(G,M_K)=H^\bullet(G,\bigoplus_{\chi}K(\chi)^{m(\chi)})=\bigoplus_{\chi}H^\bullet(G,K(\chi))^{m(\chi)}. \end{equation*} But all summands with $\chi\neq 1$ are zero, so this equals \begin{equation*} H^\bullet(G,K(1))^{m(1)}=H^0(G,M_K)\otimes_K H^\bullet(G,K) \end{equation*} and this isomorphism is induced by cup-product. We want to see that this map comes from extension of scalars from the cup-product map $\cup:H^0(G,M)\otimes_k H^\bullet(G,k)\rightarrow H^\bullet (G,M)$. This is the case, because $H^\bullet(G, N_K)=H^\bullet(G, N)\otimes_k K$ for any $G$-module $N$ compatible with cup-product. To see this take a resolution $P\rightarrow k$ by finitely generated free $k[G]$-modules. Then $H^\bullet(G, N_K)=H^\bullet(Hom_{K[G]}(P_K,N_K))$. As $k\rightarrow K$ is flat and $P$ consists of finitely generated free $k[G]$-modules one identifies \begin{equation*} H^\bullet(Hom_{K[G]}(P_K,N_K))=H^\bullet(Hom_{k[G]}(P,N))\otimes_k K=H^\bullet(G, N)\otimes_k K \end{equation*} proving the claim. Finally, the extension of scalars from $k$ to $K$ of the morphism $\cup$ is an isomorphism, therefore $\cup$ is already an isomorphism, since $k\rightarrow K$ is faithfully flat. \end{proof} \end{proposition} \begin{corollary}\label{fiber_cohom} We consider $\phi_K:\mathcal M_K\rightarrow \mathcal S_K$ and $\operatorname{Sym}^n\mathcal H$. Then cup-product induces an isomorphism $\cup:\phi_{K*}\operatorname{Sym}^n\mathcal H\otimes_{\mathbb Q} R^p\phi_{K*}({\mathbb Q})\rightarrow R^p\phi_{K*}(\operatorname{Sym}^n\mathcal H)$. \begin{proof} It is clear that we have a global cup-product map and being an isomorphism is a local problem on $\mathcal S_K$. As $\phi_K$ is a fiber bundle, it suffices to show that for all $s\in \mathcal S_K$ the induced maps on the fibers $\cup:H^0(\phi_{K}^{-1}(s),\operatorname{Sym}^n\mathcal H)\otimes_{\mathbb Q} H^p(\phi_{K}^{-1}(s),{\mathbb Q})\rightarrow H^p(\phi_{K}^{-1}(s),\operatorname{Sym}^n\mathcal H)$ are isomorphisms. Following \cite{GrT} Chapitre V we may prove this by showing that the corresponding map on the level of group cohomology $\cup:H^0(Z_K,\operatorname{Sym}^nV({\mathbb Q}))\otimes_{\mathbb Q} H^p(Z_K,{\mathbb Q})\rightarrow H^p(Z_K,\operatorname{Sym}^nV({\mathbb Q}))$ is an isomorphism. This is the case, as our situation fulfills the requirements of \Cref{cohomology_abelian_group}. $Z_K$ is a finitely generated free abelian group by Dirichlet's unit theorem and we have the field extension ${\mathbb Q}\rightarrow \overline{\mathbb Q}$ such that $\operatorname{Sym}^nV({\mathbb Q})$ decomposes: Take the isomorphism $\mathcal O\otimes_{\mathbb Z} \overline{\mathbb Q}\rightarrow \prod_{\sigma:\mathcal O\rightarrow \overline {\mathbb Q}} \overline{\mathbb Q},$ $x\otimes r\mapsto (\sigma(x)r)_\sigma$ and obtain \begin{equation*} \operatorname{Sym}^n _{\mathbb Q} V ({\mathbb Q})\otimes_{\mathbb Q} \overline{\mathbb Q}=\bigoplus_{\sum_\sigma n_\sigma =n}\bigotimes_\sigma \operatorname{Sym}^{n_\sigma} _{\overline{\mathbb Q}}\overline{\mathbb Q}^2. \end{equation*} If we take the standard basis $e^1=:X$, $e^2=:Y$ of $\overline{\mathbb Q}^2$, we get the $\overline{\mathbb Q}$-basis $\prod_\sigma X_\sigma ^{k_\sigma}Y_\sigma ^{n_\sigma -k_\sigma }$, $k_\sigma=0,...,n_\sigma$, of $\operatorname{Sym}^n _{\mathbb Q} V ({\mathbb Q})\otimes_{\mathbb Q} \overline{\mathbb Q}$, which also yields the decomposition into one-dimensional $Z_K$-representations, as $\epsilon\cdot\prod_\sigma X_\sigma ^{k_\sigma} Y_\sigma ^{n_\sigma -k_\sigma }=\prod_\sigma \sigma(\epsilon)^{n_\sigma}\prod_\sigma X_\sigma ^{k_\sigma}Y_\sigma ^{n_\sigma -k_\sigma }$ for $\epsilon \in Z_K\subset \mathcal O^{\times}$. \end{proof} \end{corollary} \begin{remark}\label{invariants} From now on we set $A:={\mathbb Z}[\frac{1}{d_FN}]$ where $d_F>0$ is the discriminant of $F$. We have the Hilbert class field $F^{\text{Hil}}$ of $F$ with ring of integers $\mathcal O^{\text{Hil}}$. Let us set $R:=\mathcal O^{\text{Hil}}[\frac{1}{Nd_F}]$. Then $A\to R$ is a faithfully flat ring extension. From now on we will always consider the sheaf $\operatorname{Sym}^k \mathcal H_{PD}$ over the ring $A$. The natural adjunction morphism \begin{equation*} \phi_K ^{-1}\phi_{K*}\operatorname{Sym}^k \mathcal H_{PD}\rightarrow \operatorname{Sym}^k \mathcal H_{PD} \end{equation*} is a split monomorphism. Indeed, the corresponding map of $G(g_f)$-modules is a split mono. To see this note first that \begin{equation*} \operatorname{Sym}^k V(g_f)_{PD} ^{Z_K}\otimes_A R=\left(\operatorname{Sym}^k V(g_f)_{PD}\otimes_A R \right)^{Z_K}, \end{equation*}
as $R$ is a free $A$-module. By faithful flatness it suffices now to find a retraction for the map \begin{equation*} \left(\operatorname{Sym}^k V(g_f)_{PD}\otimes_A R \right)^{Z_K} \rightarrow \operatorname{Sym}^k V(g_f)_{PD}\otimes_A R. \end{equation*} There are two fractional ideals $\mathfrak a$ and $\mathfrak b$ of $\mathcal O$ such that $V (g_f)=\mathfrak a\oplus \mathfrak b$. As we have $d_F\in R^{\times}$, there are again isomorphisms \begin{equation*} \mathfrak a\otimes_A R \rightarrow \prod_{\sigma:\mathcal O\rightarrow \overline {\mathbb Q}} \sigma(\mathfrak a)R,\ x\otimes r\mapsto (\sigma(x)r)_\sigma,\ \mathfrak b\otimes_A R \rightarrow \prod_{\sigma:\mathcal O\rightarrow \overline {\mathbb Q}} \sigma(\mathfrak b)R,\ x\otimes r\mapsto (\sigma(x)r)_\sigma \end{equation*} and we get \begin{equation*} \operatorname{Sym}^k V(g_f)_{PD}\otimes_A R=\bigoplus_{\sum_\sigma k_\sigma =k}\bigotimes_\sigma \operatorname{Sym}^{k_\sigma} _{R}\left(\sigma(\mathfrak a)R\oplus \sigma(\mathfrak b)R\right)_{PD}. \end{equation*} In the Hilbert class field any Ideal of $\mathcal O$ becomes principal. Therefore we get $\alpha_\sigma,\beta_\sigma\in F^{\text{Hil}}$ with $\sigma(\mathfrak a)R=\alpha_\sigma R$ and $\sigma(\mathfrak b)R=\beta_\sigma R$. If again $e^1=:X$, $e^2=:Y$ is the standard basis of $R^2$, we may split $\operatorname{Sym}^k V(g_f)_{PD}\otimes_A R$ as in \Cref{fiber_cohom} into one dimensional $Z_K$-representations $R\prod_\sigma (\alpha_\sigma X_\sigma) ^{[n_\sigma]}(\beta_\sigma Y_\sigma) ^{[k_\sigma -n_\sigma ]}$. In $\left(\operatorname{Sym}^k V(g_f)_{PD}\otimes_A R \right)^{Z_K}$ those one dimensional representations occur, on which $Z_K$ acts trivially. From this description it is clear that the inclusion of the invariants is a split mono. But we can say more. The one dimensional subrepresentations occurring in the invariants induce characters $Res_{F/{\mathbb Q}}\mathbb{G}_m(\overline {\mathbb Q})\rightarrow \mathbb{G}_m (\overline {\mathbb Q})$ which have to be trivial on $Z_K$. By \cite{Se} II.3.3 we know that these have to factor through the norm character, as our field $F$ is totally real. In other words, $R\prod_\sigma (\alpha_\sigma X_\sigma) ^{[n_\sigma]}(\beta_\sigma Y_\sigma) ^{[k_\sigma -n_\sigma ]}$ occurs in the invariants, if and only if there is an $m\in {\mathbb N}_0$ such that $m=k_\sigma$ for all $\sigma :F\rightarrow\overline {\mathbb Q}$. It follows that the module $\left(\operatorname{Sym}^k V(g_f)_{PD}\otimes_A R\right) ^{Z_K}$ does not depend on the level $K$, as soon as $Z_K\subset \mathcal O^{\times}$ does not contain an element of negative norm. By faithful flatness $\operatorname{Sym}^k V(g_f)_{PD} ^{Z_K} $ does not depend on the level $K$ in this case. As we have usually $K_f=K_N$, $N\geq 3$, we do not have elements of negative norm. \end{remark} \begin{corollary} We have $\operatorname{Sym}^k \mathcal H_{PD} ^\prime=0$, if $\xi\nmid k$. If $k=\xi m$ and $Z_K$ contains no element of negative norm then $\operatorname{Sym}^k \mathcal H_{PD} ^\prime\neq0$. \end{corollary} \begin{remark}\label{invariants_integral} We want to describe the image \begin{equation*} \operatorname{Sym}^k_ A V(g_f)_{PD}^{Z_K} \rightarrow \left(\operatorname{Sym}^k_A V(g_f)_{PD}\otimes_A R \right)^{Z_K}=\operatorname{Sym}^k_R (V(g_f)\otimes_A R)_{PD}^{Z_K} \end{equation*} explicitly. $\text{Gal}(F^{\text{Hil}}/{\mathbb Q})$ acts on $R$ and we get $R^{\text{Gal}(F^{\text{Hil}}/{\mathbb Q})}=A$ by Galois-theory. This implies \begin{equation*} \operatorname{Sym}^k _AV(g_f)_{PD} = \left(\operatorname{Sym}^k_A V(g_f)_{PD}\otimes_A R\right)^{\text{Gal}(F^{\text{Hil}}/{\mathbb Q})}=\operatorname{Sym}^k_R (V(g_f)\otimes_A R)_{PD}^{Gal(F^{\text{Hil}}/{\mathbb Q})}, \end{equation*} as $\operatorname{Sym}^k _AV(g_f)_{PD}$ is a free $A$-module. We identify \begin{equation*} \operatorname{Sym}^k _AV(g_f)_{PD} ^{Z_K}=\left(\operatorname{Sym}^k_R (V(g_f)\otimes_A R)_{PD}^{Z_K}\right)^{\text{Gal}(F^{\text{Hil}}/{\mathbb Q})}, \end{equation*} since the Galois-action and the $Z_K$-action commute. Let us suppose $k=\xi m$, otherwise $\operatorname{Sym}^k _AV(g_f)^{Z_K}=0$ by the considerations above. Write $V(g_f)=\mathfrak a \oplus \mathfrak b$ as in \Cref{invariants}. In $\operatorname{Sym}^k_R (V(g_f)\otimes_A R)_{PD}^{Z_K}$ we have $R\prod_\sigma (\beta_\sigma Y_\sigma) ^{[m]}$ as direct summand. $\text{Gal}(F^{\text{Hil}}/{\mathbb Q})$ acts trivially on $\prod_\sigma Y_\sigma ^{[m]}$, because $\tau\cdot Y_{\sigma}=Y_{\tau^{-1}\circ\sigma}$ for $\tau\in \text{Gal}(F^{\text{Hil}}/{\mathbb Q})$. Therefore \begin{equation*} \prod_\sigma \sigma(\mathfrak b)^mR\stackrel{\cong}{\rightarrow}R\prod_\sigma (\beta_\sigma Y_\sigma) ^{[m]}=R\prod_\sigma \beta_\sigma^m Y_\sigma ^{[m]},\ x\mapsto x\cdot \prod_\sigma Y_\sigma ^{[m]} \end{equation*} as Galois-modules, where on the left-hand side we have the product of ideals. \begin{lemma} \begin{equation*} \left(\prod_\sigma \sigma(\mathfrak b)R\right)^{\text{Gal}(F^{\text{Hil}}/{\mathbb Q})}=\operatorname{N}(\mathfrak b)A \end{equation*}
where $\operatorname{N}(\mathfrak b):=\prod_{\text{$\nu$ finite place of $F$}}|\kappa(\mathfrak p_\nu)|^{\nu(\mathfrak b)}$ and $\nu(\mathfrak b)$ is defined by $\mathfrak b\mathcal O_\nu=(\pi_\nu ^{\nu(\mathfrak b)})$. \begin{proof} We may assume that $\mathfrak b$ is integral, otherwise we multiply the ideal with a number $z\in {\mathbb Z}\setminus \left\{0\right\}$. As a localization of a Dedekind ring $A$ is itself a Dedekind ring. So we have unique prime factorization of ideals in $A$ and therefore our problem is local at each maximal ideal $(p)\subset A$. First we have \begin{equation*} \left(\left(\prod_\sigma \sigma(\mathfrak b)R\right)^{\text{Gal}(F^{\text{Hil}}/{\mathbb Q})}\right)_{(p)}=\left(\left(\prod_\sigma \sigma(\mathfrak b)R\right)\cap {\mathbb Q}\right)_{(p)}=\left(\prod_\sigma \sigma(\mathfrak b)R_{(p)}\right)\cap {\mathbb Q} \end{equation*} by Galois-theory. As $(p)\subset {\mathbb Q}$ and $\sigma$ is ${\mathbb Q}$-linear, we identify the last expression with $\left(\prod_\sigma \sigma(\mathfrak b_{(p)})R_{(p)}\right)\cap {\mathbb Q}$. Now $\mathcal O_{(p)}$ is a Dedekind ring with only finitely many primes and hence a principal ideal domain. So $\mathfrak b_{(p)}=b\mathcal O_{(p)}$ for $b\in \mathcal O$ and we identify our ideal with \begin{equation*} \left(\prod_\sigma \sigma(b)R_{(p)}\right)\cap {\mathbb Q}=\operatorname{N}(b)R_{(p)}\cap {\mathbb Q}. \end{equation*} $R_{(p)}$ is the integral closure of $A_{(p)}$ inside $F^{\text{Hil}}$ and as $A_{(p)}$ is integrally closed in ${\mathbb Q}$ we get $\operatorname{N}(b)R_{(p)}\cap {\mathbb Q}=\operatorname{N}(b)A_{(p)}$. We calculate \begin{equation*}
\operatorname{N}(\mathfrak b)A_{(p)}=\prod_{\text{$\nu$ finite place of $F$}}|\kappa(\mathfrak p_\nu)|^{\nu(\mathfrak b)}A_{(p)}=\prod_{\nu|p}|\kappa(\mathfrak p_\nu)|^{\nu(\mathfrak b)}A_{(p)}. \end{equation*} In $\mathcal O_{(p)}$ we have the unique prime factorization \begin{equation*}
b \mathcal O_{(p)}=\mathfrak b_{(p)}=\prod_{\text{$\mathfrak p\subset \mathcal O$ max., $\mathfrak p|p$}}\mathfrak p^{\nu_{\mathfrak p}(\mathfrak b)}\mathcal O_{(p)} \end{equation*}
and we see $\nu_{\mathfrak p}(b)=\nu_{\mathfrak p}(\mathfrak b)$ for $\mathfrak p| p$. We get \begin{equation*}
\left(\left(\prod_\sigma \sigma(\mathfrak b)R\right)^{\text{Gal}(F^{\text{Hil}}/F)}\right)_{(p)}=|\operatorname{N}( b)|A_{(p)}=\prod_{\text{$\mathfrak p\subset \mathcal O$ max.}}|\kappa(\mathfrak p)|^{\nu_{\mathfrak p}(b)}A_{(p)}= \end{equation*} \begin{equation*}
\prod_{\text{$\mathfrak p\subset \mathcal O$ max., $\mathfrak p|p$}}|\kappa(\mathfrak p)|^{\nu_{\mathfrak p}(b)}A_{(p)}=\prod_{\text{$\mathfrak p\subset \mathcal O$ max., $\mathfrak p|p$}}|\kappa(\mathfrak p)|^{\nu_{\mathfrak p}(\mathfrak b)}A_{(p)}=\operatorname{N}(\mathfrak b)A_{(p)} \end{equation*} and we are done. \end{proof} \end{lemma} \begin{proposition}\label{Y_integral}
We consider $A:={\mathbb Z}[\frac{1}{Nd_F}]$. $V(g_f)=g_f^{-1}V(\hat{\mathbb Z})\cap V({\mathbb Q})=\mathfrak a\oplus \mathfrak b$. Then $\operatorname{N}(\mathfrak b)^m\prod_\sigma Y_\sigma ^{[m]}$ is part of an $A$-basis of $\operatorname{Sym}^k _AV(g_f)_{PD} ^{Z_K}$ and $\operatorname{N}(\mathfrak b)=\left\|t_2\right\|_f $, if we write \begin{equation*} g_f=xb=x\begin{pmatrix}t_1 & t_1 u\\ 0 & t_2 \end{pmatrix}, x\in G(\hat{\mathbb Z}). \end{equation*} \begin{proof}
The first part of the proposition is clear by the preceding lemma. We have $V(g_f)=V(b)$, since $x^{-1}V(\hat{\mathbb Z})=V(\hat{\mathbb Z})$. From this description it is clear that $\mathfrak b = t_2 ^{-1} \hat\mathcal O\cap F$ and therefore $\operatorname{N}(\mathfrak b)=\left\|t_2\right\|_f $. \end{proof} \end{proposition} \end{remark} \begin{proposition}\label{global_classes} Suppose $\det(K_f)=K_f\cap Z({\mathbb A}_f)$, what we always will assume from now on. We have cohomology classes in $H^\bullet (\mathcal M_K,{\mathbb Z}[\frac{1}{2}])$ restricting for all $s\in \mathcal S_K$ to a basis of $H^\bullet(\phi_K ^{-1}(s),{\mathbb Z}[\frac{1}{2}])$. In particular, the local system $R^\bullet\phi_{K*}({\mathbb Z}[\frac{1}{2}])$ is trivial. \begin{proof} Suppose that we had the cohomology classes with the desired properties. Let $H^\bullet\subset H^\bullet (\mathcal M_K,{\mathbb Z}[\frac{1}{2}])$ be the ${\mathbb Z}[\frac{1}{2}]$-submodule generated by the elements restricting to bases of the cohomology of the fibers. The natural morphism $H^\bullet (\mathcal M_K,{\mathbb Z}[\frac{1}{2}])\rightarrow H^0(\mathcal S_K,R^\bullet\phi_{K*}({\mathbb Z}[\frac{1}{2}]))$ induces $H^\bullet \rightarrow H^0(\mathcal S_K,R^\bullet\phi_{K*}({\mathbb Z}[\frac{1}{2}]))$ and therefore an isomorphism of sheaves $\underline H^\bullet \rightarrow R^\bullet\phi_{K*}({\mathbb Z}[\frac{1}{2}])$, which would be our trivialization. So let us find these cohomology classes. We have determinant maps \begin{equation*}
\mathcal S_K\rightarrow \det(K_f)\prod_{\nu|\infty}{\mathbb R}^\times\backslash \mathbb{I}_F/F^{\times}=:Cl_F ^{\overline{K}},\ g\mapsto \det(g), \end{equation*} \begin{equation*} \mathcal M_K\rightarrow \det(K_f){\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash \mathbb{I}_F/F^{\times}=:T_K,\ g\mapsto \det(g) \end{equation*} We consider the fiber product $\mathcal S_K\times_{Cl_F ^{\overline{K}}}T_K$, where on the right hand side the structure map is the natural projection. We have the natural maps \begin{equation*} p:\mathcal M_K\rightarrow \mathcal S_K\times_{Cl_F ^{\overline{K}}}T_K,\ K^1gG({\mathbb Q})\mapsto (KgG({\mathbb Q}), \det(g)), \end{equation*} \begin{equation*} pr_2:\mathcal S_K\times_{Cl_F ^{\overline{K}}}T_K\rightarrow T_K. \end{equation*} Everything is fibered over $Cl_F ^{\overline{K}}$ so the cohomology groups decompose into direct sums corresponding to components which are cut out by elements of $Cl_F ^{\overline{K}}$, so we can treat these components separately. We consider $G(g_f)$ for some $g_f\in G({\mathbb A}_f)$. We get \begin{equation*} \phi_K:\left(\mathbb{H}^{\xi}_{\pm}\times ({\mathbb R}_{>0}\backslash Z({\mathbb R}) ^0)\right)/G(g_f)\rightarrow \mathbb{H}^{\xi}_{\pm}/G(g_f),\ (\tau,r)\mapsto \tau \end{equation*} and \begin{equation*} p:\left(\mathbb{H}^{\xi}_{\pm}\times ({\mathbb R}_{>0}\backslash Z({\mathbb R})^0)\right)/G(g_f)\rightarrow(\mathbb{H}^{\xi}_{\pm}/G(g_f))\times ({\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/\det(G(g_f))) \end{equation*} is a map of fiber bundles over $\mathbb{H}^{\xi}_{\pm}/G(g_f)$. We have the projection \begin{equation*} pr_2:(\mathbb{H}^{\xi}_{\pm}/G(g_f))\times ({\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/\det(G(g_f)))\rightarrow {\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/\det(G(g_f)) \end{equation*} and given a basis $(\eta_i)\subset H^\bullet({\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/\det(G(g_f)),{\mathbb Z}[\frac{1}{2}])$ the classes \begin{equation*} (pr_2 ^*\eta_i)\subset H^\bullet((\mathbb{H}^\xi/G(g_f))\times ({\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/\det(G(g_f))),{\mathbb Z}[\frac{1}{2}]) \end{equation*} restrict back to a basis of the fibers. $p$ respects the fibers and therefore the same is true for $(p^*pr_2^*\eta_i)=:(\overline\eta_i)$. To see this restrict $p$ to the fiber \begin{equation*}
p:{\mathbb R}_{>0}\backslash Z({\mathbb R}) ^0/Z_K\rightarrow {\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/\det(G(g_f)), x\cdot Z_K\mapsto x\cdot \det(G(g_f))
\end{equation*} We may suppose $Z_K=\det(G(g_f))$. On the left hand side $\epsilon \in Z_K$ acts by multiplication with $\epsilon ^2$, whereas $\epsilon \in \det(G(g_f))$ acts on the right hand side by multiplication with $\epsilon$. Therefore the map above is an isogeny of real tori of degree $(Z_K:Z_K^2)$, which is a power of $2$. As $2\in {\mathbb Z}[\frac{1}{2}] ^{\times}$ the map induces an isomorphism on cohomology.
Let us now denote the set of connected components of $\mathcal M_K$ by $J$. We have constructed cohomology classes $(\overline\eta_i ^j)$ for each $j\in J$ restricting to basis of the cohomology of the fibers. The family of cohomology classes $(\sum_{j\in J} \overline \eta_i ^j)$ on $\mathcal M_K$ has the desired property: If $s\in \mathcal S_K$ is given, then $\phi_K ^{-1}(s)$ will be contained in a connected component corresponding to some $j_0\in J$. We get $(\sum_{j\in J} \overline \eta_{i\ |\phi^{-1}(s)} ^j)=(\overline \eta_{i\ |\phi^{-1}(s)} ^{j_0})$ and the right-hand side is a basis of the cohomology of the fiber $\phi_K ^{-1}(s)$ by the construction above. \end{proof} \end{proposition} \begin{remark}\label{invariant_Z_cohom} The cohomology classes described above are pullback by $\det:\mathcal M_K\rightarrow \det(K_f){\mathbb R}_{>0}\left\{\pm1\right\}^\xi\backslash \mathbb{I}_F/F^{\times}=:T_K$. We have a short exact sequence of groups \begin{equation*} 0\to {\mathbb R}_{>0}\left\{\pm 1\right\}^\xi\backslash Z({\mathbb R})/\det(G(g_f))\to T_K \to Cl_F ^{\overline {K}}\to 0 \end{equation*}
The group $T_K$ acts on itself by translation and therefore also on $H^\bullet(T_K,{\mathbb Z})$ its cohomology. This action factors through the quotient $T_K\to Cl_F ^{\overline{K}}$ and we get \begin{equation*} H^\bullet (T_K,{\mathbb Z})^{T_K}=H^\bullet(T_K,{\mathbb Z})^{Cl_F ^{\overline{K}}}\cong H^\bullet({\mathbb R}_{>0}\backslash Z({\mathbb R}) ^0/\det(G(g_f)),{\mathbb Z}). \end{equation*} The pullback of these forms by $\det$ have the desired property of the proposition above and we denote the corresponding subspace by $\mathfrak H_K ^\bullet\subset H^\bullet(\mathcal M_K,{\mathbb Z})$. We have the invariant $1$-forms $\frac{dr_\nu}{r_\nu}$ on $F_{\mathbb R}^{\times}$, which we may also interpret as invariant $1$-forms on $T_K$. The invariant forms $\mathfrak H _K ^\bullet$ are ${\mathbb R}$-linear combinations of wedge-products these forms. Moreover, we set $\mathfrak H ^\bullet:=\mathfrak H_K ^\bullet\otimes{\mathbb Q}$, as this space does not depend on the level $K$. \end{remark}
\begin{theorem}\label{decomp} We have an isomorphism \begin{equation*} H^\bullet(\mathcal S_K,\operatorname{Sym}^k\mathcal H ^\prime)\otimes_{\mathbb Q} \mathfrak H ^\bullet\rightarrow H^\bullet(\mathcal M_K,\operatorname{Sym}^k\mathcal H),\ \omega\otimes\eta\mapsto \omega\cup\eta \end{equation*} \begin{proof} Let us choose a resolution $I^\bullet$ of ${\mathbb Q}$ on $\mathcal M_K$ by injective sheaves of ${\mathbb Q}$-modules. We take a basis $\eta_1,...,\eta_l$ of $\mathfrak{ H} ^\bullet$ in other words cohomology classes restricting to a basis of the cohomology of the fibers of $\phi_K$. Let us denote the cohomological degree of the class $\eta_i$ by $d_i\in {\mathbb N}_0$ ($i\geq j$, then $d_i\geq d_j$). We may interpret $\eta_i$ as a morphism $\eta_i: {\mathbb Q}[-d_i]\rightarrow I^\bullet$. These morphisms induce \begin{equation*} \operatorname{id}\otimes\eta_i=\eta_i: \operatorname{Sym}^k\mathcal H[-d_i]\rightarrow \operatorname{Sym}^k\mathcal H\otimes_{\mathbb Q} I^\bullet \end{equation*} By \Cref{injectives_local_systems} we may use $\operatorname{Sym}^k\mathcal H\otimes_{\mathbb Q} I^\bullet$ as an injective resolution of $\operatorname{Sym}^k\mathcal H$. Applying $\phi_{K *}$ and summing over all $i$ yields \begin{equation*} \sum_i \eta_i:\bigoplus_i \phi_{K *}\operatorname{Sym}^k\mathcal H [-d_i]\rightarrow R\phi_{K *}\operatorname{Sym}^k\mathcal H. \end{equation*} Now apply $H^p$ and get a map \begin{equation*} \sum_{\text{$i$ with $d_i=p$}} \eta_i:\bigoplus_{\text{$i$ with $d_i=p$}} \phi_{K *}\operatorname{Sym}^k\mathcal H[-d_i]\rightarrow R^p\phi_{K *}(\operatorname{Sym}^k\mathcal H). \end{equation*} \begin{equation*} R^p\phi_{K *}(\operatorname{Sym}^k\mathcal H)=\phi_{K *}\operatorname{Sym}^k\mathcal H\otimes_{\mathbb Q} R^p\phi_{K *}({\mathbb Q}) \end{equation*} and the properties of $\eta_i$ show that the last map is an isomorphism. In particular, \begin{equation*} \sum_i \eta_i:\bigoplus_i \phi_{K *}\operatorname{Sym}^k\mathcal H[-d_i]\rightarrow R\phi_{K *}\operatorname{Sym}^k\mathcal H \end{equation*} is a quasi-isomorphism. We conclude \begin{equation*} R\Gamma(\mathcal S_K,\phi_{K *}\operatorname{Sym}^k\mathcal H)\otimes_{\mathbb Q} \mathfrak H ^\bullet:=\bigoplus_i R\Gamma(\mathcal S_K,\phi_{K *}\operatorname{Sym}^k\mathcal H)[-d_i]= \end{equation*} \begin{equation*}
=R\Gamma(\mathcal S_K,R\phi_{K *}\operatorname{Sym}^k\mathcal H)=R\Gamma(\mathcal M_K,\operatorname{Sym}^k\mathcal H). \end{equation*} Moreover, we have $\operatorname{Sym}^k\mathcal H^\prime=\phi_{K*}\operatorname{Sym}^k\mathcal H $ completing the proof. \end{proof} \end{theorem} Let us consider $\mathfrak H ^\bullet$ as trivial $G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$-module. Set \begin{equation*} \varinjlim H^\bullet(\mathcal S_K,\operatorname{Sym}^k\mathcal H ^\prime):=H^\bullet(\mathcal S,\operatorname{Sym}^k\mathcal H ^\prime). \end{equation*}
\begin{corollary}\label{equiv_decomp} We have an isomorphism of $G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$-modules \begin{equation*} H^\bullet(\mathcal S,\operatorname{Sym}^k\mathcal H ^\prime)\otimes_{\mathbb Q} \mathfrak H ^\bullet\rightarrow H^\bullet(\mathcal M,\operatorname{Sym}^k\mathcal H) \end{equation*} \begin{proof} The isomorphism is \Cref{decomp}. $G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$-equivariance follows from the fact that $\mathfrak H ^\bullet$ consists of invariant classes. \end{proof} \end{corollary} \begin{corollary}\label{Eis^k _q} We have an $G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$-equivariant operator \begin{equation*} \operatorname{Eis}^k _q:\mathcal S(V({\mathbb A}_f),\mu^{\otimes n})^0 \otimes_{\mathbb Q} \mathfrak H ^{q*}\rightarrow H^{2\xi-1-q}(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes\mu^{\otimes n+1}) \end{equation*} $f\otimes \eta^*\mapsto \operatorname{Eis}^k(f)(\eta^*)$ for $q=0,...,\xi-1$. \begin{proof} This is just \Cref{Eis^k} and \Cref{equiv_decomp} \end{proof} \end{corollary}
\subsection{Explicit description of the decomposition of the cohomology}\label{trace_integration}
We want to calculate our polylogarithmic Eisenstein classes by differential forms in order to compare them with those of Harder. Therefore, we need to make the decomposition of the cohomology of $\mathcal M_K$ explicit. This can be done by the theory of fiber integration on de Rham cohomology.
Let us work over the coefficient ring ${\mathbb Q}$ first. We want to give the inverse map of the natural decomposition isomorphism provided by \Cref{decomp}. In other words, given a basis $\eta_1,...,\eta_l$ of $\mathfrak H ^\bullet$ and $\omega\in H^\bullet(\mathcal M_K,\operatorname{Sym}^k\mathcal H)$ we want to know explicitly $\omega_1,...,\omega_l\in H^\bullet(\mathcal S_K,\operatorname{Sym}^k\mathcal H ^\prime)$ such that $\omega=\sum_{i=1} ^l\omega_i\cup \eta_i$ holds.\\ We choose the retraction $\operatorname{Sym}^k\mathcal H\rightarrow \phi_K ^{-1}\operatorname{Sym}^k\mathcal H ^\prime$ of the canonical monomorphism $\phi_K ^{-1}\operatorname{Sym}^k\mathcal H ^\prime\rightarrow \operatorname{Sym}^k\mathcal H$ provided by \Cref{invariants}. \begin{lemma}\label{Z_proj1} The retraction induces an isomorphism \begin{equation*} H^\bullet(\mathcal M_K,\operatorname{Sym}^k\mathcal H)\rightarrow H^\bullet(\mathcal M_K,\phi_{K} ^{-1}\operatorname{Sym}^k\mathcal H ^\prime) \end{equation*} on cohomology. \begin{proof} We know by \Cref{fiber_cohom} and the projection formula \begin{equation*} R^p\phi_{K*}(\phi_K ^{-1}\operatorname{Sym}^k\mathcal H ^\prime)=\operatorname{Sym}^k\mathcal H ^\prime\otimes_{\mathbb Q} R^p \phi_{K*}({\mathbb Q})=R^p\phi_{K*}(\operatorname{Sym}^k\mathcal H) \end{equation*} so that our projection induces an isomorphism \begin{equation*} R\phi_{K*}(\operatorname{Sym}^k\mathcal H ^\prime)=R\phi_{K*}(\operatorname{Sym}^k\mathcal H )\in D^+(\mathcal S_K,{\mathbb Q}) \end{equation*} in the derived category of bounded below complexes of sheaves of ${\mathbb Q}$-modules. Applying $R^p\Gamma(\mathcal S,\ )$ yields the isomorphism in question. \end{proof} \end{lemma} \begin{remark} We may modify the proof of \Cref{decomp} to obtain an isomorphism \begin{equation*} H^\bullet(\mathcal S_K,\operatorname{Sym}^k\mathcal H^\prime)\otimes_{\mathbb Q} \mathfrak H ^\bullet\stackrel{\cup}{\rightarrow} H^\bullet(\mathcal M_K,\phi_K^{-1}\operatorname{Sym}^k\mathcal H ^\prime) \end{equation*} compatible with the isomorphism of \Cref{Z_proj1}. \end{remark} \begin{definition}\label{def_tr} Let $B$ and $F$ be topological manifolds and $f:E\rightarrow B$ a fiber bundle with typical fiber $F$. We assume $F$ to be compact of dimension $r$. So $f$ is proper. Moreover, for a ring $A$ we consider a $A$-local system $\mathcal V$ on $B$. There is the natural \textit{edge morphism} \begin{equation*}
e:H^p(E,f^{-1}\mathcal V)\rightarrow H^{p-r}(B,\mathcal V\otimes_kR^rf_*(A)). \end{equation*} Let us recall how it is constructed. $R^pf_*(f^{-1}\mathcal V)=0$, $p>r$, since \begin{equation*} R^pf_*(f^{-1}\mathcal V)_b=H^p(f^{-1}(b),f^{-1}\mathcal V)=0,\ p>r, \end{equation*} by \cite{Iv}VII 1.4, III 9.6 and 9.10. Therefore the inclusion of the truncation induces a quasi-isomorphism $\tau_{\leq r}Rf_{*}(f^{-1}\mathcal V)\rightarrow Rf_{*}(f^{-1}\mathcal V)$. We have the canonical projection \begin{equation*} \tau_{\leq r}Rf_{*}(f ^{-1}\mathcal V)\rightarrow R^{r}f_{*}(f ^{-1}\mathcal V)[-r]=\mathcal V\otimes_k R^{r}f_{*}(A)[-r], \end{equation*} where we used the projection formula on the right hand side (\cite{Iv} VII 2.4). Both maps together give us the morphism \begin{equation*} e:Rf_{*}(f ^{-1}\mathcal V)\rightarrow \mathcal V\otimes_k R^{r}f_{*}(A)[-r] \end{equation*} in $D^+(B,k)$. Applying $H^p(B,\ )$ yields the edge morphism. \end{definition} \begin{remark}\label{tr_proj} Since $\phi_K:\mathcal M_K\rightarrow \mathcal S_K$ is a fiber bundle with compact fibers, we get \begin{equation*} e:H^p(\mathcal M_K,\phi_K ^{-1}\operatorname{Sym}^k\mathcal H ^\prime)\rightarrow H^{p-(\xi-1)}(\mathcal S_K,\operatorname{Sym}^k\mathcal H ^\prime)\otimes_{\mathbb Q} \mathfrak H ^{\xi-1}. \end{equation*} Going through the proof of \Cref{decomp} we see that the composition of \begin{equation*} \bigoplus_{q} H^{p-q}(\mathcal S_K,\operatorname{Sym}^k\mathcal H ^\prime)\otimes_{\mathbb Q} \mathfrak H ^{q}\stackrel{\cup}{\rightarrow}H^p(\mathcal M_K,\phi_K ^{-1}\operatorname{Sym}^k\mathcal H ^\prime) \end{equation*} with the edge morphism is just the projection onto $H^{p-(\xi-1)}(\mathcal S_K,\operatorname{Sym}^k\mathcal H ^\prime)\otimes_{\mathbb Q} \mathfrak H ^{\xi-1}$. \end{remark} Poincaré duality $H^\bullet(\phi_K ^{-1}(s),{\mathbb Z})\cong H^{\xi-1-\bullet}(\phi_K ^{-1}(s),{\mathbb Z})^*$ induces a Poincaré duality $\mathfrak H ^\bullet\cong \mathfrak H ^{\xi-1-\bullet\ *}$. In particular, we have the isomorphism $\int:\mathfrak H ^{\xi-1}\rightarrow {\mathbb Q}=\mathfrak H ^{0}$, which is called the \textit{integration or trace morphism}, and the morphism \begin{equation*} (\operatorname{id}\otimes\int )\circ e:H^p(\mathcal M_K,\phi_K ^{-1}\operatorname{Sym}^k\mathcal H ^\prime)\rightarrow H^{p-(\xi-1)}(\mathcal S_K,\operatorname{Sym}^k\mathcal H ^\prime), \end{equation*} which we also denote by $\int$. We extend $\int$ to $\mathfrak H ^\bullet$ by setting it zero on $\mathfrak H ^p$, $p\neq \xi-1$. By Poincaré duality we get a perfect pairing $\mathfrak H ^\bullet\times \mathfrak H ^\bullet\to {\mathbb Q}$, $(\omega,\eta)\mapsto \int\omega\cup\eta$ and given our basis $\eta_1,...,\eta_l$, with cohomological degrees $deg(\eta_1)\leq...\leq deg(\eta_l)$, we choose a dual basis $\eta^* _1,...,\eta_l ^*$ with $\int\eta_i \cup\eta_j ^*=\delta_{ij}$. \begin{lemma}\label{cup_decomp} We consider $H^\bullet(\mathcal M_K,\phi_K ^{-1}\operatorname{Sym}^k\mathcal H ^\prime)$ as right $\mathfrak H ^{\bullet}$-module via cup-product. If $\omega\in H^p(\mathcal M_K,\phi_K ^{-1}\operatorname{Sym}^k\mathcal H^\prime)$ is given, we have $\omega=\sum_{i=1}^l(\int\omega\cup\eta_i^*)\cup \eta_i$. \begin{proof} To proof this we may suppose by \Cref{decomp} $\omega=\sum_{i=1}^l\omega_i\cup \eta_i$, with $\omega_i\in H^p(\mathcal S_K,\operatorname{Sym}^k\mathcal H ^\prime)$. But we see with \Cref{tr_proj} \begin{equation*} \int \omega\cup\eta_j ^*=\sum_{i=1}^l\int(\omega_i\cup \eta_i)\cup\eta_j ^*=\sum_{i=1}^l\int\omega_i\cup (\eta_i\cup\eta_j ^*)=\sum_{i=1}^l\omega_i\cup \int(\eta_i\cup\eta_j ^*)=\omega_j \end{equation*} and the claim follows. \end{proof} \end{lemma}
As we really want to calculate cohomology classes, we finally have to make $e$ explicit in de Rham cohomology. So let us consider complex coefficients and the situation $f:E\rightarrow B$ from above, where we assume now $B,E,F$ to be $\mathcal C^\infty$-manifolds and $\mathcal V$ a ${\mathbb C}$-local system on $B$. We have $Rf_{*}(f^{-1}\mathcal V)=\mathcal V\otimes f_{*}(\Omega_{E} ^\bullet)$ and the quasi-isomorphism $\mathcal V\otimes R^{r}f_{*}({\mathbb C})\rightarrow\mathcal V\otimes R^{r}f_{*}({\mathbb C})\otimes\Omega_{B} ^\bullet$. We need a chain map $e^\bullet$ such that \begin{equation*} \begin{xy} \xymatrix{ \mathcal V\otimes f_{*}(\Omega_{E} ^\bullet)\ar[d]^{e} \ar[r]^ -{e^\bullet}& \mathcal V\otimes R^{r}f_{*}({\mathbb C})\otimes\Omega_{B} ^\bullet[-r]\\ \mathcal V\otimes R^{r}f_{*}({\mathbb C})[-d]\ar[r]^{=} & \mathcal V\otimes R^{r}f_{*}({\mathbb C})[-r]\ar[u] } \end{xy} \end{equation*} commutes. It suffices to find $e^\bullet$ such that \begin{equation*} \begin{xy} \xymatrix{ f_{*}(\Omega_{E} ^\bullet)\ar[d]^{e} \ar[r]^ -{e^\bullet}& R^{r}f_{*}({\mathbb C})\otimes\Omega_{B} ^\bullet[-r]\\ R^{r}\phi_{*}({\mathbb C})[-r]\ar[r]^{=} & R^{r}\phi_{*}({\mathbb C})[-r]\ar[u] } \end{xy} \end{equation*} commutes. This is the theory of fiber integration. We assume the fiber bundle $f:E\rightarrow B$ to be orientated in the sense of \cite{HalGr}. This means that we have a $r$-form $\eta$ on $E$ restricting to a volume form on each fiber of $f$. We demand additionally $\eta$ to be closed. Therefore $\eta$ defines a trivialization of $R^{r}f_{*}({\mathbb C})$. \begin{definition}\label{fiber_integration} We define the \textit{fiber integration operator} \begin{equation*} e^\bullet:f_{*}\Omega_{E}^\bullet\rightarrow R^{r}f_{*}({\mathbb C})\otimes\Omega_{B}^\bullet[-r],\ \omega\mapsto (-1)^{r(\bullet-r)}\eta\otimes \operatorname{vol}(F)^{-1}\int_F \omega, \end{equation*} where $\int_F \omega$ is defined as in \cite{HalGr} with respect to $\eta$ and $\operatorname{vol}(F):=\int_F\eta$. \end{definition} \begin{remark} \begin{enumerate} \item $\operatorname{vol}(F)$ is a well-defined locally constant function, as $\operatorname{vol}(F)(b)=\int_{f^{-1}(b)}\eta$ is finite and $\eta$ is closed. \item The factor of $-1$ is just for the proper sign conventions, as $d[p]=(-1)^p d$ for the differential of a shifted complex. So $e^\bullet$ is a chain map. \item We have $e^p=0$ for $p<r$. \end{enumerate}
\end{remark} \begin{proposition} The map $e^\bullet$ has the desired property, in other words \begin{equation*} e=H^p(B,e^\bullet):H^p(E,f^{-1}\mathcal V)\rightarrow H^{p-r}(B,\mathcal V\otimes R^df_*({\mathbb C})) \end{equation*} is the map from \Cref{def_tr}. \begin{proof} By \Cref{def_tr} we have to show that the following diagram commutes \begin{equation*} \begin{xy} \xymatrix{ \tau_{\leq r}f_{*}(\Omega_{E} ^\bullet)\ar[d]^{e} \ar[r]^ -{e^\bullet}& R^{r}f_{*}({\mathbb C})\otimes\Omega_{B} ^\bullet[-r]\\ R^{r}\phi_{*}({\mathbb C})[-r]\ar[r]^{=} & R^{r}\phi_{*}({\mathbb C})[-r]\ar[u] } \end{xy} \end{equation*} where $e$ just comes from the canonical map $\ker(f_*d:f_*\Omega_E ^r\to f_*\Omega_E^{r+1})\to R^{r}\phi_{*}({\mathbb C})$. As $e^\bullet $ is a chain map and zero in degrees $p<r$, the only thing that has to be proven is that $e=e^r:\ker(f_*d:f_*\Omega_E ^r\to f_*\Omega_E^{r+1})\to R^{r}\phi_{*}({\mathbb C})$ and since both maps factor over exact forms we just have to prove $e^r=\operatorname{id}:R^{r}\phi_{*}({\mathbb C})\to R^{r}\phi_{*}({\mathbb C})$. The sheaf $R^{r}\phi_{*}({\mathbb C})$ is trivialized by the section $\eta$, so it suffices to prove $e^r(\eta)=\eta\otimes 1\in \Gamma(B,R^{r}f_{*}({\mathbb C})\otimes\Omega_{B} ^\bullet[-r])$. But we easily calculate \begin{equation*} e^r(\eta)=(-1)^{r(r-r)}\eta\otimes \operatorname{vol}(F)^{-1}\int_F \eta=\eta\otimes \operatorname{vol}(F)^{-1} \operatorname{vol}(F)=\eta\otimes 1. \end{equation*} \end{proof} \end{proposition} \begin{definition} Set $\left\langle n\right\rangle:=\left\{1,...,n\right\}$ considered as an ordered set. If $A$ is any ${\mathbb R}$-algebra (non-commutative), $I\subset\left\langle n\right\rangle$ an ordered subset and $x\in F_A=\prod_{i=1}^n A$, set $x_I:=\prod_{i\in I}x_i$ and $\operatorname{N}(x):=x_{\left\langle n\right\rangle}$. If $i\in \left\langle n\right\rangle$ we also write $I\setminus i$ for $I\setminus \left\{i\right\}$, $I\cup i$ for $I\cup\left\{i\right\}$ and $I^c=\left\langle n\right\rangle\setminus I$ considering them as ordered sets. Moreover, if $I\subset \left\langle n\right\rangle$ is an subset we denote by $\operatorname{sgn}(I)=\pm1$ the sign of the permutation describing $\left\langle n\right\rangle\to\left\{i_1,...,i_k,j_1,...,j_{n-k}\right\}$ with $I=\left\{i_1,...,i_k\right\}$ and $\left\langle n\right\rangle\setminus I=\left\{j_1,...,j_{n-k}\right\}$. \end{definition} \begin{remark}\label{int_decomp}
Let us come back to $\mathcal M_K$ and summarize what it means to do fiber integration to decompose the cohomology. First of all it is enough to know how things work on connected components of $\mathcal M_K$, so let us consider $(\mathbb{H}^{\xi}_{\pm}\times ({\mathbb R}_{>0}\left\{\pm1\right\}^\xi\backslash Z({\mathbb R})))/G(g_f)$. If we consider $F_{\mathbb R} ^1:=\left\{t\in F_{\mathbb R} ^{\times}:|\operatorname{N}(t)|=1 \right\}$, we may also parametrize the space above by $(\mathbb{H}^{\xi}_{\pm}\times (\left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1))/G(g_f)$. We denote the typical coordinate on $F_{\mathbb R} ^1$ by $\tilde{r}:=r_{|F_{\mathbb R} ^1}$. The cohomology of the fiber $\left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K)$ may be generated as a ring by $\frac{d\tilde{r}}{\tilde{r}}_i$, $i=1,...,\xi$. The $\frac{d\tilde{r}}{\tilde{r}}_i$ are not linear independent, because $0=\frac{d(|\operatorname{N}(\tilde{r})|)}{|\operatorname{N}(\tilde{r})|}=\sum_{i=1}^\xi\frac{d\tilde{r}}{\tilde{r}}_i$. But $\frac{d\tilde{r}}{\tilde{r}}_i$, $i=1,....,\xi-1$, are linear independent and therefore the elements $\frac{d\tilde{r}}{\tilde{r}}_I$, $I\subset \left\langle \xi-1\right\rangle$, form a basis of $\mathfrak H_{\mathbb C}^\bullet=H^\bullet(\left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K),{\mathbb C})$. We get the volume form
$\operatorname{vol}_{F_{\mathbb R} ^1}:=\frac{d\tilde{r}}{\tilde{r}}_{\left\langle \xi-1\right\rangle}$ satisfying the formula $\operatorname{vol}_{F_{\mathbb R} ^1 }\wedge\frac{dt}{t}=\frac{dr}{r}_{\left\langle \xi\right\rangle}=:\operatorname{vol}_{F_{\mathbb R} ^\times}$ for $t:=|\operatorname{N}(r)|$. Note that the volume form $\operatorname{vol}_{F_{\mathbb R} ^1 }$ allows integration on $F_{\mathbb R}^1$ and induces a Haar measure $d^\times\tilde{r}$ on $F_{\mathbb R}^1$. We fix the measure $\frac{dt}{t}$ on ${\mathbb R}^\times$ and the product measure on $\prod_{\nu|\infty}{\mathbb R}^\times={\mathbb F}_{\mathbb R}^\times$, which we denote by $d^\times r$. The Haar measure $d^\times\tilde{r}$ is now uniquely determined by the Fubini formula $d^\times r=d^\times \tilde{r}\frac{dt}{t}$. The fiber $\left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K)$ is also oriented with respect to the volume form $\operatorname{vol}_{F_{\mathbb R} ^1}$ and we define integration on this space with respect to this volume form. Set $R_K:=\operatorname{vol}(\left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K))$. The trace morphism $\int:\mathfrak H_{\mathbb C} ^{\xi-1}\to {\mathbb C}$ is given by integration of top forms and the basis which under Poincaré duality is dual to $(\frac{d\tilde{r}}{\tilde{r}}_I)_{I\subset \left\langle \xi-1\right\rangle}$ is $(\frac{(-1)^{\operatorname{sgn}(I)}}{R_K}\frac{d\tilde{r}}{\tilde{r}}_{\left\langle \xi-1\right\rangle\setminus I})_{I\subset \left\langle \xi-1\right\rangle}$.
Now we want to make \cref{cup_decomp} explicit. Suppose we are given a differential form $\omega=\sum_{J\subset \left\langle \xi\right\rangle}\omega_J(r)\wedge\frac{dr}{r}_J$, such that $\omega_J(r)$ is a form of degree zero with respect to $r$, representing a cohomology class in $H^\bullet(\mathcal M_K,\phi_K ^{-1}\operatorname{Sym}^k\mathcal H ^\prime)$. First we may pullback the form to $(\mathbb{H}^{\xi}_{\pm}\times (\left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1))/G(g_f)$ giving the form $\sum_{J\subset \left\langle \xi\right\rangle}\omega_J(\tilde{r})\wedge\frac{d\tilde{r}}{\tilde{r}}_J$. We rewrite this form as \begin{equation*} \sum_{J\subset \left\langle \xi\right\rangle}\omega_J(\tilde{r})\wedge\frac{d\tilde{r}}{\tilde{r}}_J=\sum_{I\subset\left\langle \xi-1\right\rangle}\tilde{\omega}(\tilde{r})_I\wedge \frac{d\tilde{r}}{\tilde{r}}_{I} \end{equation*} where the $\tilde{\omega}_I$ are just linear combinations of the $\omega_J$. So we get by \cref{cup_decomp} \begin{equation*} \omega =\sum_I \int(\tilde{\omega}(\tilde{r})_I\wedge \frac{d\tilde{r}}{\tilde{r}}_{I}\wedge \frac{(-1)^{\operatorname{sgn} (I)}}{R_K}\frac{d\tilde{r}}{\tilde{r}}_{\left\langle \xi-1\right\rangle\setminus I})\wedge \frac{d\tilde{r}}{\tilde{r}}_{I}=\sum_I \frac{1}{R_K}\int(\tilde{\omega}(\tilde{r})_I\wedge \operatorname{vol}_{F_{\mathbb R} ^1})\wedge \frac{d\tilde{r}}{\tilde{r}}_{I} \end{equation*} But we have $\int(\tilde{\omega}(\tilde{r})_I\wedge \operatorname{vol}_{F_{\mathbb R} ^1})=\int_{\left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K)}\tilde{\omega}(\tilde{r})_Id^\times\tilde{r}$ by the definition of fiber integration, see \cite{HalGr}. So we get \begin{equation*} \omega =\sum_{I} \frac{1}{R_K}\int_{\left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K)}\tilde{\omega}(\tilde{r})_Id^\times\tilde{r}\wedge \frac{d\tilde{r}}{\tilde{r}}_{I}=\frac{1}{R_K}\sum_{J}\int_{\left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K)}\omega_J(\tilde{r})d^\times\tilde{r}\wedge\frac{d\tilde{r}}{\tilde{r}}_J. \end{equation*}
\end{remark}
\chapter{Comparison with Harder's Eisenstein classes}
\cite{Ki1} and \cite{Bl1} have shown that polylogarithmic Eisenstein classes may restrict non-trivially to cohomology classes of the boundary of the Borel-Serre compactification of Hilbert-Blumenthal varieties. This is done by evaluating the polylogarithmic Eisenstein classes on the boundary.\\ Harder on the other hand started with cohomology classes on the boundary and constructed an operator from the cohomology of the boundary into the cohomology of the Hilbert-Blumenthal variety, which is a section for the restriction map. The image of this operator is called the Eisenstein cohomology. \\ Now we want to represent the polylogarithmic Eisenstein classes by differential forms so that we can compare them with Harder's Eisenstein cohomology classes finally.
\section{Nori's calculation of the polylogarithm as current}
One of the big advantages of the polylogarithm is that it can actually be calculated. For example, this has been done by \cite{No} and A. Levin in \cite{L} with ${\mathbb C}$-coefficients by using currents. We will follow Nori's approach. We recall a well known lemma which justifies the extension of coefficients to ${\mathbb C}$. \subsection{Integral cohomology classes} \begin{lemma}\label{ext_scalars} Let $X$ be a topological manifold which has a finite good cover $\mathfrak U=(U_i)_{i=1,...,m}$, in other words a cover by finitely many open sets each homeomorphic to ${\mathbb R}^n$ such that all finite intersections $U_{i_0}\cap...\cap U_{i_p}$ are again homeomorphic to ${\mathbb R}^n$. Moreover, we consider $\mathcal V$ a locally constant sheaf of $A$-modules on $X$ and $A\rightarrow R$ a flat ring extension. Then the natural morphism $H^\bullet(X,\mathcal V)\otimes_A R\rightarrow H^\bullet(X,\mathcal V\otimes_A R)$ is an isomorphism. \begin{proof} Let $\mathfrak U=(U_i)$ be a finite good cover of $X$.
We have the the $\check C$ech to derived functor spectral sequence \cite{Go} Théorème 5.4.1. $E_2 ^{p,q} =\check{H}^p(\mathfrak U, \underline H^q(\mathcal V))\Rightarrow H^{p+q}(X,\mathcal V)$, with the presheaf $\underline H^q(\mathcal V)(V)=H^q(V,\mathcal V)$, $V\subset X$ open. Consider $U_{i_0},...,U_{i_p}\in \mathfrak U$. As $\mathfrak U$ is good, we get that $U_{i_0}\cap...\cap U_{i_p}$ is homeomorphic to ${\mathbb R}^n$ and in particular contractible. It follows that $\mathcal V_{|U_{i_0}\cap...\cap U_{i_p}}$ is constant and therefore \begin{equation*} \underline H^q(\mathcal V)(U_{i_0}\cap...\cap U_{i_p})=H^q(U_{i_0}\cap...\cap U_{i_p},\mathcal V)=0,\ q>0, \end{equation*} by homotopy invariance of sheaf cohomology with constant coefficients \cite{Iv} IV. Theorem 1.1. We conclude $C^p(\mathfrak U, \underline H^q(\mathcal V))=0$, if $q>0$, and therefore $E_2 ^{p,q}=0$, $q\neq 0$. The spectral sequence degenerates and the edge morphisms yield isomorphisms \begin{equation*} H^p(X,\mathcal V)=\check{H}^p(\mathfrak U, \underline H^0(\mathcal V))=\check{H}^p(\mathfrak U, \mathcal V). \end{equation*} If $A\rightarrow R$ is a flat ring extension, we see \begin{equation*} H^p(X,\mathcal V\otimes_A R)=\check{H}^p(\mathfrak U,\mathcal V\otimes_A R)=\check{H}^p(\mathfrak U,\mathcal V)\otimes_A R=H^p(X,\mathcal V)\otimes_A R, \end{equation*} as the covering $\mathfrak U$ is finite and $\Gamma(U_{i_0}\cap...\cap U_{i_p},\mathcal V\otimes_A R)=\Gamma(U_{i_0}\cap...\cap U_{i_p},\mathcal V)\otimes_A R$, since $\mathcal V$ is constant on $U_{i_0}\cap...\cap U_{i_p}$. \end{proof} \end{lemma} \begin{remark} Let $X$ a topological manifold with a finite good cover and $\mathcal V$ an $A$-local system. By induction on the cardinality of good covers one easily shows using the Mayer-Vietoris sequence that $H^\bullet(X,\mathcal V)$ is a finitely generated $A$-module. Let now $A\subset {\mathbb C}$ be a principal ideal domain. By \Cref{ext_scalars} we may embed our cohomology groups of local $A$-systems - up to torsion - in the cohomology groups with ${\mathbb C}$-coefficients. More precisely, $H^\bullet(X,\mathcal V)$ decomposes into a direct sum of a free $A$-module $H^\bullet(X,\mathcal V)_{\text{free}}$ and a torsion $A$-module $H^\bullet(X,\mathcal V)_{\text{tor}}$. We have isomorphisms \begin{equation*} H^\bullet(X,\mathcal V)_{\text{free}}\otimes_A{\mathbb C}\cong H^\bullet(X,\mathcal V)\otimes_A{\mathbb C} \cong H^\bullet(X,\mathcal V\otimes_A{\mathbb C}) \end{equation*} and $H^\bullet(X,\mathcal V)_{\text{free}}$ is isomorphic to $\operatorname{im}(H^\bullet(X,\mathcal V)\stackrel{can}{\rightarrow}H^\bullet(X,\mathcal V\otimes_A{\mathbb C}))$. We call the latter group the $A$-integral or simply the \textit{integral classes} inside $H^\bullet(X,\mathcal V\otimes_A{\mathbb C})$. If $X$ is a manifold, integral classes may be calculated by differential forms or currents with values in these local systems. In particular, we will represent the integral part of the polylogarithm by a current as proposed in \Cref{differential_equ}. \end{remark}
\subsection{Solving the differential equation}
Let us consider $\pi_W:\mathcal T_W\rightarrow \mathcal M_K$. Let us suppose $K=K_N$ and $W=V(\hat{\mathbb Z})\rtimes K_N$. To find a representative for the polylogarithm we may treat each connected component of $\mathcal M_{K_N}$ separately. So our topological situation is of the form \begin{equation*}
\prod_{\nu|\infty}SO(2) \backslash G({\mathbb R}) \stackrel{\cong}{\rightarrow}M:=\left\{(\tau,r)\in\mathbb{H}_{\pm}^\xi\times Z({\mathbb R}):\text{Im}(r\tau)\in Z({\mathbb R})^0\right\},\ g\mapsto (i\cdot g, \det(g)), \end{equation*} \begin{equation*} \pi:\left(V({\mathbb R})/V(g_f)\times M\right)/G(g_f) \rightarrow M/G(g_f) \end{equation*} and $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in G({\mathbb R})$ acts on $M$ by $(\tau,r)\gamma:=\left(\frac{b+d\tau}{a+c\tau},r\cdot \det(\gamma)\right)$. We write as before $V(g_f)=\mathfrak a\oplus \mathfrak b$ as a sum of two fractional ideals. Set $L:=V(g_f)$, $\Gamma:=G(g_f)$ and $\tilde{\pi}: \tilde T:=V({\mathbb R})/L\times M\rightarrow M$ the projection. Moreover, we consider ${\mathbb R}_{>0}\subset Z({\mathbb R})$ acting on $M$ by multiplication. \\ By \Cref{invariant_functionals} we have \begin{equation*} \mathcal R^\infty\otimes_{\mathcal C^\infty_{\tilde T/\Gamma}}\otimes \mathscr{D}_{\tilde T/\Gamma}=q_*\left(\prod_{k\geq 0}\operatorname{Sym}^kV({\mathbb C})\otimes \mathscr{D}_{\tilde T}\right)^\Gamma, \end{equation*} where $q:\tilde T\rightarrow \tilde T/\Gamma$ is the canonical projection. We will construct a $\Gamma$-invariant $2\xi-1$-current on $\tilde T$, which descends to the right current $\operatorname{pol}(f)$ on $\tilde T/\Gamma$. Let us fix coordinates first. We have $F_{\mathbb R} \stackrel{\cong}{\rightarrow}\prod_\sigma {\mathbb R}$, $x\otimes r\rightarrow (\sigma(x)r)_\sigma$ and after fixing an ordering of the set of embeddings $\sigma:F\rightarrow {\mathbb R}$ we get an isomorphism $\prod_\sigma {\mathbb R} \rightarrow {\mathbb R}^\xi$. Since $V({\mathbb R})=F_{\mathbb R}^2$, we take the coordinates $w=(w^1,w^2)$, where $w^{i}=(w^{i}_\sigma)_\sigma$, $i=1,2$. $\mathbb{H}_{\pm}^\xi$ is considered as an open subspace of the complex manifold $F_{\mathbb C}$ and we take the induced coordinates $\tau=(\tau_\sigma)_\sigma$. The corresponding real coordinates are denoted by $(x,y)$, i.e. $\tau=x+iy$. The ($F_{\mathbb C}$-valued) differential forms $dw^1,dw^2$ are globally defined on $\tilde T$ making the algebra of global differentials on $\tilde T$ into a free $\mathcal C^\infty(\tilde T)$-module of finite rank. We trivialize $or^{-1}_{M/\Gamma}\otimes {\mathbb C}$ and $or^{-1}_{{\mathbb R}_{>0}\backslash M/\Gamma}\otimes {\mathbb C}$ by choosing volume forms $\operatorname{vol}_{M/\Gamma}$ and $\operatorname{vol}_\mathcal M:=\operatorname{vol}_{{\mathbb R}_{>0}\backslash M/\Gamma}$. We also need an orientation of the fibers of our family of tori. We take $\theta:=\frac{1}{\operatorname{vol}(T)}\bigwedge_\sigma dw^1_\sigma\wedge dw^2_\sigma$, where the volume is calculated with respect to $\operatorname{vol}_V:=\bigwedge_\sigma dw^1_\sigma\wedge dw^2_\sigma$, but the current we are going to construct will not depend on the particular choice of $\theta$.
We can identify distributions $\phi$ on $T$ with $0$-currents, when we use our volume form $\theta$: Given a form $\eta$ with compact support of top degree on $T$ we may write it uniquely as $f\theta$ with a compactly supported $\mathcal C^\infty$-function $f$ on $T$. Then we define $\phi(\eta)=:\phi(f)$. For example, we have the distribution $\delta_f=\sum_{t\in T}f(t)\delta_t$, where $f:T\rightarrow {\mathbb C}$ is a map of finite support and $\delta_t$ is the Delta distribution in $t$.\\ Currents $\phi$ on $T$ can be considered as linear functionals $\phi:\Omega_c(\tilde T)\rightarrow \Omega_c(M)$; compare \cite{DR} Th\'{e}or\`{e}me 9. In this manner we can consider $\delta_f \theta$ as a $2\xi$-current on $\tilde T$ with values in forms on $M$. Given a $\eta\in \Gamma_c(\tilde T, \mathcal C^\infty _{\tilde T}\otimes_{\tilde\pi ^{-1}\mathcal C^\infty _M}\tilde\pi ^{-1}\Omega_M)$ we have explicitly \begin{equation*} \delta_f \theta(\eta)=\delta_f(w)(\theta\wedge\eta)=\sum_{t\in T}f(t)\eta(t,-) \end{equation*} where $\delta_f(w)$ means that $\delta_f$ just acts on the coordinates of $T$. Moreover, we have $1\otimes\delta_f \theta=\sum_{t\in T} f(t)\int_{\left\{t\right\}\times M}$ where $1\otimes\delta_f \theta$ means the tensor product of currents on $M$ and $T$ and integration is defined with respect to our fixed orientations. In general, all currents on $T$ can be identified with currents on $\tilde T$ with values in forms on $M$ and these can be identified via integration over $M$ with currents on $\tilde T$. So we may identify $1\otimes\delta_f \theta$ with $\delta_f \theta$.
With this overview at hand we can say in which space we want to look for $\operatorname{pol}(f)$. We consider the space generated by forms $\phi \wedge\omega$, where $\phi$ is a $0$-current on $T$ and $\omega$ a smooth form on $\tilde T$. Then $\phi\wedge \omega(\eta)=\phi(w)(\omega\wedge \eta)$ as currents on $\tilde T$ with values in forms on $M$. Consider the the pullback of $Log^\infty$ to $\tilde T$. We identify it with the trivial pro vector bundle $\prod_{k\geq0} \operatorname{Sym}^k V({\mathbb C})\otimes \mathcal C_{\tilde T} ^\infty$. We have the connection $\nabla=d-\kappa$ and are interested in solving the equation $\nabla(\phi)=\delta_{f} \theta$ of currents on $\tilde T$ with values in forms on $M$, where the support of $f$ consists of points of order $N$ on the torus $T$ and $\operatorname{aug}(f)=\sum_{t\in T}f(t)=f(0)=0$. \begin{remark}\label{delta_int} Our arithmetic group $\Gamma$ guarantees that $t:(\tau,r)\mapsto (t,\tau,r)$ defines a torsion section of $\tilde T/\Gamma\stackrel{\pi}{\rightarrow}M/\Gamma$ so that $1\otimes\delta_{f} \theta$ descends to a current on $\tilde T/\Gamma$, which can be identified with the functional $\sum_{t\in T} f(t)\int_{M/\Gamma} t^*=\int_f$ which does not depend on the choice of $\theta$. \end{remark} Currents $\phi$ on $T$ have a Fourier expansion. Since each current $\phi$ on $T$ can be uniquely written as differential form with distributional coefficients in the basis $dw^1,dw^2$, we may define Fourier expansion coefficientwise. Therefore it suffices to settle the case, when $\phi$ itself is a distribution. But then we have $\phi=\sum_{\lambda\in L^*}\phi_\lambda \exp(2\pi i \lambda(w))$, where $Hom(L,{\mathbb Z})=:L^*$ and $\phi_\lambda=\phi(\exp(-2\pi i\lambda(w)))$. Now we may evaluate $\phi$ as \begin{equation*} \phi(g)= \sum_{\lambda\in L^*}\phi_\lambda g_{-\lambda},\ \text{with }g=\sum_{\lambda\in L^*}g_\lambda \exp(2\pi i \lambda(w))\in \mathcal C^\infty (T). \end{equation*} For this see \cite{S} VII 1. \begin{remark}\label{eval_distr} If $\phi\in \mathcal C^\infty(T)$ and $T_\phi$ denotes the distribution $T_\phi(f):=\int_Tf\phi dx$, we have $\phi=\operatorname{vol}(T)^{-1}\sum_\lambda T_{\phi,\lambda}\exp(2\pi i \lambda(w))$. \end{remark} So we have a Fourier expansion \begin{equation*} \delta_{f} \theta=\sum_{\lambda\in L^*}\sum_{t\in T}f(t)\exp(2\pi i \lambda(w-t))\theta=\sum_{0\neq\lambda\in L^*}\sum_{t\in T}f(t)\exp(2\pi i \lambda(w-t))\theta, \end{equation*} as $\operatorname{aug}(f)=0$. In this spirit let us write $\phi=\sum_{\lambda\in L^*} \phi_{\lambda}\exp(2\pi i \lambda(w))$ as Fourier series where $\phi_{\lambda}$ are smooth forms on $M$ with values in the completed symmetric algebra tensored with closed invariant forms on $T$. Applying $\nabla$ we get the differential equation \begin{equation*} d(\phi_\lambda)+A_\lambda \wedge \phi_\lambda=\sum_{t\in T}f(t)\exp(2\pi i \lambda(-t))\theta,\ \lambda\in L^*,\ A_\lambda := 2\pi i\, d\lambda - \kappa \end{equation*} for each Fourier coefficient. We try to solve this equation in \begin{equation*} R=\prod_k \operatorname{Sym}^k V({\mathbb C})\otimes\Omega(M)\otimes_{\mathbb C} \bigwedge ^\bullet V({\mathbb C})^*\subset \prod_k \operatorname{Sym}^k V({\mathbb C})\otimes\Omega_{\tilde T}(\tilde T) \end{equation*} $R=\bigoplus R^{p,q}$ is bigraded: $R^{p,q}:=\prod_k \operatorname{Sym}^k V({\mathbb C})\otimes\Omega^p(M)\otimes_{\mathbb C} \bigwedge ^q V({\mathbb C})^*$. Let us first consider the case $\lambda= 0$. We get the equation $d(\phi_0)-\kappa \wedge \phi_0=0$, as $\delta_f$ has $0$ zeroth Fourier coefficient. So we simply set $\phi_0=0$. We endow $V({\mathbb R})$ with a symplectic structure: \begin{equation*} \left\langle .,.\right\rangle:V({\mathbb R})\wedge V({\mathbb R})\rightarrow {\mathbb R},\ w_1\wedge w_2 \rightarrow \operatorname{Tr}( \det(w_1,w_2)) \end{equation*} with $\operatorname{Tr}(x\otimes y):=\operatorname{Tr}_{F/{\mathbb Q}}(x\otimes y)=\sum_\sigma \sigma(x)y$ for $x\otimes y\in F_{\mathbb R}$. Next we fix global ($F_{\mathbb C}$-valued) vector fields $u:=\frac{\overline{\tau}e^1-e^2}{\overline{\tau}-\tau}$, $\overline{u} :=-\frac{\tau e^1-e^2}{\overline{\tau}-\tau}$ on $\tilde T$, where the tangent bundle of $T$ is trivialized by $e^1\mapsto \partial_{w_1}$, $e^2\mapsto \partial_{w_2}$. These vector fields reflect the fact that we have a decomposition into types $\mathcal H_{\mathbb Z}\otimes\mathcal C^\infty _{M/\Gamma}=\mathcal H^{0,-1}\oplus \mathcal H^{-1,0}$ coming from the Hodge theory of the fibers. The two subspaces are Lagrangian with respect to $\left\langle .,.\right\rangle_{\mathbb C}$ and we have $\det( u,\overline u)= \frac{1}{\overline \tau-\tau}$. Whenever $s$ is a local section of $\mathcal H_{\mathbb Z}\otimes\mathcal C^\infty _{M/\Gamma}$ denote by $s^{0,-1}$, $s^{-1,0}$ the decomposition into types. In this spirit $u$ defines a frame for $\mathcal H^{-1,0}$ and $\overline{u}$ for $\mathcal H^{0,-1}$ and, since we have \begin{equation*} w=w^1e^1+w^2e^2=(w^1+\tau w^2)u+(w^1+\overline{\tau} w^2)\overline{u} \end{equation*} for $ w\in V({\mathbb R})$, we see $w^{-1,0}=(w^1+w^2\tau)u$. Moreover, the decomposition into types is $\Gamma$-equivariant and we get \begin{equation*} (\gamma w)^{-1,0}=\gamma w^{-1,0},\ \gamma u =(a+c\tau)u,\ \gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}. \end{equation*} Since $\left\langle .,.\right\rangle$ is non-degenerate, we may express $L^*$ by it. If we set $\mathfrak d\subset \mathcal O$ to be the different-ideal of $\mathcal O$, we have an isomorphism \begin{equation*} \mathfrak b ^{-1} \mathfrak d^{-1}\oplus \mathfrak a ^{-1}\mathfrak d^{-1} \rightarrow L^*,\ l\mapsto \left\langle l,.\right\rangle:=\lambda \end{equation*} of $\mathcal O$-modules. We will stick to the vector fields $\frac{l^{-1,0}}{r}$, $0\neq l =(l_1,l_2)\in L^*\subset V({\mathbb R})$. We have $\gamma\frac{l^{-1,0}}{r}=\frac{\left(\det\gamma^{-1}\gamma l\right)^{-1,0}}{r}$ compatible with the natural $\Gamma$ action on $L^*$ \begin{equation*} \gamma\lambda(w):=\lambda(\gamma^{-1}w)=\left\langle l,\gamma^{-1}w\right\rangle=\left\langle \det\gamma^{-1} \gamma l,w\right\rangle,\ \lambda=\left\langle l,.\right\rangle. \end{equation*} Now we follow Nori's construction of $\operatorname{pol}(f)$. The vector field $\frac{l^{-1,0}}{r}$ on $\tilde T$ defines a linear operator on the differential algebra $R$ onto itself by contraction. Let us denote this operator by $i_l$. Furthermore, define the operator $C_l:=d+A_l\wedge$ on $R$, where $A_l$ is the $A_\lambda$ from above. Then $C_l\circ C_l=0$, $i_l\circ i_l=0$ and $C_l\circ i_l+i_l\circ C_l$ is an isomorphism. The first assertion is immediately clear. For the second write $F_l=(d i_l +i_l d) +(A_l i_l+i_l A_l)$. The first summand is the Lie derivative $\mathcal L_{\frac{l^{-1,0}}{r}}$ with respect to the vector field $\frac{l^{-1,0}}{r}$. It is a sum of maps $R^{p,q}\rightarrow R^{p+1,q-1}$ and therefore $\mathcal L_{\frac{l^{-1,0}}{r}}^{2\xi+1}=0$ and $\mathcal L_{\frac{l^{-1,0}}{r}}$ is nilpotent on $R$. \begin{equation*} (A_l i_l+ i_l A_l)(\omega)=A_l(i_l\omega)+i_l(A_l\wedge\omega)=A_l(i_l\omega)+i_l(A_l)\wedge\omega-A_l\wedge i_l(\omega) \end{equation*} by the rules of interior multiplication and therefore \begin{equation*} (A_l i_l+ i_l A_l)(\omega)=i_l(A_l)\wedge\omega=i_l(A_l)\omega \end{equation*} is just multiplication by \begin{equation*} i_l(A_l)=2\pi i\, d\lambda(\frac{l^{-1,0}}{r})-\kappa(\frac{l^{-1,0}}{r})=2\pi i\left\langle l,\frac{l^{-1,0}}{r}\right\rangle - \frac{l^{-1,0}}{r} \end{equation*} a function with values in the completed symmetric algebra of $V({\mathbb C})$. We calculate \begin{equation*} \left\langle l,\frac{l^{-1,0}}{r}\right\rangle=\left\langle l^{0,-1},\frac{l^{-1,0}}{r}\right\rangle, \end{equation*} since the Hodge decomposition is Lagrangian, and endow $F_{\mathbb C}$ with the natural Hermitian metric induced by $\operatorname{Tr}$ to get \begin{equation*}
2\pi i\left\langle l^{0,-1},\frac{l^{-1,0}}{r}\right\rangle=\pi \left\|\frac{l_1+l_2\tau}{\sqrt{r\text{Im}(\tau)}}\right\|^2=F_l (\tau,r). \end{equation*} Note $\gamma F_l(\tau,r)=F_l(\tau\gamma,r\det\gamma)=F_{\det\gamma^{-1}\gamma l}(\tau,r)$. Since $l\neq 0$, we have $F_l(\tau,r)\in {\mathbb C}^{\times}$ and therefore $i_l(A_l)$ is certainly invertible in $\prod_k \operatorname{Sym}^k V({\mathbb C})\otimes \mathcal C^\infty_M $, because the leading coefficient is invertible. It follows that $A_l i_l+ i_l A_l$ is invertible.\\ Now $\mathcal L_{\frac{l^{-1,0}}{r}}$ and $i_l(A_l)$ commute, since the vector field $\frac{l^{-1,0}}{r}$ takes its values in the tangent space of $T$ and therefore its Lie derivative is $\mathcal C_M ^\infty$-linear. We conclude that the sum $\mathcal L_{\frac{l^{-1,0}}{r}}+i_l(A_l)$ is also invertible. Explicitly, \begin{equation*} (i_lC_l+C_li_l)^{-1}=(\mathcal L_{\frac{l^{-1,0}}{r}}+i_l(A_l))^{-1}=\sum_{k\geq0}(-1)^k i_l(A_l)^{-(k+1)}\mathcal L_{\frac{l^{-1,0}}{r}}^k. \end{equation*} This is a finite sum. Set $\omega_{t,l}:=\exp(-2\pi i\lambda(t))\theta$. Then $i_l(i_lC_l+C_li_l)^{-1}(\omega_{t,l})$ is a solution for $d(\phi_\lambda)+A_\lambda \wedge \phi_\lambda=C_l(\phi_\lambda)=\exp(2\pi i \lambda(-t))\theta$. To prove this note that $C_l$ and $i_lC_l+C_li_l$ commute. Therefore $C_l$ and $(i_lC_l+C_li_l)^{-1}$ commute and \begin{equation*} \omega_{t,l}=(i_lC_l+C_li_l)(i_lC_l+C_li_l)^{-1}(\omega_{t,l})=C_li_l(i_lC_l+C_li_l)^{-1}(\omega_{t,l}), \end{equation*} since $i_lC_l(i_lC_l+C_li_l)^{-1}(\omega_{t,l})=i_l(i_lC_l+C_li_l)^{-1}C_l(\omega_{t,l})=0$, as $C_l\omega_{t,l}=0$. Because $i_l\mathcal L_{\frac{l^{-1,0}}{r}}=i_ldi_l$ we get \begin{equation*} \phi_l=\phi_\lambda=\sum_{t\in T}f(t)\sum_{k\geq0}(-1)^k i_l(A_l)^{-(k+1)}i_l(di_l)^k\omega_{t,l} \end{equation*} So we should have \begin{equation*}
\phi= \sum_{k\geq0} \sum_{0\neq l\in L^*}(-1)^k \frac{\sum_{t\in T}f(t)\exp(2\pi i\left\langle l,-t\right\rangle)i_l(d i_l)^k\theta}{\left(\pi \left\|\frac{l_1+l_2\tau}{\sqrt{r\text{Im}(\tau)}}\right\|^2-\frac{l^{-1,0}}{r}\right)^{k+1}}\exp(2\pi i\left\langle l,w\right\rangle) \end{equation*} The sum over $k$ only has non-trivial summands between zero and $2\xi-1$. We rewrite \begin{equation*}
\left(\pi \left\|\frac{l_1+l_2\tau}{\sqrt{r\text{Im}(\tau)}}\right\|^2-\frac{l^{-1,0}}{r}\right)^{-(k+1)} \end{equation*} as binomial series \begin{equation*}
\sum_{n\geq0} \frac{(k+n)!}{k!n!} \left\|\frac{\sqrt{\pi}(l_1+l_2\tau)}{\sqrt{r\text{Im}(\tau)}}\right\|^{-2(k+n+1)}\left(\frac{l^{-1,0}}{r}\right)^{\otimes n}. \end{equation*} So in total $\phi$ should be given by \begin{equation*}
\sum_{n,k\geq0}\sum_{0\neq l\in L^*} \frac{(k+n)!(-1)^{k}\left(\frac{l^{-1,0}}{r}\right)^{\otimes n}\sum_{t\in T}f(t)\exp(2\pi i\left\langle l,w-t\right\rangle)i_l(d i_l)^k\theta}{k!n!\left\|\frac{\sqrt{\pi}(l_1+l_2\tau)}{\sqrt{r\text{Im}(\tau)}}\right\|^{2(k+n+1)}} \end{equation*} We want to see that this expression defines a current with values in the completed symmetric algebra. To do so we have to control the coefficients of the Fourier series. The first part is to control \begin{equation*}
\frac{1}{\left\|\frac{\sqrt{\pi}(l_1+l_2\tau)}{\sqrt{r\text{Im}(\tau)}}\right\|^{2(k+n+1)}}. \end{equation*}
Consider $\frac{\sqrt{\pi}(l_1+l_2\tau)}{\sqrt{r\text{Im}(\tau)}}$ as a continuous map $S:V({\mathbb R})\times M\rightarrow F_{\mathbb C}$, which is $F_{\mathbb R}$-linear in the first argument. Take any compact set $K$ in $M$ and $K^\prime$ the one-sphere in $V({\mathbb R})$. The product is again compact and $\left\|S\right\|$ has a minimal value on $K^\prime\times K$. Since $S(.,\tau,r)$ is injective for each $(\tau,r)$, this minimum has to be bigger than zero. Therefore, there is a constant $C>0$ such that $\left\|S(v,\tau,r)\right\|>C\left\|v\right\|$ for all $0\neq v\in V({\mathbb R})$ and $(\tau,r)\in K$. It follows that \begin{equation*}
\frac{1}{\left\|\frac{\sqrt{\pi}(l_1+l_2\tau)}{\sqrt{r\text{Im}(\tau)}}\right\|^{2(k+n+1)}}\leq \frac{1}{C\left\|l\right\|^{2(k+n+1)}} \end{equation*}
for all $l$ and for any $(\tau,r)$ in a compact set $K$ with $C$ only depending on $K$. Next we have to consider $\left(\frac{l^{-1,0}}{r}\right)^{\otimes n}i_l(d i_l)^k\theta$. We have to estimate each coefficient in front of an $\mathcal C^\infty (M)$-basis of $\operatorname{Sym}^k V({\mathbb C})\hat{\otimes}\Omega^p(M)\otimes_{\mathbb C} \bigwedge ^q V({\mathbb C})^*$. Each coefficient can be controlled by $C^\prime\left\|l\right\|^{n+k+1}$ for $(\tau,r)$ in a compact set.\\
Now we want to see that each coefficient of $\phi$, which is given as a formal Fourier series, actually converges to a distribution on $\tilde T$. But we have just shown that the coefficients can be controlled by $C\left\|l\right\|^{n+k+1}$ for all $(\tau,r)\in K$ compact. \cite{S} VII.1 tells us that $\phi$ is a current, since our estimates are locally uniform in $(\tau,r)$. The next step is to prove that the current $\phi$ is $\Gamma$-invariant. First we have $\gamma^*\phi_l= \phi_{\det\gamma^{-1}\gamma l}$, as \begin{equation*} \gamma^*i_l(d i_l)^k\theta=i_{\det\gamma^{-1}\gamma l}d(i_{\det\gamma^{-1}\gamma l})^k \theta, \end{equation*} since $\gamma^*\circ i_l=i_{\det\gamma^{-1}\gamma l}\circ \gamma^*$ and $\theta$ is $\Gamma$-invariant. Therefore \begin{equation*} \gamma^*\sum_l \phi_l \exp(2\pi i\left\langle l,w\right\rangle)=\sum_l(\gamma^*\phi_l) \exp(2\pi i\left\langle l,\gamma^{-1}w\right\rangle)= \end{equation*} \begin{equation*} \sum_l\phi_{\det\gamma^{-1}\gamma l} \exp(2\pi i\left\langle \det\gamma^{-1}\gamma l,w\right\rangle). \end{equation*} Evaluating on a test form and using locally uniform convergence we conclude that this equals $\sum_l\phi_{ l} \exp(2\pi i\left\langle l,w\right\rangle)$ again. Finally, we have thanks to \cite{No}. \begin{theorem} \begin{equation*}
\phi= \sum_{n,k\geq0}\sum_{0\neq l\in L^*} \frac{(k+n)!(-1)^{k}\left(\frac{l^{-1,0}}{r}\right)^{\otimes n}\sum_{t\in T}f(t)\exp(2\pi i\left\langle l,w-t\right\rangle)i_l(d i_l)^k\theta}{k!n!\left\|\frac{\sqrt{\pi}(l_1+l_2\tau)}{\sqrt{r\text{Im}(\tau)}}\right\|^{2(k+n+1)}} \otimes \operatorname{vol}_{\mathcal M} \end{equation*} defines a ${\mathbb R}_{>0}\times \Gamma$-invariant current with values in the completed symmetric algebra on $\tilde T$ that satisfies $\nabla(\phi)=\delta_f\theta$. In particular, $\phi$ induces a current on ${\mathbb R}_{>0}\backslash\tilde T/\Gamma$ with values in $Log^\infty\otimes \pi^{-1}or^{-1}_{{\mathbb R}_{>0}\backslash M/\Gamma}$, which equals $\operatorname{pol}(f)$ as cohomology class. \begin{proof}
See \Cref{differential_equ}, \Cref{invariant_functionals} and \Cref{delta_int}.
\end{proof} \end{theorem}
\section{Global description of the polylogarithm in adelic coordinates}
We already have an explicit description of the polylogarithm as a current on each connected component of $\mathcal T_W$. However, the choice of a connected component of $\mathcal M_K$ always means a choice of a specific lattice $L$ inside $V({\mathbb R})$, which represents this connected component. We want a description of the polylogarithm, which does not depend on a particular choice of a lattice. To achieve this we use adelic coordinates. We start by recalling Fourier analysis on $V({\mathbb A})$.
\subsection{Fourier analysis}\label{Fourier}
Let us take the skew-symmetric non-degenerate bilinear Form \begin{equation*} V({\mathbb A})\times V({\mathbb A})\rightarrow {\mathbb A},\ \left\langle x,y\right\rangle:=\operatorname{Tr}_{F/{\mathbb Q}}(\det(x,y)) \end{equation*} Let us fix a character $\psi_0:{\mathbb A}\rightarrow{\mathbb C}^{\times}$ trivial on ${\mathbb Q}$. We define $\psi_{0}=\exp(2\pi i \lambda) $ where we take $\lambda: {\mathbb A}\rightarrow {\mathbb R}/{\mathbb Z}$, $\lambda =\sum_{\nu\text{ place of }{\mathbb Q}} \lambda_\nu$, with
\begin{equation*}
\lambda_p: {\mathbb Q}_p\stackrel{can.}{\rightarrow}{\mathbb Q}_p/{\mathbb Z}_p \subset {\mathbb Q}/{\mathbb Z}\subset {\mathbb R}/{\mathbb Z} \text{ and }\lambda_\infty: {\mathbb R}\rightarrow {\mathbb R}/{\mathbb Z},\ x\mapsto -x\ \text{mod}\ {\mathbb Z}.
\end{equation*}
We identify $V({\mathbb A})$ with its Pontryagin dual by $V({\mathbb A})\rightarrow \widehat{V({\mathbb A})}$, $x\rightarrow \psi_{0}\left\langle x,\ \right\rangle:=\psi_{0}(\left\langle x,\ \right\rangle)$. Next we need to fix a Haar measure $dv$ on $V({\mathbb A})={\mathbb A}_F^2$. To do so we fix a Haar measure $dx$ on ${\mathbb A}_F$ and take $dv$ to be the product measure. We take Haar measures $dx_\nu$ on $F_\nu$ for each place $\nu$. If $\nu|\infty$, we have $F_\nu={\mathbb R}$ and we take the Lebesgue measure. If $\nu|p$ we fix $dx_\nu$ by $dx_\nu(\mathcal O_\nu)=N(\mathfrak d_\nu)^{-\frac{1}{2}}$, where $\mathfrak d_\nu\subset \mathcal O_\nu$ is the different of $\mathcal O_\nu/{\mathbb Z}_p$. We take then $dx=\prod_{\nu\text{ place of }F} dx_\nu$ in the sense of Tate, see \cite{T} XV 3.3. In a similar manner we get the Haar measure $dv$ on $V({\mathbb A}_f)$ and if we restrict $\psi_{0}$ to $V({\mathbb A}_f)$, we get the topological isomorphism $V({\mathbb A}_f)\rightarrow \widehat{V({\mathbb A}_f)}$, $x\rightarrow \psi_{0}\left\langle x,\ \right\rangle$.
For any integrable function $\varphi:V({\mathbb A})\rightarrow{\mathbb C}$ and any $g\in G({\mathbb A})$ we have the transformation formula \begin{equation*}
\int_{v\in V({\mathbb A})}\varphi(gv)dv=\left\|\det(g)\right\|^{-1}\int_{v\in V({\mathbb A})}\varphi(v)dv,
\end{equation*}
with $\left\|\ \right\|:=\prod_{\nu\text{ place of }F} |\ |_\nu$ the Tate-character and similarly \begin{equation*}
\int_{v\in V({\mathbb A}_f)}\varphi(g_fv)dv=\left\|\det(g_f)\right\|_f ^{-1}\int_{v\in V({\mathbb A}_f)}\varphi(v)dv
\end{equation*} for integrable $\varphi:V({\mathbb A}_f)\rightarrow{\mathbb C}$. We define the Fourier transform of a Schwartz-Bruhat function $\varphi:V({\mathbb A})\rightarrow {\mathbb C}$ by \begin{equation*} \hat\varphi(x):=\int_{V({\mathbb A})}\varphi(v)\overline{\psi_{0}\left\langle x,v\right\rangle} dv. \end{equation*}
The measure $dv$ is self dual, in other words, the Fourier inversion formula holds: $\hat{\hat\varphi}=\varphi$. There is no twist by $-1^*$. We have the formula $\widehat{(\varphi\cdot g)}(x)=\left\|\det(g)\right\|^{-1}\hat\varphi (\hat g^{-1}x)$, for $\varphi \in \mathcal S(V({\mathbb A}))$ and $g\in G({\mathbb A})$. Here $\hat g$ denotes the adjoint operator of $g$ with respect to $\left\langle \ ,\ \right\rangle$. It is explicitly given by $\hat {g}=\det g \cdot g^{-1}$. Similarly, we get $\widehat{(\varphi\cdot g_f)}(x)=\left\|\det(g_f)\right\|_f^{-1}\hat\varphi (\hat g_f^{-1}x)$, for $\varphi \in \mathcal S(V({\mathbb A}_f),{\mathbb C})$ and $g_f\in G({\mathbb A}_f)$.
We want to examine $\hat \varphi$ of a Schwartz-Bruhat function $\varphi:V({\mathbb A}_f)\rightarrow {\mathbb C}$. Let us assume that $\text{supp}(\varphi)\subset N^{-1}V(\hat{\mathbb Z})$ and $\varphi$ is well-defined modulo $NV(\hat {\mathbb Z})$. We may write \begin{equation*} \varphi=\sum_{\overline u \in N^{-1}V(\hat{\mathbb Z})/NV(\hat{\mathbb Z}) } \varphi(u)\chi_{\overline u}, \end{equation*} where $\overline u=u+NV(\hat {\mathbb Z})$ and $\chi_M$ always denotes the characteristic function of a subset $M$. We calculate \begin{equation*} \hat \varphi(x)=\sum_{\overline u \in N^{-1}V(\hat{\mathbb Z})/NV(\hat{\mathbb Z}) } \varphi(u)\int_{V({\mathbb A}_f)}\chi_{\overline u}(v)\overline{\psi_{0}\left\langle x,v\right\rangle} dv \end{equation*} The latter integral is \begin{equation*} \int_{V({\mathbb A}_f)}\chi_{\overline u}(v)\overline{\psi_{0}\left\langle x,v\right\rangle} dv=\int_{V({\mathbb A}_f)}\chi_{\overline u}(v+u)\overline{\psi_{0}\left\langle x,v+u\right\rangle} dv=\int_{V({\mathbb A}_f)}\chi_{\overline u-u}(v)\overline{\psi_{0}\left\langle x,v+u\right\rangle} dv \end{equation*} \begin{equation*} =\int_{V({\mathbb A}_f)}\chi_{NV(\hat{\mathbb Z})}(v)\overline{\psi_{0}\left\langle x,v+u\right\rangle} dv=\overline{\psi_{0}\left\langle x,u\right\rangle} \int_{NV(\hat {\mathbb Z})}\overline{\psi_{0}\left\langle x,v\right\rangle} dv. \end{equation*} The last integral is now easy to compute. $\int_{NV(\hat {\mathbb Z})}\overline{\psi_{0}\left\langle x,v\right\rangle} dv$ is zero, whenever $\overline{\psi_{0}\left\langle x,\ \right\rangle} $ is not the trivial character on $NV(\hat{\mathbb Z})$. This is exactly the case, when $x\notin N^{-1}\mathfrak d^{-1} V(\hat{\mathbb Z})$. Otherwise the integral is just $\int_{NV(\hat {\mathbb Z})}dv=(N^{2\xi}d_F)^{-1}$. This gives the final formula \begin{equation*} \hat \varphi(x)=(N^{2\xi}d_F)^{-1}\sum_{\overline u \in N^{-1}V(\hat{\mathbb Z})/NV(\hat{\mathbb Z}) } \varphi(u)\overline{\psi_{0}\left\langle x,u\right\rangle} \chi_{N^{-1}\mathfrak d^{-1} V(\hat{\mathbb Z})}(x). \end{equation*}
\subsection{The adelic polylogarithm}
Let us give the description of the polylogarithm current on $\mathcal T_W$ in global adelic coordinates. First we trivialize the orientation bundle by the section \begin{equation*}
\operatorname{vol}_{\mathcal T/\mathcal M}^*:=\left\| \det g\right\|_f \operatorname{sgn}(\operatorname{N}(\det g))^{-1}\bigwedge_\sigma X_\sigma\wedge Y_\sigma\in H^0(\mathcal M,\mu)=H^0(\mathcal M,\bigwedge^{2\xi}\mathcal H) \end{equation*} which is the element dual to the form \begin{equation*}
\operatorname{vol}_{\mathcal T/\mathcal M}:=\frac{\bigwedge_\sigma dw^1_\sigma\wedge dw^2_\sigma}{\left\| \det g\right\|_f \operatorname{sgn}(\operatorname{N}(\det g))^{-1}}=\frac{\operatorname{vol}_V}{ \left\| \det g\right\|_f \operatorname{sgn}(\operatorname{N}(\det g))^{-1}}. \end{equation*}
Here we extended the norm character $\operatorname{N}$ and $\left\|\ \right\|_f$ to the whole of $\mathbb{I}_F$ by setting $\operatorname{N}=1$ on the finite ideles and $\left\|.\right\|_f=1$ on $F_{\mathbb R} ^{\times}$.
Suppose we are given a function in $\mathcal S(V({\mathbb A}_f),\mu^{\otimes n})^0 $. Of course, any such function may be written as \begin{equation*}
(v,g)\mapsto \left(\left\|\det(g)\right\|_f \operatorname{sgn}(\operatorname{N}(\det(g)))^{-1}\right)^n f(v,g)=\varphi(v,g), \end{equation*} with $f\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes 0})^0 $ such that $f$ factors in the second argument over $K_f=K_{N^2}$, $\text{supp}(f(\ ,g))\subset N^{-1}V(\hat {\mathbb Z})$ and $f(\ ,g)$ is well-defined modulo $NV(\hat{\mathbb Z})$ for all $g\in G({\mathbb A})$. Let us recall the main steps in the calculation of $\operatorname{pol}(f)=\operatorname{pol}_{W}(f)$, $W_f=NV(\hat{\mathbb Z})\rtimes K_{N^2}$. First of all we did our calculations in the coordinates $(w,g)$, see \Cref{connected_component}. This means that we associated to $f$ on each fiber \begin{equation*} V({\mathbb R})/V({\mathbb Q})\cap (g^{-1}NV(\hat{\mathbb Z})+V({\mathbb R}))\cong g^{-1}NV(\hat{\mathbb Z})\backslash V({\mathbb A})/V({\mathbb Q}) \end{equation*} the delta distribution \begin{equation*} \sum_{w\in g^{-1}N^{-1}V(\hat {\mathbb Z})/g^{-1}NV(\hat {\mathbb Z})}f(g_fw,g)\delta_{w}=\sum_{v\in N^{-1}V(\hat {\mathbb Z})/NV(\hat {\mathbb Z})}f(v,g)\delta_{g_f^{-1}v} . \end{equation*} We fixed the lattice $L$ given by $ V({\mathbb Q})\cap(Ng^{-1}V(\hat {\mathbb Z})+V({\mathbb R}))$ and the dual lattice $L^*=V({\mathbb Q})\cap (N^{-1}\mathfrak d^{-1}\hat{g}V(\hat{{\mathbb Z}})+V({\mathbb R}))$ in $V({\mathbb R})$ with respect to $\left\langle \ ,\ \right\rangle$. The canonical map \begin{equation*} N^{-2}L/L=g^{-1}N^{-1}V(\hat {\mathbb Z})\cap V({\mathbb Q})/g^{-1}NV(\hat {\mathbb Z})\cap V({\mathbb Q})\rightarrow g^{-1}N^{-1}V(\hat {\mathbb Z})/g^{-1}NV(\hat {\mathbb Z}) \end{equation*} is an isomorphism. We constructed the polylogarithm $\operatorname{pol}(f)$ with respect to \\ $\sum_{w\in N^{-2}L/L}f(g_fw,g)\delta_{w}$ and $\theta=\frac{\operatorname{vol}_{\mathcal T/\mathcal M}}{d_F N^{2\xi}}$ as the current \begin{equation*}
\sum_{j,k\geq0} \sum_{0\neq l\in L^*} \frac{(j+k)! (-1)^{k}\tilde f(l,g)\left(\frac{l^{-1,0}}{r}\right)^{\otimes j}i_l(d i_l)^k\frac{\operatorname{vol}_{\mathcal T/\mathcal M}}{d_F N^{2\xi}}}{j!k!\left\|\frac{\sqrt{\pi}(l_1+l_2\tau)}{\sqrt{r\text{Im}(\tau)}}\right\|^{2(j+k+1)}} \exp(2\pi i\left\langle l,w_\infty\right\rangle)\otimes \operatorname{vol}_\mathcal M, \end{equation*} where $\tilde f(l,g)$ was actually given by \begin{equation*} \sum_{w\in N^{-2}L/L}f(g_fw,g)\exp(2\pi i\left\langle l,-w\right\rangle)=\sum_{v\in N^{-1}V(\hat {\mathbb Z})/NV(\hat {\mathbb Z})}f(v,g)\psi_{0}\left\langle l,-g_f^{-1}v\right\rangle,\ l\in L^*. \end{equation*} On the left-hand side of the last equation we interpret $w\in N^{-2}L \subset V({\mathbb R})$ in the infinite component, whereas on the right-hand side we have $v\in N^{-1}V(\hat {\mathbb Z})\subset V({\mathbb A}_f)$. If we have Fourier analysis for Schwartz-Bruhat functions on $V({\mathbb A}_f)$ in mind, we see $(N^{2\xi}d_F)^{-1}\tilde f(l,g)=\hat f(\hat g^{-1} l,g)$. Here $\hat f$ means that we do Fourier transformation in the first argument. More precisely, $\hat{g}^{-1\ *}\chi_{N^{-1}{\mathfrak{d}}^{-1}V(\hat{\mathbb Z})}=\chi_{N^{-1}{\mathfrak{d}}^{-1}\hat{g}V(\hat{\mathbb Z})}$, so $\hat{g}^{-1\ *}\chi_{N^{-1}{\mathfrak{d}}^{-1}V(\hat{\mathbb Z})}(l)$ is not equal to zero for $l\in V({\mathbb Q})$ if and only if $l\in L^*$. With this notation at hand we may write $\operatorname{pol}(f)$ as \begin{equation*}
\sum_{j,k\geq0} \sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}} \frac{(j+k)!(-1)^{k}\hat f(\hat g^{-1}l,g)\left(\frac{l^{-1,0}}{r}\right)^{\otimes j}i_l(d i_l)^k\operatorname{vol}_{\mathcal T/\mathcal M}}{j!k!\left\|\frac{\sqrt{\pi}(l_1+l_2\tau)}{\sqrt{r\text{Im}(\tau)}}\right\|^{2(j+k+1)}} \overline{\psi_0\left\langle l,w_\infty\right\rangle}\otimes \operatorname{vol}_\mathcal M. \end{equation*} If we multiply with our trivialization $\operatorname{vol}_{\mathcal T/\mathcal M}^{*n}$, we may write $\operatorname{pol}(\varphi)$ as \begin{equation*}
\sum_{j,k\geq0}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}} \frac{(j+k)!(-1)^{k}\hat f(\hat g^{-1}l,g)\left(\frac{l^{-1,0}}{r}\right)^{\otimes j}i_l(d i_l)^k\operatorname{vol}_{\mathcal T/\mathcal M}\otimes\operatorname{vol}_{\mathcal T/\mathcal M}^{*n}}{j!k!\left\|\frac{\sqrt{\pi}(l_1+l_2\tau)}{\sqrt{r\text{Im}(\tau)}}\right\|^{2(j+k+1)}} \overline{\psi_0\left\langle l,w_\infty\right\rangle}\otimes \operatorname{vol}_\mathcal M. \end{equation*}
\subsection{Analytic continuation of the polylogarithm}
If we want to specialize the polylogarithm along the zero section, we need to know, whether this current may be represented by a smooth differential form on $U$. This is what Levin proved in \cite{Bl1}. Let us quickly recall how things work. We have identifications $\mathcal O\otimes{\mathbb R}^2\cong \mathcal O\otimes{\mathbb C}$, $e^1\mapsto 1$, $e^2\mapsto i$, \begin{equation*}
(\prod_{\nu|\infty}SO(2))\backslash G({\mathbb R})\rightarrow M,\ g_\infty\mapsto (ig_\infty,\det g_\infty)=(\tau,r), \end{equation*} \begin{equation*}
\left\|\frac{\sqrt{\pi}(l_1+l_2\tau)}{\sqrt{r\text{Im}(\tau)}}\right\|=\pi\left\|\hat g_\infty ^{-1}l\right\|,\ \text{and } \frac{\Gamma(s)}{x^s}=\int_{{\mathbb R}_{>0}}\exp(-ux){u^s}\frac{d u}{u},\ x\in{\mathbb R}_{>0}. \end{equation*} This yields \begin{equation*}
\left\|\frac{\sqrt{\pi}(l_1+l_2\tau)}{\sqrt{r\text{Im}(\tau)}}\right\|^{-2(j+k+1)}=\int_{{\mathbb R}_{>0}}\exp(-u\pi\left\|\hat g_\infty ^{-1}l\right\|^2)\frac{u^{j+k+1}}{(j+k)!}\frac{du}{u}. \end{equation*} Given $\varphi_f\in \mathcal S(V({\mathbb A}_f),{\mathbb C})$ and $\varphi_\infty\in \mathcal S(V({\mathbb R}),{\mathbb C})$ we define $\varphi_f\otimes \varphi_\infty \in\mathcal S(V({\mathbb A}),{\mathbb C})$ by \begin{equation*} \varphi_f\otimes \varphi_\infty(v):=\varphi_f(v_f)\varphi_\infty(v_\infty),\ v\in V({\mathbb A}). \end{equation*}
$w\mapsto \exp(-\pi \left\|w\right\|^2)$ is fixed by Fourier transformation, so we have a $(V({\mathbb A}_f)\rtimes G({\mathbb A}_f))\times\pi_0(G_\infty)$-equivariant linear map \begin{equation*}
\phi:\mathcal S (V({\mathbb A}_f))\rightarrow \mathcal S(V({\mathbb A})),\ f\mapsto f\otimes \exp(-\pi\left\|x\right\|^2), \end{equation*} which is compatible with Fourier transformation, in other words, $\widehat{\phi(f)}=\phi(\hat f)$. We can write the polylogarithm as \begin{equation*} \sum_{j,k\geq0}\frac{(j+k)!}{j!k!}(-1)^{k}\phi^{(j,k)}\otimes \operatorname{vol}_{\mathcal T/\mathcal M}^{*n}\otimes\operatorname{vol}_\mathcal M , \end{equation*} \begin{equation*} \phi^{(j,k)}:= \sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}} \int_{u\in {\mathbb R}_{>0}}\frac{\phi(\hat{f})(\sqrt{u}\hat g^{-1}l,g)\left(\frac{l^{-1,0}}{r}\right)^{\otimes j}u^{j+k+1}\frac{du}{u}i_l(d i_l)^k \operatorname{vol}_{\mathcal T/\mathcal M}}{(j+k)!} \overline{\psi_0\left\langle l,w_\infty\right\rangle} \end{equation*} The first observation is that for $k+j+1>2\xi$ we already have honest differential forms, because the sum over $l$ converges locally uniformly. For these big $k+j+1$ we may also put the integral over $u\in {\mathbb R}_{>0}$ in front of all sums by uniform convergence. Let us consider the differential form \begin{equation*} \phi_s ^{(j,k)}:=\int_{u\in {\mathbb R}_{>0}}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{\phi(\hat{f})(\sqrt{u}\hat g^{-1}l,g)\left(\frac{l^{-1,0}}{r}\right)^{\otimes j}u^{s}\frac{du}{u}i_l(d i_l)^k \operatorname{vol}_{\mathcal T/\mathcal M}}{(j+k)!}\cdot \overline{\psi_0\left\langle l,w_\infty\right\rangle} \end{equation*} for complex $s\in {\mathbb C}$, $\operatorname{Re}(s)>2\xi$. Levin proves by using Poisson summation that $\phi^{(j,k)}_s$ has an analytic continuation as a holomorphic function in $s\in{\mathbb C}$ on $U$. This analytic continuation yields the representation of the polylogarithmic current as a differential form on $U$, see also \cite{BKL} Theorem 2.3.6.: $\operatorname{pol}(\varphi)=\sum_{j,k\geq 0}\phi^{(j,k)}_{j+k+1}\otimes \operatorname{vol}_{\mathcal T/\mathcal M}^*$. The additional $\operatorname{vol}_{\mathcal T/\mathcal M}^*$ and the missing $\operatorname{vol}_\mathcal M$ are due to the fact that differential forms resolve the constant sheaf ${\mathbb C}$, whereas the currents resolve the orientation bundle of $\mathcal T_W$, and due to the definition of the Fourier series of our currents, see \cref{eval_distr}. Now we may specialize our polylogarithm to get the representation of $\operatorname{Eis}^k(\varphi)$ as \begin{equation*} \lim{s\rightarrow 2\xi+k}\int_{u\in{\mathbb R}_{>0}}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{-\phi(\hat{f})(\sqrt{u}\hat g^{-1}l,g)}{k!(2\xi-1)!}\left(\frac{l^{-1,0}}{r}\right)^{\otimes k}u^{s}\frac{du}{u}i_l(d i_l)^{2\xi-1} \operatorname{vol}_{\mathcal T/\mathcal M}\otimes\operatorname{vol}_{\mathcal T/\mathcal M}^{*n+1} \end{equation*} for $k\geq0$ and the limit exists by analytic continuation.
\section{The decomposition isomorphism as Mellin transform}
Next we are going to apply \Cref{trace_integration} to $\operatorname{Eis}^k(\varphi)$. The fiber integral of $\operatorname{Eis}^k(\varphi)$ turns out to be a Mellin transform of a theta series. Given this description of $\operatorname{Eis}^k(\varphi)$ and having \cite{Wi} in mind the connection to Eisenstein series and Harder's Eisenstein classes is immediately apparent. But before we can apply the fiber integral, we need to express $\operatorname{Eis}^k(\varphi)$ by $\mathfrak H^\bullet _{\mathbb C}$ which is generated by $\frac{dr_i}{r_i}$, $i=1,...,\xi$.
\begin{lemma}\label{calculating_vol} \begin{equation*} \frac{i_l(d i_l)^{2\xi-1}\operatorname{vol}_V}{(2\xi-1)!}= \operatorname{N}\left(\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\right)\sum_{i=1} ^\xi\prod_{i\neq j}\left(d\left(\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\right)_j \wedge d\overline{\tau}_j\right)\wedge\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}_i d\overline{\tau}_i \end{equation*} \begin{proof} We follow the computation of \cite{Bl2} 4.7. Set $dw^1+\tau dw^2=:\eta^1$, $\frac{\overline\eta^1}{\overline\tau-\tau}=:\eta^2$. We have \begin{equation*} \eta^1(u)=1,\ \overline\eta^1(u)=0,\ d\eta^1=d\tau\wedge dw^2=d\tau\wedge\frac{\overline \eta^1-\eta^1}{\overline\tau-\tau},\ \operatorname{vol}_V=\prod_{i=1}^\xi\eta^1_i\wedge\eta^2_i. \end{equation*} \begin{equation*} i_l(d i_l)^{2\xi-1}\operatorname{vol}_V=i_l\mathcal L_{\frac{l^{-1,0}}{r}}^{2\xi-1}\operatorname{vol}_V \end{equation*} and we calculate \begin{equation*} \mathcal L_{\frac{l^{-1,0}}{r}}^{2\xi-1}\operatorname{vol}_V=\mathcal L_{\frac{l^{-1,0}}{r}}^{2\xi-1}\prod_{i=1}^\xi\eta^1_i\wedge\eta^2_i= \end{equation*} \begin{equation*} (2\xi-1)!\sum_{i=1}^\xi\prod_{j\neq i}\left(\mathcal L_{\frac{l^{-1,0}}{r}}\eta^1_j\wedge\mathcal L_{\frac{l^{-1,0}}{r}}\eta^2_j\right)\wedge\left(\eta^1_i\wedge\mathcal L_{\frac{l^{-1,0}}{r}}\eta^2_i+\mathcal L_{\frac{l^{-1,0}}{r}}\eta^1_i\wedge \eta^2 _i\right), \end{equation*} as $\mathcal L_{\frac{l^{-1,0}}{r}}^2\eta^1_i=\mathcal L_{\frac{l^{-1,0}}{r}}^2\eta^2_i=0$ for $i=1,...,\xi$ and $\mathcal L_{\frac{l^{-1,0}}{r}}$ satisfies the Leibniz rule. Moreover, \begin{equation*} \mathcal L_{\frac{l^{-1,0}}{r}}\eta^1=(di_l+i_l d)(\eta^1)=d(\frac{l_1+\tau l_2}{r})+\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}d\tau, \end{equation*} \begin{equation*} \mathcal L_{\frac{l^{-1,0}}{r}}\eta^2=(di_l+i_l d)(\eta^2)=i_ld(\eta^2)=\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\frac{d\overline\tau}{\overline{\tau}-\tau} \end{equation*} and $i_l\mathcal L_{\frac{l^{-1,0}}{r}}\eta^1_i=i_l\mathcal L_{\frac{l^{-1,0}}{r}}\eta^2_i=i_l\eta^2_i=0$, $i=1,...,\xi$. Now we get \begin{equation*} \frac{i_l(d i_l)^{2\xi-1}\operatorname{vol}_V}{(2\xi-1)!}=\sum_{i=1}^\xi\prod_{j\neq i}\left(\mathcal L_{\frac{l^{-1,0}}{r}}\eta^1_j\wedge\mathcal L_{\frac{l^{-1,0}}{r}}\eta^2_j\right)\wedge\left(i_l\eta^1_i\wedge\mathcal L_{\frac{l^{-1,0}}{r}}\eta^2_i\right)= \end{equation*} \begin{equation*} \sum_{i=1}^\xi\prod_{j\neq i}\left(\left(d(\frac{l_1+\tau l_2}{r})+\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}d\tau\right)\wedge\left(\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\frac{d\overline\tau}{\overline{\tau}-\tau}\right)\right)_j\wedge\left(\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\right)_i^2d\overline\tau_i= \end{equation*} \begin{equation*} \sum_{i=1}^\xi\prod_{j\neq i}\left(\left(\frac{d(\frac{l_1+\tau l_2}{r})}{\overline{\tau}-\tau}+\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\frac{d\tau}{\overline{\tau}-\tau}\right)\wedge\left(\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}d\overline\tau\right)\right)_j\wedge\left(\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\right)_i^2d\overline\tau_i= \end{equation*} \begin{equation*} \sum_{i=1}^\xi\prod_{j\neq i}\left(d\left(\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\right)\wedge\left(\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}d\overline\tau\right)\right)_j\wedge\left(\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\right)_i^2d\overline\tau_i \end{equation*} \end{proof} \end{lemma}
\begin{lemma}\label{calc_vol} \begin{equation*} \operatorname{N}\left(\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\right)^{-1}\frac{i_l(d i_l)^{2\xi-1}\operatorname{vol}_V}{(2\xi-1)!}= \end{equation*} \begin{equation*}
\sum_{i=1} ^\xi\sum_{I\subset \left\langle \xi\right\rangle\setminus i}(-1)^{|I|}\frac{l_1+\overline\tau l_2}{r(\overline{\tau}-\tau)}_{(i\cup I)^c}\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}_{I\cup i} \frac{d\tau\wedge d\overline\tau}{\overline\tau-\tau}_{(i\cup I)^c}\wedge \frac{dr\wedge d \overline\tau}{r}_I\wedge d\overline\tau_i \end{equation*} \begin{proof} We have \begin{equation*} d\left(\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\right) \wedge d\overline{\tau}=\left(-\frac{l_1+\tau l_2}{r(\overline{\tau}-\tau)}\frac{dr}{r}+d\left(\frac{l_1+\tau l_2}{\overline{\tau}-\tau}\right)r^{-1}\right)\wedge d\overline{\tau} \end{equation*} and \begin{equation*} d\left(\frac{l_1+\tau l_2}{\overline{\tau}-\tau}\right)\wedge d\overline\tau=d\left(\frac{l_1+\overline\tau l_2 -\overline\tau l_2+\tau l_2}{\overline{\tau}-\tau}\right)\wedge d\overline\tau= \end{equation*} \begin{equation*} d\left(\frac{l_1+\overline\tau l_2}{\overline{\tau}-\tau}-l_2\right)\wedge d\overline\tau=d\left(\frac{l_1+\overline\tau l_2}{\overline{\tau}-\tau}\right)\wedge d\overline\tau=\frac{l_1+\overline\tau l_2}{(\overline{\tau}-\tau)}\frac{d\tau\wedge d\overline\tau}{\overline{\tau}-\tau} \end{equation*} If we plug these formulas in \Cref{calculating_vol}, we get the desired formula. \end{proof} \end{lemma} \begin{corollary}\label{dr_invariant} The coefficients in front of the invariant forms $\frac{dr}{r}_I$, $I\subset\left\langle \xi\right\rangle$, of \\ $\operatorname{N}(r)^2i_l(di_l)^{2\xi-1}\operatorname{vol}_V$ do not depend on $r$. \end{corollary} \begin{lemma}\label{Z_proj} We set $\omega:=(\overline\tau-\tau)u$ and $\operatorname{N}(\omega):=\prod_{i=1}^\xi\omega_i$. The projection of $\frac{1}{k!}\left(\frac{l^{-1,0}}{r}\right)^{\otimes k}$ onto the $\operatorname{Sym}^k V({\mathbb C})^{Z_K}$-part is \begin{equation*} \operatorname{N}\left(\frac{l_1+\tau l_2}{r(\overline\tau-\tau)}\right)^m\frac{\operatorname{N}(\omega)^m}{(m!)^\xi},\ \text{if}\ k=\xi m, \end{equation*} and zero otherwise. \begin{proof} We set $x^{[k]}:=\frac{x^k}{k!}$ for $x$ in any ${\mathbb Q}$-Algebra $R$. We have $(x+y)^{[k]}=\sum_{l=0}^kx^{[l]}y^{[k-l]}$. Doing induction on $n\in {\mathbb N}$ we may easily proof the formula \begin{equation*} \left(\sum_{i=1}^nx_i\right)^{[k]}=\sum_{(k_i)\in{\mathbb N}_0 ^n:\sum k_i=k}\prod_{i=1}^nx^{[k_i]} \end{equation*} Write $\frac{l^{-1,0}}{r}=\sum_{i=1}^\xi\frac{l^{-1,0}}{r}_i$ and get \begin{equation*} \frac{1}{k!}\left(\frac{l^{-1,0}}{r}\right)^{\otimes k}=\left(\frac{l^{-1,0}}{r}\right)^{\otimes [k]}=\left(\sum_{i=1}^\xi\frac{l^{-1,0}}{r}_i\right)^{\otimes [k]}= \sum_{(k_i)\in {\mathbb N}_0^{\xi}:\sum k_i=k} \prod_{i=1}^\xi\left(\frac{l^{-1,0}}{r}\right)_i ^{\otimes [k_i]}= \end{equation*} \begin{equation*} \sum_{(k_i)\in {\mathbb N}_0^{\xi}:\sum k_i=k} \prod_{i=1}^\xi \frac{1}{k_i !}\left(\frac{l^{-1,0}}{r}\right)_i ^{\otimes k_i}=\sum_{(k_i)\in {\mathbb N}_0^{\xi}:\sum k_i=k} \prod_{i=1}^\xi \frac{1}{k_i !}\left(\frac{l_1+\tau l_2}{r(\overline\tau -\tau)}\omega\right)_i ^{\otimes k_i} \end{equation*} The projection of this element onto the $\operatorname{Sym}^k V({\mathbb C})^{Z_K}$-part are those summands where the $Z_K$ action factors through the norm character. Thus the only summand living in $\operatorname{Sym}^k V({\mathbb C})^{Z_K}$ is the one corresponding to $k_i =m$ for $i=1,...,\xi$, see \Cref{invariants}. \end{proof} \end{lemma}
Before we can go on, we need to fix measures again. Let us consider the ideles $\mathbb{I}_F$. If $\nu\nmid\infty$, we take $d^{\times}x_\nu:=\frac{|\kappa(\mathfrak p_\nu)|}{|\kappa(\mathfrak p_\nu)|-1}\frac{dx_\nu}{|x_\nu|_\nu}$, as Haar measure of the multiplicative group $F_\nu ^{\times}$, where $dx_\nu$ is the Haar measure on the additive group $F_\nu$, which we already have fixed. If $\nu|\infty$, we have $F_\nu={\mathbb R}$ and we take $d^{\times}x_\nu:=\frac{dx_\nu}{|x_\nu|}$, where on the right-hand side $dx_\nu$ is the usual Lebesgue measure on ${\mathbb R}$. These local measures induce global measures $d^{\times}x$ on $\mathbb{I}_F$ and $\mathbb{I}_{F,f}$ and $F_{\mathbb R} ^{\times}$ as in \Cref{Fourier}. \begin{remark} \begin{itemize} \item Let $G$ be a locally compact group and $H\subset G$ a normal and closed subgroup. Suppose we have fixed (left) Haar measures $dg$ on $G$ and $dh$ on $H$. Then there is a uniquely determined (left) Haar measure $dgH$ on $G/H$ such that \begin{equation*} \int_Gf(g)dg=\int_{G/H}\left(\int_{H}f(gh)dh\right)dgH \end{equation*} holds for all integrable functions $f$ on $G$. \item If $H\subset G$ is an open subgroup and we have fixed a Haar measure $dg$ on $G$, we always take on $H$ the restriction of $dg$ as Haar measure on $H$. \item We have $d^{\times}x({\mathbb R}_{>0}\backslash F_{\mathbb R} ^{\times,0}/\det(Z_K))=R_K$ (see \cref{int_decomp}), where the measure $d^{\times}x$ is induced by the measure $\frac{dt}{t}$ on ${\mathbb R}_{>0}$, $d^{\times}x$ on $F_{\mathbb R} ^{\times,0}$ and the counting measure on $\det(Z_K)$. \item In the following calculations the choice of the Haar measures will be clear from the context. \end{itemize} \end{remark} Set \begin{equation}\label{Theta_1} \Theta_l:=\operatorname{N}\left(\frac{l_1+\tau l_2}{\overline\tau-\tau}\right)^m \operatorname{N}(\omega)^m\operatorname{N}(r)^2i_l(di_l)^{2\xi-1}\operatorname{vol}_{\mathcal T/\mathcal M}\otimes \operatorname{vol}_{\mathcal T/\mathcal M}^{*n+1} \end{equation} a form constant in $r$ and $C(k):=\frac{-1}{(m!)^\xi(2\xi-1)!\xi}$.
\begin{lemma}\label{decomp_Eis} We have \begin{equation*} \operatorname{Eis}^k(\varphi)=
\lim{s\rightarrow 0}\frac{C(k)2^\xi}{R_K}\int_{t\in\left\{\pm1\right\}^\xi\backslash Z({\mathbb R})/Z_K}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{\phi(\hat{f})(t\hat g ^{-1}l,g) |\operatorname{N}(t)|^{2(m+2+s)}}{\operatorname{N}(\det(g))^{m+2}} d^{\times}t\cdot \Theta_l \end{equation*} for $k=\xi m\geq0$. \begin{proof} From \cref{Z_proj1}, \cref{Z_proj} and \cref{Theta_1} we already know that we may identify the cohomology class $\operatorname{Eis}^k(\varphi)$ with \begin{equation*} \lim{s\rightarrow 2\xi+k}\xi C(k)\int_{u\in {\mathbb R}_{>0}}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{\phi(\hat{f})\left(\sqrt{u}\hat g^{-1}l,g\right) u^{s}}{\operatorname{N}(r)^{m+2}}\frac{du}{u}\cdot\Theta_l \end{equation*} We write everything in base and fiber coordinates and have to work with $\tilde{r}\in F_{\mathbb R} ^1$: \begin{equation*}
\begin{pmatrix}1 & x\\ 0 & y\end{pmatrix}=:(1,\tau)\ \text{and } \sqrt{u}\hat g^{-1}=\frac{\sqrt{u}\hat g_f ^{-1}(1,\tau)}{\sqrt{\sqrt[\xi]{|N(r)|}|\tilde{r}\text{Im}(\tau)|}}. \end{equation*} and we get for $\operatorname{Eis}^k(\varphi)$ \begin{equation*} \lim{s\rightarrow 2\xi+k}\xi C(k)\int_{u\in {\mathbb R}_{>0}}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{\phi(\hat{f})\left(\frac{\sqrt{u}\hat g_f ^{-1}(1,\tau)}{\sqrt{\tilde{r}\text{Im}(\tau)}}l ,g\right) u^{s}}{\operatorname{sgn}\operatorname{N}(\det(g))^{m+2}}\frac{du}{u}\cdot\Theta_l \end{equation*} Note that $\Theta_l$ is just built up by the invariant forms $\frac{dr}{r}_i$, $i=1,...,\xi$, and is constant in $r$, therefore we may apply \cref{int_decomp} to identify $\operatorname{Eis}^k(\varphi)$ with \begin{equation*} \lim{s\rightarrow 2\xi+k}\frac{\xi C(k)}{R_K}\int_{\tilde{r}\in \left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K)}\int_{u\in {\mathbb R}_{>0}}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{\phi(\hat{f})\left(\frac{\sqrt{u}\hat g_f ^{-1}(1,\tau)}{\sqrt{\tilde{r}\text{Im}(\tau)}}l,g\right) u^{s}}{ \operatorname{sgn} \operatorname{N}((\det(g)))^{m+2}}\frac{du}{u}d^\times\tilde{r}\cdot\Theta_l \end{equation*} Here we already have interchanged the limit and $\int_{\tilde{r}\in \left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K)}$. This is possible by the dominated convergence theorem, as the limit in $s$ gives a continuous function in $r$, which is integrable over the compact space $\left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K)$. We continue our calculation by changing variables $\tilde{r}\mapsto \tilde{r}^{-1}$ and changing the order of integration with Fubini to get for $\operatorname{Eis}^k(\varphi)$ \begin{equation*}
\lim{s\rightarrow 2\xi+k}\frac{\xi C(k)}{R_K}\int_{u\in {\mathbb R}_{>0}}\int_{\tilde{r}\in \left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K)}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{\phi(\hat{f})\left(\frac{\sqrt{|u\tilde{r}|}\hat g_f ^{-1}(1,\tau)}{\sqrt{|\text{Im}(\tau)|}}l,g\right) u^{s}}{ \operatorname{sgn} \operatorname{N}((\det(g)))^{m+2}}d^\times\tilde{r}\frac{du}{u}\cdot\Theta_l= \end{equation*} \begin{equation*}
\lim{s\rightarrow 2\xi+k}\frac{C(k)}{R_K}\int_{v\in {\mathbb R}_{>0}}\int_{\tilde{r}\in \left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^1/\det(Z_K)}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{\phi(\hat{f})\left(\frac{\sqrt{|\sqrt[\xi]{v}\tilde{r}|}\hat g_f ^{-1}(1,\tau)}{\sqrt{|\text{Im}(\tau)|}}l,g\right) v^{\frac{s}{\xi}}}{ \operatorname{sgn} \operatorname{N}((\det(g)))^{m+2}}d^\times\tilde{r}\frac{dv}{v}\cdot\Theta_l \end{equation*} with $v=u^\xi$ and $\frac{dv}{v}=\xi\frac{du}{u}$. We have $d^\times r=d^\times\tilde{r}\frac{dv}{v}$ using the isomorphism (the trivialization of the fiber bundle $F_{\mathbb R} ^\times$) \begin{equation*} F_{\mathbb R} ^1\times {\mathbb R}_{>0}\to F_{\mathbb R} ^\times,\ (\tilde{r},v)\mapsto \sqrt[\xi]{v}\tilde{r} \end{equation*} as remarked in \cref{int_decomp} and therefore $\operatorname{Eis}^k(\varphi)$ equals \begin{equation*}
\lim{s\rightarrow 2\xi+k}\frac{C(k)}{R_K}\int_{r\in \left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^\times/\det(Z_K)}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{\phi(\hat{f})\left(\frac{\sqrt{|r|}\hat g_f ^{-1}(1,\tau)}{\sqrt{|\text{Im}(\tau)|}}l,g\right) |\operatorname{N}(r)|^{\frac{s}{\xi}}}{ \operatorname{sgn} \operatorname{N}((\det(g)))^{m+2}}d^\times r\cdot\Theta_l \end{equation*} Using the isomorphism \begin{equation*} \varphi:\left\{\pm1\right\}^\xi\backslash Z({\mathbb R})/Z_K\rightarrow \left\{\pm1\right\}^\xi\backslash F_{\mathbb R} ^{\times}/\det(Z_K),\ t\mapsto t^2=r,\ \varphi^*d^{\times}r=2^{\xi} d^{\times}t. \end{equation*} we get for $\operatorname{Eis}^k(\varphi)$ \begin{equation*}
\lim{s\rightarrow 2\xi+k}\frac{2^{\xi}C(k)}{R_K}\int_{t\in \left\{\pm1\right\}^\xi\backslash Z({\mathbb R})/Z_K}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{\phi(\hat{f})\left(\frac{t\hat g_f ^{-1}(1,\tau)}{\sqrt{|\text{Im}(\tau)|}}l,g\right) |\operatorname{N}(t)|^{\frac{2s}{\xi}}}{ \operatorname{sgn} \operatorname{N}((\det(g)))^{m+2}}d^\times t\cdot\Theta_l= \end{equation*} \begin{equation*}
=\lim{s\rightarrow 0}\frac{2^{\xi}C(k)}{R_K}\int_{t\in \left\{\pm1\right\}^\xi\backslash Z({\mathbb R})/Z_K}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{\phi(\hat{f})\left(t\hat g ^{-1}l,g\right) |\operatorname{N}(t)|^{2(m+2+s)}}{ \operatorname{N}((\det(g)))^{m+2}}d^\times t\cdot\Theta_l \end{equation*} \end{proof} \end{lemma} \begin{remark} From now on we assume $K_f$ is so small that $Z_K\subset Z({\mathbb R})^0$. Then we may write \begin{equation*} \operatorname{Eis}^k(\varphi)= \frac{C(k)2^\xi}{R_K}\lim{s\rightarrow 0}\int_{t\in Z({\mathbb R})^0/Z_K}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\frac{\phi(\hat{f})(t\hat g ^{-1}l,g)\operatorname{N}(t)^{2(m+2+s)} d^{\times}t}{\operatorname{N}(\det(g_\infty))^{m+2}}\cdot\Theta_l. \end{equation*} \end{remark} The next step is to interpret this integral as a global Tate integral. To do this we decompose $\operatorname{Eis}^k(\varphi)$ into several summands.\\ There is a short exact sequence \begin{equation*} 1\rightarrow Z({\mathbb R})^0/Z_K\rightarrow K_f ^Z\backslash Z({\mathbb A})/Z({\mathbb Q})\rightarrow Cl_F ^K\rightarrow 1, \end{equation*} with $K ^Z:=K\cap Z({\mathbb A})$ and $Cl_F ^K=K_f ^Z Z({\mathbb R})^0\backslash Z({\mathbb A})/Z({\mathbb Q})$ (we always suppose $K_f ^Z=\det(K_f)$). Denote by $\widehat{Cl_F ^K}$ the Pontryagin dual of the last group and let $h_{K}$ be its cardinality. Moreover, we define \begin{equation*} \widehat{Cl_F ^K}(m):=\left\{\chi\in \widehat{Cl_F ^K}:\chi(t)=\operatorname{sgn}(\operatorname{N}(t))^m,\ t\in Z({\mathbb R}) \right\}. \end{equation*} \begin{remark}\label{Iwasawa}
Recall the Iwasawa decomposition on $G_0({\mathbb R})=G_0(F_\nu)$ for $\nu|\infty$: \begin{equation*} g=\begin{pmatrix}a&b\\c&d\end{pmatrix}=k(g)a(g)n(g): \end{equation*} \begin{equation*} k(g)=\frac{1}{\sqrt{a^2+c^2}}\begin{pmatrix}a&-c\\c&a\end{pmatrix}=\begin{pmatrix}\cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta)\end{pmatrix}\in SO(2), \end{equation*} \begin{equation*} a(g)=\begin{pmatrix}\sqrt{a^2+c^2}&0\\0&\frac{\det(g)}{\sqrt{a^2+c^2}}\end{pmatrix}=\begin{pmatrix}t_1&0\\0& t_2\end{pmatrix}\in T_0(F_\nu), \end{equation*} \begin{equation*} n(g)=\begin{pmatrix}1&\frac{ab+dc}{a^2+c^2}\\0&1\end{pmatrix}=\begin{pmatrix}1&x\\0& 1\end{pmatrix}\in U_0(F_\nu). \end{equation*} We can consider $k,a,n$ as continuous functions $G(F_\nu)\rightarrow {\mathbb C}$. Moreover, we set $b:=an$.
One also has a Iwasawa decomposition for $g\in G_0(F_\nu)$ and $\nu$ a finite place: \begin{equation*} g=kb \text{ with } k\in G_0(\mathcal O_\nu) \text{ and } b=\begin{pmatrix}t_1&t_1x\\0&t_2\end{pmatrix}\in B_0(F_\nu). \end{equation*}
This decomposition is not unique, as $b$ is just well-defined up to elements from $B_0(\mathcal O_\nu)$. Nevertheless, the Iwasawa decomposition often suffices to define functions on $G_0(F_\nu)$. For example, if $\chi:F_\nu^{\times}\to{\mathbb C}^{\times}$ is an unramified quasi-character, say $\chi(x)=|x|_\nu ^s$ for $s\in {\mathbb C}$, we have well-defined continuous functions $G_0(F_\nu)\to {\mathbb C}^{\times}$, $g\mapsto \chi(t_1)$ or $\chi(t_2)$. Finally, we get the Iwasawa decomposition for $g\in G({\mathbb A})$. Of course, it is defined place by place and is given by \begin{equation*}
\begin{pmatrix}a&b\\c&d\end{pmatrix}=g=kb \text{ with } k\in G(\hat{\mathbb Z})\cdot\prod_{\nu|\infty}SO(2) \text{ and } b=\begin{pmatrix}t_1&t_1x\\0&t_2\end{pmatrix}\in B({\mathbb A}). \end{equation*}
Again, this decomposition is just well-defined up to elements from $B(\hat{\mathbb Z})$. For the infinite places $\nu|\infty$ we also have the functions $\theta_\nu:G_0(F_\nu)\to{\mathbb R}/2\pi{\mathbb Z}$ defined by $e^{i\theta}_\nu:=e^{i\theta_\nu}=\frac{(a_\nu+ic_\nu)}{\sqrt{a_\nu^2+c_\nu^2}}$. \end{remark} For $i\in \left\langle \xi\right\rangle$ and $I\subset \left\langle \xi\right\rangle\setminus i$ we define the form \begin{equation*}
\Theta_{I,i}:=\frac{(-1)^{|I|}(2\xi-1)! \operatorname{N}(e^{-(m+1)i\theta})e^{-i\theta}_{I\cup i}e^{i\theta}_{(I\cup i)^c}}{(-2i)^{\xi(m+2)} \left\| \det g\right\|_f \operatorname{sgn}(\operatorname{N}(\det g))^{-1}{\operatorname{N}(t_{2}) ^{m+2}}}\cdot \end{equation*} \begin{equation}\label{Theta_2} \operatorname{N}(\omega)^m\frac{d\tau\wedge d\overline\tau}{\overline\tau-\tau}_{\left\langle \xi\right\rangle\setminus (i\cup I)}\wedge \frac{dr\wedge d \overline\tau}{r}_I\wedge d\overline\tau_i\otimes \operatorname{vol}_{\mathcal T/\mathcal M}^{*n+1} \end{equation} We have Schwartz-Bruhat-functions \begin{equation*} \phi(\hat{f})^m_{I\cup i}(v,g):=\phi(\hat{f})(v,g)\operatorname{N}(v_\infty ^1+iv_\infty ^2)^{m+1}(v_\infty ^1+iv_\infty ^2)_{I\cup i}(v_\infty ^1-iv_\infty ^2)_{(I\cup i)^c} \end{equation*} on $V({\mathbb A})$. Write $\phi(\hat{f})^m_{I\cup i}=\phi(\hat{f})^m_{I\cup i,\infty}\otimes \phi(\hat{f})^m_{I\cup i,f}$ with $\phi(\hat{f})^m_{I\cup i,f}=\hat f$. For $\chi\in \widehat{Cl_F ^K}$ and $s\in{\mathbb C}$ we may define differential forms \begin{equation*} \operatorname{Eis}^k _{I,i}(\varphi,\chi,s)= \sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q}))}\int_{t\in Z({\mathbb A})}
\phi(\hat{f})^m _{I\cup i}(tg\gamma e^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{s} d^{\times}t\cdot \Theta_{I,i}. \end{equation*} \begin{lemma} If $Re(s)>2$, $\operatorname{Eis}^k _{I,i}(\varphi,\chi,s)$ is an honest differential form and a holomorphic function in $s$. $\operatorname{Eis}^k _{I,i}(\varphi,\chi,s)$ has an analytic continuation as meromorphic function in $s$ to the whole complex plane and does not have a pole at $s=2$. In particular, the limit \begin{equation*} \lim {s\rightarrow 0}\operatorname{Eis}^k _{I,i}(\varphi,\chi,m+2+2s)=:\operatorname{Eis}^k _{I,i}(\varphi,\chi) \end{equation*} exists and gives a well-defined differential form on $\mathcal M_K$. \begin{proof} The first statement is \cite{Wi} III Proposition 6. Wielonsky considers functions in the variable $\det(t)$, whereas we have the variable $t$. Therefore Wielonsky has the statement for $\sigma>1$ and we have our statement for $Re(s)>2$. The second statement is \cite{Wi} III Proposition 9, as we have $\widehat{\phi(\hat{f})^m _{I\cup i}}(0)=0$. \end{proof} \end{lemma} Let us set $\kappa_F:=d^{\times}t({\mathbb R}_{>0}\backslash Z({\mathbb A})/Z({\mathbb Q}))$. \begin{theorem}\label{Wi-series} Suppose we are given a function in $\varphi\in\mathcal S(V({\mathbb A}_f),\mu^{\otimes n})^0$. Write \begin{equation*}
(v,g)\mapsto \left(\left\|\det(g)\right\|_f \operatorname{sgn}(\operatorname{N}(\det(g)))^{-1}\right)^n f(v,g)=\varphi(v,g), \end{equation*} with $f\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes 0})^0$ such that $f$ factors in the second argument over $K_f$ with $Z_K\subset Z({\mathbb R})^0$. The polylogarithmic Eisenstein class associated to $\varphi$ \begin{equation*} \operatorname{Eis}^k(\varphi)\in H^{2\xi-1}(\mathcal M,\operatorname{Sym}^k\mathcal H\otimes\mu^{\otimes n+1})=\bigoplus_{p+q=2\xi-1}H^p(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes\mu^{\otimes n+1})\otimes \mathfrak H^q \end{equation*} may be represented by the differential form \begin{equation*} \frac{ 2C(k)}{\kappa_F}\sum_{i=1}^\xi\sum_{I\subset\left\langle \xi\right\rangle\setminus i}\sum_{\chi\in \widehat{Cl_F ^K}(m)}\operatorname{Eis}^k_{I,i}(\varphi,\chi) \end{equation*} \begin{proof}
Define $\operatorname{sgn}(r):=(\operatorname{sgn}(r)_\nu)_{\nu| \infty}:=(\operatorname{sgn}(r_\nu))_{\nu|\infty}$, $r\in F_{\mathbb R} ^{\times}$ and $S:V({\mathbb R})\to F_{\mathbb C}$, $v=(v^1,v^2)\mapsto v^1+iv^2$. For $g\in G({\mathbb R})$ we have $S(k(g)v)=e^{i\theta(g)}S(v)$ and \begin{equation*} \operatorname{sgn}(\det(g))k(\hat g^{-1})=\widehat{k(g)}^{-1},\ b(\hat g^{-1})=\operatorname{sgn}(\det(g))\widehat{b(g)}^{-1}=\frac{1}{\sqrt{ry}}\begin{pmatrix}1& x\\ 0& y\end{pmatrix}. \end{equation*} Therefore \begin{equation*} \operatorname{N}(e^{-(m+1)i\theta})e^{-i\theta}_{I\cup i}e^{i\theta}_{(I\cup i)^c}\frac{\operatorname{sgn}(\operatorname{N}(r))^{m+2}}{\operatorname{N}(\sqrt{ry})^{m+2}}\phi(\hat{f})^m_{I\cup i,\infty}(\hat{g}^{-1}v,g)= \end{equation*} \begin{equation*} \phi(\hat{f})(\hat{g}^{-1}v,g)\operatorname{N}\left(\frac{v_\infty ^1+\tau v_\infty ^2}{ry}\right)^{m+1}\left(\frac{v_\infty ^1+\tau v_\infty ^2}{ry}\right)_{I\cup i}\left(\frac{v_\infty ^1+\overline\tau v_\infty ^2}{ry}\right)_{(I\cup i)^c}. \end{equation*}
We see $\operatorname{sgn}(r)\sqrt{ry}=\operatorname{sgn}(r)|t_2|=t_2$ and with \Cref{calc_vol}, the definition of $\Theta_l$ (\Cref{Theta_1}) and $\Theta_{I,i}$ (\Cref{Theta_2}) we conclude for $t\in Z({\mathbb R})^0$ \begin{equation*}
\sum_{i=1}^\xi\sum_{I\subset\left\langle \xi\right\rangle\setminus i}\phi(\hat{f})^m _{I\cup i}(t\hat{g}^{-1}l,g) |\operatorname{N}(t)|^{m+2+2s} \Theta_{I,i}= \phi(\hat{f})(t\hat g ^{-1}l,g)\operatorname{N}(t)^{2(m+2+s)}\operatorname{N}(r)^{-(m+2)}\Theta_l. \end{equation*} We express the characteristic function of $Z({\mathbb R})^0/Z_K\subset {K_f ^Z}\backslash Z({\mathbb A})/Z({\mathbb Q})$ by $h_K^{-1}\sum_{\chi\in \widehat{Cl_F ^K}}\chi$ by considering a character $\chi$ as a function on $Z({\mathbb A})$. We get \begin{equation*} \frac{ d^{\times}t(K_f ^Z)R_K h_K}{2^{\xi}C(k)}\operatorname{Eis}^k(\varphi)= \end{equation*} \begin{equation*} \sum_{i=1}^\xi\sum_{I\subset\left\langle \xi\right\rangle\setminus i}\sum_{\chi\in \widehat{Cl_F ^K}}
\lim{s\rightarrow0}\int_{t\in Z({\mathbb A})/Z({\mathbb Q})}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\phi(\hat{f})^m_{I\cup i}(t\hat g ^{-1}l,g) \chi(t)\left\|t\right\|^{m+2+2s} d^{\times}t\cdot \Theta_{I,i}. \end{equation*} Using short exact sequences one calculates $2^{-(\xi-1)}d^{\times}t(K_f ^Z)R_K h_K=\kappa_F$. For $Re(s)>0$ the inner sum converges absolutely and locally uniformly. Therefore we may rearrange summation $\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}=\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}/Z({\mathbb Q})}\sum_{z\in Z({\mathbb Q})}$, use the bijection $G({\mathbb Q})/B({\mathbb Q})\rightarrow V({\mathbb Q})\setminus \left\{0\right\}/Z({\mathbb Q})$, $\gamma\mapsto \gamma e^1$, and interchange $\int_{Z({\mathbb A})/{\mathbb Z}({\mathbb Q})}$ and $\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q})}$. We get in total \begin{equation*}
\lim{s\rightarrow0}\int_{t\in Z({\mathbb A})/Z({\mathbb Q})}\sum_{l\in V({\mathbb Q})\setminus \left\{0\right\}}\phi(\hat{f})^m_{I\cup i}(t\hat g ^{-1}l,g) \chi(t)\left\|t\right\|^{m+2+2s} d^{\times}t\cdot \Theta_{I,i} = \end{equation*} \begin{equation*}
\lim{s\rightarrow0}\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q})}\int_{t\in Z({\mathbb A})}\phi(\hat{f})^m_{I\cup i}(t\hat g ^{-1}\gamma e^1,g) \chi(t)\left\|t\right\|^{m+2+2s} d^{\times}t\cdot \Theta_{I,i}= \end{equation*} \begin{equation*}
\lim{s\rightarrow0}\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q})}\int_{t\in Z({\mathbb A})}\phi(\hat{f})^m_{I\cup i}(t g \gamma e^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2+2s} d^{\times}t\cdot \Theta_{I,i}. \end{equation*} For this see also \cite{Wi} III Proposition 7. So we already have \begin{equation*} \operatorname{Eis}^k(\varphi)=\frac{2 C(k)}{\kappa_F}\sum_{i=1}^\xi\sum_{I\subset\left\langle \xi\right\rangle\setminus i}\sum_{\chi\in \widehat{Cl_F ^K}}\operatorname{Eis}^k_{I,i}(\varphi,\chi). \end{equation*} To prove the theorem it suffices to show that $\operatorname{Eis}^k_{I,i}(\varphi,\chi)=0$, whenever $\chi\notin \widehat{Cl_F ^K}(m)$. So take a $\chi\notin \widehat{Cl_F ^K}(m)$ and an $\epsilon\in \left\{\pm 1\right\}^\xi\subset Z({\mathbb R})$ with $\chi(\epsilon)\operatorname{sgn}(\operatorname{N}(\epsilon))^m\neq 1$. Then \begin{equation*} \int_{t\in Z({\mathbb A})}
\phi(\hat{f})^m _{I\cup i}(t g \gamma e^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{s} d^{\times}t= \end{equation*} \begin{equation*} \int_{t\in Z({\mathbb A})}
\phi(\hat{f})^m _{I\cup i}(\epsilon t g \gamma e^1,g) \chi(\epsilon \det(g) t)\left\|\epsilon \det(g) t\right\|^{s} d^{\times}t= \end{equation*} \begin{equation*}
\chi(\epsilon)\operatorname{sgn}(\operatorname{N}(\epsilon))^m\int_{t\in Z({\mathbb A})}\phi(\hat{f})^m _{I\cup i}(t g \gamma e^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{s} d^{\times}t, \end{equation*} since \begin{equation*} \phi(\hat{f})^m _{I\cup i}(\epsilon t g \gamma e^1,g)=\operatorname{sgn}(\operatorname{N}(\epsilon))^m\phi(\hat{f})^m _{I\cup i}(tg\gamma e^1,g). \end{equation*} So we conclude \begin{equation*}
\int_{t\in Z({\mathbb A})}\phi(\hat{f})^m _{I\cup i}(t g \gamma e^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{s} d^{\times}t=0 \end{equation*} and therefore $\operatorname{Eis}^k _{I,i}(\varphi,\chi)=0$. \end{proof} \end{theorem}
\section{Cohomology of the boundary}\label{boundary}
As Harder's Eisenstein classes are by definition determined by their restriction to the cohomology of the boundary of the Borel-Serre compactification of $\mathcal S_K$, we need a thorough understanding of the cohomology of the boundary to compare the polylogarithmic Eisenstein classes with those of Harder. In this section we quickly recall Harder's description of the cohomology of the boundary as a module induced from $B({\mathbb A}_f)$ to $G({\mathbb A}_f)$.
From now on we consider ${\mathbb Q}$-coefficients, but we will emphasize, when things can be defined integrally. We introduce the spaces $\partial \mathcal S_K:=K\backslash G({\mathbb A})/B({\mathbb Q})$ and $\partial \mathcal M_K:=K^1\backslash G({\mathbb A})/B({\mathbb Q})$. Denote by $\overline{\mathcal S_K}$ the Borel-Serre compactification of the space $\mathcal S_K$ . It is a manifold with corners and we have that the boundary $\overline{\mathcal S_K}\setminus\mathcal S_K$ is homotopy equivalent to $\partial\mathcal S_K$, see \cite{Ha1} 2.1. Therefore, we refer to $H^\bullet(\partial \mathcal S_K,\operatorname{Sym}^k\mathcal H ^\prime\otimes \mu^{\otimes n})$ as the \textit{cohomology of the boundary}. \\ The natural map $j_{\mathcal S_K}:\partial \mathcal S_K\rightarrow\mathcal S_K$ induces by pullback a restriction map on cohomology \begin{equation*} \operatorname{res}_{\mathcal S_K}:H^\bullet(\mathcal S_K,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n})\rightarrow H^\bullet(\partial\mathcal S_K,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n}) \end{equation*} and performing the colimits over $K$ these maps glue to $G({\mathbb A}_f)\times \pi_0(G_\infty)$-equivariant maps $\operatorname{res}_{\mathcal S}:H^\bullet(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n})\rightarrow H^\bullet(\partial\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n})$ of the colimits of the groups above. \\ Analogously we define $\operatorname{res}_{\mathcal M}:H^\bullet(\mathcal M,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n})\rightarrow H^\bullet(\partial\mathcal M,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n})$. The cohomology groups of the boundary are induced $G({\mathbb A}_f)\times \pi_0(G_\infty)$-modules. More precisely, set $K^B:=K\cap B({\mathbb A})$, $K^{1,B}:=K^1\cap B({\mathbb A})$ and get the natural inclusions $\mathcal S_{K}^B:=K ^B\backslash B({\mathbb A})/B({\mathbb Q})\rightarrow \partial \mathcal S_K$ and $\mathcal M_{K}^B:=K^{1,B}\backslash B({\mathbb A})/B({\mathbb Q})\rightarrow \partial \mathcal M_K$, which are actually inclusions of several connected components. As usual we set \begin{equation*} \varinjlim_K H^\bullet(\mathcal S_{K}^B,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n})=:H^\bullet(\mathcal S^B,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n}). \end{equation*} These groups are $B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))$-modules and we have \begin{equation*} \operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))} ^{G({\mathbb A}_f)\times \pi_0(G_\infty)}H^\bullet(\mathcal S^B,\operatorname{Sym}^k\mathcal H^\prime \otimes \mu^{\otimes n})=H^\bullet(\partial\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n}) \end{equation*} as $G({\mathbb A}_f)\times \pi_0(G_\infty)$-modules (\cite{Ha2}, p. 117.).
Next consider $W_0=\mathbb G_a\subset V_0=\mathbb G_a^2$ embedded in the first component. If $g_f\in G({\mathbb A}_f)$ is given, we define $W(g_f):=W({\mathbb Q})\cap V(g_f)$. If we write $V(g_f)=\mathfrak a\oplus \mathfrak b$, we have $W(g_f)=\mathfrak a$. We recognize $W({\mathbb Q})$ as a $B({\mathbb Q})$-invariant submodule of $V({\mathbb Q})$ and $W(g_f)$ as a $B(g_f):=G(g_f)\cap B({\mathbb Q})$-invariant submodule. Remember \begin{equation*} \operatorname{Sym}^kV(g_f)_{PD}=\operatorname{Sym}^k(\mathfrak a\oplus \mathfrak b)_{PD}=\bigoplus_{l=0}^k\operatorname{Sym}^{k-l}(\mathfrak a)_{PD}\otimes \operatorname{Sym}^l(\mathfrak b)_{PD}. \end{equation*} So we get the $B(g_f)$-invariant submodule \begin{equation*} W_k(g_f):=\bigoplus_{l=1}^k\operatorname{Sym}^{l}(\mathfrak a)_{PD}\otimes \operatorname{Sym}^{k-l}(\mathfrak b)_{PD}\subset \operatorname{Sym}^kV(g_f)_{PD}. \end{equation*} We set again $A:={\mathbb Z}[\frac{1}{Nd_F}]$ and $R:=\mathcal O_{F^{\text{Hil}}}[\frac{1}{Nd_F}]$. When coefficients are extended to $A$ the submodule $W_k(g_f)_A^{Z_K}\subset \operatorname{Sym}^k _AV(g_f)_{PD}^{Z_K}$ is a direct summand. To see this we may do the faithfully flat ring extension $A\to R$. If we are in the non-trivial case $k=\xi m$, we have by \Cref{invariants} the elements $\prod_\sigma (\alpha_\sigma X_\sigma )^{[n_\sigma] }(\beta_\sigma Y_\sigma)^{[m-n_\sigma]}$, $n_\sigma=0,...,m$, as $R$-basis of $\operatorname{Sym}^k _AV(g_f)_{PD}^{Z_K}$ and we see immediately that the direct summand generated by those basis elements where $n_\sigma\geq1$ equals $W_k(g_f)_R^{Z_K}$. In particular, we conclude using \Cref{Y_integral} that $\operatorname{Sym}^{k} _A V(g_f)_{PD}^{Z_K}/W_k(g_f)_A^{Z_K}$ is a free $A$-module of rank one generated by the residue class of \begin{equation*}
\left\|t_2\right\|_f^m \prod_\sigma Y^{[m]} _\sigma,\ g_f=x\begin{pmatrix}t_1& t_1u\\0&t_2 \end{pmatrix},\ x\in G(\hat{\mathbb Z}), \end{equation*} if $k=\xi m$ and zero otherwise. Note that the $B(g_f)$-action factors over $T(g_f)$ the image of $B(g_f)$ in $T({\mathbb Q})$.
Let us consider the locally constant sheaf $\underline\omega^k$ on $\partial\mathcal S_K$, which is associated to the $T({\mathbb Q})$-module $\operatorname{Sym}^{k}V({\mathbb Q})^{Z_K}/(\operatorname{Sym}^{k-1}V({\mathbb Q})\cdot W({\mathbb Q}))^{Z_K}$. The sheaf associated to the $T(g_f)$-module $\operatorname{Sym}^{k} _A V(g_f)_{PD}^{Z_K}/W_k(g_f)_A^{Z_K}$ defines an integral structure $\underline\omega^k_{PD}$ for the sheaf $\underline\omega^k$. We have the natural projection map $\operatorname{Sym}^k\mathcal H^\prime\rightarrow \underline\omega^k$ of sheaves on $\mathcal S_{K}^B$. Set $K^T:=\text{image of $K^B$ in $T({\mathbb A})$}$.
The natural map $B\rightarrow T$ induces a fiber bundle \begin{equation*} p:\mathcal S_{K}^B\rightarrow K ^T\backslash T({\mathbb A})/T({\mathbb Q})=:\mathcal S_{K}^T \end{equation*} with fiber $p^{-1}(1)=K\cap U({\mathbb A})\backslash U({\mathbb A})/U({\mathbb Q})$, which is a topological torus of dimension $\xi$. As described in \Cref{trace_integration} we have the trace morphism \begin{equation*} Rp_*(\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n})\rightarrow Rp_*(\underline\omega^k\otimes \mu^{\otimes n})\stackrel{tr}{\rightarrow} R^{\xi}p_*({\mathbb Q})\otimes\underline\omega^k \otimes \mu^{\otimes n}[-\xi]. \end{equation*} Here we used the projection formula and that $\underline\omega ^k$ is pullback by $p$. Of course, the trace morphism is defined integrally. Now we have the following result due to Harder. \begin{theorem}\label{boundary_fiber} The trace morphism \begin{equation*} H^p(\mathcal S_{K}^B,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n})\rightarrow H^{p-\xi}(\mathcal S_{K}^T,R^{\xi}p_*({\mathbb Q})\otimes\underline\omega^k \otimes \mu^{\otimes n}),\ p\geq \xi, \end{equation*} is an isomorphism. \begin{proof} By faithfully flatness and \Cref{ext_scalars} we may extend coefficients from ${\mathbb Q}$ to $\overline{\mathbb Q}$. In this situation see \cite{Ha1} II. Important is \cite{Ha1} 2.8 for the occurring weights. \end{proof} \end{theorem} Harder gives a complete description of the cohomology group on the right-hand side, that we will recall now. First we trivialize the sheaf $R^\xi p_*({\mathbb Q})$. We fix an ordering $\Sigma=\left\{\sigma_1,...,\sigma_\xi\right\}$. We write elements \begin{equation*} b=\begin{pmatrix}t_1& t_1 x\\0 & t_2\end{pmatrix}\in B({\mathbb R})=B_0(F\otimes_{\mathbb Q} {\mathbb R}) \end{equation*} and $x=(x_1,...,x_\xi)\in F\otimes_{\mathbb Q}{\mathbb R}={\mathbb R}^\xi$. Set \begin{equation*}
\operatorname{vol}_{\mathcal S^B/\mathcal S^T}(t):=\frac{dx_1\wedge...\wedge dx_\xi}{\sqrt{d_F}\left\|t_2 ^{-1}t_1 \right\|_f \operatorname{sgn}(\operatorname{N}(t_2 ^{-1}t_1))^{-1}}. \end{equation*} This form is $T({\mathbb Q})$ invariant, closed and gives each fiber $p^{-1}(p(b))$ a rational volume, therefore we may interpret $\operatorname{vol}_{\mathcal S^B/\mathcal S^T}$ as a cohomology class in $H^0(\mathcal S^T _K,R^\xi p_*({\mathbb Q}))$. If we have $K=K_N$, then $N^{-\xi}\operatorname{vol}_{\mathcal S^B/\mathcal S^T}$ gives each fiber exactly volume one and defines an integral trivialization. For $k=\xi m$ we have \begin{equation*}
\left\|t_2\right\|_f ^m \operatorname{sgn}(t_2)^{-m}\prod_\sigma (-Y_\sigma)^{[m]} \in H^0(\mathcal S^T _K,\underline\omega^k_{PD}) \end{equation*} in the cohomology with $A$-coefficients. It defines an $A$-integral trivialization of $\underline\omega^k_{PD}$. Finally, \begin{equation*}
\left\|\det(g)\right\|_f \operatorname{sgn}(\operatorname{N}(\det(g)))^{-1}\operatorname{vol}_V^*=\operatorname{vol}_{\mathcal T/\mathcal M}^*,\ \operatorname{vol}_V^*:=\bigwedge_\sigma X_\sigma\wedge Y_\sigma \end{equation*} is a trivialization of $\mu$. If we have $K=K_N$, then it even defines a trivialization over the ring $A$, as one sees by considering the volumes of the fibers of $\mathcal T_W\rightarrow \mathcal M_{K_N}$, $W_f=V(\hat{\mathbb Z})\rtimes K_N$. We suppose $K^T=Z({\mathbb A})\cap K \times Z({\mathbb A})\cap K$ . We have a short exact sequence of groups \begin{equation*} 1\rightarrow Z({\mathbb R})^0/ Z_K\stackrel{t\mapsto (s,t)}{\rightarrow}\mathcal S_K ^T\rightarrow \pi_0(\mathcal S_K ^T)=K^T T({\mathbb R})^0\backslash T({\mathbb A})/T({\mathbb Q})\rightarrow 1. \end{equation*} The cohomology of a connected component of $\mathcal S_K ^T$ is therefore isomorphic to the cohomology of $Z({\mathbb R})^0/Z_K$. Because $Z_K$ acts properly discontinuously and fixpoint free on $Z({\mathbb R})^0$, we get $H^\bullet( Z({\mathbb R})^0/Z_K,{\mathbb Z})=H^\bullet(Z_K,{\mathbb Z})$. We may interpret these classes as invariant classes on $\mathcal S_K ^T$ as described in \Cref{invariant_Z_cohom} and as such they span a free ${\mathbb Z}$-submodule $\mathcal H^\bullet(T/Z)_K\subset H^\bullet(\mathcal S_K ^T,{\mathbb Z})$ isomorphic to $H^\bullet(Z_K,{\mathbb Z})$. We set $\mathcal H^\bullet(T/Z):=\mathcal H^\bullet(T/Z)_K\otimes_{\mathbb Z}{\mathbb Q}$ as this space does not depend on the level $K$.
For $m\in {\mathbb N}_0$ and $n\in {\mathbb Z}$ we fix characters \begin{equation*} \gamma_{m,n}:T({\mathbb R})\rightarrow {\mathbb R}^{\times},\ (t_1,t_2)\mapsto \operatorname{N}(t_2)^{-(m+n+1)}\operatorname{N}(t_1)^{-(n-1)}. \end{equation*} Harder considers algebraic Hecke characters of type $\gamma_{m,n}$ (\cite{Ha1} (2.5.2)), in other words, continuous homomorphisms \begin{equation*}
\phi:T({\mathbb A})/T({\mathbb Q})\rightarrow {\mathbb C}^{\times}, \ \phi_{|T({\mathbb R})^0}=\gamma_{m,n|T({\mathbb R})^0}^{-1}, \end{equation*}
to decompose the cohomology of the boundary. We have $\phi_{|T({\mathbb R})}=\gamma_{m,n}\operatorname{sgn}(\phi)$ for a certain character $\operatorname{sgn}(\phi):T({\mathbb R})^0\backslash T({\mathbb R})\rightarrow \left\{\pm 1\right\}$. We set $\tilde{\phi}_f:=\phi_f\cdot \operatorname{sgn}(\phi)$, where as usual $\phi_f=\phi_{|T({\mathbb A}_f)}$. Then $\tilde{\phi}_f(xa)=\tilde{\phi}_f(x)\gamma_{m,n}(a)$ for all $a\in T({\mathbb Q})$ and $x\in T({\mathbb A})$. Moreover, $\tilde{\phi}_f$ takes its values in $\overline {\mathbb Q}$ and therefore $\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})$ acts on the set of $\tilde\phi_f$ by $ \sigma\cdot\tilde\phi_f(x):=\sigma(\tilde\phi_f(x))$, when we have $\sigma \in \text{Gal}(\overline{\mathbb Q}/{\mathbb Q})$ and $x\in T({\mathbb A})$.
\begin{theorem}\label{cohom_boundary} If $k=\xi m$ and $p\in {\mathbb N}_0$, we have an isomorphism of $\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})$-modules \begin{equation*}
\bigoplus_{\phi:\ type(\phi)=\gamma_{m,n},\tilde{\phi}_{f|Z({\mathbb R})=1}}\overline{\mathbb Q} \cdot\tilde\phi_f\otimes \mathcal H^p(T/Z)\cong H^{p+\xi}(\mathcal S^B,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n}\otimes \overline{\mathbb Q}) \end{equation*} \begin{equation*} \psi\otimes \eta\mapsto \frac{\psi \cdot \prod_\sigma (-Y_\sigma)^{[m]}\otimes \operatorname{vol}_V^{*n} dx_1\wedge...\wedge dx_{\xi }}{\sqrt{d_F}}\cup\eta \end{equation*} \begin{proof} \cite{Ha1} Theorem 1. \end{proof} \end{theorem} \begin{remark} Harder's ${\mathbb Q}$-structure on cohomology, compare \cite{Ha1} 1.3, 2.4 and (2.7.1), agrees with our natural ${\mathbb Q}$-structure on $H^\bullet(\mathcal S_K,\operatorname{Sym}^k\mathcal H^\prime \otimes\mu^{\otimes n})$. The point is that Harder's ${\mathbb Q}$-structure on cohomology is $H^\bullet(\mathcal S_K,\operatorname{Sym}^k\mathcal H^\prime \otimes\mu^{\otimes n}\otimes \overline{\mathbb Q})^{\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})}$, where the Galois action comes from functoriality, compare \Cref{invariants_integral} and \cite{Ha1} 1.3, 1.4. We have by \Cref{ext_scalars} \begin{equation*} H^\bullet(\mathcal S_K,\operatorname{Sym}^k\mathcal H^\prime \otimes\mu^{\otimes n}\otimes \overline{\mathbb Q})^{\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})}=\left(H^\bullet(\mathcal S_K,\operatorname{Sym}^k\mathcal H^\prime \otimes\mu^{\otimes n})\otimes \overline{\mathbb Q}\right)^{\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})}= \end{equation*} \begin{equation*} H^\bullet(\mathcal S_K,\operatorname{Sym}^k\mathcal H^\prime \otimes\mu^{\otimes n}). \end{equation*} \end{remark} \begin{remark} Even though Harder started with more general local systems than $\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n}$, see \cite{Ha1} 1.4, he does not obtain more cohomology groups in degree $p=\xi,...,2\xi-1$ on the boundary than we do. \cite{Ha1} (2.8.2) tells us that we get all weights occurring in cohomological degrees $p=\xi,...,2\xi-1$. \end{remark}
\section{Restriction of polylogarithmic Eisenstein classes to the boundary}
The next step is to consider the restriction of our polylogarithmic Eisenstein classes to the boundary. We have the isomorphism of groups \begin{equation*} {\mathbb A}_F\rightarrow U({\mathbb A}),\ x\mapsto \begin{pmatrix}1&x\\0&1 \end{pmatrix}=u \end{equation*} inducing on $U({\mathbb A})$ a Haar measure $du$ corresponding to our Haar measure $dx$ on ${\mathbb A}_F$. \begin{proposition}\label{boundary_residueI} The cohomology class \begin{equation*} \operatorname{res}_{\mathcal S}(\operatorname{Eis}^k(\varphi))\in \bigoplus_{p+q=2\xi-1}H^{p}(\partial\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1})\otimes \mathfrak H^q \end{equation*} for $\varphi\in\mathcal S(V({\mathbb A}_f),\mu^{\otimes n})^0 $ may be represented by the differential form \begin{equation*} \frac{ 2C(k)}{\kappa_F}\sum_{i=1}^\xi\sum_{I\subset\left\langle \xi\right\rangle\setminus i}\sum_{\chi\in \widehat{Cl_F ^K}(m)}
\int_{t\in Z({\mathbb A})}\phi(\hat{f})^m _{I\cup i}(tge^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2} d^{\times}t\cdot \Theta_{I,i} \end{equation*} In other words, $\operatorname{res}_{\mathcal S}(\operatorname{Eis}^k(\varphi))$ is determined by the zero Fourier coefficient of the $U({\mathbb Q})$-invariant form $\operatorname{Eis}^k(\varphi)$. \begin{proof} Let us set for a moment \begin{equation*} F_{I \cup i}(\chi,s,g):=\int_{t\in Z({\mathbb A})}
\phi(\hat{f})^m _{I\cup i}(tge^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2+2s} d^{\times}t. \end{equation*} By \Cref{cohom_boundary} and \Cref{boundary_fiber} we know that we may perform fiber integration, see \Cref{trace_integration}, to simplify our cohomology classes. The connected components of $\partial\mathcal S_K$ are fiber bundles and the fiber passing through $g\in G({\mathbb A})$ is \begin{equation*} U({\mathbb R})/g_fK_fg_f^{-1}\cap U({\mathbb Q})=gKg^{-1}\cap U({\mathbb A})\backslash U({\mathbb A})/U({\mathbb Q})\hookrightarrow K\backslash G({\mathbb A})/B({\mathbb Q})=\partial \mathcal S_K, \end{equation*} $u\mapsto gu$. As we have \begin{equation*} \operatorname{Eis}^k(\varphi)=\frac{2 C(k)}{\kappa_F}\sum_{i=1}^\xi\sum_{I\subset\left\langle \xi\right\rangle\setminus i}\sum_{\chi\in \widehat{Cl_F ^K}(m)}\operatorname{Eis}^k_{I,i}(\varphi,\chi), \end{equation*} we have to do fiber integration for each $\operatorname{Eis}^k_{I,i}(\varphi,\chi)$ and for those we have \begin{equation*} du(g_fK_fg_f^{-1})^{-1}\int_{u\in U({\mathbb R})/g_fK_fg_f^{-1}\cap U({\mathbb Q})}\lim{s\rightarrow 0}\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q}))} F_{I \cup i}(\chi,s,tg_f k_\infty a_\infty u \gamma)du\cdot \Theta_{I,i}, \end{equation*} where we have used the Iwasawa decomposition of $g_\infty=k_\infty a_\infty n_\infty\in G({\mathbb R})$ and that the forms $\Theta_{I,i}$ are constant in $u$ direction. If we use translation invariance of $du$, we may write the last integral as \begin{equation*} \int_{u\in U({\mathbb A})/U({\mathbb Q})}\lim{s\rightarrow 0}\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q}))} F_{I \cup i}(\chi,s,tgu\gamma)du\cdot \Theta_{I,i}. \end{equation*}
By dominated convergence we may interchange the limit and integration, as the limit is a continuous function in $u$ and therefore integrable over the compact space $U({\mathbb A})/U({\mathbb Q})$. We obtain \begin{equation*} \lim{s\rightarrow 0}\int_{u\in U({\mathbb A})/U({\mathbb Q})}\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q}))} F_{I \cup i}(\chi,s,gu\gamma)du\cdot \Theta_{I,i}. \end{equation*} We write our integral as a sum \begin{equation*} \lim{s\rightarrow 0}\int_{u\in U({\mathbb A})/U({\mathbb Q})}F_{I \cup i}(\chi,s,gu)du\cdot \Theta_{I,i}+ \end{equation*} \begin{equation*}
\lim{s\rightarrow 0}\int_{u\in U({\mathbb A})/U({\mathbb Q})}\sum_{1\neq\gamma\in G({\mathbb Q})/B({\mathbb Q}))}
F_{I \cup i}(\chi,s,gu\gamma) du\cdot \Theta_{I,i}. \end{equation*} The first summand does not depend on $u$, as $ue^1=e^1$, so it equals \begin{equation*}
\int_{t\in Z({\mathbb A})}\phi(\hat{f})^m _{I\cup i}(tge^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2} d^{\times}t\cdot \Theta_{I,i} \end{equation*} and this Tate integral exists by \cite{Wi} III Proposition 5. To treat the second summand we consider the bijection \begin{equation*} U({\mathbb Q})\rightarrow \left\{\gamma B({\mathbb Q})\in G({\mathbb Q})/B({\mathbb Q}): \gamma\notin B({\mathbb Q})\right\} \end{equation*} \begin{equation*} u=\begin{pmatrix} 1 & \alpha\\ 0& 1\end{pmatrix}\mapsto uJ=\begin{pmatrix} \alpha & -1\\ 1& 0\end{pmatrix},\ J=\begin{pmatrix} 0 & -1\\ 1& 0\end{pmatrix} \end{equation*} and obtain the integral \begin{equation*}
\lim{s\rightarrow 0}\int_{u\in U({\mathbb A})}\int_{t\in Z({\mathbb A})}\phi(\hat{f})^m _{I\cup i}(tguJe^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2+2s} d^{\times}t du\cdot \Theta_{I,i}. \end{equation*} This integral is zero by the following lemma and our proposition is proved. \end{proof} \end{proposition} \begin{lemma} \begin{equation*}
\lim{s\rightarrow 0}\int_{u\in U({\mathbb A})}\int_{t\in Z({\mathbb A})}\phi(\hat{f})^m _{I\cup i}(tguJe^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2+2s} d^{\times}t du=0 \end{equation*} \begin{proof} By Fubini's theorem we may switch the order of integration and obtain the integral \begin{equation*}
\lim{s\rightarrow 0}\int_{t\in Z({\mathbb A})}\int_{u\in U({\mathbb A})}\phi(\hat{f})^m _{I\cup i}(tguJe^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2+2s} du d^{\times}t= \end{equation*} \begin{equation*}
\lim{s\rightarrow 0}\int_{t\in Z({\mathbb A})}\int_{u\in U({\mathbb A})}\phi(\hat{f})^m _{I\cup i|g}(tuJe^1,g) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2+2s} du d^{\times}t, \end{equation*}
where as usual $\psi_{|g}(v)=\psi(gv)$ for a Schwartz-Bruhat-function $\psi\in \mathcal S(V({\mathbb A}))$. The advantage of this integral is that we may interpret the inner integral as a Fourier transform, whereas the exterior integral may be examined with the theory of Tate integrals. We had $u=\begin{pmatrix}1&x\\0&1 \end{pmatrix}$ and get \begin{equation*}
\lim{s\rightarrow 0}\int_{t\in Z({\mathbb A})}\int_{x\in {\mathbb A}_F}\phi(\hat{f})^m _{I\cup i|g}((tx,t),g) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2+2s} dxd^{\times}t= \end{equation*} \begin{equation*}
\lim{s\rightarrow 0}\int_{t\in \mathbb{I}_F}\left\|t\right\|^{-1}\int_{x\in {\mathbb A}_F}\phi(\hat{f})^m _{I\cup i|g}((x,t),g) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2+2s} dx d^{\times}t= \end{equation*} \begin{equation*}
\chi(\det(g))\left\|\det(g)\right\|^{m+2} \lim{s\rightarrow 0}\int_{t\in \mathbb{I}_F}\int_{x\in {\mathbb A}_F}\phi(\hat{f})^m _{I\cup i|g}((x,t),g) \chi(t)\left\|t\right\|^{m+1+2s} dx d^{\times}t \end{equation*} To examine the integral \begin{equation*}
\lim{s\rightarrow 0}\int_{t\in \mathbb{I}_F}\int_{x\in {\mathbb A}_F}\phi(\hat{f})^m _{I\cup i|g}((x,t),g) \chi(t)\left\|t\right\|^{m+1+2s} dx d^{\times}t \end{equation*} we treat the finite and the infinite places separately. Let us start with the finite part. It is of the form \begin{equation*}
\int_{t\in \mathbb{I}_{F,f}}\int_{x\in {\mathbb A}_{F,f}}\hat{f}_{|g}((x,t),g)dx\left\|t\right\|^{1+m+2s}\chi(t) d^{\times}t. \end{equation*} We define the Schwartz-Bruhat function \begin{equation*}
\varphi:{\mathbb A}_{F,f}\rightarrow {\mathbb C},\ \varphi(u):=\int_{x\in {\mathbb A}_{F,f}}\hat{f}_{|g}((x,u),g)dx. \end{equation*} We denote the Fourier transform of Schwartz-Bruhat functions $\varphi:{\mathbb A}_F\rightarrow {\mathbb C}$ by \begin{equation*} P\varphi(v):=\int_{x\in {\mathbb A}_F}\phi(x)\overline{\psi_0(\operatorname{Tr}_{F/{\mathbb Q}}(xv))}dx, \end{equation*} where $\psi_0:{\mathbb A}/{\mathbb Q}\rightarrow {\mathbb C}^{\times}$ is our fixed non-trivial character. We see \begin{equation*}
P\varphi(0)=\int_{x\in {\mathbb A}_{F,f}}\int_{u\in {\mathbb A}_{F,f}}\hat{f}_{|g}((u,x),g)dudx=\int_{v\in V({\mathbb A}_f)}\hat{f}_{|g}(v,g)dv= \end{equation*} \begin{equation*}
\left\|\det(g)\right\|_f ^{-1}\hat{\hat{f}}(0,g)=\left\|\det(g)\right\|_f ^{-1}f(0,g)=0. \end{equation*}
We add an appropriate Euler factor at infinity. For a place $\nu|\infty$ we set $\varphi^{p(m)}_\nu(u):=\exp(-\pi u^2)$, if $2|m$, and $\varphi^{p(m)}_\nu(u):=\exp(-\pi u^2)u$, if $2\nmid m$. We define \begin{equation*}
\varphi^{p(m)}_\infty:F\otimes_{\mathbb Q} {\mathbb R}=\prod_{\nu|\infty} {\mathbb R}\rightarrow {\mathbb C},\ (u_\nu)\mapsto \prod_{\nu|\infty}\varphi^{p(m)}_\nu(u_\nu), \end{equation*} \begin{equation*} \varphi^{p(m)}:{\mathbb A}_F \rightarrow {\mathbb C}, u=(u_f,u_\infty)\mapsto \varphi^{p(m)}_\infty(u_\infty)\varphi(u_f), \end{equation*}
and consider the Tate integral $\lim{s\rightarrow 0}\int_{t\in \mathbb{I}_F}\varphi^{p(m)}(t)\chi(t)\left\|t\right\|^{m+1+2s}d^{\times}t$. The integral exists in any case, since $P\varphi^{p(m)}(0)=0$ and we do not have a pole at $m+1+2s=1$, see \cite{T} Main Theorem 4.4.1. Moreover, the integral over the infinite places exists and is non-zero as computed in \cite{T} 2.5. We conclude that the finite integral $\lim{s\rightarrow 0}\int_{t\in \mathbb{I}_{F,f}}\varphi(t)\chi(t)\left\|t\right\|^{m+1+2s}d^{\times}t$ exists.
So it suffices to to show that \begin{equation*}
\lim{s\rightarrow 0}\int_{t\in F_{\mathbb R}^{\times}}\int_{x\in F_{\mathbb R}}\phi(\hat{f})^m _{I\cup i|g,\infty}((x,t),g) \chi(t)\left\|t\right\|^{m+1+2s} dx d^{\times}t=0. \end{equation*} We transform the integral back and and have to show the vanishing of \begin{equation*}
\lim{s\rightarrow 0}\int_{t\in F_{\mathbb R}^{\times}}\int_{x\in F_{\mathbb R}}\phi(\hat{f})^m _{I\cup i|g,\infty}((tx,t),g) \chi(t)\left\|t\right\|^{m+2+2s} dx d^{\times}t= \end{equation*} \begin{equation*}
2^\xi\lim{s\rightarrow 0}\int_{x\in F_{\mathbb R}}\int_{t\in F_{\mathbb R}^{\times,0}}\phi(\hat{f})^m _{I\cup i|g,\infty}((tx,t),g) \left\|t\right\|^{m+2+2s} d^{\times}t dx. \end{equation*} The last equality holds, since we have $\chi\in \widehat{Cl_F ^K}(m)$. We use again the Iwasawa decomposition for $g\in G({\mathbb R})$ and calculate the last integral as \begin{equation*}
\frac{2^\xi\operatorname{N}(e^{(m+1)i\theta})e^{i\theta}_{I\cup i}e^{-i\theta}_{(I\cup i)^c}}{\operatorname{N}(t_1)^{m+2}}\lim{s\rightarrow 0}\int_{x\in F_{\mathbb R}}\int_{t\in F_{\mathbb R}^{\times,0}}\phi(\hat{f})^m _{I\cup i,\infty}(t(x,y),g) \left\|t\right\|^{m+2+2s} d^{\times}t dx, \end{equation*} with $y=\frac{t_2}{t_1}$. To end the proof we show \begin{equation*}
\lim{s\rightarrow 0}\int_{x\in F_{\mathbb R}}\int_{t\in F_{\mathbb R}^{\times,0}}\phi(\hat{f})^m _{I\cup i,\infty}(t(x,y),g) \left\|t\right\|^{m+2+2s} d^{\times}t dx=0. \end{equation*} We have \begin{equation*} \phi(\hat{f})^m_{I\cup i,\infty}(t(x,y),g)=
\exp(-\pi\left\|t(x,y)\right\|^2)\operatorname{N}(t(x+iy))^{m+1}(t(x+iy))_{I\cup i}(t(x-iy))_{(I\cup i)^c}, \end{equation*} so \begin{equation*}
\int_{t\in F_{\mathbb R}^{\times,0}}\phi(\hat{f})^m _{I\cup i,\infty}(t(x,y),g) \left\|t\right\|^{m+2+2s} d^{\times}t= \end{equation*} \begin{equation*}
\frac{\Gamma(m+2+s)^\xi}{2^\xi\pi^{\xi(m+2+s)}}\frac{\operatorname{N}(x+iy)^{m+1}(x+iy)_{I\cup i}(x-iy)_{(I\cup i)^c}}{|\operatorname{N}(x+iy)|^{2(m+2+s)}} \end{equation*} and the integral in question is therefore \begin{equation*}
\frac{\Gamma(m+2)^\xi}{2^\xi\pi^{\xi(m+2)}}\lim{s\rightarrow 0}\int_{x\in F_{\mathbb R}}\frac{dx}{\operatorname{N}(x-iy)^{(m+1)}(x-iy)_{I\cup i}(x+iy)_{(I\cup i)^c}|\operatorname{N}(x+iy)|^{2s}}. \end{equation*} This can be calculated as \begin{equation*} \frac{\Gamma(m+2)^\xi}{\pi^{\xi(m+2)}}\int_{x\in F_{\mathbb R}}\frac{dx}{\operatorname{N}(x-iy)^{(m+1)}(x-iy)_{I\cup i}(x+iy)_{(I\cup i)^c}} \end{equation*} by dominated convergence, because the integral is dominated by \begin{equation*}
c\int_{x\in F_{\mathbb R}}\frac{dx}{|\operatorname{N}(x+iy)|^{(m+2)}},\ c\in {\mathbb R}_{>0}, \end{equation*}
which exists by Fubini, as the corresponding one dimensional integrals $\int_{x\in{\mathbb R}}\frac{dx}{|x+iy|^{(m+2)}}$ exist for $y\in {\mathbb R}^{\times}$. This is seen as follows. The integral equals $\int_{x\in{\mathbb R}}\frac{dx}{(x^2+y^2)^{(\frac{m}{2}+1)}}$. We split it up as \begin{equation*} \int_{x\in{\mathbb R}}\frac{dx}{(x^2+y^2)^{(\frac{m}{2}+1)}}=2\int_0 ^1 \frac{dx}{(x^2+y^2)^{(\frac{m}{2}+1)}} +2\int_1 ^\infty\frac{dx}{(x^2+y^2)^{(\frac{m}{2}+1)}}. \end{equation*} Because $\frac{1}{(x^2+y^2)^{(\frac{m}{2}+1)}}$ is as function in $x$ continuous on $[0,1]$, there is a constant $C>0$ such that $\frac{1}{(x^2+y^2)^{(\frac{m}{2}+1)}}<C$, $x\in [0,1]$. Therefore we may estimate our integral by $2C +2\int_1 ^\infty\frac{dx}{(x^2+y^2)^{(\frac{m}{2}+1)}}$. Moreover, we have $\frac{1}{(x^2+y^2)^{(\frac{m}{2}+1)}}\leq \frac{1}{x^{m+2}}$, $x\in [1,\infty[$, and our integral may be estimated by $2C +2\int_1 ^\infty\frac{dx}{x^{m+2}}$, which is finite as $m\geq 0$. Let us come back to \begin{equation*} \frac{\Gamma(m+2)^\xi}{2^\xi\pi^{\xi(m+2)}}\int_{x\in F_{\mathbb R}}\frac{dx}{\operatorname{N}(x-iy)^{(m+1)}(x-iy)_{I\cup i}(x+iy)_{(I\cup i)^c}}, \end{equation*} which is a product of integrals. If we want to show that it vanishes, it suffices to show the vanishing of a single factor, for example the $i$th-factor. This factor is essentially the integral \begin{equation*} \int_{x\in{\mathbb R}}\frac{1}{(x-iy)^{(m+2)}}dx=\left[\frac{1}{-(m+1)(x-iy)^{m+1}}\right]_{-\infty} ^{\ \infty}=0. \end{equation*} \end{proof} \end{lemma} Consider the Iwasawa decomposition \begin{equation*}
g =kb\in G({\mathbb A}) \text{ with } b=\begin{pmatrix}t_1 & t_1x\\ 0 & t_2 \end{pmatrix} \text{ and } k\in G(\hat{\mathbb Z})\cdot\prod_{\nu|\infty} SO(2). \end{equation*} We define a differential form on $K^1\backslash G({\mathbb A})$: \begin{equation*} \operatorname{vol}_P:=\sum_{i=1} ^\xi \frac{(-2)^{\xi}}{\xi\kappa_F}\prod_{j\neq i}\left(\frac{dt_{2,j}}{t_{2,j}}\wedge dx_j\right) \wedge dx_i. \end{equation*} \begin{lemma}\label{Theta_vol_P} We have \begin{equation*}
\frac{\xi\kappa_F\operatorname{vol}_P}{\left\|t_2 ^{-1}t_1\right\|_f\operatorname{sgn}(\operatorname{N}(t_2 ^{-1}t_1))} =
\sum_{i=1} ^\xi\sum_{I\subset \left\langle \xi\right\rangle\setminus i}\frac{-2(-1)^{|I|}}{\left\|t_2 ^{-1}t_1\right\|_f\operatorname{sgn}(\operatorname{N}(t_2 ^{-1}t_1))} \frac{d\tau\wedge d\overline\tau}{\overline\tau-\tau}_{(i\cup I)^c}\wedge \frac{dr\wedge d \overline\tau}{r}_I\wedge d\overline\tau_i \end{equation*} as cohomology classes in $H^{2\xi-1}(\partial\mathcal M_K,{\mathbb C})=\bigoplus_{p+q=2\xi-1}H^{p}(\partial\mathcal S_K,{\mathbb C})\otimes \mathfrak H^q$. More precisely, if $A={\mathbb Z}[\frac{1}{Nd_F}]$, $K=K_N$ and $N\geq 3$ \begin{equation*}
\frac{d^{\times}t(K_{N,f} ^Z)h_{K_N}\operatorname{vol}_P}{(-2)^\xi\sqrt{d_F}\left\|t_2 ^{-1}t_1\right\|_f\operatorname{sgn}(\operatorname{N}(t_2 ^{-1}t_1))} \end{equation*} is in the image of the cohomology with $A$-coefficients and part of an $A$-basis of the $A$-integral classes. \begin{proof} We have $\frac{d\tau\wedge d\overline\tau}{\overline\tau-\tau}=\frac{dx\wedge dy}{y}$ and we see that the top $x$-component of \begin{equation*}
\sum_{i=1} ^\xi\sum_{I\subset \left\langle \xi\right\rangle\setminus i}(-1)^{|I|}\frac{d\tau\wedge d\overline\tau}{\overline\tau-\tau}_{(i\cup I)^c}\wedge \frac{dr\wedge d \overline\tau}{r}_I\wedge d\overline\tau_i \end{equation*} equals \begin{equation*}
\sum_{i=1} ^\xi\sum_{I\subset \left\langle \xi\right\rangle\setminus i}(-1)^{|I|}\frac{dx\wedge dy}{y}_{(i\cup I)^c}\wedge \frac{dr\wedge dx}{r}_I\wedge dx_i. \end{equation*} By \Cref{boundary_fiber} we already know that these two classes have to be cohomologous. We have the formulas \begin{equation*} ry\, d\left(\frac{1}{ry}\right)\wedge dx=-\frac{d(ry)}{ry}\wedge dx=\frac{dx\wedge dy}{y}-\frac{dr}{r}\wedge dx,\ ry\, d\left(\frac{1}{ry}\right)\wedge dx=(-2)\frac{dt_{2}}{t_{2}}\wedge dx \end{equation*} from which the claimed equality of forms follows. What remains to be shown is the integrality statement. Consider components $\left\{\pm1\right\}^\xi{\mathbb R}_{>0}\backslash B({\mathbb R})/B(g_f)\subset\partial\mathcal M_{K_N}$. Set $P_0:=\left\{g\in G_0:ge^1=e^1\right\}$ and get $P:=Res_{\mathcal O/{\mathbb Z}}P_0$. Define \begin{equation*} P({\mathbb R})^1:=\left\{x\in P({\mathbb R}):\ x\in P({\mathbb R})^0,\ \operatorname{N}(\det(x))=1\right\}. \end{equation*} This gives the submanifold \begin{equation*} i:P({\mathbb R})^1/P(g_f)\rightarrow \left\{\pm1\right\}^\xi{\mathbb R}_{>0}\backslash B({\mathbb R})/B(g_f)\text{ with }P(g_f):=B(g_f)\cap P({\mathbb R}) ^0. \end{equation*} The form \begin{equation*}
\frac{d^{\times}t(K_{N,f} ^Z)h_{K_N}}{(-2)^\xi\sqrt{d_F}\left\|t_2 ^{-1}t_1\right\|_f\operatorname{sgn}(\operatorname{N}(t_2 ^{-1}t_1))}\operatorname{vol}_P \end{equation*} is a volume form of the compact space on the left-hand side giving it volume $N^\xi\in A^{\times}$. So the form is (up to a factor of $N^\xi$) dual to the fundamental class under Poincaré duality and hence defines an $A$-integral class on $P({\mathbb R})^1/P(g_f)$. As we also have a right-inverse \begin{equation*}
p:\left\{\pm1\right\}^\xi{\mathbb R}_{>0}\backslash B({\mathbb R})/B(g_f)\rightarrow P({\mathbb R})^1/P(g_f),\ \begin{pmatrix}t_1&t_1x\\0&t_2 \end{pmatrix}\mapsto\begin{pmatrix}1 & x\\0&\frac{|t_2t_1^{-1}|}{\sqrt[n]{|\operatorname{N}(t_2t_1^{-1})|}} \end{pmatrix} \end{equation*} for the map $i$, we may conclude that our form is part of an $A$-basis of the cohomology of $\partial \mathcal M_{K_N}$. \end{proof} \end{lemma}
\begin{proposition}\label{boundary_residueII} For $\varphi\in\mathcal S(V({\mathbb A}_f),\mu^{\otimes n})^0 $ the $K_f$-invariant cohomology class \begin{equation*} \operatorname{res}_{\mathcal S}(\operatorname{Eis}^k(\varphi))\in \bigoplus_{p+q=2\xi-1}H^{p}(\partial\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1})\otimes \mathfrak H^q \end{equation*} equals \begin{equation*}
\sum_{\chi\in \widehat{Cl_F ^K}(m)}\frac{\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb A}_f)}\hat{f} (tg_fe^1,g) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t\cdot \end{equation*} \begin{equation*}
\frac{\operatorname{N}(\omega)^m \operatorname{vol}_P\otimes \operatorname{vol}_V^{*n+1}}{(m!)^\xi\left(\left\|\det(g)\right\|_f\operatorname{sgn}(\operatorname{N}(\det(g)))^{-1}\right)^{-n}} \end{equation*} In particular, \begin{equation*}
\sum_{\chi\in \widehat{Cl_F ^K}(m)}\frac{\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb A}_f)}\hat{f} (tg_fe^1,g) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t \end{equation*} is a rational number.
\begin{proof} We calculate \begin{equation*}
\int_{t\in Z({\mathbb R})}\phi(\hat{f})^m _{I\cup i,\infty}(tg_\infty e^1,g_\infty) \chi(\det(g_\infty)t)\left\|\det(g_\infty)t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*} \operatorname{N}(e^{(m+1)i\theta})e^{i\theta}_{I\cup i}e^{-i\theta}_{(I\cup i)^c}\frac{\operatorname{N}(t_{2}) ^{m+2}\Gamma(m+2)^{\xi}}{\pi^{\xi(m+2)}} \end{equation*} Remembering the definition of the $\Theta_{I,i}$ (\Cref{Theta_2}) the formula for $\operatorname{res}_{\mathcal S}(\operatorname{Eis}^k(\varphi))$ follows from \Cref{boundary_residueI} and \Cref{Theta_vol_P}. The rationality statement follows, since \begin{equation*}
\left\|t_2\right\|_f ^{m+2}\operatorname{sgn}(\operatorname{N}(t_2))^{m+2}\left\|\det(g)\right\|_f ^{n}\operatorname{sgn}(\operatorname{N}(\det g))^{n}\operatorname{N}(\omega)^m \operatorname{vol}_P\otimes \operatorname{vol}_V^{*n+1}= \end{equation*} \begin{equation*}
\left\|t_2\right\|_f ^{m+2}\operatorname{sgn}(\operatorname{N}(t_2))^{m+2}\left\|\det(g)\right\|_f ^{n}\operatorname{sgn}(\operatorname{N}(\det g))^{n}\prod_\sigma(-Y_\sigma)^m \operatorname{vol}_P\otimes \operatorname{vol}_V^{*n+1} \end{equation*} as cohomology classes in $ H^\bullet(\partial\mathcal S,\underline\omega^k\otimes \mu^{\otimes n+1})\otimes \mathfrak H^\bullet$. The latter defines a rational class, compare \Cref{boundary}, \Cref{Theta_vol_P} and recall $d^{\times}t(K_f ^Z)\in {\mathbb Q}^{\times}\frac{1}{\sqrt{d_F}}$, since $K_f ^Z\subset \hat\mathcal O ^{\times}$ is a subgroup of finite index and $d^{\times}t(\hat\mathcal O^{\times})=\frac{1}{\sqrt{d_F}}$. \end{proof} \end{proposition} Of course, we can refine this statement, if we use the integral structures on cohomology, which we have chosen. \begin{proposition}\label{integral_L_value} Let $N\geq 3$ be an integer, $A:={\mathbb Z}[\frac{1}{d_FN}]$ and $f\in \text{map}(V({\mathbb Z}/N{\mathbb Z}),A)$ with $f(0)=\sum_{v\in V(Z/N{\mathbb Z})}f(v)=0$. We denote by $\mathcal O^{\times}(N)^+\subset \mathcal O^{\times}$ the group of totally positive units which are congruent to one modulo $N$. We write \begin{equation*} g=k\begin{pmatrix}t_1& t_1x\\ 0& t_2 \end{pmatrix}\in G({\mathbb A}_f)\times \pi_0(G({\mathbb R})),\ k\in G(\hat{\mathbb Z}), \end{equation*} and get a well-defined function in $map(K_N\backslash G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))/B({\mathbb Q}),A)$ \begin{equation*}
g\mapsto \frac{(-1)^\xi\sqrt{d_F}\Gamma(m+2)^\xi }{(-2\pi i)^{\xi(m+2)}\operatorname{sgn}(\operatorname{N}(t_2))^{m+2}\left\|t_2\right\|_f^{m+2}}\sum_{l\in F^{\times}/\mathcal O^{\times}(N)^+}\frac{\hat{f}(\hat{g}_fle^1)}{ \operatorname{N}(l)^{m+2}}\in A \end{equation*} \begin{proof} We apply the construction of polylogarithmic Eisenstein classes to $f$ and obtain by \Cref{pol} \begin{equation*} \operatorname{Eis}^k(f)\in H^{2\xi-1}(\mathcal M_{K_N}, \operatorname{Sym}^k\mathcal H_{PD}\otimes \mu) \end{equation*} with $A$-coefficients. Since $\operatorname{Sym}^k\mathcal H^\prime _{PD}\subset \operatorname{Sym}^k\mathcal H_{PD}$ is a direct summand, we obtain by restricting $\operatorname{Eis}^k(f)$ to the boundary $\partial \mathcal M_{K_N}$ \begin{equation*} \operatorname{res}_{\mathcal M}(\operatorname{Eis}^k(f))\in H^{2\xi-1}(\partial\mathcal M_{K_N}, \underline{\omega}^k _{PD}\otimes \mu) \end{equation*} with $A$-coefficients. We have calculated $\operatorname{res}_{\mathcal M}(\operatorname{Eis}^k(f))$ with ${\mathbb C}$-coefficients as \begin{equation*}
\sum_{\chi\in \widehat{Cl_F ^{K_N}}(m)}\frac{\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb A}_f)}\hat{f} (tg_fe^1) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t \cdot \end{equation*} \begin{equation*}
\frac{\operatorname{N}(\omega)^m \operatorname{vol}_P\otimes \operatorname{vol}_V^{*}}{(m!)^\xi} \end{equation*}
Consider $\varphi_\infty\in \mathcal S(F_{\mathbb R})$ defined by $\varphi_\infty(v):=\exp(-\left\|v\right\|^2)\operatorname{N}(v)^{m+2}$, $v\in F_{\mathbb R}$. We have for $\chi\in \widehat{Cl_F ^{K_N}}(m)$ \begin{equation*}
\int_{t\in F_{\mathbb R}^{\times}}\varphi_\infty(t)\chi_\infty(t)|\operatorname{N}(t)|^{m+2}d^{\times}t=\Gamma(m+2)^\xi. \end{equation*} Define $\varphi\in \mathcal S({\mathbb A}_F)$ by $\varphi(v):=\varphi_\infty(v_\infty)\hat{f}(g_fv_fe^1)$ for $v\in {\mathbb A}_F$. We calculate the integral \begin{equation*}
\sum_{\chi\in \widehat{Cl_F ^{K_N}}(m)}\frac{\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb A}_f)}\hat{f} (tg_fe^1) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*}
\sum_{\chi\in \widehat{Cl_F ^{K_N}}(m)}\frac{1}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb A})}\varphi(t)\chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*}
\sum_{\chi\in \widehat{Cl_F ^{K_N}}}\frac{1}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb A})}\varphi(t)\chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*}
\sum_{\chi\in \widehat{Cl_F ^{K_N}}}\frac{d^{\times}t(K_{N,f} ^Z)}{(-2\pi i)^{\xi(m+2)}}\int_{t\in K_{N,f} ^Z\backslash Z({\mathbb A})/F^{\times}}\sum_{l\in F^{\times}}\varphi(tl)\chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*}
\frac{d^{\times}t(K_{N,f} ^Z)h_{K_N}}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb R})^0/\mathcal O^{\times}(N)^+}\sum_{l\in F^{\times}}\varphi(\det(g_f)^{-1}tl)\left\|t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*}
\frac{d^{\times}t(K_{N,f}^Z)h_{K_N}}{(-2\pi i)^{\xi(m+2)}}\sum_{l\in F^{\times}/\mathcal O^{\times}(N)^+}\int_{t\in Z({\mathbb R})^0}\varphi(\det(g_f)^{-1}tl)\left\|t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*} \frac{d^{\times}t(K_{N,f}^Z)h_{K_N}\Gamma(m+2)^\xi}{2^\xi(-2\pi i)^{\xi(m+2)}}\sum_{l\in F^{\times}/\mathcal O^{\times}(N)^+}\frac{\hat{f}(\hat{g}_fle_1)}{\operatorname{N}(l)^{m+2}}. \end{equation*} The form \begin{equation*}
\frac{d^{\times}t(K_{N,f} ^Z)h_{K_N}}{\sqrt{d_F}(-2)^\xi\left\|\frac{t_1}{t_2}\right\|_f\operatorname{sgn}(\operatorname{N}(\frac{t_1}{t_2}))}\operatorname{vol}_P \end{equation*} is part of an $A$-basis of the cohomology of $\partial\mathcal M_{K_N}$ as seen in \Cref{Theta_vol_P}. We already have mentioned in \Cref{boundary} that \begin{equation*}
\left\|t_2\right\|_f^{m}\operatorname{sgn}(\operatorname{N}(t_2))^{m}\prod_{\sigma}(-Y_\sigma)^{[m]}=\frac{\left\|t_2\right\|_f^{m}\operatorname{sgn}(\operatorname{N}(t_2))^{m}\operatorname{N}(\omega)^m}{(m!)^\xi}\in H^0(\partial\mathcal M_{K_N},\underline\omega^k_{PD}) \end{equation*} and \begin{equation*}
\left\|\det(g)\right\|_f\operatorname{sgn}(\operatorname{N}(\det(g)))\operatorname{vol}_V^*\in H^0(\partial\mathcal S_{K_N},\mu) \end{equation*} define $A$-trivializations and therefore \begin{equation*}
\frac{d^{\times}t(K_{N,f} ^Z)h_{K_N}\left\|t_2\right\|_f^{m}\operatorname{sgn}(\operatorname{N}(t_2))^{m}\left\|\det(g)\right\|_f\operatorname{sgn}(\operatorname{N}(\det(g)))\operatorname{N}(\omega)^m\operatorname{vol}_P\otimes \operatorname{vol}_V^* }{\sqrt{d_F}(m!)^\xi(-2)^\xi\left\|\frac{t_1}{t_2}\right\|_f\operatorname{sgn}(\operatorname{N}(\frac{t_1}{t_2}))} \end{equation*} is part of an $A$-basis of \begin{equation*} \operatorname{im}(H^{2\xi-1}(\partial\mathcal M_{K_N}, \underline{\omega}^k _{PD}\otimes \mu)\rightarrow H^{2\xi-1}(\partial\mathcal M_{K_N}, \underline{\omega}^k _{PD}\otimes \mu\otimes {\mathbb C})). \end{equation*} Now we may conclude that \begin{equation*}
\frac{(-1)^\xi\sqrt{d_F}\Gamma(m+2)^\xi }{(-2\pi i)^{\xi(m+2)}\operatorname{sgn}(\operatorname{N}(t_2))^{m+2}\left\|t_2\right\|_f^{m+2}}\sum_{l\in F^{\times}/\mathcal O^{\times}(N)^+}\frac{\hat{f}(\hat{g}_fle^1)}{ \operatorname{N}(l)^{m+2}}\in A, \end{equation*} as $\operatorname{res}_{\mathcal M}(\operatorname{Eis}^k(f))$ is $A$-integral. \end{proof} \end{proposition} \begin{definition} Let us define the \textit{horospherical map} \begin{equation*}
\rho_{m,n}^0:\mathcal S(V({\mathbb A}),\mu^{\otimes n}\otimes \overline{\mathbb Q})^0\rightarrow \operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}\bigoplus_{\phi:\ type(\phi)=\gamma_{m,n+1},\tilde{\phi}_{f|Z({\mathbb R})=1}}\overline{\mathbb Q} \cdot\phi, \end{equation*} by \begin{equation*}
\rho_{m,n}^0(\varphi)(g):= \sum_{\chi\in \widehat{Cl_F ^{K}}(m)}\frac{\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}\frac{\int_{t\in Z({\mathbb A}_f)}\hat{f} (tg_fe^1,g) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t}{\left(\left\|\det(g)\right\|_f\operatorname{sgn}(\operatorname{N}(\det(g)))^{-1}\right)^{-n}}, \end{equation*} where we have written \begin{equation*}
\varphi(v,g)=f(v,g)\left(\left\|\det(g)\right\|_f\operatorname{sgn}(\operatorname{N}(\det(g)))^{-1}\right)^{n} \end{equation*} with $f\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes 0}\otimes \overline{\mathbb Q})^0$ and $K_f$-invariant. \end{definition} \begin{remark} Let us quickly recall why $\rho_{m,n}^0$ is well-defined. Take $K^\prime_f \subset K_f$, $\chi\in Cl_F ^{K^\prime}(m)$ and $\epsilon \in K_f$ with $\chi(\epsilon)\neq 1$. Then \begin{equation*}
\int_{t\in Z({\mathbb A}_f)}\hat{f} (tg_fe^1,g) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*}
\int_{t\in Z({\mathbb A}_f)}\hat{f} (\epsilon tg_fe^1,g) \chi(\det(g_f)\epsilon t)\left\|\det(g_f)\epsilon t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*}
\chi(\epsilon)\int_{t\in Z({\mathbb A}_f)}\hat{f} ( tg_fe^1,g) \chi(\det(g_f) t)\left\|\det(g_f) t\right\|^{m+2} d^{\times}t, \end{equation*} so we conclude for such $\chi$ \begin{equation*}
0=\int_{t\in Z({\mathbb A}_f)}\hat{f} ( tg_fe^1,g) \chi(\det(g_f) t)\left\|\det(g_f) t\right\|^{m+2} d^{\times}t \end{equation*} and the definition of $\rho_{m,n}^0$ is independent of the chosen $K_f$. To see that $\rho_{m,n}^0$ takes its values in $\overline{{\mathbb Q}}$-valued functions we can actually imitate the proof of \cref{boundary_residueII}. Given a $\varphi\in\mathcal S(V({\mathbb A}),\mu^{\otimes n}\otimes \overline{\mathbb Q})^0$ we get that \begin{equation*} \operatorname{res}_{\mathcal S}(\operatorname{Eis}^k(\varphi))\in \bigoplus_{p+q=2\xi-1}H^{p}(\partial\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1})\otimes \mathfrak H^q\otimes \overline{\mathbb Q} \end{equation*} \begin{equation*}
\cong\bigoplus_{\phi:\ type(\phi)=\gamma_{m,n},\tilde{\phi}_{f|Z({\mathbb R})=1}}\operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}\overline{\mathbb Q} \cdot\tilde\phi_f\otimes \mathcal H^p(T/Z) \end{equation*} equals \begin{equation*}
\sum_{\chi\in \widehat{Cl_F ^K}(m)}\frac{\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb A}_f)}\hat{f} (tg_fe^1,g) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t\cdot \end{equation*} \begin{equation*}
\frac{\operatorname{N}(\omega)^m \operatorname{vol}_P\otimes \operatorname{vol}_V^{*n+1}}{(m!)^\xi\left(\left\|\det(g)\right\|_f\operatorname{sgn}(\operatorname{N}(\det(g)))^{-1}\right)^{-n}} \end{equation*} and therefore \begin{equation*}
\sum_{\chi\in \widehat{Cl_F ^K}(m)}\frac{\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb A}_f)}\hat{f} (tg_fe^1,g) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t \end{equation*} is an algebraic number, since the cohomology class \begin{equation*}
\frac{\operatorname{N}(\omega)^m \operatorname{vol}_P\otimes \operatorname{vol}_V^{*n+1}}{(m!)^\xi\left(\left\|\det(g)\right\|_f\operatorname{sgn}(\operatorname{N}(\det(g)))^{-1}\right)^{-n}} \end{equation*} is rational. We may even conclude that $\rho_{m,n}^0$ is $\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})$-equivariant. \end{remark} \begin{proposition}\label{hor} We have \begin{equation*} \operatorname{im}(\operatorname{res}_\mathcal S\circ \operatorname{Eis}^k _q)\cong \operatorname{im}(\rho_{m,n}^0)\otimes\mathcal H(T/Z)^{\xi-1-q}. \end{equation*} \begin{proof} We prove the proposition with ${\mathbb C}$-coefficients and all of the following constructions will respect the chosen ${\mathbb Q}$-structures. By \Cref{boundary_residueII} \begin{equation*} \operatorname{res}_\mathcal S\circ \operatorname{Eis}^k(\varphi)=\rho_{m,n}^0(\varphi)\cdot\frac{\operatorname{N}(\omega)^m}{(m!)^\xi} \operatorname{vol}_P\otimes \operatorname{vol}_V^{*n+1}\in H^\bullet(\partial\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1})\otimes \mathfrak H^\bullet _{\mathbb C}. \end{equation*} Set \begin{equation*} \operatorname{vol}_{T_2}=\frac{(-1)^{\frac{\xi(\xi-1)}{2}}(-2)^{\xi}\sqrt{d_F}}{\xi\kappa_F}\sum_{i=1} ^\xi (-1)^{i-1}\frac{dt_{2}}{t_{2}}_{\left\langle \xi\right\rangle\setminus i},\ \operatorname{vol}_{U}=\frac{dx_1\wedge...\wedge dx_n}{\sqrt{d_F}} \end{equation*} and get $\operatorname{vol}_P=\operatorname{vol}_{T_2}\wedge \operatorname{vol}_{U}$. Dividing $\operatorname{res}_\mathcal S\circ \operatorname{Eis}^k(\varphi)$ by $\frac{\operatorname{N}(\omega)^m}{(m!)^\xi}\operatorname{vol}_U\otimes\operatorname{vol}_V^{*n+1}$ we get \begin{equation*} \rho_{m,n}^0(\varphi)\cdot \operatorname{vol}_{T_2}\in
\operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}\bigoplus_{\phi:\ type(\phi)=\gamma_{m,n+1},\tilde{\phi}_{f|Z({\mathbb R})=1}}{\mathbb C} \cdot\phi \cdot \mathcal H^\bullet (T/Z)\otimes \mathfrak H^\bullet_{\mathbb C} \end{equation*} So what remains to be shown is that $\mathfrak H^{\bullet,*}_{\mathbb C}\rightarrow \mathcal H^\bullet (T/Z)_{\mathbb C}$, $v \mapsto \operatorname{vol}_{T_2}(v)$ is surjective. Clearly, $\mathcal H^\bullet (T/Z)\otimes \mathfrak H^{\bullet}_{\mathbb C}=H^\bullet(T({\mathbb R})^0/T({\mathbb Q})\cap (K_f\cdot T({\mathbb R})^0,{\mathbb C})$ and we may interpret $\operatorname{vol}_{T_2}$ as rational volume form in $t_2$-direction. Therefore \begin{equation*} \operatorname{vol}_{T_2}=c^\prime\cdot \frac{dt_{2}}{t_2}_{\left\langle \xi-1\right\rangle} \in H^\bullet(T({\mathbb R})^0/T({\mathbb Q})\cap (K_f\cdot T({\mathbb R})^0,{\mathbb C}) \end{equation*} for a $c^\prime\in {\mathbb C}^{\times}$. We take $\frac{dr}{r}_i$, $i=1,...,\xi-1$, as basis for $\mathfrak H_{\mathbb C} ^1$ and $\frac{dy}{y}_i$, $i=1,...,\xi-1$, as basis for $\mathcal H^1 (T/Z)_{\mathbb C}$. We have $\frac{dt_{2,i}}{t_{2,i}}=\frac{1}{2}\left(\frac{dy}{y}_i+\frac{dr}{r}_i\right)$, as $t_{2,i}=\sqrt{y_ir_i}$, and therefore \begin{equation*}
\operatorname{vol}_{T_2}=c^\prime\cdot \frac{1}{2}\left(\frac{dy}{y}_1+\frac{dr}{r}_1\right)\wedge...\wedge \frac{1}{2}\left(\frac{dy}{y}_{\xi-1}+\frac{dr}{r}_{\xi-1}\right)= \end{equation*} \begin{equation*}
\frac{c^\prime}{2^{\xi-1}}\cdot\sum_{I\subset \left\langle \xi-1\right\rangle}\operatorname{sgn}(I)\frac{dy}{y}_I\wedge\frac{dr}{r}_{\left\langle \xi-1\right\rangle\setminus I}. \end{equation*} If we take the dual basis $V(r)_i\in \mathfrak H_{\mathbb C} ^{1\ *}$ of $\frac{dr}{r}_i$, $i=1,...,\xi-1$, we may see \begin{equation*}
\operatorname{vol}_{T_2}(V(r)_J)= \frac{c^\prime}{2^{\xi-1}}\cdot \operatorname{sgn}(\left\langle \xi-1\right\rangle\setminus J)\frac{dy}{y}_{\left\langle \xi-1\right\rangle\setminus J},\ J\subset \left\langle \xi-1\right\rangle. \end{equation*} In particular, $\mathfrak H^{\bullet\ *}_{\mathbb C}\rightarrow \mathcal H^\bullet (T/Z)_{\mathbb C}$, $v \mapsto \operatorname{vol}_{T_2}(v)$, is surjective. \end{proof} \end{proposition} \begin{remark}\label{im_res_Eis} The last proposition shows that for any $\varphi\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes n}\otimes {\mathbb C})^0$ \begin{equation*} \rho_{m,n}^0(\varphi)\frac{\operatorname{N}(\omega)^m}{(m!)^\xi}\otimes \operatorname{vol}_V^{*n+1}\otimes\frac{d\tau\wedge d\overline \tau}{\overline\tau-\tau}_{I^c}\wedge d\overline\tau_{I},\ \emptyset\neq I\subset\left\langle \xi\right\rangle, \end{equation*} is in the image of $\sum_q \operatorname{res}_\mathcal S\circ \operatorname{Eis}^k_{q}$ with ${\mathbb C}$-coefficients. \end{remark}
\section{Comparison with Harder's Eisenstein classes}
The reason why we talk about polylogarithmic Eisenstein classes is that these classes are actually Eisenstein cohomology classes in the sense of Harder. This is what we prove now.\\ \begin{theorem}\label{Harder} Define $\tilde{H}^\bullet(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1}):=$ \begin{equation*} \ker\left(\operatorname{res}_{\mathcal S}:H^\bullet(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1})\rightarrow H^\bullet(\partial\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1})\right). \end{equation*} One has a $G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$-equivariant operator \begin{equation*} \operatorname{Eis}_{Harder}:\operatorname{im}(\operatorname{res}_\mathcal S)\rightarrow H^\bullet(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1}), \end{equation*} which is a section for $\operatorname{res}_{\mathcal S}$. We denote the image of the operator $\operatorname{Eis}_{Harder}$ by \begin{equation*} H^\bullet_{\operatorname{Eis}}(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1}) \end{equation*} and get a decomposition of $G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$-modules \begin{equation*} H^\bullet(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1})=\tilde{H}^\bullet(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1})\oplus H^\bullet_{\operatorname{Eis}}(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1}) \end{equation*} \begin{proof} \cite{Ha1} Theorem 2. \end{proof} \end{theorem} The aim of this section is to prove the following \begin{theorem}\label{pol_Eis} \begin{equation*} \operatorname{im}(\operatorname{Eis}^k_q)\subset H^\bullet_{\operatorname{Eis}}(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1}) \end{equation*} \end{theorem} We generated $\mathfrak H^\bullet _{\mathbb C}\subset H^\bullet(\mathcal M_K,{\mathbb C})$ by the invariant differential forms $\frac{dr}{r}_i$, $i\in \left\langle \xi-1\right\rangle$. Let $V(r)_i$ be the invariant vector fields dual to $\frac{dr}{r}_i$ and set \begin{equation*} V(r)_I:=V(r)_{i_1}\wedge...\wedge V(r)_{i_q},\ I=\left\{i_1,...,i_{q}\right\}\subset \left\langle \xi-1\right\rangle. \end{equation*} \begin{lemma} To prove \Cref{pol_Eis} it suffices to show \begin{equation*} \operatorname{Eis}^k_q(\varphi)(V(r)_I)\in H^\bullet_{\operatorname{Eis}}(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1}\otimes {\mathbb C}), \end{equation*} for all $I\subset\left\langle \xi-1\right\rangle$ and \begin{equation*}
\varphi(v,g)=f(v)\cdot\eta(\det(g))\left(\left\|\det(g)\right\|_f\operatorname{sgn}(\operatorname{N}(\det(g)))\right)^{n}, \end{equation*}
for $f\in\mathcal S(V({\mathbb A}_f),{\mathbb C})^0$ and $\eta:\prod_{\nu|\infty}{\mathbb R}_{>0}\backslash\mathbb{I}_{F}/F^{\times}\rightarrow {\mathbb C}$ a continuous character. \begin{proof} To prove \Cref{pol_Eis} we may extend coefficients to ${\mathbb C}$. The $\varphi$ described above generate $\mathcal S(V({\mathbb A}_f),\mu^{\otimes n}\otimes{\mathbb C})^0$. As $(V(r)_I)_{I\subset\left\langle \xi-1\right\rangle}$ is a basis of $\mathfrak H_{\mathbb C} ^{\bullet *}$, the lemma follows. \end{proof} \end{lemma} From now on we suppose that $\varphi$ is of the form above. For $\emptyset \neq I\subset \left\langle \xi\right\rangle$ we define $\tilde{\Theta}_{I}$ as the form \begin{equation*}
\frac{(-2i)^{-\xi(m+2)}\operatorname{N}(e^{-(m+1)i\theta})e^{-i\theta}_{I}e^{i\theta}_{I^c}}{\operatorname{N}(t_{2}) ^{m+2}\left(\left\| \det g_f\right\|_f \operatorname{sgn}(\operatorname{N}(\det g_\infty))^{-1}\right)^{-n}}\operatorname{N}(\omega)^m\frac{d\tau\wedge d\overline\tau}{\overline\tau-\tau}_{I^c}\wedge d \overline\tau_{I}\otimes \operatorname{vol}_V^{*n+1} \end{equation*} and set $\widetilde{\operatorname{Eis}}^k _{I}(\varphi,\chi):=$ \begin{equation*} \lim{s\rightarrow 0}\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q}))}\int_{t\in Z({\mathbb A})}
\phi(\hat{f})^m _{I}(tg\gamma e^1,g)\eta(\det(g)) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2+2s} d^{\times}t\cdot \tilde{\Theta}_{I}. \end{equation*} \begin{lemma} To prove \Cref{pol_Eis} it suffices to show that the differential form $\sum_{\chi \in\widehat{Cl_F ^{K}}(m)}\widetilde{\operatorname{Eis}}^k _{I}(\varphi,\chi)$ defines a class in $H^\bullet_{\operatorname{Eis}}(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1}\otimes {\mathbb C})$ for all $\emptyset\neq I\subset\left\langle \xi\right\rangle$. \begin{proof} $\operatorname{Eis}^k_q(\varphi)(V(r)_I)$ are ${\mathbb C}$-linear combinations of $\sum_{\chi \in\widehat{Cl_F ^{K}}(m)}\widetilde{\operatorname{Eis}}^k _{I}(\varphi,\chi)$. \end{proof} \end{lemma} To prove this last step we have to rewrite our classes in $(\mathfrak g,K)$-cohomology, as Harder constructed his operator $\operatorname{Eis}_{Harder}$ in this setting. \begin{remark}\label{g_K} The local system $\operatorname{Sym}^k \mathcal H^\prime\otimes \mu ^{\otimes n+1}$ corresponds to the $G({\mathbb Q})$-representation $\operatorname{Sym}^k _{\mathbb Q} V({\mathbb Q})\otimes \operatorname{vol}_V^{*n+1}$ on which $G({\mathbb Q})$ acts by \begin{equation*} \rho(g)v_1 ...v_k \operatorname{vol}_V^{*n+1}:= \operatorname{N}(\det(g))^{n+1}gv_1... gv_k \operatorname{vol}_V ^{*n+1}, \end{equation*} for $v_1,...,v_n\in V({\mathbb Q})$, $g\in G({\mathbb Q})$ and $gv_i$ the standard action by matrix multiplication. Set \begin{equation*} \mathfrak g:=Lie(G({\mathbb R}))=T_1 G({\mathbb R}),\ \mathfrak k:=Lie(K_\infty)=T_1K_{\infty}. \end{equation*}
Of course, we have $\mathfrak g=\bigoplus_{\nu|\infty}Lie(G_0(F_\nu))$ and $\mathfrak k=\bigoplus_{\nu|\infty}Lie({\mathbb R}_{>0}\cdot SO(2))$. Moreover, $\mathfrak g$ and $\mathfrak k$ are $F_{\mathbb R}$-modules. We get the isomorphism $\Psi$ \begin{equation*} \Gamma(\mathcal S_K,\Omega_{\mathcal S_K} \otimes_{\mathbb Q} \operatorname{Sym}^n\mathcal H^\prime\otimes\mu^{\otimes n+1})= \end{equation*} \begin{equation*} H^0(G({\mathbb Q}),\Gamma(K\backslash G({\mathbb A}),\Omega_{K\backslash G({\mathbb A})} \otimes_{\mathbb Q} \operatorname{Sym}^nV({\mathbb Q})\otimes \operatorname{vol}_V^{*n+1}))\rightarrow \end{equation*} \begin{equation*} Hom_{K_\infty}(\bigwedge ^\bullet\mathfrak g/\mathfrak k,\mathcal C^\infty(K_f\backslash G({\mathbb A})/G({\mathbb Q}))\otimes \operatorname{Sym}^n V({\mathbb Q})\otimes \operatorname{vol}_V^{*n+1}) \end{equation*} as described in \cite{B-W} chapter VII, proposition 2.5. Given a differential form $\eta$ on the left hand side and $X\in \bigwedge ^\bullet\mathfrak g/\mathfrak k$ this means explicitly \begin{equation*} \Psi(\eta)(X)(g):=\rho(g_\infty)\eta(p(g))(dp_{g}dr_{g,1} X)\text{ where} \end{equation*} \begin{equation*} r_{g}:K_f\backslash G({\mathbb A})\rightarrow K_f\backslash G({\mathbb A}),\ x\mapsto xg,\ \text{and } p:K_f \backslash G({\mathbb A})\rightarrow K\backslash G({\mathbb A}) \end{equation*} is the canonical map. Unlike \cite{B-W} we consider right translation instead of left translation due to the fact that $G({\mathbb Q})$ acts from the right. We can make everything explicit by giving a basis for $\mathfrak g/\mathfrak k\otimes {\mathbb C}$. We take \begin{equation*} P_{\pm}:=\frac{1}{2}\begin{pmatrix}1& \pm i\\ \pm i & -1\end{pmatrix}=\frac{1}{2}\begin{pmatrix}1 & 0\\ 0& -1 \end{pmatrix} \pm\frac{1}{2}\begin{pmatrix}0 & 1\\ 1& 0\end{pmatrix}\otimes i \end{equation*} as a basis for $Lie(GL_2({\mathbb R}))/Lie({\mathbb R}_{>0}SO(2))_{\mathbb C}$, compare \cite{Ha4} p. 130. We have for the adjoint representation \begin{equation*} Ad(k)P_{\pm}=kP_{\pm} k^{-1}=(a\mp ic)^2P_{\pm}, \ k=\begin{pmatrix}a& -c\\c&a\end{pmatrix}\in SO(2). \end{equation*} We have $\mathbb{H}_{\pm}:={\mathbb C}\setminus {\mathbb R}$ and the isomorphism \begin{equation*} {\mathbb R}^\times SO(2)\backslash GL_2({\mathbb R})\rightarrow \mathbb{H}_{\pm},\ g\mapsto z=ig=\frac{b+id}{a+ic}=\frac{w_2}{w_1}, \end{equation*} \begin{equation*} \text{ for } g=\begin{pmatrix}a&b\\c&d\end{pmatrix},\ w_1:=a+ic,\ w_2:=b+id, \end{equation*} and we get \begin{equation*}
dr_{g,1}P_+=(w_1\frac{\partial}{\partial\overline w_1}+w_2\frac{\partial}{\partial \overline w_2})_{|(w_1,w_2)=(1,i)g},\
dr_{g,1}P_-=(\overline w_1\frac{\partial}{\partial w_1}+\overline w_2\frac{\partial}{\partial w_2})_{|(w_1,w_2)=(1,i)g}, \end{equation*} \begin{equation*} d\overline z(p(g))(dr_{g,1}P_-)=0,\ d\overline z(p(g))(dr_{g,1}P_+)=2i\frac{\det(g)}{(a-ic)^2}=2i\frac{t_2}{t_1}e^{i2\theta}\text{ and} \end{equation*} \begin{equation*} dz(p(g))(dr_{g,1}P_+)=0,\ dz(p(g))(dr_{g,1}P_-)=-2i\frac{\det(g)}{(a+ic)^2}=-2i\frac{t_2}{t_1}e^{-i2\theta} \end{equation*} where we used the Iwasawa decomposition of $g\in GL_2({\mathbb R})$, see \Cref{Iwasawa}. Finally, consider $\omega_0:=(e^1\overline z-e^2)\in \mathcal C^\infty(\mathbb{H}_{\pm},{\mathbb C}^2)$ and rewrite it using the $GL_2({\mathbb R})$-action on ${\mathbb C}^2$ as \begin{equation*} \omega_0=g^{-1}(\frac{-i\det(g)}{a-ic}(e^1-ie^2))=g^{-1}(-it_2e^{i\theta}(e^1-ie^2)). \end{equation*} \end{remark} Remember that we had \begin{equation*}
\varphi(v,g)=f(v)\cdot\eta(\det(g))\left(\left\|\det(g)\right\|_f\operatorname{sgn}(\operatorname{N}(\det(g)))\right)^{n}, \end{equation*}
for $f\in\mathcal S(V({\mathbb A}_f))^0$ and $\eta:\prod_{\nu|\infty}{\mathbb R}_{>0}\backslash\mathbb{I}_{F}/F^{\times}\rightarrow {\mathbb C}^{\times}$ a continuous character. For $\chi\in \widehat{Cl_F ^K}(m)$ we consider the algebraic Hecke character \begin{equation*}
\phi:T({\mathbb A})/T({\mathbb Q})\rightarrow {\mathbb C}^{\times},\ (t_1,t_1)\mapsto \eta(t_1t_2)\chi(t_2)\left\|t_2\right\|^{m+n+2}\left\|t_1\right\|^n. \end{equation*}
$\phi$ is of type $\gamma_{m,n+1}$ and $\tilde\phi_{f|Z({\mathbb R})}=1$, therefore $\phi$ is a character contributing to the cohomology of the boundary. Write \begin{equation*} g=kb,\ k\in SL_2(\hat\mathcal O)\cdot K^1,\ b=\begin{pmatrix}t_1& t_1 x\\ 0 & t_2\end{pmatrix}\in B({\mathbb A}). \end{equation*} Given $\eta$, $\chi$ and $f$ as above we have well-defined functions $F_{I,\infty} ^m(\chi,s,g):=$ \begin{equation*}
|\operatorname{N}(\frac{t_2}{t_1})|^s\eta_\infty(\det(g)) \operatorname{N}(-it_2e^{i\theta})^m \operatorname{N}(2i\frac{t_2}{t_1})(e^{i2\theta})_{I}\operatorname{N}(\det(g))^{n+1}\operatorname{sgn}(\operatorname{N}(\det(g)))^{n} \end{equation*} on $G({\mathbb R})$, $F_{I,f} ^m(\chi,s,g_f):=$ \begin{equation*}
\frac{\Gamma(m+2)^\xi\left\|\frac{t_2}{t_1}\right\|_f ^s\eta_f(\det(g))\int_{t \in Z({\mathbb A}_f)}\hat{f}(t g_fe_1)\chi_f(\det(g_f)t)\left\|\det(g_f)t\right\|_f ^{2+m}d^{\times}t}{(-2\pi i)^{\xi(m+2)}\left(\left\|\det(g_f)\right\|_f \right)^{-n}} \end{equation*} on $G({\mathbb A}_f)$ and $F_{I} ^m(\chi,s,\ ):=F_{I,f} ^m(\chi,s,\ )\otimes F_{I,\infty} ^m(\chi,s,\ )$ on $G({\mathbb A})$. In other words, \begin{equation*} F_{I} ^m(\chi,s,g):=F_{I,f} ^m(\chi,s,g_f)\cdot F_{I,\infty} ^m(\chi,s,g_\infty). \end{equation*}
Now we need the Harish-Chandra modules $V^* _{\phi\left\|\frac{t_2}{t_1}\right\|^s}$ as defined in \cite{Ha1} 3.3 and p. 79. \begin{lemma} We define \begin{equation*}
\omega_{I} ^m(\varphi,\chi,s)\in Hom_{K_\infty}(\bigwedge^\bullet\mathfrak g /\mathfrak k,V^* _{\phi\left\|\frac{t_2}{t_1}\right\|^s}\otimes \operatorname{Sym}^k_{\mathbb C} V({\mathbb C})\otimes \operatorname{vol}_V^{*n+1}) \end{equation*} by \begin{equation*} \left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I}\mapsto F_{I} ^m(\chi,s,\ )\operatorname{N}(e_1-e_2 i)^m \operatorname{vol}_V^{*n+1} \end{equation*} and all elements not collinear with $\left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I}$ mapping to zero. \begin{proof} We have to show that $\omega_{I} ^m(\varphi,\chi,s)$ actually defines a form as claimed. For $b=\begin{pmatrix}t_1& t_1 x\\ 0 & t_2\end{pmatrix}\in B({\mathbb A})$ and $g\in G({\mathbb A})$ we have \begin{equation*}
F_{I} ^m(\chi,s,gb)=\left\|\frac{t_2}{t_1}\right\|^s \phi(t_1,t_2)F_{I} ^m(\chi,s,g). \end{equation*}
$K_f$ and $K_\infty$-finiteness of $F_{I} ^m(\chi,s, )$ are clear and therefore $F_{I} ^m(\chi,s,\ )\in V^* _{\phi\left\|\frac{t_2}{t_1}\right\|^s}$. So what remains to be shown is that \begin{equation*} \left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I}\mapsto F_{I} ^m(\chi,s,\ )\operatorname{N}(e_1-e_2 i)^m \operatorname{vol}_V^{*n+1} \end{equation*} is $K_\infty$-equivariant. Consider \begin{equation*} k=t\begin{pmatrix}x& -y\\ y& x\end{pmatrix}\in K_\infty\text{ with }t\in Z({\mathbb R})^0\text{ and }\begin{pmatrix}x& -y\\ y& x\end{pmatrix}_\nu=\begin{pmatrix}\cos\theta_\nu & -\sin \theta_\nu \\ \sin\theta_\nu & \cos\theta_\nu\end{pmatrix}. \end{equation*} We have \begin{equation*} Ad(k^{-1})\left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I}=Ad\begin{pmatrix}x& -y\\ y& x\end{pmatrix}^{-1}\left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I}= \end{equation*} \begin{equation*} e^{2i\theta}_{I}\left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I} \end{equation*} and on the other hand \begin{equation*} k^{-1}\omega_{I} ^m(\varphi,\chi,s)(\left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I})(kg)=k^{-1}F_{I} ^m(\chi,s,kg )\operatorname{N}(e_1-e_2 i)^m \operatorname{vol}_V^{* n+1}= \end{equation*} \begin{equation*} \operatorname{N}(t^{-1})^{2(n+1)}\operatorname{N}(t^{-1}e^{-i\theta})^m F_{I} ^m(\chi,s,kg )\operatorname{N}(e_1-e_2 i)^m\operatorname{vol}_V^{* n+1}= \end{equation*} \begin{equation*} \operatorname{N}(t^{-1})^{2(n+1)}\operatorname{N}(t^{-1}e^{-i\theta})^mF_{I,f} ^m(\chi,s,g_f )F_{I,\infty} ^m(s,kg_\infty ) \operatorname{N}(e_1-e_2 i)^m\operatorname{vol}_V^{* n+1}= \end{equation*} \begin{equation*} e^{2i\theta} _{I}F_{I} ^m(\chi,s,g )\operatorname{N}(e_1-e_2 i)^m\operatorname{vol}_V^{* n+1} \end{equation*} by the definition of $F_{I,\infty} ^m(\chi,s,kg )$ and it follows that $\omega_{I} ^m(\varphi,\chi,s)$ is (right)-$K_\infty$-equivariant. \end{proof} \end{lemma} Harder defines \begin{equation*} \operatorname{Eis}^*(\omega_{I} ^m(\varphi,\chi,s))\in Hom_{K_\infty}(\bigwedge^\bullet\mathfrak g /\mathfrak k,\mathcal C^\infty(K_f\backslash G({\mathbb A})/G({\mathbb Q}))\otimes \operatorname{Sym}^k_{\mathbb C} V({\mathbb C})\otimes \operatorname{vol}_V^{*n+1}) \end{equation*} by \begin{equation*} \left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I}\mapsto\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q})} F_{I} ^m(\chi,s,g\gamma)\operatorname{N}(e_1-e_2 i)^m \operatorname{vol}_V^{*n+1} \end{equation*} and all elements not collinear with $\left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I}$ mapping to zero, see \cite{Ha1} (4.2.2). \begin{proposition}\label{comparison} \begin{equation*} \sum_{\chi \in\widehat{Cl_F ^{K}}(m)}\lim{s\rightarrow 0}\operatorname{Eis}^*(\omega_{I} ^m(\varphi,\chi,s))=\operatorname{Eis}_{Harder}(\rho_{m,n}^0(\varphi)\operatorname{N}(\omega)^m\frac{d\tau\wedge d\overline \tau}{\overline\tau-\tau}_{I^c}\wedge d\overline\tau_{I}\otimes \operatorname{vol}_V^{*n+1}), \end{equation*} \begin{equation*} \lim{s\rightarrow 0}\operatorname{Eis}^*(\omega_{I} ^m(\varphi,\chi,s))=\Psi(\widetilde{\operatorname{Eis}}^k_{I}(\varphi,\chi)) \end{equation*} and \Cref{pol_Eis} is proved. \begin{proof} We have \begin{equation*} \rho_{m,n}^0(\varphi)\operatorname{N}(\omega)^m\frac{d\tau\wedge d\overline \tau}{\overline\tau-\tau}_{I^c}\wedge d\overline\tau_{I}\otimes \operatorname{vol}_V^{*n+1}= \end{equation*} \begin{equation*} \sum_{\chi \in\widehat{Cl_F ^{K}}(m)}F_{I,f} ^m(\chi,0,g)\eta_\infty(\det(g))\operatorname{sgn}(\operatorname{N}(\det(g)))^n \operatorname{N}(\omega)^m\frac{d\tau\wedge d\overline\tau}{\overline\tau-\tau}_{I^c}\wedge d \overline\tau_I\otimes \operatorname{vol}_V^{*n+1}. \end{equation*} Following \Cref{g_K} and \cite{Ha1} p.79 and p.80 we see that \begin{equation*} \sum_{\chi \in\widehat{Cl_F ^{K}}(m)}\lim{s\rightarrow 0}\operatorname{Eis}^*(\omega_{I} ^m(\varphi,\chi,s))=\operatorname{Eis}_{Harder}(\rho_{m,n}^0(\varphi)\operatorname{N}(\omega)^m\frac{d\tau\wedge d\overline\tau}{\overline\tau-\tau}_{I^c}\wedge d\overline\tau_{I}\otimes \operatorname{vol}_V^{*n+1}) \end{equation*} by the very definition of Harder. The right hand side of the equation is defined, as we know by \Cref{im_res_Eis} that \begin{equation*} \rho_{m,n}^0(\varphi)\operatorname{N}(\omega)^m\frac{d\tau\wedge d\overline \tau}{\overline\tau-\tau}_{I^c}\wedge d\overline\tau_{I}\otimes \operatorname{vol}_V^{*n+1}\in \operatorname{im}(\operatorname{res}_\mathcal S). \end{equation*} So what remains to be shown is $\lim{s\rightarrow 0}\operatorname{Eis}^*(\omega_{I} ^m(\varphi,\chi,s))=\Psi(\widetilde{\operatorname{Eis}}^k_{I}(\varphi,\chi))$. We start with the calculation of $\Psi(\widetilde{\operatorname{Eis}}^k_{I}(\varphi,\chi))$. Using \Cref{g_K} we see that it is determined by \begin{equation*} \lim{s\rightarrow 0}\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q}))}\int_{t\in Z({\mathbb A})}
\phi(\hat{f})^m _{I}(tg\gamma e^1)\eta(\det(g)) \chi(\det(g)t)\left\|\det(g)t\right\|^{m+2+2s} d^{\times}t= \end{equation*} \begin{equation*} \lim{s\rightarrow 0}\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q}))}\int_{t\in Z({\mathbb A})}
\phi(\hat{f})^m _{I}(tg\gamma e^1)\eta(\det(g\gamma)) \chi(\det(g\gamma)t)\left\|\det(g\gamma)t\right\|^{m+2+2s} d^{\times}t= \end{equation*} \begin{equation*} \lim{s\rightarrow 0}\sum_{\gamma\in G({\mathbb Q})/B({\mathbb Q}))}\int_{t\in Z({\mathbb A}_f)}
\hat{f}(tg_f\gamma e^1)\eta_f(\det(g_f\gamma)) \chi_f(\det(g_f\gamma)t)\left\|\det(g_f\gamma)t\right\|_f^{m+2+2s} d^{\times}t\cdot \end{equation*} \begin{equation*} \int_{t\in Z({\mathbb R})}
\phi(\hat{f})^m_{I,\infty}(tg_\infty\gamma e^1)\eta_\infty (\det(g_\infty\gamma)) \chi_\infty(\det(g_\infty\gamma)t)|\operatorname{N}(\det(g_\infty\gamma)t)|^{m+2+2s} d^{\times}t \end{equation*} Let us set \begin{equation*}
F_\infty(s,g):=\int_{t\in Z({\mathbb R})}\phi(\hat{f})^m_{I,\infty}(tg_\infty e^1)\eta_\infty (\det(g_\infty)) \chi_\infty(\det(g_\infty)t)|\operatorname{N}(\det(g_\infty)t)|^{m+2+2s} d^{\times}t, \end{equation*} \begin{equation*}
F_f(s,g):=\int_{t\in Z({\mathbb A}_f)}\hat{f}(tg_f e^1)\eta_f(\det(g_f)) \chi_f(\det(g_f)t)\left\|\det(g_f)t\right\|_f^{m+2+2s} d^{\times}t \end{equation*} for the moment. Consider \begin{equation*} G(g_f):=(g_f^{-1}K_fg_f\cdot G({\mathbb R}) ^0)\cap G({\mathbb Q}),\ B(g_f)=:G(g_f)\cap B({\mathbb A}). \end{equation*} $G(g_f)$ acts on $G({\mathbb Q})/B({\mathbb Q})$ by left translation and we decompose the latter into $G(g_f)$ orbits. Now we may write our sum above as \begin{equation*} \lim{s\rightarrow 0}\sum_{\alpha\in G(g_f)\backslash G({\mathbb Q})/B({\mathbb Q})}\sum_{\gamma\in G(g_f\alpha)/B(g_f\alpha)}F_f(s,g\alpha\gamma)F_\infty(s,g\alpha\gamma) \end{equation*} $G(g_f)\backslash G({\mathbb Q})/B({\mathbb Q})$ is finite, as $G({\mathbb Z})\backslash G({\mathbb Q})/B({\mathbb Q})\cong Cl_F$ is the class group of $F$ and the groups $G(g_f)$ and $G({\mathbb Z})$ are commensurable. Moreover, $G(g_f)$ does not affect $F_f(s,\ )$: By construction of $\operatorname{Eis}^k$ we have chosen the level $K_f$ for $f$ and $\eta$ such that $f(kv)=f(v)$ and $\eta(\det(k))=1$ for all $k\in K_f$ and $v\in V({\mathbb A}_f)$, see \Cref{Eis^k}. Therefore we may write our sum as \begin{equation*} \sum_{\alpha\in G(g_f)\backslash G({\mathbb Q})/B({\mathbb Q})}F_f(0,g\alpha) \lim{s\rightarrow 0} \sum_{\gamma\in G(g_f \alpha)/B(g_f \alpha)}F_\infty (s,g\alpha\gamma) \end{equation*} Let us take care of $F_\infty$. We write \begin{equation*} g_\infty=kb=k t_1\begin{pmatrix}1& x \\ 0& y\end{pmatrix}\text{ and }\alpha\gamma =\begin{pmatrix} a& b\\ c& d \end{pmatrix}. \end{equation*} We get \begin{equation*}
F_\infty(s,g\alpha\gamma)=\operatorname{N}(e^{i\theta})^{m+1}e^{i\theta}_{ I}e^{-i\theta}_{I^c}\eta_\infty (\det(g_\infty \alpha\gamma)) \operatorname{N}(t_2)^{m+2} |\operatorname{N}(t_2)|^{2s}\cdot \end{equation*} \begin{equation*}
\frac{\operatorname{N}(\det(\alpha\gamma))^{m+2}|\operatorname{N}(\det(\alpha\gamma))|^{2s}\operatorname{N}(a+c\tau)^{m+1}(a+c\tau)_{I}(a+c\overline\tau)_{I^c}\Gamma(m+2+s)^\xi}{\pi^{\xi(m+2)+{\xi s}}|\operatorname{N}(a+c\tau)|^{2(m+2+s)}}. \end{equation*}
Using $|\operatorname{N}(\det(\gamma))|=1$ we calculate \begin{equation*} \lim{s\rightarrow 0} \sum_{\gamma\in G(g_f \alpha)/B(g_f \alpha)}F_\infty(s,g\alpha\gamma)= \frac{\Gamma(m+2)^\xi}{\pi^{\xi(m+2)}}\operatorname{N}(e^{i\theta})^{m+1}e^{i\theta}_{I}e^{-i\theta}_{I^c}\operatorname{N}(t_2)^{m+2}\cdot \end{equation*} \begin{equation*}
\lim{s\rightarrow 0} \sum_{\gamma\in G(g_f \alpha)/B(g_f \alpha)}\frac{\eta_\infty (\det(g \alpha\gamma))\operatorname{N}(\det(\alpha\gamma ))^{m+2} \operatorname{N}(a+c\tau)^{m+1}(a+c\tau)_{I}(a+c\overline\tau)_{I^c}}{|\operatorname{N}(a+c\tau)|^{2(m+2+s)}}= \end{equation*} \begin{equation*}
\lim{s\rightarrow 0} \sum_{\gamma\in G(g_f \alpha)/B(g_f \alpha)}\frac{\eta_\infty (\det(\alpha\gamma))|\operatorname{N}(\det(\alpha\gamma))|^{s}\operatorname{N}(\det(\alpha\gamma ))^{m+2} \operatorname{N}(a+c\tau)^{m+1}(a+c\tau)_{I}(a+c\overline\tau)_{I^c}}{|\operatorname{N}(a+c\tau)|^{2(m+2+s)}}\cdot \end{equation*} \begin{equation*}
\frac{\Gamma(m+2)^\xi}{\pi^{\xi(m+2)}}\eta_\infty(\det(g))\operatorname{N}(e^{i\theta})^{m+1}e^{i\theta}_{I}e^{-i\theta}_{I^c}\operatorname{N}(t_2)^{m+2}|\operatorname{N}(\frac{t_2}{t_1})|^s \end{equation*} Looking carefully at the formulas in \Cref{g_K} and recalling \begin{equation*}
\left\|\det(g_f\alpha)\right\|^{-n}_f|\operatorname{N}(\det(\alpha\gamma))|^{-n}=\left\|\det(g_f)\right\|_f^{-n} \end{equation*} we may conclude that $\Psi(\widetilde{\operatorname{Eis}}_{I}^k(\varphi,\chi))$ is the form that maps $\left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I}$ to \begin{equation*} \lim{s\rightarrow 0}\sum_{\alpha\in G(g_f)\backslash G({\mathbb Q})/B({\mathbb Q})}F_{I,f}^m(\chi,0,g_f \alpha) \end{equation*} \begin{equation*}
\sum_{\gamma\in G(g_f \alpha)/B(g_f \alpha)}\frac{\eta_\infty (\det(\alpha\gamma))|\operatorname{N}(\det(\alpha\gamma))|^s \operatorname{N}(\det(\alpha\gamma ))^{m+2} \operatorname{N}(a+c\tau)^{m+1}(a+c\tau)_{I}(a+c\overline\tau)_{I^c}}{|\operatorname{N}(\det(\alpha\gamma))|^{-n}|\operatorname{N}(a+c\tau)|^{2(m+2+s)}}\cdot \end{equation*} \begin{equation*}
\operatorname{sgn}(\operatorname{N}(\det(g)))^{n}\eta_\infty(\det(g))|\operatorname{N}(\frac{t_2}{t_1})|^s \operatorname{N}(-it_2e^{i\theta}(e_1-ie_2))^m \operatorname{N}(2i\frac{t_2}{t_1})(e^{i2\theta})_{I}\operatorname{N}(\det(g))^{n+1} \operatorname{vol}_V^{*n+1} \end{equation*} and all the elements not collinear with $\left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I}$ to zero. The last sum is nothing else but \begin{equation*} \lim{s\rightarrow 0}\sum_{\gamma \in G({\mathbb Q})/B({\mathbb Q})}F_{I,f}^m(\chi,s,g_f \gamma)F_{I,\infty}^m(\chi,s,g)\operatorname{N}(e_1-e_2 i)^m \operatorname{vol}_V^{*n+1}\cdot \end{equation*} \begin{equation*}
\frac{\eta_\infty (\det(\alpha\gamma))\operatorname{N}(\det(\alpha\gamma ))^{m+2}|\operatorname{N}(\det(\alpha\gamma))|^s \operatorname{N}(a+c\tau)^{m+1}(a+c\tau)_{I}(a+c\overline\tau)_{I^c}}{|\operatorname{N}(\det(\alpha\gamma))|^{-n}|\operatorname{N}(a+c\tau)|^{2(m+2+s)}} \end{equation*} Finally, \begin{equation*}
t_2(g\alpha\gamma)=\frac{\det(\alpha\gamma)}{|a+c \tau|}t_2(g)\text{ and } e^{i\theta(g\alpha\gamma)}=\frac{a+c \tau}{|a+c \tau|}e^{i\theta(g)} \end{equation*} and therefore $F_{I,\infty}^m(\chi,s,g \alpha\gamma)$ equals \begin{equation*} F_{I,\infty}^m(\chi,s,g)\cdot
\frac{\eta_\infty (\det(\alpha\gamma))\operatorname{N}(\det(\alpha\gamma ))^{m+2}|\operatorname{N}(\det(\alpha\gamma))|^s \operatorname{N}(a+c\tau)^{m+1}(a+c\tau)_{I}(a+c\overline\tau)_{I^c}}{|\operatorname{N}(\det(\gamma))|^{-n}|\operatorname{N}(a+c\tau)|^{2(m+2+s)}} \end{equation*} Consequently $\Psi(\widetilde{\operatorname{Eis}}_{I}^k(\varphi,\chi))$ is the form that maps \begin{equation*} \left(P_{-}\wedge P_{+}\right)_{I^c}\wedge P_{+ I}\mapsto \lim{s\rightarrow 0}\sum_{\gamma \in G({\mathbb Q})/B({\mathbb Q})}F_{I}^m(\chi,s,g \gamma)\operatorname{N}(e_1-e_2 i)^m \operatorname{vol}_V^{*n+1}. \end{equation*} This form is exactly $\lim{s\rightarrow 0}\operatorname{Eis}^*(\omega_{I}^m(\varphi,\chi,s))$. \end{proof} \end{proposition}
\section{The image of the polylogarithmic Eisenstein operator}
The polylogarithmic Eisenstein classes are built up by Eisenstein series associated to Schwartz-Bruhat functions on $V({\mathbb A})$, while Harder's Eisenstein series are associated to functions on $G({\mathbb A})$, which are induced from algebraic Hecke characters on $T({\mathbb A})/T({\mathbb Q})$. Eisenstein series associated to induced functions on $G({\mathbb A})$ can be represented as finite sums of Eisenstein series coming from Schwartz-Bruhat functions on $V({\mathbb A})$, see \cite{J-Z} Lemma. So we have a good reason to believe that the image of the polylogarithmic Eisenstein operator is quite big. In this section we determine $\operatorname{im}(\operatorname{Eis}^k_q)\subset H^\bullet_{\operatorname{Eis}}(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1})$ completely. As
\begin{equation*} \operatorname{res}_\mathcal S:H^\bullet_{\operatorname{Eis}}(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1})\rightarrow H^\bullet(\partial \mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1}) \end{equation*} is an isomorphism of $G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$-modules onto its image, it suffices to examine the image of $\operatorname{res}_\mathcal S\circ \operatorname{Eis}^k_q$. This last operator has already been calculated in \Cref{boundary_residueII} and we know by \Cref{hor} that it suffices to understand the image of the horospherical map $\rho_{m,n}^0$. As we may multiply our functions by \begin{equation*}
\left\|\det(g)\right\|_f ^{-n}\operatorname{sgn}(\operatorname{N}(\det(g)))^{-n},\ n\in {\mathbb Z}, \end{equation*} it even suffices to understand the image of $\rho_{m,0}^0$. But first we want to understand a more general map. \begin{definition} Define the \textit{horospherical map} for $m\geq 0$ \begin{equation*}
\rho_{m}:\mathcal S(V({\mathbb A}_f),\mu^{\otimes 0}\otimes {\mathbb C})\rightarrow \operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}\bigoplus_{\phi:\ type(\phi)=\gamma_{m,1},\tilde{\phi}_{f|Z({\mathbb R})=1}}{\mathbb C} \cdot\tilde\phi_f \end{equation*} by \begin{equation*} \rho_{m}(f)(g):=
\sum_{\chi\in \widehat{Cl_F ^K}(m)}\frac{\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb A}_f)}\hat{f} (tg_fe^1,g) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t , \end{equation*} if $f$ is $K_f$-invariant. \end{definition} \subsection{Surjectivity of the horospherical map}
\begin{proposition}\label{hor_surjective} The horospherical map $\rho_m$ is surjective.
\begin{proof} The idea how to prove this proposition is already in \cite{J-Z} Lemma. Nevertheless, we give the proof for the sake of completeness. Set for $N\in {\mathbb N}$ \begin{equation*}
U_N:=\ker(\hat\mathcal O^{\times}\rightarrow (\mathcal O/N\mathcal O) ^{\times}),\ Cl_F ^{(N)}:=U_N \prod_{\nu|\infty}{\mathbb R}_{>0}\backslash \mathbb{I}_F/F^{\times}. \end{equation*} Let $\phi$ be an algebraic Hecke character as above. We may write \begin{equation*}
\phi(t_1,t_2)=\eta(t_1t_2)\chi^\prime(t_2)\left\|t_2\right\|^{m+2}, (t_1,t_2)\in T({\mathbb A}), \end{equation*} for characters $\eta,\chi^\prime \in \widehat{Cl_F ^{(N)}}$ for some $N\in {\mathbb N}$. Then \begin{equation*}
\tilde\phi_f(t_1,t_2)=\eta(t_1t_2)\chi^\prime(t_2)\left\|t_2\right\|_f^{m+2}\operatorname{sgn}(\operatorname{N}(t_2))^{m+2}, (t_1,t_2)\in T({\mathbb A}). \end{equation*}
$\tilde\phi_{f|Z({\mathbb R})}=1$ means $\chi^\prime(t)=\operatorname{sgn}(\operatorname{N}(t))^{m+2},\ t\in F_{\mathbb R} ^{\times}$, in other words $\chi\in Cl_F ^{(N)}(m):=Cl_F ^{K_N}(m)$. Now take \begin{equation*} \psi\in \operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}{\mathbb C} \cdot\tilde\phi_f. \end{equation*} By choosing $N$ big enough we may also assume that $\psi(kx)=\psi(x)$ holds for all $x\in G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$ and $k\in K_N$. Define $\overline \psi$ by \begin{equation*} \overline \psi(x)=\eta(\det(x))\psi(\det(x)^{-1}\cdot x),\ x\in G({\mathbb A}_f)\times \pi_0(G({\mathbb R})). \end{equation*} As $\det(K_N)=U_N$, we still have $\overline\psi(kx)=\overline\psi(x)$ for all $x\in G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))$ and $k\in K_N$, but now we have for $b=\begin{pmatrix}t_1&t_1 u\\ 0& t_2\end{pmatrix}\in B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))$ \begin{equation*}
\overline\psi(xb)=\overline\psi(x)\chi^\prime(t_1^{-1})\left\|t_1 ^{-1}\right\|_f^{m+2}\operatorname{sgn}(\operatorname{N}(t_1^{-1}))^{m+2}=
\overline\psi(x)\chi^\prime_f(t_1)^{-1}\left\|t_1 \right\|_f^{-(m+2)}. \end{equation*} This means that $\overline \psi$ is a function \begin{equation*} K_N\backslash G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))/P({\mathbb A}_f)\times \pi_0(P_\infty)\rightarrow {\mathbb C}. \end{equation*} We restrict $\overline \psi $ to $G(\hat{\mathbb Z})$ and obtain \begin{equation*} \overline\psi:K_N\backslash G(\hat {\mathbb Z})/P(\hat{\mathbb Z})=G({\mathbb Z}/N{\mathbb Z})/P({\mathbb Z}/NZ)\rightarrow {\mathbb C}. \end{equation*} We consider the embedding $i:G({\mathbb A}_f)/P({\mathbb A}_f)\hookrightarrow V({\mathbb A}_f),\ x\mapsto xe^1$, and make $\overline \psi $ to a function on $V({\mathbb A}_f)$ by extending it by zero outside $G( {\mathbb A}_f)/P({\mathbb A}_f)$. When we restrict $\overline\psi$ to $V(\hat{\mathbb Z})$, it even induces a function on $V({\mathbb Z}/N{\mathbb Z})$ and may also be interpreted as a Schwartz-Bruhat function $s\overline\psi\in\mathcal S(V({\mathbb A}_f),{\mathbb C})$. Explicitly $s\overline\psi$ can be described as follows: Take a complete set of representatives $x\in G(\hat{\mathbb Z})$ of $G( {\mathbb Z}/N{\mathbb Z})/P({\mathbb Z}/N{\mathbb Z})$. Then \begin{equation*} s\overline\psi:=\sum_{x\,\text{mod}N\in G( {\mathbb Z}/N{\mathbb Z})/P({\mathbb Z}/N{\mathbb Z})}\overline \psi(x)\chi_{xe^1+NV(\hat{\mathbb Z})}=\otimes_{\nu\nmid \infty}s\overline\psi_\nu \end{equation*} where the local functions $s\overline\psi_\nu:F_\nu\to{\mathbb C}$ are defined by $s\overline\psi_\nu:=\chi_{\mathcal O_\nu ^2}$, if $\nu\nmid N$, and \begin{equation*}
s\overline\psi_\nu:=\sum_{x_\nu\in G_0(\mathcal O_\nu/N\mathcal O_\nu)/P_0(\mathcal O_\nu/N\mathcal O_\nu)}
\overline\psi(x_\nu)\chi_{x_\nu e^1+N\mathcal O_\nu^2}=\overline\psi\cdot \chi_{G_0(\mathcal O_\nu)/P_0(\mathcal O_\nu)},\text{ if }\nu| N. \end{equation*} Take a full set of representatives $u\in \mathbb{I}_{F}$ of $Cl_F ^{(N)}$. We define $\varphi\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes 0}\otimes {\mathbb C})$ by its Fourier transform with respect to $v$ \begin{equation*}
\hat\varphi(v,g):=\sum_u \chi^\prime_f(u)\left\|u\right\|_f ^{m+2}\eta(\det(g))s\overline\psi(uv). \end{equation*} Let us calculate $\rho_m(\varphi)$. We have by definition \begin{equation*} \rho_m(\varphi)(g)= \sum_{\chi\in \widehat{Cl_F ^{(N)}}(m)}\frac{\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}
\int_{t\in Z({\mathbb A}_f)}\hat{\varphi} (tg_fe^1,g) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t = \end{equation*} \begin{equation*}
\sum_u\sum_{\chi\in \widehat{Cl_F ^{(N)}}(m)}\chi^\prime_f(u)\left\|u\right\|_f ^{m+2}\eta(\det(g))\frac{\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}\cdot \end{equation*} \begin{equation*}
\int_{t\in Z({\mathbb A}_f)}s\overline\psi(utg_fe^1) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t. \end{equation*} So we need to understand the last integral. Analogous to $s\overline \psi$ we define the Schwartz-Bruhat function $s\chi^{\prime-1}\in \mathcal S({\mathbb A}_{F,f},{\mathbb C})$. Extend $\chi^{\prime-1}$ by zero to a function on ${\mathbb A}_{F,f}$ and choose again a full set of representatives $x\in \hat\mathcal O^{\times}$ of $(\mathcal O/N\mathcal O)^{\times}=U_N\backslash\hat\mathcal O^{\times}$. Then \begin{equation*} s\chi^{\prime-1}:=\sum_{x\in (\mathcal O/N\mathcal O)^{\times}} \chi^{\prime-1}(x)\chi_{x+N\hat\mathcal O}=\otimes_{\nu\mid\infty}s\chi_\nu^{\prime-1} \end{equation*} and the local functions on $F_\nu$ are defined by \begin{equation*}
s\chi_\nu^{\prime-1}:=\chi_{\mathcal O_\nu}, \text{ if }\nu\nmid N,\text{ and } s\chi_\nu^{\prime-1}:=\sum_{x_\nu\in (\mathcal O_\nu/N\mathcal O_\nu)^{\times}}\chi^{\prime -1}(x_\nu)\chi_{x_\nu+N\mathcal O_\nu}=\chi^{\prime-1}\cdot\chi_{\mathcal O_\nu ^{\times}}, \text{ if }\nu|N. \end{equation*} Write $g_f=xb$ for $x\in G(\hat{\mathbb Z})$ and $b\in B({\mathbb A}_f)$ and see \begin{equation*}
s\overline \psi(tug_fe_1)=s\overline\psi(xtut_1e^1)=\prod_{\nu\nmid N}\chi_{\mathcal O_\nu^2}((xtut_1e^1)_\nu)\prod_{\nu|N} \overline\psi((xtut_1)_\nu)\chi_{G_0(\mathcal O_\nu)/P_0(\mathcal O_\nu)}((xtut_1e^1)_\nu). \end{equation*} As $x\in G(\hat{\mathbb Z})$, we may write this as \begin{equation*}
\prod_{\nu\nmid N}\chi_{\mathcal O_\nu}((tut_1)_\nu)\prod_{\nu|N} \overline\psi((xtut_1)_\nu)\chi_{\mathcal O_\nu ^{\times}}((tut_1)_\nu). \end{equation*} Now \begin{equation*}
\overline\psi((xtut_1)_\nu)=\left\|(tut_1)_\nu\right\|_f^{-(m+2)}\chi^{\prime-1}((tut_1)_\nu)\overline\psi(x_\nu)=\chi^{\prime-1}((tut_1)_\nu)\overline\psi(x_\nu) \text{ for } (tut_1)_\nu\in \mathcal O_\nu ^{\times} \end{equation*}
and we get using $\overline\psi_{|K_N}=1$ \begin{equation*}
\prod_{\nu\nmid N}\chi_{\mathcal O_\nu}((tut_1)_\nu)\prod_{\nu|N} \overline\psi(x_\nu)\chi^{\prime-1}((tut_1)_\nu)\chi_{\mathcal O_\nu ^{\times}}((tut_1)_\nu)= \end{equation*} \begin{equation*}
\overline\psi(x)\prod_{\nu\nmid N}\chi_{\mathcal O_\nu}((tut_1)_\nu)\prod_{\nu|N} \chi^{\prime-1}((tut_1)_\nu)\chi_{\mathcal O_\nu ^{\times}}((tut_1)_\nu)=\overline\psi(x)s\chi^{\prime-1}(tut_1). \end{equation*} We may write our integral as \begin{equation*}
\int_{t\in Z({\mathbb A}_f)}s\overline\psi(utg_fe^1) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*}
\overline\psi (x)\int_{t\in Z({\mathbb A}_f)}s\chi^{\prime-1} (utt_1) \chi(\det(g_f)t)\left\|\det(g_f)t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*}
\overline\psi (x)\chi_f (u t_1)^{-1}\left\|ut_1\right\|_f^{-(m+2)}\chi_f(\det(g_f))\left\|\det(g_f)\right\|_f^{m+2}\int_{t\in Z({\mathbb A}_f)}s\chi^{\prime-1} (t) \chi(t)\left\|t\right\|^{m+2} d^{\times}t \end{equation*}
Remembering $\overline\psi(g)=\overline\psi (x)\chi^\prime _f(t_1)^{-1}\left\|t_1\right\|_f ^{-(m+2)}$ and \begin{equation*}
\overline\psi(g)=\eta(\det(g))\psi(g\det(g)^{-1})=\psi(g)\eta(\det(g))^{-1}\chi_f^\prime(\det(g)^{-1})\left\|\det(g)\right\|_f ^{-(m+2)} \end{equation*} we see now that $\frac{(-2\pi i)^{\xi(m+2)}}{\Gamma(m+2)^{\xi}}\rho_m(\varphi)(g)$ equals \begin{equation*}
\psi (g)\sum_u\sum_{\chi\in \widehat{Cl_F ^{(N)}}(m)}\chi^\prime_f\cdot\chi_f^{-1}(u t_1 \det(g)^{-1})\cdot\int_{t\in Z({\mathbb A}_f)}s\chi^{\prime-1} (t) \chi(t)\left\|t\right\|^{m+2} d^{\times}t= \end{equation*} \begin{equation*}
\psi (g)\sum_u\sum_{\chi\in \widehat{Cl_F ^{(N)}}(m)}\chi^\prime\cdot\chi^{-1}(u t_1 \det(g)^{-1})\cdot\int_{t\in Z({\mathbb A}_f)}s\chi^{\prime-1} (t) \chi(t)\left\|t\right\|^{m+2} d^{\times}t. \end{equation*} Of course, $\sum_u\chi^\prime\cdot\chi^{-1}(u t_1 \det(g)^{-1})=0$, whenever $\chi\neq \chi^\prime$, and therefore \begin{equation*}
\rho_m(\varphi)=\frac{|Cl_F^{(N)}|\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb A}_f)}s\chi^{\prime-1} (t) \chi^\prime(t)\left\|t\right\|^{m+2} d^{\times}t\cdot\psi. \end{equation*} To complete our proof we need to show that \begin{equation*}
\int_{t\in Z({\mathbb A}_f)}s\chi^{\prime-1} (t) \chi^\prime(t)\left\|t\right\|^{m+2} d^{\times}t\neq 0. \end{equation*} But following \cite{T} p.342 f.f. and using $s\chi^{\prime-1}=\otimes_{\nu\nmid \infty}s\chi_\nu^{\prime-1}$ we calculate \begin{equation*}
\int_{t\in Z({\mathbb A}_f)}s\chi^{\prime-1} (t) \chi^\prime(t)\left\|t\right\|^{m+2} d^{\times}t=\frac{1}{\sqrt{d_F}}\prod_{\nu\nmid N\infty}\frac{1}{1-\chi^\prime(\pi_\nu)\operatorname{N}(\mathfrak p_\nu)^{-(m+2)}} \end{equation*} an convergent Euler product and therefore non-zero. \end{proof} \end{proposition} \begin{remark}\label{preimage} We keep the notations from above. Let us set \begin{equation*}
\Lambda_N(\chi,m+2)=\frac{|Cl_f ^{(N)}|\Gamma(m+2)^{\xi}}{(-2\pi i)^{\xi(m+2)}}\int_{t\in Z({\mathbb A}_f)}s\chi^{\prime-1} (t) \chi^\prime(t)\left\|t\right\|^{m+2} d^{\times}t \end{equation*} For \textit{any} set of representatives $\left\{u\right\}$ the function $\varphi_{\left\{u\right\}}\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes 0}\otimes{\mathbb C})$ defined by its Fourier transform \begin{equation*}
\hat\varphi_{\left\{u\right\}}(v,g):=\Lambda_N(\chi,m+2)^{-1}\sum_u \chi^\prime_f(u)\left\|u\right\|_f ^{m+2}\eta(\det(g))s\overline\psi(uv) \end{equation*} is a preimage of $\psi$ under $\rho_m$. \end{remark} Let us set $T^1:=\ker(\det:T\rightarrow Res_{\mathcal O/{\mathbb Z}}\mathbb G_m)$. \begin{lemma} \begin{equation*}
\operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}\bigoplus_{\phi:\ type(\phi)=\gamma_{m,1},\tilde{\phi}_{f|Z({\mathbb R})=1}}{\mathbb C} \cdot\tilde\phi_f \end{equation*}
is in the image of $\rho_{m,0} ^0$, if $\phi_{|T^1({\mathbb A})}\neq \left\|\ \cdot\ \right\|^2$. \begin{proof}
Write $\phi(t_1,t_2)=\eta(t_1t_2)\chi(t_2)\left\|t_2\right\|^{m+2}$. The assumption $\phi_{|T^1({\mathbb A})}\neq \left\|\ \cdot\ \right\|^2$ tells us that $m\geq1$ or $\chi\neq1$ and if $m=0$ we know $\chi_{|F_{\mathbb R}^{\times}}=1$. So there is an $u_0\in \mathbb{I}_{F}$ such that $\chi_f(u_0)\left\|u_0\right\|_f^m\neq 1$. We calculate \begin{equation*}
\chi_f(u_0)\left\|u_0\right\|_f^m\int_{v\in V({\mathbb A}_f)}\hat\varphi_{\left\{u\right\}}(v,g)dv= \end{equation*} \begin{equation*}
\Lambda_N(\chi,m+2)^{-1}\sum_u \chi_f(u_0u)\left\|uu_0\right\|_f ^{m}\left\|u\right\|_f ^{2}\eta(\det(g))\int_{v\in V({\mathbb A}_f)}s\overline\psi(uv)dv \end{equation*} \begin{equation*}
\Lambda_N(\chi,m+2)^{-1}\sum_u \chi_f(u_0u)\left\|u_0u\right\|_f ^{m}\left\|u\right\|_f ^{2}\eta(\det(g))\int_{v\in V({\mathbb A}_f)}s\overline\psi(u_0u(u_0^{-1}v))dv \end{equation*} \begin{equation*}
\Lambda_N(\chi,m+2)^{-1}\sum_u \chi_f(u_0u)\left\|u_0u\right\|_f ^{m}\left\|uu_0\right\|_f ^{2}\eta(\det(g))\int_{v\in V({\mathbb A}_f)}s\overline\psi(u_0uv)dv= \end{equation*} \begin{equation*} \int_{v\in V({\mathbb A}_f)}\hat\varphi_{\left\{u_0u\right\}}(v,g)dv. \end{equation*} In other words, \begin{equation*}
\chi_f(u_0)^{-1}\left\|u_0\right\|_f^{-m}\hat\varphi_{\left\{u_0u\right\}}-\hat\varphi_{\left\{u\right\}}\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes 0}\otimes{\mathbb C})^0 \end{equation*} and by Fourier inversion \begin{equation*}
\chi_f(u_0)^{-1}\left\|u_0\right\|_f^{-m}\varphi_{\left\{u_0u\right\}}-\varphi_{\left\{u\right\}}\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes 0}\otimes{\mathbb C})^0. \end{equation*} Now we see \begin{equation*}
\rho_{m,0}^0\left(\frac{\chi_f(u_0)^{-1}\left\|u_0\right\|_f^{-m}\varphi_{\left\{u_0u\right\}}-\varphi_{\left\{u\right\}}}{\chi_f(u_0)^{-1}\left\|u_0\right\|_f^{-m}-1}\right)=
\rho_m\left(\frac{\chi_f(u_0)^{-1}\left\|u_0\right\|_f^{-m}\varphi_{\left\{u_0u\right\}}-\varphi_{\left\{u\right\}}}{\chi_f(u_0)^{-1}\left\|u_0\right\|_f^{-m}-1}\right)= \end{equation*} \begin{equation*}
\frac{\chi_f(u_0)^{-1}\left\|u_0\right\|_f^{-m}\rho_m(\varphi_{\left\{u_0u\right\}})-\rho_m(\varphi_{\left\{u\right\}})}{\chi_f(u_0)^{-1}\left\|u_0\right\|_f^{-m}-1}=
\frac{\chi_f(u_0)^{-1}\left\|u_0\right\|_f^{-m}\psi-\psi}{\chi_f(u_0)^{-1}\left\|u_0\right\|_f^{-m}-1}=\psi \end{equation*} \end{proof} \end{lemma} \subsection{The spherical functions}\label{spherical} So what is left to be considered are \begin{equation*}
\operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}{\mathbb C} \cdot\tilde\phi_f,\ \phi_{|T_1({\mathbb A})}=\left\|\ \cdot\ \right\|^2, \end{equation*} in particular $m=0$. We choose the unique right-invariant Haar measure $ds$ on $SL_2({\mathbb A}_{F,f})$ with $ds(SL_2(\hat\mathcal O))=1$. The group $SL_2({\mathbb A}_{F,f})$ is unimodular. \begin{definition}
Let $\phi:T({\mathbb A})/T({\mathbb Q})\to {\mathbb C}^{\times}$ be an algebraic Hecke character of type $\gamma_{0,n}$ with $\phi_{|T^1({\mathbb A})}=\left\|\ \cdot \ \right\|^2$. Then we have the \textit{spherical function} \begin{equation*} S(\phi)\in \operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}{\mathbb C} \cdot\tilde\phi_f \end{equation*} defined by \begin{equation*} S(\phi)(g):=\tilde\phi_f(b):=\tilde\phi_f(t_1,t_2),\ \text{if } g=xb,\ x\in SL_2(\hat\mathcal O),\ b=\begin{pmatrix}t_1& t_1u\\ 0& t_2\end{pmatrix}\in B({\mathbb A}) \end{equation*} \end{definition} \begin{remark} If we have a non-trivial $K_N$-invariant $\psi\in \operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}{\mathbb C} \cdot\tilde\phi_f$, then $\tilde\phi_f$ already has to be $K_N\cap T({\mathbb A}_f)$-invariant and, in particular, the function $S(\phi)$ is $K_N$-invariant. \end{remark} \begin{lemma}
Let $\phi:T({\mathbb A})\rightarrow {\mathbb C}^{\times}$ be an algebraic Hecke character of type $\gamma_{0,n}$ with $\phi_{|T^1({\mathbb A})}=\left\|\ \cdot \ \right\|^2$. Then we have a linear projection operator \begin{equation*} \Psi_\phi:\operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}{\mathbb C} \cdot\tilde\phi_f\rightarrow {\mathbb C}\cdot S(\phi) \end{equation*} defined by \begin{equation*} \Psi_\phi(\psi)(g):=\int_{s\in SL_2(\hat\mathcal O)}\psi(sg)ds \end{equation*} \begin{proof} Write $g=xb$, $x\in SL_2(\hat\mathcal O)$, as above. Then \begin{equation*} \Psi_\phi(\psi)(g):=\int_{s\in SL_2(\hat\mathcal O)}\psi(sg)ds=\int_{s\in SL_2(\hat\mathcal O)}\psi(sxb)ds= \end{equation*} \begin{equation*} \tilde\phi_f(b)\int_{s\in SL_2(\hat\mathcal O)}\psi(sx)ds=S(\phi)(g)\int_{s\in SL_2(\hat\mathcal O)}\psi(s)ds=S(\phi)(g)\Psi_\phi(\psi)(1). \end{equation*} \end{proof} \end{lemma} \begin{remark} $\Psi_\phi(S(\phi))=S(\phi)$, so $\Psi_\phi$ is a split epimorphism, and $\psi\in \ker(\Psi_\phi)$ if and only if \begin{equation*} \int_{s\in SL_2(\hat\mathcal O)}\psi(s)ds=\Psi_\phi(\psi)(1)=0. \end{equation*} \end{remark} \begin{lemma}
Let $\phi:T({\mathbb A})\rightarrow {\mathbb C}^{\times}$ be an algebraic Hecke character of type $\gamma_{0,1}$ with $\tilde\phi_{f|Z({\mathbb R})} =1$ and $\phi_{|T^1({\mathbb A})}=\left\|\ \cdot \ \right\|^2$. Then $\ker(\Psi_\phi)$ is in the image of $\rho_{0,0}^0$. \begin{proof} Let $\psi\in \ker(\Psi_\phi)$ be given. We suppose $\psi$ is left $K_N$-invariant and $\phi$ is $U_N$ invariant. For any set of representatives $\left\{u\right\}\subset \mathbb{I}_F$ of $Cl_F ^{(N)}$ the function $\varphi_{\left\{u\right\}}\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes 0}\otimes{\mathbb C})$ defined by its Fourier transform \begin{equation*}
\hat\varphi_{\left\{u\right\}}(v,g):=\Lambda_N(\chi,m+2)^{-1}\sum_u \chi^\prime_f(u)\left\|u\right\|_f ^{m+2}\eta(\det(g))s\overline\psi(uv) \end{equation*} is a preimage of $\psi$ under $\rho_m$. To prove the lemma we show that $\varphi_{\left\{u\right\}}\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes 0}\otimes{\mathbb C})^0$. By Fourier inversion we have to show that $\hat\varphi_{\left\{u\right\}}\in \mathcal S(V({\mathbb A}_f),\mu^{\otimes 0}\otimes{\mathbb C})^0$ and therefore it suffices to prove \begin{equation*} \int_{v\in V({\mathbb A}_f)}s\overline\psi(uv)dv=0. \end{equation*} Recall how we defined $s\overline\psi$ in the course of the proof of \cref{hor_surjective} and calculate \begin{equation*}
\int_{v\in V({\mathbb A}_f)}s\overline\psi(uv)dv=\left\|u\right\|_f ^{-2}\int_{v\in V({\mathbb A}_f)}s\overline\psi(v)dv=\left\|u\right\|_f ^{-2}dv(NV(\hat{\mathbb Z}))\sum_{v\in V({\mathbb Z}/N{\mathbb Z})}s\overline\psi(v)= \end{equation*} \begin{equation*}
\left\|u\right\|_f ^{-2}dv(NV(\hat{\mathbb Z}))\sum_{x\in G({\mathbb Z}/N{\mathbb Z})/P({\mathbb Z}/N{\mathbb Z})}\overline\psi(x)= \end{equation*} \begin{equation*}
\left\|u\right\|_f ^{-2}dv(NV(\hat{\mathbb Z}))\sum_{s\in SL_2(\mathcal O/N\mathcal O)/U_0(\mathcal O/N\mathcal O)}\overline\psi(s)=\frac{\left\|u\right\|_f ^{-2}dv(NV(\hat{\mathbb Z}))}{|U_0(\mathcal O/N\mathcal O)|}\sum_{s\in SL_2(\mathcal O/N\mathcal O)}\overline\psi(s). \end{equation*} If we set $K_N\cap SL_2(\hat\mathcal O):=K_N ^{(1)}$, we may write this as \begin{equation*}
\frac{\left\|u\right\|_f ^{-2}dv(NV(\hat{\mathbb Z}))}{|U_0(\mathcal O/N\mathcal O)|}\sum_{s\in K_N^{(1)}\backslash SL_2(\hat\mathcal O)}\overline\psi(s)= \end{equation*} \begin{equation*}
\frac{\left\|u\right\|_f ^{-2}dv(NV(\hat{\mathbb Z}))}{|U_0(\mathcal O/N\mathcal O)|ds(K_N ^{(1)})}\int_{s\in SL_2(\hat\mathcal O)}\overline\psi(s)ds=\frac{\left\|u\right\|_f ^{-2}dv(NV(\hat{\mathbb Z}))}{|U_0(\mathcal O/N\mathcal O)|ds(K_N ^1)}\int_{s\in SL_2(\hat\mathcal O)}\psi(s)ds=0, \end{equation*}
as $\psi_{|SL_2(\hat\mathcal O)}=\overline\psi_{|SL_2(\hat\mathcal O)}$. \end{proof} \end{lemma} Now we may define the operator $\Psi_{m,n}$ \begin{equation*}
\operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}\bigoplus_{\phi:\ type(\phi)=\gamma_{m,n},\tilde{\phi}_{f|Z({\mathbb R})=1}}{\mathbb C} \cdot\tilde\phi_f\rightarrow
\bigoplus_{\phi:\ type(\phi)=\gamma_{m,n},\\ \tilde{\phi}_{f|Z({\mathbb R})=1},\phi_{|T^1({\mathbb A})}=\left\|\ \cdot\ \right\|^2}{\mathbb C}\cdot S(\phi), \end{equation*}
where $\Psi_{m,n}$ is zero on those summands with $\phi_{|T^1({\mathbb A})}\neq\left\|\ \cdot\ \right\|^2$ and $\Psi_{m,n}$ equals $\Psi_\phi$ defined above on those summands with $\phi_{|T^1({\mathbb A})}=\left\|\ \cdot\ \right\|^2$. \begin{remark} $\Psi_{m,n}$ is defined over $\overline{\mathbb Q}$. To see this consider a function \begin{equation*} \psi\in \operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}\overline{\mathbb Q} \cdot\tilde\phi_f\ \text{and}\ \Psi_\phi(\psi)=\int_{s\in SL_2(\hat\mathcal O)}\psi(s)ds\cdot S(\phi). \end{equation*} There is some compact open subgroup $K_f\subset SL_2(\hat\mathcal O)$ such that \begin{equation*} \int_{s\in SL_2(\hat\mathcal O)}\psi(s)ds=\sum_{s\in K_f\backslash SL_2(\hat\mathcal O)}\psi(s)ds(K_f). \end{equation*} As the index of $K_f\subset SL_2(\hat\mathcal O)$ is finite and $ds(SL_2(\hat\mathcal O))=1$, we have $ds(K_f)\in {\mathbb Q}^{\times}$, and therefore \begin{equation*} \int_{s\in SL_2(\hat\mathcal O)}\psi(s)ds\text{ and } S(\phi)(g)\in \overline {\mathbb Q}^{\times}\text{ for }g\in G({\mathbb A}). \end{equation*} So the function $\Psi_\phi(\psi)$ has its values in $\overline{\mathbb Q}$. Moreover, $\Psi_{m,n}$ is Galois-equivariant. This follows from \begin{equation*}
\int_{s\in SL_2(\hat\mathcal O)}\sigma \cdot\psi(s)ds=\sum_{s\in K_f\backslash SL_2(\hat\mathcal O)}\sigma\cdot\psi(s)ds(K_f)=\sigma\left(\int_{s\in SL_2(\hat\mathcal O)}\psi(s)ds\right),\ \sigma\in \text{Gal}(\overline{\mathbb Q}/{\mathbb Q}), \end{equation*}
and we conclude that the vector space $\ker(\Psi_{m,n})$ has a natural ${\mathbb Q}$ structure which we denote by $\ker(\Psi_{m,n|{\mathbb Q}})$. \end{remark}
\begin{corollary}\label{ker(Psi)}
$\ker(\Psi_{m,n+1|{\mathbb Q}})$ is in the image of $\rho_{m,n}^0$. \end{corollary}
\subsection{The long exact sequence for the cohomology of the boundary}
The next step is to show that $\ker(\Psi_{m,n+1|{\mathbb Q}})$ is exactly the image of $\rho_{m,n}^0$. To do so we use a long exact cohomology sequence relating the cohomology with compact supports, the usual cohomology of $\mathcal S_K$ and the cohomology of the boundary.
We denote again by $\overline{\mathcal S_K}$ the Borel-Serre compactification of $\mathcal S_K$. Consider $j:\mathcal S_K\to \overline{\mathcal S_K}$ the open inclusion and $i:\overline{\mathcal S_K}\setminus\mathcal S_K\to \overline{\mathcal S_K}$ the closed inclusion of the boundary. For any abelian sheaf $\mathcal F$ on $\overline{\mathcal S_K}$ we have the exact sequence \begin{equation*}
0\to j_!j^{-1}\mathcal F\to\mathcal F\to i_* i^{-1}\mathcal F\to 0. \end{equation*} If we apply $H^p_c(\overline{\mathcal S_K},\ )=H^p(\overline{\mathcal S_K},\ )$ to this sequence, we obtain a long exact cohomology sequence \begin{equation*} \cdots\to H^p_c(\overline{\mathcal S_K},j_!{\mathbb Q})\to H^p(\overline{\mathcal S_K},{\mathbb Q})\to H^p(\overline{\mathcal S_K}\setminus\mathcal S_K,{\mathbb Q})\to H^{p+1}_c(\overline{\mathcal S_K},j_!{\mathbb Q})\to\cdots \end{equation*} $H^p_c(\overline{\mathcal S_K},j_!\ )=H^p_c(\mathcal S_K,\ )$, the space $\overline{\mathcal S_K}\setminus\mathcal S_K$ is homotopy equivalent to $\partial \mathcal S_K$ and $\overline{\mathcal S_K}$ is homotopy equivalent to $\mathcal S_K$ by \cite{Ha1} 2.1, so we get the long exact cohomology sequence \begin{equation*} \cdots\to H^p_c(\mathcal S_K,{\mathbb Q})\to H^p(\mathcal S_K,{\mathbb Q})\to H^p(\partial\mathcal S_K,{\mathbb Q})\to H^{p+1}_c(\mathcal S_K,{\mathbb Q})\to\cdots \end{equation*} \begin{lemma} We have an exact sequence \begin{equation*} H^{2\xi-1}(\mathcal S_{K_N},{\mathbb Q})\stackrel{\operatorname{res}_{\mathcal S_{K_N}}}{\longrightarrow} H^{2\xi-1}(\partial\mathcal S_{K_N},{\mathbb Q})\rightarrow H^{2\xi}_c(\mathcal S_{K_N},{\mathbb Q})\rightarrow 0 \end{equation*}
and $\operatorname{im}(\operatorname{res}_{\mathcal S_{K_N}})$ has codimension $|Cl_F ^{K_N}|=|Cl_F ^{(N)}|$ in $ H^{2\xi-1}(\partial\mathcal S_{K_N},{\mathbb Q})$. \begin{proof} We start with the exact sequence \begin{equation*} H^{2\xi-1}(\mathcal S_{K_N},{\mathbb Q})\stackrel{\operatorname{res}_{\mathcal S_{K_N}}}{\longrightarrow} H^{2\xi-1}(\partial\mathcal S_{K_N},{\mathbb Q})\rightarrow H^{2\xi}_c(\mathcal S_{K_N},{\mathbb Q})\rightarrow H^{2\xi}(\mathcal S_{K_N},{\mathbb Q}) \end{equation*} We use Poincaré duality to see \begin{equation*} H^{2\xi}_c(\mathcal S_{K_N},{\mathbb C})=H^0(\mathcal S_{K_N},{\mathbb C}),\ H^{2\xi}(\mathcal S_{K_N},{\mathbb C})=H^0 _c(\mathcal S_{K_N},{\mathbb C})=0. \end{equation*} The latter is zero, because $\mathcal S_{K_N}$ is not compact. The first group is non-zero, we even conclude \begin{equation*}
dim_{\mathbb Q} H^{2\xi}_c(\mathcal S_{K_N},{\mathbb Q})=dim_{\mathbb C} H^0(\mathcal S_{K_N},{\mathbb C})=|Cl_F ^{K_N}|=|Cl_F ^{(N)}|. \end{equation*} So we have the exact sequence \begin{equation*} H^{2\xi-1}(\mathcal S_{K_N},{\mathbb Q})\stackrel{\operatorname{res}_{\mathcal S_{K_N}}}{\longrightarrow} H^{2\xi-1}(\partial\mathcal S_{K_N},{\mathbb Q})\rightarrow H^{2\xi}_c(\mathcal S_{K_N},{\mathbb Q})\rightarrow 0 \end{equation*} and the codimension of $\operatorname{im}(\operatorname{res}_{\mathcal S_{K_N}})$ in $H^{2\xi-1}(\partial\mathcal S_{K_N},{\mathbb Q})$ equals \begin{equation*}
dim_{\mathbb Q} H^{2\xi}_c(\mathcal S_{K_N},{\mathbb Q})=|Cl_F ^{(N)}|. \end{equation*} \end{proof} \end{lemma}
This lemma tells us that $\operatorname{im}(\operatorname{Eis}_0 ^0)$ cannot give more than a subspace of codimension $|Cl_F ^{(N)}|$ in the cohomology of the boundary in cohomological degree $2\xi-1$. In particular, $\rho_{0,-1}^0$ cannot be surjective.
\subsection{Determination of the image}
Now we can completely determine the image of our polylogarithmic Eisenstein operator. \begin{lemma}\label{im_hor}
$\operatorname{im}(\rho_{m,n}^0)=\ker(\Psi_{m,n+1|{\mathbb Q}})$ \begin{proof} It suffices to prove the case $m=0$ and $n=-1$ with complex coefficients. Consider the epimorphism (the map is split) \begin{equation*}
\Psi^{K_N}_{0,0}=\Psi_{0,0}:\left(\operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}\bigoplus_{\phi:\ type(\phi)=\gamma_{0,0},\tilde{\phi}_{f|Z({\mathbb R})=1}}{\mathbb C} \cdot\tilde\phi_f\right)^{K_N}\rightarrow \end{equation*} \begin{equation*}
\left(\bigoplus_{\phi:\ type(\phi)=\gamma_{0,0},\tilde{\phi}_{f|Z({\mathbb R})=1},\phi_{|T^1({\mathbb A})}=\left\|\ \cdot\ \right\|^2}{\mathbb C}\cdot S(\phi)\right)^{K_N} \end{equation*} Any $\phi $ on the right-hand side is of the form \begin{equation*}
\phi(t_1,t_2)=\eta(t_1t_2)\left\|\frac{t_2}{t_1}\right\| \end{equation*}
and $\eta$ has to be trivial on $U_N$. Therefore $\eta\in \widehat{Cl_F ^{K_N}}$ and the number of different $\phi$ on the right-hand side is exactly $|Cl_F^{K_N}|$. So the codimension of $\ker(\Psi^{K_N})$ inside \begin{equation*}
\left(\operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}\bigoplus_{\phi:\ type(\phi)=\gamma_{0,0},\tilde{\phi}_{f|Z({\mathbb R})=1}}{\mathbb C} \cdot\tilde\phi_f\right)^{K_N} \end{equation*}
is $|Cl_F^{K_N}|$. By \Cref{ker(Psi)} we know that $\ker(\Psi^{K_N}_{0,0})\subset \operatorname{im}(\rho_{0,-1}^0)^{K_N}$, but \begin{equation*} \operatorname{im}(\rho_{0,-1}^0)^{K_N}\subset \operatorname{im}(\operatorname{res}_{\mathcal S_{K_N}}) \end{equation*}
and $\operatorname{im}(\operatorname{res}_{\mathcal S_{K_N}})$ has already codimension $|Cl_F^{K_N}|$ inside \begin{equation*}
H^{2\xi-1}(\partial\mathcal S_{K_N},{\mathbb C})\cong\left(\operatorname{Ind}_{B({\mathbb A}_f)\times \pi_0(B({\mathbb R}))}^{G({\mathbb A}_f)\times \pi_0(G({\mathbb R}))}\bigoplus_{\phi:\ type(\phi)=\gamma_{0,0},\tilde{\phi}_{f|Z({\mathbb R})=1}}{\mathbb C} \cdot\tilde\phi_f\right)^{K_N}. \end{equation*} Therefore \begin{equation*} \ker(\Psi^{K_N}_{0,0})=\operatorname{im}(\rho_{0,-1}^0)^{K_N}= \operatorname{im}(\operatorname{res}_{\mathcal S_{K_N}}) \end{equation*} and consequently $\operatorname{im}(\rho_{0,-1}^0)= \ker(\Psi_{0,0})$. \end{proof} \end{lemma} \begin{theorem}\label{im_Eis} Let \begin{equation*} \operatorname{Eis}^k_q:\mathcal S(V({\mathbb A}_f),\mu^{\otimes n})\otimes \mathfrak H^{q*}\rightarrow H^{2\xi-1-q}(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1}) \end{equation*}
be the polylogarithmic Eisenstein operator defined in \Cref{Eis^k _q}. The image of this operator is isomorphic to $\ker(\Psi_{m,n+1|{\mathbb Q}})\otimes\mathcal H(T/Z)^{\xi-1-q}$. \begin{proof} By \Cref{pol_Eis} and \Cref{Harder} we know that $\operatorname{res}_\mathcal S: \operatorname{im}{\operatorname{Eis}^k_q}\rightarrow \operatorname{im}(\operatorname{res}_\mathcal S\circ \operatorname{Eis}^k _q)$ is an isomorphism. By \Cref{hor} we have the isomorphism \begin{equation*}
\operatorname{im}(\operatorname{res}_\mathcal S\circ \operatorname{Eis}^k _q)\cong \operatorname{im}(\rho_{m,n}^0)\otimes\mathcal H(T/Z)^{\xi-1-q}. \end{equation*}
\Cref{im_hor} tells us $\ker(\Psi_{m,n+1|{\mathbb Q}})=\operatorname{im}(\rho_{m,n}^0)$, from which the theorem follows. \end{proof} \begin{corollary}
If $k>0$, then $\operatorname{im}(\operatorname{Eis}^k_q)=H^{2\xi-1-q}_{\operatorname{Eis}}(\mathcal S,\operatorname{Sym}^k\mathcal H^\prime\otimes \mu^{\otimes n+1})$. Moreover, $\operatorname{im}(\operatorname{Eis}^0_0)=H^{2\xi-1}_{\operatorname{Eis}}(\mathcal S,\mu^{\otimes n+1})$ and $\operatorname{im}(\operatorname{Eis}^0_q)^{K_N}\subset H^{2\xi-1-q}_{\operatorname{Eis}}(\mathcal S_{K_N},\mu^{\otimes n+1})$ is a subspace of codimension $|Cl_F ^{K_N}|$, if $q>0$. \begin{proof} This is just \Cref{im_Eis} together with \cite{Ha1} Theorem 2. For the calculation of the codimension see also the proof of \Cref{im_hor}. \end{proof} \end{corollary} \end{theorem}
\cleardoublepage \addcontentsline{toc}{chapter}{Bibliography}
\end{document} | arXiv |
The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$?
The harmonic mean of $x$ and $y$ is equal to $\frac{1}{\frac{\frac{1}{x}+\frac{1}{y}}2} = \frac{2xy}{x+y}$, so we have $xy=(x+y)(3^{20}\cdot2^{19})$, and by SFFT, $(x-3^{20}\cdot2^{19})(y-3^{20}\cdot2^{19})=3^{40}\cdot2^{38}$. Now, $3^{40}\cdot2^{38}$ has $41\cdot39=1599$ factors, one of which is the square root ($3^{20}2^{19}$). Since $x<y$, the answer is half of the remaining number of factors, which is $\frac{1599-1}{2}= \boxed{799}$. | Math Dataset |
The number of invariant subspaces under a linear operator on finite vector spaces
AMC Home
Codes from the incidence matrices and line graphs of Hamming graphs $H^k(n,2)$ for $k \geq 2$
May 2011, 5(2): 395-406. doi: 10.3934/amc.2011.5.395
On the weight distribution of codes over finite rings
Eimear Byrne 1,
School of Mathematical Sciences, University College Dublin, Springfield, MO 65801-2604, United States
Received May 2010 Revised November 2010 Published May 2011
Let $R>S$ be finite Frobenius rings for which there exists a trace map $T:$ S$R \rightarrow$S$R$. Let $C$f,s$:=\{x \mapsto T(\alpha x + \beta f(x)) : \alpha, \beta \in R \}$. $C$f,s is an $S$-linear subring-subcode of a left linear code over $R$. We consider functions $f$ for which the homogeneous weight distribution of $C$f,s can be computed. In particular, we give constructions of codes over integer modular rings and commutative local Frobenius that have small spectra.
Keywords: Ring-linear code, weight distribution, character module., homogeneous weight.
Mathematics Subject Classification: Primary: 11T71; Secondary: 14G5.
Citation: Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395
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\begin{document}
\title{The Message of the Quantum?}
\classification{
03.65.Ta.
} \keywords{
Reality and information; no-hidden-variables theorems;
determinism and quantum mechanics.
}
\author{Martin Daumer}{address={Sylvia Lawry Centre for
Multiple Sclerosis Research, Hohenlindenerstr.\ 1,
81677 M\"unchen, Germany.\\
e-mail: [email protected].}} \author{Detlef D\"urr}{address={Mathematisches Institut,
Ludwig-Maximilians-Universit\"at, Theresienstr.\ 39, 80333
M\"unchen, Germany.\\ e-mail: [email protected]}} \author{Sheldon Goldstein}{address={Departments of Mathematics and Physics,
Hill Center, Rutgers, The State University of New
Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA.\\
e-mail: [email protected]}} \author{Tim Maudlin}{address={Department of Philosophy, Davison Hall,
Rutgers, The State University of New Jersey, 26 Nichol Avenue,
New Brunswick, NJ 08901-1411, USA.\\
e-mail: [email protected]}} \author{Roderich Tumulka}{address={Mathematisches Institut,
Eberhard-Karls-Universit\"at, Auf der Morgenstelle 10, 72076
T\"ubingen, Germany.\\ e-mail:
[email protected]}} \author{Nino Zangh\`\i}{address={Dipartimento di Fisica dell'Universit\`a di
Genova and INFN sezione di Genova, Via Dodecaneso 33, 16146
Genova, Italy.\\ e-mail: [email protected]}}
\begin{abstract} We criticize speculations to the effect that quantum mechanics is fundamentally about information. We do this by pointing out how unfounded such speculations in fact are. Our analysis focuses on the dubious claims of this kind recently made by Anton Zeilinger. \end{abstract}
\maketitle
Quantum theory has always invited rather extreme speculations about the nature of physical reality. John Wheeler \cite{wheeler}, for example, famously conjectured that quantum mechanics suggests a ``participatory universe'' in which the present observations of experimentalists can give ``tangible reality'' to the distant past, that current actions can somehow \textit{produce} the past physical structure of the universe, rather than merely \textit{inform} us about it. It is a bold speculation, but one underpinned by nothing in quantum mechanics itself. Neither the experimental consequences of quantum mechanics nor any presently existing precise understanding of the theory support this astonishing suggestion. ``Quantum mechanics'' cannot justify this speculation because precisely formulated theories that recover the quantum mechanical predictions do not posit any such backward effect. So one sure check on claims about ``the lesson of quantum theory'' can be obtained by considering precise theories that recover all of the quantum predictions that have ever been verified. Several such theories exist.
Wheeler's rather obscure ideas appear to live on today in the remarkable suggestion that physics is only about information or, even more astonishingly, that the physical world itself just \emph{is} information. The first suggestion would seem to contradict the everyday belief that physics is concerned with the physical structure of objects and the laws governing that structure: it is, therefore, about molecules and atoms and stars and electrons, among other things. Electrons can be \emph{used} in systems that convey information, as in telephone lines, but that is a rather specialized set of circumstances, and physics should cover all of the universe, not just special systems. As for the suggestion that the physical world just \emph{is} information, the suggestion sounds more mystical than scientific. If it were to be put forward as a serious proposal it would need both clarification and powerful justification. Unfortunately, when the topic of discussion is quantum theory, basic standards of clarity and argumentation seem to be abandoned, even in the most prestigious journals.
A conspicuous example has recently been published by Anton Zeilinger. He has put forward, in his essay ``The message of the quantum'' \cite{Zei05}, some thoughts about the relationship between reality and information, very much in the spirit of the traditional ``Copenhagen'' view of quantum mechanics. He has accompanied his opinion with his personal summary of the conclusions that one should draw from quantum mechanics at the centennial of Einstein's annus mirabilis. Unfortunately, his claims are at best dubious, and most of them are simply wrong.
Zeilinger writes that ``The discovery that individual events are irreducibly random is probably one of the most significant findings of the twentieth century.'' He claims, in other words, that determinism has been refuted, that it has been proven that (some) individual events in the quantum world are irreducibly random, rather than merely seeming random because of our ignorance. This conclusion has been challenged, most famously by Einstein. On what basis does Zeilinger conclude that Einstein was wrong? He presumably relies on the various no-go or no-hidden-variables theorems---of von Neumann, Bell, Kochen and Specker, and the like---which are supposed to show that quantum randomness can't be regarded as arising merely from ignorance. However, in his seminal paper \cite{Bell87b} ``On the problem of hidden variables in quantum mechanics,'' John Bell has shown that these theorems involve unwarranted assumptions, and thus don't justify the rejection of Einstein's view about the origin of quantum randomness.
Perhaps Zeilinger merely means that the predictions of quantum theory---or at least the experimental facts on which quantum mechanics is based---strongly support a non-deterministic formulation. But this view is easily refuted by the counter-example provided by Bohmian mechanics \cite{Bohm52,Gol01}, a theory describing the deterministic evolution of particles that accounts for all of Zeilinger's examples and indeed all of the phenomena of nonrelativistic quantum mechanics, from spectral lines to the two-slit experiment and random decay times. Thus, the experimental facts of quantum mechanics do not establish indeterminism. At best, which explanation of the experimental facts to prefer, Bohm's simple deterministic one or the convoluted indeterministic one of the Copenhagen view, remains our theoretical choice.
Zeilinger suggests that ``one could find comfort'' in the idea of determinism, if it were tenable. This suggestion, whatever its merits, gives entirely the wrong impression of the main motivation of the critics of the Copenhagen view. David Bohm and John Bell \cite{Bell87b}, two of its leading critics, did not hesitate to use stochastic theories---with irreducible randomness---when that served a purpose. Even Einstein, often inappropriately depicted as a stubborn adherent of determinism claiming that ``God does not play dice,'' made it clear in his ``Reply to Critics'' \cite{Ein49} that he rejected the Copenhagen view not because of its indeterminism but because it fails to describe quantum phenomena in terms of objective events occurring independently of subjective perceptions.
Zeilinger writes that ``John Bell showed that the quantum predictions for entanglement are in conflict with local realism.'' In fact, realism was not among the assumptions Bell used for deriving the conflict with quantum mechanics, even though realism about spin observables---which is much more than realism in general---occurred in the argument as an implication of locality. What Bell proved is that the predictions of quantum theory for spin correlations are incompatible with locality, i.e., that quantum mechanics is irreducibly \emph{nonlocal}, a point that Bell \cite{Bell87b} repeatedly stressed.
In Zeilinger's view, Bell's result suggests, not that there is nonlocality, but that ``the concept of reality itself is at stake.'' What that is supposed to mean is left vague, but it is hard to see what the meaning could be that does not ultimately lead to the view that nothing exists objectively outside our minds. That is not a scientific view and there is nothing in Bell's result to support it. And contrary to Zeilinger's claim, it is not supported by the Kochen--Specker theorem either, as Bell was the first to point out: It cannot be maintained that the Kochen-Specker paradox supports the notion that there is a problem with ``the concept of reality itself'' when there are perfectly realistic theories, such as Bohmian mechanics or the Ghirardi--Rimini--Weber version of quantum mechanics involving spontaneous random collapse of the wave function \cite{grw}, that account for all of the experimental facts on which that paradox is based.
Interestingly, Zeilinger draws the correct moral from the Kochen--Specker paradox when he writes: ``even for single particles, it is not always possible to assign definite measurement outcomes independently of and prior to the selection of specific measurement apparatus in the specific experiment.'' That is, in giving a physical account of a measurement, we must take account of the exact physics of the experimental situation. Indeed, Bohmian mechanics and the Ghirardi--Rimini--Weber version of quantum mechanics allow us to do precisely this, since they do not postulate some special physics for measurements. And once done, all the predictions come out right.
Most baffling, Zeilinger suggests that ``the distinction between reality and our knowledge of reality, between reality and information, cannot be made.'' After such a counterintuitive assertion, we naturally expect to find a mighty argument in its behalf. Here, however, is Zeilinger's: ``There is no way to refer to reality without using the information we have about it.'' In other words, what we can say about reality, or better what we can know about reality, must correspond to our information about reality. In other words, what we know about reality must conform to what we know about reality. Does Zeilinger really believe that a tautology such as this can have interesting consequences?
The very concepts of knowledge and information imply a special kind of relationship between different things, appropriate correlations between a knower and what is known. Thus ``the distinction between reality and our knowledge of reality'' not only can be made; {\em it must be made if the notions of knowledge and information are to have any meaning in the first place.}
At a time when the forces of obfuscation in America are engaged in a campaign against the theory of evolution on behalf of Intelligent Design, it is perhaps worth asking Zeilinger how the idea that there is no difference between information and reality can be compatible with the emergence of information processing systems such as we are from a lifeless reality. And it is perhaps also worth asking the editors of Nature how, at a time when, rightly, papers on Intelligent Design are consistently rejected by peer-reviewed journals, an essay like Zeilinger's is not.
\begin{theacknowledgments}
M.~Daumer is supported in part by the German National Science Foundation
DFG, SFB 386. S.~Goldstein is supported in part by NSF Grant DMS-0504504.
N.~Zangh\`\i\ is supported in part by INFN. \end{theacknowledgments}
\end{document} | arXiv |
Spectroscopy of short-lived radioactive molecules
The ultrafast X-ray spectroscopic revolution in chemical dynamics
Peter M. Kraus, Michael Zürch, … Stephen R. Leone
Rotational spectroscopy of cold and trapped molecular ions in the Lamb–Dicke regime
S. Alighanbari, M. G. Hansen, … S. Schiller
Photo-excitation of long-lived transient intermediates in ultracold reactions
Yu Liu, Ming-Guang Hu, … Kang-Kuen Ni
Detection of metastable electronic states by Penning trap mass spectrometry
R. X. Schüssler, H. Bekker, … K. Blaum
Direct observation of a Feshbach resonance by coincidence detection of ions and electrons in Penning ionization collisions
Baruch Margulis, Julia Narevicius & Edvardas Narevicius
Spectroscopic probes of quantum gases
Chris J. Vale & Martin Zwierlein
Observation of the proton emitter $${}_{\,57}^{116}$$ 57 116 La59
Wei Zhang, Bo Cederwall, … Robert Wadsworth
Determining the nature of quantum resonances by probing elastic and reactive scattering in cold collisions
Prerna Paliwal, Nabanita Deb, … Edvardas Narevicius
Laser spectroscopic characterization of the nuclear-clock isomer 229mTh
Johannes Thielking, Maxim V. Okhapkin, … Ekkehard Peik
R. F. Garcia Ruiz ORCID: orcid.org/0000-0002-2926-55691,2,
R. Berger ORCID: orcid.org/0000-0002-9107-27253,
J. Billowes4,
C. L. Binnersley4,
M. L. Bissell4,
A. A. Breier ORCID: orcid.org/0000-0003-1086-90955,
A. J. Brinson ORCID: orcid.org/0000-0002-9551-52982,
K. Chrysalidis ORCID: orcid.org/0000-0003-2908-84241,
T. E. Cocolios ORCID: orcid.org/0000-0002-0456-78786,
B. S. Cooper4,
K. T. Flanagan4,7,
T. F. Giesen5,
R. P. de Groote8,
S. Franchoo9,
F. P. Gustafsson6,
T. A. Isaev10,
Á. Koszorús6,
G. Neyens1,6,
H. A. Perrett4,
C. M. Ricketts4,
S. Rothe ORCID: orcid.org/0000-0001-5727-77541,
L. Schweikhard11,
A. R. Vernon4,
K. D. A. Wendt12,
F. Wienholtz1,11,
S. G. Wilkins ORCID: orcid.org/0000-0001-8897-72271 &
X. F. Yang ORCID: orcid.org/0000-0002-1633-400013
Electronic structure of atoms and molecules
Exotic atoms and molecules
Experimental nuclear physics
Molecular spectroscopy offers opportunities for the exploration of the fundamental laws of nature and the search for new particle physics beyond the standard model1,2,3,4. Radioactive molecules—in which one or more of the atoms possesses a radioactive nucleus—can contain heavy and deformed nuclei, offering high sensitivity for investigating parity- and time-reversal-violation effects5,6. Radium monofluoride, RaF, is of particular interest because it is predicted to have an electronic structure appropriate for laser cooling6, thus paving the way for its use in high-precision spectroscopic studies. Furthermore, the effects of symmetry-violating nuclear moments are strongly enhanced5,7,8,9 in molecules containing octupole-deformed radium isotopes10,11. However, the study of RaF has been impeded by the lack of stable isotopes of radium. Here we present an experimental approach to studying short-lived radioactive molecules, which allows us to measure molecules with lifetimes of just tens of milliseconds. Energetically low-lying electronic states were measured for different isotopically pure RaF molecules using collinear resonance ionisation at the ISOLDE ion-beam facility at CERN. Our results provide evidence of the existence of a suitable laser-cooling scheme for these molecules and represent a key step towards high-precision studies in these systems. Our findings will enable further studies of short-lived radioactive molecules for fundamental physics research.
Molecular systems provide a versatile physical environment in which to study the fundamental symmetries of nature and the interactions and properties of subatomic particles1,2,12,13. Among the four known fundamental forces, the weak force is the only one that is known to violate symmetry with respect to spatial inversion of all particle coordinates (known as parity violation), giving rise to various intriguing phenomena. Some of these parity-violating effects have been measured with high accuracy in atomic systems13,14,15, contributing to the most stringent low-energy tests of the Standard Model of particle physics. In certain molecules, effects resulting from both parity violation (P-odd) and time-reversal violation (T-odd) are considerably enhanced with respect to atomic systems5,7,8,13,16, offering the means to explore unknown aspects of the fundamental laws of physics. The strengths of these interactions scale with atomic number, nuclear spin and nuclear deformation, and so molecular compounds of heavy radioactive nuclei are predicted to exhibit unprecedented sensitivity, with an enhancement of more than two orders of magnitude for effects that are P-odd or simultaneously P- and T-odd5,6,7,8,17,18,19,20.
However, the experimental knowledge of radioactive molecules is scarce21, and quantum chemistry calculations often constitute the only source of information. Molecules possess complex quantum level structures, which renders spectroscopy of their structure considerably more challenging compared to atoms. Moreover, major additional experimental challenges must be overcome to study molecules containing heavy and deformed nuclei, which can have lifetimes of just a few milliseconds. These radioactive nuclei are very rare in nature or do not occur naturally and so must be produced artificially at specialized facilities, such as at the Isotope Separator On-line Device (ISOLDE) at CERN. Furthermore, molecules containing short-lived isotopes can only be produced in quantities smaller than 10−8 g (typically with rates of less than 106 particles s−1). Thus, spectroscopic studies require particularly sensitive experimental techniques adapted to the properties of radioactive ion beams and the conditions present at radioactive-beam facilities. Here, we present an approach for performing laser spectroscopy of short-lived radioactive molecules, using the highly sensitive collinear resonance ionization method22. These results provide the first spectroscopic information of RaF, including isotopologues composed of radioactive isotopes with lifetimes as short as a few days. To our knowledge, this is the first laser spectroscopy study performed on a molecule containing a short-lived isotope. Moreover, this experimental scheme can be applied to study other radioactive molecules, even those composed of isotopes with lifetimes as short as a few tens of milliseconds.
Since the direct cooling of diatomic molecules with lasers23 was experimentally demonstrated24, there has been a wealth of studies on laser-cooling techniques and applications in molecular physics25,26,27,28,29,30,31. In contrast to other heavy-atom molecules, RaF is predicted to have highly closed excitation and re-emission optical cycles, which would make it ideal for laser cooling and trapping6. Moreover, owing to the recently discovered pear-shaped nuclear deformation of certain radium isotopes11, the interactions of the electrons with the P-odd nuclear anapole moment as well as with the P,T-odd nuclear Schiff and magnetic quadrupole moments are predicted to be enhanced by more than two orders of magnitude4,5,19,32. Hence, these molecules could provide a unique environment in which to measure these symmetry-violating nuclear moments.
Figure 1 shows a diagram of the experimental setup used to produce and study the RaF molecules. As a first step, radium isotopes were produced by diffusion out of an irradiated target (see Methods section 'Production of RaF molecules'). RaF+ molecular ions were formed upon injection of CF4 gas into the target environment. The molecular ions were extracted from the ion source by applying an electrostatic field, and molecules containing one specific radium isotope were selected with a high-resolution magnetic mass separator (Δm/m ≈ 1/2,000). The ions were collisionally cooled in a radio-frequency quadrupole (RFQ) trap filled with helium gas at room temperature (about 300 K). After up to 10 ms of cooling time, bunches of RaF+ with a 4-μs temporal width were released and accelerated to 39,998(1) eV, before entering into the Collinear Resonance Ionisation Spectroscopy (CRIS) setup22,33,34. At the CRIS beam line, the ions were first neutralized in-flight by passing through a collision cell filled with a sodium vapour, inducing charge exchange according to the reaction RaF+ + Na → RaF + Na+. As the ionization energy of RaF is estimated to be close to that of sodium (5.14 eV)35, the neutralization reaction dominantly populates the RaF X2Σ+ electronic ground state. Molecular pseudo-orbitals obtained from one-component open-shell (neutral) or closed-shell (ion) restricted Hartree–Fock calculations with an energy-consistent effective core potential on radium are shown schematically in Fig. 1 (bottom). The lowest unoccupied molecular orbital in RaF+, which is mainly of non-bonding character, becomes occupied by an unpaired electron (symbolized in Fig. 1 by a red sphere together with an arrow representing the electron spin) upon neutralization. This is shown schematically as an isodensity, with lobes in slightly transparent blue and transparent red indicating different relative phases of the single-electron wavefunction.
Fig. 1: Experimental scheme for the production and study of short-lived radioactive molecules.
Radioactive radium isotopes were created by impinging 1.4-GeV protons from the CERN Proton Synchrotron Booster (PSB) on a uranium carbide (UCx) target. Radium monofluoride cations (RaF+) were produced by passing tetrafluoromethane (CF4) gas through the activated UCx target at 1,300 °C. Molecular ions were extracted from the source, mass-selected and injected into a helium-filled RFQ trap, where they were accumulated for 10 ms. Bunches of molecular ions were extracted and neutralized in flight by charge exchange with neutral sodium atoms. Neutral RaF molecules were overlapped with different laser beams (step 1, TiSa, Dye1 and Dye2, and step 2, a 355-nm laser; see Methods section 'Laser setup') in a collinear geometry. Resonantly reionized molecules were deflected onto a particle detector. The resonance ionization scheme is shown at top right. At bottom, molecular orbitals are shown schematically. Nuclear positions within the molecules are coarsely indicated by a grey sphere (Ra) and green sphere (F), and the sigma bond between the atoms is indicated by the grey cylinders. Further details are provided in 'Experimental scheme'.
After the charge-exchange reaction, non-neutralized RaF+ ions were deflected out of the beam, and the remaining bunch of neutral RaF molecules was overlapped in time and space by several (pulsed) laser beams in a collinear arrangement, along the ultrahigh-vacuum (10−10 mbar) interaction region of 1.2-m length. Laser pulses (step 1) of tunable wavelength were used to resonantly excite the transition of interest, and a high-power 355-nm laser pulse (step 2) was used to subsequently ionize the excited RaF molecules into RaF+ (see Fig. 1, top). The resonantly ionized molecules were then separated from the non-ionized molecules by deflecting the ions onto a particle detector. When the excitation laser is on resonance with a transition in the molecule (step 1 in Fig. 1), the second laser pulse ionizes the molecule, producing a signal at the detector. Molecular excitation spectra were obtained by monitoring the ion counts as a function of the wavenumber of the first laser.
Only theoretical predictions were available for the excitation energies of RaF, and so finding the transition experimentally required scanning a large wavelength range (>1,000 cm−1). The prediction for the A2Π1/2−X2Σ+ (0, 0) transition, for example, was 13,300 cm−1, with an accuracy estimated to be within 1,200 cm−1 (refs. 6,32). Given the bandwidths of the commonly available lasers (<0.3 cm−1), the scan of such a large wavelength region on samples produced at rates below 106 molecules s−1 represented a major experimental challenge. To optimize the search of molecular transitions, three broadband lasers were scanned simultaneously and both collinearly and anti-collinearly (see Methods section 'Laser setup').
The predicted region for the A2Π1/2 ← X2Σ+ transition was scanned at a speed of 0.06 cm−1 s−1, covering a range of 1,000 cm−1 in about 5 h, using the six simultaneously applied scanning regions. After a few hours of scanning on a beam of 226RaF, a clear sequence of vibronic absorption signals was recorded. The measured spectrum assigned to the (v′, v″) vibrational transitions (0, 0), (1, 1), (2, 2), (3, 3) and (4, 4) of the A2Π1/2−X2Σ+ band system is shown in Fig. 2a. Weaker band structures, that were found at about +440 cm−1 and −440 cm−1 with respect to the (0, 0) band, were assigned to the Δv = ±1 transitions (v′, v″) = (1, 0), (2, 1), (3, 2), (4, 3), (5, 4) and (v′, v″) = (0, 1), (1, 2), respectively (Fig. 2b, c). The quantum number assignment for Δv = −1 is tentative, owing to the highly dense structure of overlapping vibronic bands.
Fig. 2: Examples of vibronic spectra measured for 226RaF.
a–f, The counts on the particle detector were measured as a function of the laser wavenumber of the resonant step. A fixed wavelength (355 nm) was used for the ionization step. a, The observed peaks corresponding to the vibronic spectra of the Δv = 0 band system of v″ = 0, 1, 2, 3, 4, scanned by the grating Ti:sapphire laser. b, c, The pulsed dye laser was used to scan electronic transitions in different wavelength ranges: the Δv = +1 band system of the A2Π1/2 ← X2Σ+ transition with v″ = 0, 1, 2, 3, 4 (b) and the (v′, v″) = (0, 1) and (1, 2) band. d–f, The corresponding transitions to other electronic states: A2Π3/2 ← X2Σ+ (d), B2Δ3/2 ← X2Σ+ (tentatively assigned; e) and C2Σ+ ← X2Σ+ (f). The shape of the spectra is due to population distribution of different rotational states. The solid lines show the fit with skewed Voigt profiles. g, Scheme of the molecular energy levels. The estimated upper limit of the ionization potential (IP) is indicated. Three essential properties for laser cooling of RaF molecules were identified: 1) the short lifetime of the excited states 2Π1/2 (T1/2 < 50 ns), which will allow for the application of strong optical forces; 2) dominant diagonal transitions, (Δv = 0)/(Δv = ±1, Δv = 0) > 0.97, indicating a large diagonal Franck–Condon factor; and 3) the expected low-lying electronic states B2Δ3/2, A2Π3/2 and C2Σ+ were found to be above the A2Π1/2 states, which will enable efficient optical-cooling cycles. Wavenumbers in the spectra are given in the rest frame of the molecule. In a–f, the error bars show the statistical uncertainties (1 standard deviation) for the number of resonantly ionized molecules obtained within each laser frequency interval.
In addition to the A2Π1/2−X2Σ+ band system, we found spectroscopic signatures of electronic transitions to higher-lying states. Some examples of recorded spectra are shown in Fig. 2d–f, along with the energy-level scheme. We assign the observed transitions as follows: 1) The band system around 15,325 cm−1 (Fig. 2d) is attributed to the A2Π3/2−X2Σ+ transition, owing to the complex rovibrational structure expected to arise from the intense satellites that are possible in these transitions. Because the bands are comparatively strong, they are assigned to the Δv = 0 band system. Although the individual assignments to vibrational transitions must be considered to be tentative, as per the congested structure of the Franck–Condon profile, the Δv = 0 assignment is substantiated because no additional structure was located within a relative range of −400 to +400 cm−1. The band system located around 15,143 cm−1 (Fig. 2e) is tentatively assigned to the B2Δ3/2−X2Σ+ transition by virtue of the good agreement with the computed excitation energies to the Ω = 3/2 state of mixed Δ/Π character6,32. This mixing provides intensity to the one-photon transition from a Σ state into the Δ manifold. The computed Born–Oppenheimer potentials for this Ω = 3/2 state and the electronic ground state are, however, highly parallel, which would suggest a sparser Franck–Condon profile than was observed experimentally. However, we note that the related B2Δ3/2−X2Σ+ transition in BaH and BaD was reported to have a perturbed character owing to mixing between electronic levels36. Thus, in the present case, a vibrational profile that is richer than expected from adiabatic potentials cannot be ruled out a priori. The band system with origin at 16,175 cm−1 (Fig. 2f) is assigned to the C2Σ+−X2Σ+ transition on the basis of the observed Franck–Condon profile, which is in good agreement with the computed harmonic vibrational energy spacings as well as the expected intensity distribution, and is in a wavenumber region that is only slightly lower than predicted6,32. All measured and assigned vibronic bands of the four electronic transitions are listed in Table 1.
Table 1 Measured vibronic transitions of 226RaF from the X2Σ+ electronic ground state to the excited A2Π and B2Δ states
The measured A2Π1/2−X2Σ+ (0, 0) band centre, \({\tilde{{\mathscr{T}}}}_{{\rm{e}}}\) = 13,287.8(1) cm−1 is in excellent agreement with the ab initio calculated value of 13,300(1,200) cm−1 (ref. 32). In accordance with theoretical predictions6, we found vibronic transitions with Δv = 0 to be much stronger than those with Δv = ±1. For most of the measurements, the power density used for the resonant step was 100(5) μJ cm−2 per pulse, as measured at the entry window of the beam line. Reducing the power by 50% did not reduce the resonant ionization rate, indicating that these transitions were measured well above saturation. The much weaker vibrational transitions with Δv = ±1 were scanned with a pulsed dye laser of 500(5) μJ cm−2 power density per pulse (bandwidth of 0.1 cm−1). The Δv = ±1 transitions were measured well above saturation and with laser beams of different characteristics, and so a precise estimation of the Franck–Condon factors could not be obtained. Instead, a lower limit of 0.97 for the peak intensity ratio I(0, 0)/I(0, 1) was derived, indicating highly diagonal Franck–Condon factors, an essential property for laser cooling6.
By measuring the resonant ionization rate for different time delays between the excitation and ionization laser pulses, we obtained an upper limit for the lifetime of the excited state 2Π1/2 (v′ = 0): T1/2 ≤ 50 ns. The measurements were performed with the wavenumber of the resonant laser fixed at the resonance value of the transition (v′, v″) = (0, 0). The resonant ionization rate dropped by more than 70% for delays above 50 ns. This short lifetime corresponds to a large spontaneous decay rate (>2 × 107 s−1), which would allow for the application of strong optical forces for laser cooling. An additional concern for the suitability of laser cooling is related to the existence of metastable states lying energetically below the 2Π1/2 level, which could prevent the application of a closed optical-cooling loop, a major problem encountered for BaF (ref6.). In contrast to BaF, all other predicted electronic states (2Π3/2, 2Δ3/2 and 2Σ) in RaF were found to be energetically above the 2Π1/2 state, indicating that its electronic structure will allow for efficient optical-cooling cycles.
From combination differences of energetically low-lying vibronic transitions in the band system A2Π1/2−X2Σ+, we have derived experimental values for the harmonic frequency, \({\tilde{\omega }}_{{\rm{e}}}\), and the dissociation energies, \({\tilde{{\mathscr{D}}}}_{{\rm{e}}}\), using a Morse potential approximation. Results are given in Table 2, and further details of the analysis can be found in Methods section 'Spectroscopic analysis'.
Table 2 226RaF Morse potential parameters for X2Σ+ electronic ground and A2Π1/2 excited states
Furthermore, we measured the A2Π1/2 ← X2Σ+ vibronic spectra of 226RaF and the short-lived isotopologues 223RaF, 224RaF, 225RaF, and 228RaF (Fig. 3). All vibrational transitions were clearly observed, including those of the molecule with the shortest-lived radium isotope studied, 224RaF (T1/2 = 3.6 d). An on-line irradiation of the target material will enable the study of molecules containing isotopes with lifetimes as short as a few tens of milliseconds. The main limitation is dictated by the release from the target and the time spent in the RFQ trap (>5 ms). Future high-resolution measurements will enable studies of nuclear structure changes resulting from different isotopes and nuclear spins.
Fig. 3: Vibronic spectra measured for different isotopologues of RaF.
Measured vibronic absorption spectra for the A2Π1/2 ← X2Σ+ transition are shown for the isotopologues 223RaF, 224RaF, 225RaF, 226RaF and 228RaF. Wavenumber values are relative to the transition (0, 0) of 226RaF.
Conclusions and future perspectives
In summary, this Article presents an experimental approach for performing laser spectroscopy studies of molecules containing radioactive nuclei, which are typically produced at rates lower than 106 molecules s−1. Our results have established the energetically low-lying electronic structure of RaF, providing experimental evidence for the suitability of this diatomic molecule in a laser-cooling scheme. These findings are a pivotal step towards precision measurements in this system, which are expected to provide a highly sensitive environment for the exploration of physics beyond the Standard Model of particle physics.
Our experimental scheme can also be used to perform laser spectroscopy of a wide variety of neutral molecules and molecular ions, including those composed of isotopes with lifetimes of a few tens of milliseconds. Radioactive molecules can be precisely tailored to enhance their sensitivity to parity- and time-reversal-violating effects by introducing heavy and octupole-deformed nuclei. Moreover, by systematically replacing their constituent nuclei with different isotopes of the same element, both nuclear-spin-independent and nuclear-spin-dependent effects can be comprehensively studied. In addition, the present technique is applicable to other molecules of interest in studies of fundamental physics that are as yet experimentally unexplored, such as RaOH (ref. 37), RaO (ref. 18), RaH (ref. 17), AcF (ref. 38) and 229ThO (ref. 5).
In addition to the impact of our findings on quantum chemistry, nuclear structure and fundamental physics research, the ability to produce, mass-select and spectroscopically study short-lived radioactive molecules is of importance to other fields of research such as radiochemistry21 and astrophysics39,40. Laboratory measurements of the spectra of radioactive molecules of astrophysical interest will allow their unambiguous identification in future astronomical observations. Furthermore, the possibility of performing spectroscopy on fast molecular beams will enable sub-Doppler spectroscopy to be performed even on molecules created at high temperatures (>600 K). Thus, we expect our results will motivate further avenues of research at the increasingly capable radioactive-ion-beam facilities around the world.
Production of RaF molecules
Ra isotopes were produced 33 d before the laser-spectroscopy measurements by impinging 1.4 GeV protons on the cold UCx target material. The target was exposed to pulses of 1013 protons per pulse over a period of 2 d. After irradiation with a total of 8 × 1017 protons, the target was kept in a sealed chamber filled with Ar gas. After day 33, the target was connected to the High-Resolution Separator (HRS) front-end at ISOLDE. FLUKA41 simulations predicted 2 × 1013 atoms of 226Ra in the target material (7.5 × 10−9 g), following proton irradiation of a cold target. The target was pumped down to pressures below 10−5 mbar, and the target holder and ion source were gradually heated up to about 1,300 °C, in order for the Ra isotopes to diffuse towards the surface of the target material. A leak valve attached to the target was used to inject CF4 into the target environment. The CF4 molecules dissociate and react with atoms and molecules on the target surface until an equilibrium is reached. RaF molecules were formed by reactive collisions of CF4 molecules with Ra atoms present inside the irradiated target material.
According to thermodynamic equilibrium calculations42, RaF2 or RaF are expected to form, depending on the local temperature. Within the temperature gradient between the target (1,300 °C) and the ion source (2,000 °C), RaF2 fully reacts to form RaF. A measured ratio of the ion-beam intensity of Ra+ to RaF+ of less than 0.05 indicates that more than 95% of the Ra isotopes released from the target material are converted and extracted as molecules.
The 226RaF+(A = 245) beam extracted from the ISOLDE target unit was sent to the ISOLTRAP setup43, where the molecular ions were captured, cooled and bunched by a different RFQ trap and subsequently analysed using a multi-reflection time-of-flight mass spectrometer44. A measured mass spectrum is shown in Extended Data Fig. 1. After 1,000 revolutions in the device, a mass resolving power (R = m/Δm) of 1.7 × 105 was achieved, which allowed the isobaric beam composition to be analysed. The only mass peak detected was identified as the signal of 226Ra19F+, confirming the purity of the beam from ISOLDE.
The intensity of RaF+ molecules depends strongly on the target and ion source temperature. For a target temperature of 1,300 °C, a mean value of 2 × 107 molecules s−1 of 226RaF+ was measured after the mass separator. Depending on the molecular mass and beam intensity, the transmission efficiency through the RFQ trap varied from 15% to 30%. The ion-beam transmission from the ion trap to the interaction region was measured to be 25(5)%. The charge exchange cell vapour was heated to produce a measured neutralization rate of 30(5)%. Thus, we estimate that on average 5 × 104 neutral 226RaF molecules s−1 were delivered to be resonantly excited. From the analysis of the measured spectra it was concluded that the neutral molecules populate the low-lying vibrational states ν = 0, 1, 2, 3, 4 following a relative population of 0.47:0.29:0.13:0.05:0.03. Resonantly ionized molecules with rates of the order of 103 counts s−1 at the peak of the 0 ← 0 transition were measured at the particle detector. Future production of RaF+ molecular rates of the order of 109–1010 molecules s−1 is feasible using active proton irradiation45.
Laser setup
The resonance ionization schemes used for the study of RaF molecules are shown in Fig. 1. Three different laser systems were prepared to cover the scanning range from 12,800 cm−1 to 13,800 cm−1: 1) A dye-laser system (Dye1; Spectrolase 4000, Spectron) provided pulses of 100(5) μJ with a linewidth of 10 GHz (0.3 cm−1). 2) A dye laser (Dye2; Cobra, Sirah) with a narrower linewidth of 2.5 GHz (0.09 cm−1) produced pulses of similar energy. The lasers were loaded with either Styryl 8 or DCM dyes to provide wavenumber ranges 12,800–14,000 cm−1 and 15,150–16,600 cm−1, respectively. Both dye lasers were pumped by 532-nm pulses at 100 Hz, obtained from two different heads of a twin-head Nd:YAG laser (LPY 601 50–100 PIV, Litron). 3) A grating Ti:sapphire laser system with a linewidth of 2 GHz (0.07 cm−1) produced pulses of 20(1) μJ, pumped by 532-nm pulses at 1 kHz from a Nd:YAG laser (LDP-100MQ, LEE Laser). The non-resonant ionization step was obtained by 355-nm pulses of 30 mJ at 100 Hz, produced by the third-harmonic output of a high-power Nd:YAG laser (TRLi 250-100, Litron).
The release of the ion bunch was synchronized with the laser pulses by triggering the flash-lamps and Q-switch of the pulsed lasers with a digital delay pulse generator (Quantum Composers 9528).
The dye-laser wavelengths were measured with a wavelength meter (WS6-600 HighFinesse) and the Ti:sapphire laser wavelengths were measured by a wavelength meter (WSU-2 HighFinesse) calibrated by measuring a reference wavelength provided by a stabilized diode laser (DLC DL PRO 780, Toptica).
Collinear and anti-collinear excitation
For the initial peak searching, a zero-degree mirror at the end of the beam line was used to reflect the laser light anti-collinearly with respect to the travelling direction of the RaF bunch. Thus, each scanning laser covered two different wavenumber regions in the molecular rest frame, owing to the Doppler shift present for the fast RaF molecules. For a molecule travelling at velocity v, the laser wavenumber in the laboratory frame, \({\tilde{\nu }}_{0}\), is related to the wavenumber in the molecule rest frame, \(\tilde{\nu }\), by the expression \(\tilde{\nu }=\tfrac{1+\beta \cos \,\theta }{\sqrt{1-{\beta }^{2}}}{\tilde{\nu }}_{0}\), with β = v/c (c, speed of light in vacuum) and where θ is the angle between the direction of the laser beam and the velocity of the molecule. For RaF molecules at 39,998(1) eV (v ≈ 0.18 m μs−1), a difference of 15.7 cm−1 is obtained between the laser pulse sent out collinearly (cosθ = 1) and anti-collinearly (cosθ = −1) with respect to the direction of the velocity of the molecule.
Spectroscopic analysis
The peaks in the different spectra were identified by rebinning the spectra using coarse bin sizes with values up to 1 cm−1. Only groups of data points that were consistently observed with a 5-sigma significance above background were considered as candidates for transitions. The vibrational transitions in Fig. 2 show asymmetric line profiles with a maximum located towards higher wavenumbers. The band centres cannot be determined directly from the measured line profiles, and so we used the wavenumber positions of the maxima in our data analysis. Extended Data Table 1 lists the maximum peak positions and estimated uncertainties are given in parentheses. The wavenumber difference, \(\Delta \tilde{\nu }\), of vibrational levels in the electronic 2Σ+ ground state and in the 2Π1/2 excited state were derived from combination differences of the recorded 226RaF spectra (see Extended Data Table 1).
In our analysis we used vibrational energy terms Ev/(hc) of a Morse potential according to:
$${E}_{v}/(hc)={\tilde{\omega }}_{{\rm{e}}}\left(v+\frac{1}{2}\right)-\frac{{\tilde{\omega }}_{{\rm{e}}}^{2}}{4{\tilde{{\mathscr{D}}}}_{{\rm{e}}}}{\left(v+\frac{1}{2}\right)}^{2}$$
Energy-level differences
$$({E}_{v+1}-{E}_{v})/(hc)={\tilde{\omega }}_{{\rm{e}}}-\frac{{\tilde{\omega }}_{{\rm{e}}}^{2}}{2{\tilde{{\mathscr{D}}}}_{{\rm{e}}}}(v+1)$$
were used to derive the Morse potential parameters \({\tilde{\omega }}_{{\rm{e}}}\) and \({\tilde{{\mathscr{D}}}}_{{\rm{e}}}\) from a least-squares fit analysis. The derived energy-level differences are given in Extended Data Table 1, whereas Extended Data Table 2 contains the molecular parameters from the fit. The harmonic vibration frequencies \({\tilde{\omega }}_{{\rm{e}}}\) of the 2Σ+ and 2Π1/2 states are almost identical and correspond well to the theoretical predictions with a deviation of less than 5%; see Extended Data Table 2. The same holds for the estimated dissociation energy \({\tilde{{\mathscr{D}}}}_{{\rm{e}}}\), which is in better agreement with the values of ref. 6, as therein also the low-energy part of the potentials was used to estimate the dissociation energy.
In the case of the two low-lying 2Π fine-structure levels, the observed origins T0,0 agree well with the calculated values based on the Relativistic Correlation Consistent – Atomic Natural Orbital (RCC-ANO) basis set. From the energy difference of the fine-structure components the effective spin-orbital coupling parameter A is derived. For the 2Π states, the experimental value of 2,068(5) cm−1 is in good agreement with the calculated value. The band origins are in reasonable agreement with results from the RCC-ANO basis set calculation, if one attributes the Ω = 3/2 levels, which were computationally found to be of mixed Π3/2 and Δ3/2 character in this order of energies. A reverse assignment also gives better agreement with experiment. Calculations of the gas-phase bond lengths, dissociation energies and additional properties of RaF molecules have been reported8,32,35.
Examples of vibronic spectra measured for RaF molecules are included as source data with this Article. All other relevant data supporting the findings of these studies are available from the corresponding author upon request.
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This work was supported by the ERC Consolidator grant no. 648381 (FNPMLS); Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project no. 328961117 – SFB 1319 ELCH; STFC grants ST/L005794/1, ST/L005786/1 and ST/P004423/1, and Ernest Rutherford grant no. ST/L002868/1; projects from FWO-Vlaanderen, GOA 15/010 from KU Leuven and BriX IAP research programme no. P7/12; European Union grant agreement no. 654002 (ENSAR2); the Russian Science Foundation under grant no. 18-12-00227; and BMBF grants 05P15HGCIA and 05P18HGCIA. We thank J. P. Ramos, J. Ballof and T. Stora for their support in the production of RaF molecules. A.J.B. was suppported by the Henry W. Kendall (1955) Fellowship. We would also like to thank the ISOLDE technical group for their support and assistance. We thank D. Budker for comments and suggestions as well as A. Petrov for discussions on Δ states. R.B. acknowledges I. Tietje for early discussions on various experiments at CERN and thanks A. Welker for sharing knowledge on isotope production and separation as well as for initial discussions on the RaF studies. R.B. acknowledges discussions with K. Gaul on molecular properties and with D. Andrae on finite nuclear size effects. R.B. and T.A.I. acknowledge S. Hoekstra and L. Willmann for early discussions on the production of RaF. T.A.I. thanks A. Zaitsevskii for discussions on the coupled-cluster method.
CERN, Geneva, Switzerland
R. F. Garcia Ruiz, K. Chrysalidis, G. Neyens, S. Rothe, F. Wienholtz & S. G. Wilkins
Massachusetts Institute of Technology, Cambridge, MA, USA
R. F. Garcia Ruiz & A. J. Brinson
Fachbereich Chemie, Philipps-Universität Marburg, Marburg, Germany
R. Berger
Department of Physics and Astronomy, The University of Manchester, Manchester, UK
J. Billowes, C. L. Binnersley, M. L. Bissell, B. S. Cooper, K. T. Flanagan, H. A. Perrett, C. M. Ricketts & A. R. Vernon
Laboratory for Astrophysics, Institute of Physics, University of Kassel, Kassel, Germany
A. A. Breier & T. F. Giesen
KU Leuven, Instituut voor Kern- en Stralingsfysica, Leuven, Belgium
T. E. Cocolios, F. P. Gustafsson, Á. Koszorús & G. Neyens
Photon Science Institute, The University of Manchester, Manchester, UK
K. T. Flanagan
Department of Physics, University of Jyväskylä, Jyväskylä, Finland
R. P. de Groote
Institut de Physique Nucleaire d'Orsay, Orsay, France
S. Franchoo
NRC 'Kurchatov Institute'-PNPI, Gatchina, Russia
T. A. Isaev
Institut für Physik, Universität Greifswald, Greifswald, Germany
L. Schweikhard & F. Wienholtz
Institut für Physik, Johannes Gutenberg-Universität Mainz, Mainz, Germany
K. D. A. Wendt
School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing, China
X. F. Yang
R. F. Garcia Ruiz
J. Billowes
C. L. Binnersley
M. L. Bissell
A. A. Breier
A. J. Brinson
K. Chrysalidis
T. E. Cocolios
B. S. Cooper
T. F. Giesen
F. P. Gustafsson
Á. Koszorús
G. Neyens
H. A. Perrett
C. M. Ricketts
S. Rothe
L. Schweikhard
A. R. Vernon
F. Wienholtz
S. G. Wilkins
R.F.G.R. led the experiments and R.B. led the theoretical support for this work. R.F.G.R., R.B., C.L.B., M.L.B., K.C., B.S.C., K.T.F., R.P.d.G., S.F., F.P.G., Á.K., H.A.P., C.M.R., S.R., A.R.V., F.W. and S.G.W. performed the experiment. R.F.G.R., R.B., A.A.B., A.J.B. and T.F.G. performed the data analysis. R.F.G.R. prepared the figures. R.B. and T.A.I. performed theoretical predictions that motivated the experimental proposal and analysis of the results. R.F.G.R. and R.B. prepared the initial draft of the manuscript with input from A.A.B., A.J.B., K.T.F., T.F.G., T.A.I., G.N. and S.G.W. All authors discussed the results and contributed to the manuscript at different stages.
Correspondence to R. F. Garcia Ruiz or R. Berger.
Peer review information Nature thanks Michael Patzschke and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Extended data figures and tables
Extended Data Fig. 1 Time-of-flight spectrum measured at mass A = 245.
The time-of-flight spectrum of the 226RaF+ (A = 245) beam as delivered from ISOLDE after 1,000 revolutions in the multi-reflection time-of-flight mass spectrometer. A mass resolving power of 1.7 × 105 was achieved, which allowed the isobaric beam composition to be analysed. Only 226Ra19F+ ions were detected. The positions of the most probable accompanying ions are highlighted by dotted vertical lines.
Extended Data Table 1 226RaF vibrational transitions in the electronic X2Σ+ ground state and the A2Π1/2 excited state derived from combination differences
Extended Data Table 2 Molecular parameters of RaF from vibrational analysis of the electronic ground state (X2Σ+) and excited states (A2Π, B2Δ and C2Σ+)
Source Data Extended Data Fig. 1
Garcia Ruiz, R., Berger, R., Billowes, J. et al. Spectroscopy of short-lived radioactive molecules. Nature 581, 396–400 (2020). https://doi.org/10.1038/s41586-020-2299-4
Issue Date: 28 May 2020
Relativistic coupled-cluster study of SrF for low-energy precision tests of fundamental physics
Kaushik Talukdar
Haimyapriya Buragohain
Sourav Pal
Theoretical Chemistry Accounts (2023)
Laser spectroscopy of indium Rydberg atom bunches by electric field ionization
Scientific Reports (2020) | CommonCrawl |
Research | Open | Open Peer Review | Published: 22 March 2016
Identifying potential differences in cause-of-death coding practices across Russian regions
Inna Danilova1,2,
Vladimir M. Shkolnikov1,3,
Dmitri A. Jdanov1,3,
France Meslé4 &
Jacques Vallin4
Population Health Metricsvolume 14, Article number: 8 (2016) | Download Citation
Reliable and comparable data on causes of death are crucial for public health analysis, but the usefulness of these data can be markedly diminished when the approach to coding is not standardized across territories and/or over time. Because the Russian system of producing information on causes of death is highly decentralized, there may be discrepancies in the coding practices employed across the country. In this study, we evaluate the uniformity of cause-of-death coding practices across Russian regions using an indirect method.
Based on 2002–2012 mortality data, we estimate the prevalence of the major causes of death (70 causes) in the mortality structures of 52 Russian regions. For each region-cause combination we measured the degree to which the share of a certain cause in the mortality structure of a certain region deviates from the respective inter-regional average share. We use heat map visualization and a regression model to determine whether there is regularity in the causes and the regions that is more likely to deviate from the average level across all regions. In addition to analyzing the comparability of cause-specific mortality structures in a spatial dimension, we examine the regional cause-of-death time series to identify the causes with temporal trends that vary greatly across regions.
A high level of consistency was found both across regions and over time for transport accidents, most of the neoplasms, congenital malformations, and perinatal conditions. However, a high degree of inconsistency was found for mental and behavioral disorders, diseases of the nervous system, endocrine disorders, ill-defined causes of death, and certain cardiovascular diseases. This finding suggests that the coding practices for these causes of death are not uniform across regions. The level of consistency improves when causes of death can be grouped into broader diagnostic categories.
This systematic analysis allows us to present a broader picture of the quality of cause-of-death coding at the regional level. For some causes of death, there is a high degree of variance across regions in the likelihood that these causes will be chosen as the underlying causes. In addition, for some causes of death the mortality statistics reflect the coding practices, rather than the real epidemiological situation.
Data on mortality by causes of death are important for monitoring epidemiological patterns. These data, which are widely used in demographic and medical research, often provide crucial information for identifying public health problems and developing health care strategies. However, the usefulness and the interpretability of mortality data depend largely on their quality, which varies between countries [1–3].
The quality of cause-of-death reporting in a certain country or territory is often assessed by examining the prevalence of obvious flaws in cause-specific mortality data (such as the use of unspecified or "garbage" causes, or the violation of the logical correspondence between the causes of death and the age or sex of the deceased) [1, 4]. But even if the prevalence of such obvious problems is shown to be moderate, the quality of cause-of-death reporting can be considered imperfect; even if these problems are not immediately apparent, it is possible that certain causes of death are being systematically misclassified [5–11], and that the cause-specific mortality data being generated are therefore of limited utility for the purposes of public health decision-making and research.
Although the International Classification of Diseases (ICD) manuals provide clear and detailed instructions on the coding process, for a variety of reasons the actual coding practices in a country may not follow these rules. Studies that have compared mortality from specific diseases across countries have found that coding practices can vary substantially, and that this lack of consistency has the potential to distort the comparability of the cause-specific mortality data for different populations [12–18]. Medical concepts and coding practices may also vary within a single country, especially if it has a large and diverse population spread over a vast territory. Thus, the consistency of cause-specific mortality data at the subnational level also determines the usability and the interpretability of national and subnational cause-of-death statistics [19, 20].
As Russia covers a very large territory, mortality levels – as well as general socioeconomic, ethno-cultural, and climatic conditions – vary substantially across the country's regions. In 2013, life expectancy at birth for the Russian population as a whole (both sexes) was 70.8 years. The regional disparities for the same year were substantial: the standard deviation across regions was 2.6 years and the maximum vs. minimum range exceeded 17 years. Reliable data on cause-specific mortality at the subnational level can help to explain the origins of health inequality in Russia, and may prove useful for designing interventions to reduce it.
The Russian system for cause-of-death diagnostics and coding
The World Health Organization (WHO) has rated the coverage and the quality of cause-of-death mortality data in Russia as "medium" [21]. The Russian system for registering and coding deaths is characterized by almost full coverage of civil registration of the underlying causes of death (estimated completeness in 2006–2012 was 99 %) [22]. Moreover, the share of deaths for which post-mortem autopsies are performed is larger in Russia than in many other countries [23, 24]. However, recent studies on the quality of Russian cause-specific mortality statistics have shown that a large proportion of deaths in Russia are assigned to the ICD codes for various ill-defined and "unspecified" conditions. These codes provide poor information for the purposes of developing health policy [25–27]. In particular, N. Gavrilova and co-authors have made the claim that since the 1990s, there has been a general deterioration over time in the quality of cause-specific mortality statistics in Russia [26]. One piece of evidence that supports this assertion is the increase in the number of deaths in which the deceased's identity could not be established; this trend was observed throughout the 1990s and the first half of the 2000s [28]. In 2005, an approximate age of death could not be specified for 0.76 % of all deaths in Russia (1.15 % for males and 0.31 % for females), even after a forensic autopsy. Since 2005, this trend has reversed, and by 2013 the share of such cases had been reduced to 0.27 % (0.42 % for males and 0.11 % for females).
Before 1999, detailed ICD was not used in Russia. The Central Statistical Office of the USSR developed brief Soviet Classifications that were roughly based on contemporary versions of the ICD. The Classification of 1981, which was modified in 1988 (hereinafter SC-1988), was the last Soviet Classification, and was in use until 1998. The SC-1988's list of causes of death consisted of 184 aggregated items based on all of the codes of the ICD-9 (plus 10 items for the double classification of external causes of death by the character of the injury).
Russia implemented the ICD-10 in 1999, and since then all death certificates issued in the country must be filled in with the original ICD-10 codes. Even though all medical death certificates have four-digit ICD-10 codes indicating the causes of death, data at this level of detail are unavailable for research purposes. The Russian State Statistics Service (Rosstat) publishes information on causes of death in aggregate form only. In these published data tables, deaths are tabulated in accordance with the Russian Abridged Classification (hereinafter RC-1999) launched in 1999, which consists of 234 items (plus 10 additional items for the double classification of external causes of death) that correspond to groups of detailed ICD-10 codes. In the routine data tables, the age of death is given in categories: 0, 1–4, 5–9, 10–14, …, 85+. In most cases, researchers have access to aggregate data only. These aggregate data tables are provided to the WHO by the Rosstat and the Russian Ministry of Health.
For Russia, the transition to the ICD-10 in 1999 represented not just a move to a new cause-of-death classification, but also entailed changes in the basic principles of coding and gathering information about the causes of death [29]. Before 1999, medical professionals in Russia had no responsibility to assign codes to causes of death. Their main duty was to fill in the medical death certificate by writing down the sequence of medical causes that contributed to the death. These medical death certificates were then submitted by the decedent's relatives or by the responsible institution to the respective district office of the Registration of Acts of Civil Status (ZAGS), a government body that was (and still is) responsible for the civil registration of deaths and for issuing civil death certificates. A civil death certificate was (and still is) needed for burial and for legal purposes. While the medical death certificate specified the cause of death in detail, the civil death certificate (the document that is given to the relatives as a final document confirming the death event) did not contain any information on the cause of death. The cause-of-death information was excluded from this document starting in 1997. The district offices of ZAGS passed the medical death certificates received from the decedent's relatives to the Regional Statistics Services, where trained statisticians coded the underlying cause of death according to the contemporary version of the Soviet Abridged Classification. The corresponding cause-of-death data were then computerized and sent to the Central Statistical Administration.
The data processing system has changed since 1999, with one of the main differences being that the coding procedure now takes place not at the end of the cycle described above, but at the very beginning. Medical professionals in Russia are now responsible not only for certifying the death, making the diagnosis, and indicating the sequence of the causes that contributed to death; but also for coding the death in accordance with the ICD-10 rules. There is no centralized and/or automated coding system to assist the medical professionals in choosing and coding the underlying causes. The statisticians in the Regional Statistics Services are now only responsible for checking the ICD codes for obvious mistakes, and for aggregating these codes into RC-1999 items.
The Russian system of cause-of-death statistics is thus highly decentralized. Since 1999, each medical practitioner in charge of issuing death certificates constitutes a separate coding unit. Such a system may have certain advantages for Russia, as the country has a huge territory and a large population distributed very unevenly across space. However, this system may also result in coding discrepancies across territorial units. We should note here that although the coding process in Russia is performed at the level of individual medical workers, it is likely that some territorial "schools" of coding exist. In most of the Russian regions, the majority of medical death certificates are issued by a limited number of parties: the physicians responsible for coding in a few large hospitals, the autopsy departments in these large hospitals, and several forensic examination bureaus (in 2012, 50.6 % of all deaths were subject to autopsy). Thus, the coding practices in a limited number of institutions may largely determine each region's approach to coding. These institutions are in turn accountable to the regional Ministry of Health, and are expected to follow the decrees and instructions issued by this authority. All of the practitioners working in the medical institutions that are in charge of filling in the medical death certificates are expected to follow the same guidelines. We can therefore expect that the medical practitioners in a given region will tend to use similar approaches to diagnosis and coding.
Prior research on the topic
There have been a number of studies that have addressed the quality of cause-specific mortality statistics in the USSR and Russia (Table 1). First, there are a few audits conducted in the USSR to assess the accuracy of death certification. The general atmosphere of conspiracy and the desire to avoid announcing unfavorable mortality trends in the USSR led to the keeping data on all-cause and cause-specific mortality in secrecy, and to scarcity of statistical publications on mortality. The three studies shown in Table 1 were identified by a team of French and Russian demographers in the 1990s [30]. These studies had similar designs, and were based on a re-inspection of samples of medical death certificates. Following their analysis of the results of the three surveys, the researchers concluded – despite previous expectations to the contrary – that there was no evidence of significant overestimation of deaths from all cardiovascular diseases combined in the USSR and Russia. Specifically, they found that even though the error rates were high for specific circulatory conditions, these problems were compensated for within the ICD chapter "circulatory diseases." When the researchers looked at the other groups of causes, they found the largest shares of diagnosis and coding errors for digestive and respiratory diseases.
Table 1 Prior research on the topic
When mortality data for Russia were again made available for research and publication at the end of the 1980s, researchers showed a strong interest in investigating various aspects of Russian mortality, and the issue of the validity of cause-of-death data in particular.
The first and the most comprehensive attempt to analyze different aspects of Russian mortality for a longer time period was the project by the French-Russian team mentioned above [30–32]. The outcomes of this project, which were published in 1996, included observations about the quality of cause-specific mortality statistics in Russia (USSR). Certain indirect findings on the validity of cause-of-death data in Russia were based on a visual inspection of cause-specific mortality trends. The analysis was also unable to confirm that there was an overestimation of entire class of cardiovascular diseases in Russia. Indeed, the results indicated that mortality from cardiovascular diseases might have even been underestimated among the elderly. A special decree by the Soviet Ministry of Health in 1989 had resulted in a massive artificial transfer of deaths from the chapter "diseases of the circulatory system" to the ICD chapter "symptoms, signs, and ill-defined conditions." Moreover, in line with the results of the comparison of the three Soviet surveys, some misclassification was discovered among the causes related to cardiovascular diseases. In particular, the analysis found that Soviet (and later Russian) coding practices tended to assign the excess number of deaths to atherosclerotic heart disease, which resulted in an underestimation of mortality from other ischemic and non-ischemic heart diseases.
Like the earlier surveys of the Soviet era, subsequent studies that assessed the quality of Russian cause-of-death data used direct techniques. Two of these studies were within the framework of the Udmurt and the Izhevsk studies, which were conducted in the Udmurt Republic and its capital, the city of Izhevsk. The aim of these studies was to identify the reasons for the high premature male mortality rates in Russia, and specifically to clarify the link between premature male mortality and hazardous alcohol drinking. The research included the hypothesis that deaths from acute alcohol poisoning may have been misattributed to circulatory diseases. Based on necropsy records and information obtained from medical files, medical experts checked whether the officially recorded underlying cause of death in the medical death certificates was credible. The findings indicated that the ICD chapter assignments in the cause-specific mortality statistics were quite reliable, but that the incidence of misattribution was higher when the recorded cause and the actual cause were in the same chapter [33]. The hypothesis that a significant fraction of deaths from acute alcohol poisoning were being hidden behind the mask of cardiovascular disease was not confirmed [33–35].
Other studies that investigated the possibility that deaths from acute alcohol poisoning were misclassified as deaths from circulatory diseases were also carried out in two other Russian cities, Barnaul [36] and Arkhangelsk [23], by other groups of researchers.
The authors of the Barnaul Study argued that the abrupt changes in the death rates from many circulatory causes (especially from other forms of ischemic heart disease) in the 1990s were caused by the misclassification of deaths from alcohol poisonings. In particular, they based this hypothesis on their finding that between 1990–2004 in Barnaul, the post-mortem blood alcohol concentration (BAC) was lethal (>4 g/L) for 14 % of the autopsied deaths of males aged 35–69 that were officially recorded as deaths from cardiovascular causes. However, these results contradict the outcomes of the Arkhangelsk study and of the earlier Izhevsk study. The authors of the Arkhangelsk study inspected death certificates issued in this city between January 2008 and August 2009 and found no cases in which the death of a man aged 30–49 with BAC > 4 g/l was certified as circulatory disease [23]. The results for Izhevsk indicated that 0 % of deaths of males aged 20–55 in the years 1998–1999 [34] and 5 % of deaths of males aged 25–54 in the years 2003–5 [35] with BAC > 4 g/l were recorded as deaths from circulatory disease.
It is interesting to note that studies that had similar designs, but were conducted in three different cities, produced such a wide range of results. The large discrepancies in the findings of these studies could be at least partly attributable to differences in the approaches to the certification of causes of death in different sites in Russia.
Studies that specifically addressed the issue of the possible misattribution of different causes of death were also conducted in a few other regions of Russia [26, 37–40]. These studies relied primarily on the examination of medical death certificates. In most of these investigations, researchers tried to check whether the underlying cause of death was reported correctly by consulting other information presented in the death record (such as the immediate and contributing causes of death and the place of death). In addition, some researchers used the medical files of the deceased to check the diagnosis [38, 40].
Studies based on the re-inspection of death certificates were performed in only a very few Russian regions and at a few points in time. While it appears that these studies accurately reported the types and the origins of miscoding, it is still not clear to what extent specific regional findings can be generalized to the national level. Thus, these studies do not provide us with any conclusive insights into how the quality of coding varies across regions in Russia.
A comprehensive evaluation of the comparability of cause-of-death mortality data reporting by Russian regions has not yet been conducted. We are aware of only two papers that specifically examined regional peculiarities in the coding of some specific causes of death in Russia. The first is the 2003 study conducted by W. Pridemore that investigated the comparability of two sources of homicide estimates in Russia: data from the vital statistics registration system and data from the Ministry of the Interior [41]. The results showed that there were certain disparities across regions in the reporting of homicides in the mortality and crime data. The second study on coding discrepancies across Russian regions, by A. Nemtsov, examined spatial-temporal variations in alcohol-related mortality in Russia [42]. Comparing mortality levels from acute alcohol poisonings and alcohol psychoses, Nemtsov showed that the mortality levels and the dynamics of these two causes did not correspond to each other in many regions; this finding contradicts our current understanding of the link between these causes, and can be regarded as a statistical artifact. Although the studies by Pridemore and Nemtsov exclusively examined the statistics on, respectively, homicides and alcohol-related causes, they provide us with some important insights into the different regional approaches to cause-of-death reporting.
Specific objective of the current study
Here we present the first study that systematically addresses the problem of the comparability of cause-specific mortality statistics across Russian regions. Our purpose in this study is to use the indirect tools and the limited data available to provide a snapshot of the quality of cause-of-death mortality statistics in Russia at the regional level. Our overview offers some instantaneous, easy-to-interpret results, and can also serve as a starting point for more in-depth investigations. Specifically, we aim to:
evaluate the regional cause-specific mortality data published in official statistics;
examine how the prevalence of particular causes of death in the mortality structure changes across Russian regions; and
identify the most obvious discrepancies across different regions.
Using the available tools and data, we provide an indirect estimation of the uniformity of cause-of-death coding practices across Russia, and seek to identify the most problematic points of disagreement between different regions. Furthermore, we present a broader picture of the quality of cause-of-death coding practices at the subnational level in Russia.
Regional data on causes of death
Regional death counts and mid-year population estimates by sex and age were obtained from the Russian Federal State Statistics Service. Age-standardized death rates (SDRs) for both sexes combined were calculated with the European Population Standard [43].
We use data for the period from 2002 (the year when RC-1999 was de facto implemented throughout Russia) to 2012 for a sub-sample of 52 Russian regions. To avoid the random fluctuations caused by small numbers of death events, we have limited our analysis to the 52 regions of Russia in which the annual population exposure (average for the period) was one million person-years or higher. We also excluded the Chechen Republic because death counts for this territory were only available from 2004 onward. This sample of 52 regions is presented in Table 2. In 2002–2012, 88.4 % of the total population and 88.5 % of all deaths in Russia were in these regions.
Table 2 Regions under study, by federal district of Russia
For the same reason – i.e., to eliminate biases generated by small numbers – we assigned some items of the RC-1999 to broader diagnostic groups of causes of death. Moreover, we had to exclude some ICD-10 chapters from our analysis (Chapters III, VII, VIII, XII, XIII, XV) because the numbers of deaths from the causes that constitute these chapters were too low, and no meaningful grouping with the other chapters could be done. The final list of selected causes of death includes 70 items (Table 3).
Table 3 Causes of death under study
To estimate the inter-regional variability of mortality from specific cause of death we used the cause-specific share of the all-cause age-standardized death rate:
$$ {S}_{r,c,t}=\frac{SD{R}_{r,c,t}}{SD{R}_{r,t}}100\%, $$
where SDR r,c,t is the age-standardized death rate for cause c in region r in year t, and SDR r,t is the all-cause age-standardized death rate in region r in year t. We used the indicators S r,c,t instead of the cause-specific rates in order to eliminate the influence of variation in overall mortality levels across regions and over time.
Next, for each possible combination region/cause we calculated the indicator measuring the deviation from the cross-regional mean (period average) (2):
$$ {V}_{r,c}=\frac{1}{T}{\displaystyle {\sum}_{t=1}^T\left|\frac{S_{r,c,t}-\overline{{S}_{\bullet, c,t}}}{S_{\bullet, c,t}}\right|}\;100\%, $$
where \( \overline{{S}_{\bullet, c,t}} \) is the mean of regional \( {S}_{r,c,{t}^1} \), T – the length of time series. We thereby obtained a data set of scores in which each percentage score V r,c shows how much on average (with respect to time) the share of cause c in the all-cause SDR of region r differs from an average of the inter-regional share of the same cause. The total size of the data set is equal to the number of regions multiplied by the number of causes of death.
After computing indicators V r,c according to equation (2), we obtained a matrix that had 52 columns (the number of regions) and 70 rows (the number of causes of death). To present this matrix in an intelligible form, we plotted a heatmap in which each row corresponds to a particular cause of death and each column represents a specific region. The cells are colored based on the values of V r,c using the yellow-red gradient palette. The points with a light yellow color have the lowest levels of deviation, and the points become darker in color as the degree of deviation increases. We set up our system of color gradation so that only the cases that deviated significantly from the average are clearly detectable on the heatmap. Deviations of less than 40 % from the average are not obviously recognizable on the heatmap, and are seen as low values. We used this color gradation deliberately in order to identify the cases for which the degree of deviation is so high that it is likely that they are attributable to differences in coding practices, rather than to real differences in regional epidemiological patterns.
Statistical analysis of variability
To determine whether there is a certain regularity in the causes and the regions that are more likely than others to deviate from an average inter-regional level, we applied a least squares regression model (3) with two sets of dummy variables for regions and for causes of death:
$$ {V}_{r,c}=a+{b}_r{I}_r+{d}_c{I}_c+{\upvarepsilon}_{r,c} $$
where a is a constant term; I r and I c are, respectively, independent regional and cause-specific dummy variables; b r and d c are the coefficients on these variables; and ε r,c is an error term.
We used "Kaluga Oblast" as the reference category for variable I r , and "trachea, bronchus, and lung cancers" as the reference category for variable I c . This region and this group of causes were chosen as the omitted reference units as these categories appeared to deviate from the average less than the others on our heatmap. Thus, the total number of estimated coefficients through a regression equals 120 (51 for regional and 69 for cause-specific dummies). We then checked the sustainability of these results using sensitivity analysis.
Before presenting our results, we will briefly describe a finding that was obtained outside of the framework of the current study. The initial impulse to conduct this study arose from a finding that emerged while we were engaged in a reconstruction of coherent cause-specific mortality time series in Russia. The introduction of the ICD-10 and the RC-1999 classification systems in 1999 resulted in inconsistencies between the mortality series coded under the RC-1999 and the series coded under the previous SC-1988 classification. Before we could compare mortality over a time period that was covered by several classifications, we had to reconstruct the cause-specific data series so that the full period was covered by the same classification. The reconstruction process was done using the method developed by J. Vallin and F. Meslé (for a description of the original method, see: Vallin and Meslé, 1988; Meslé et al., 1992; Meslé and Vallin, 1996) [44–46]. While performing this work, we discovered indirect indications that there could be significant discrepancies in cause-of-death coding practices across subnational entities in Russia. First, we found that the transitions to the ICD-10 and the RC-1999 classification systems at the regional level were not done simultaneously: four regions (the city of Moscow, Stavropol Kray, the Republic of Ingushetia, and the Sverdlovsk Oblast) postponed the transition for up to three years. While all of the deaths in the aforementioned regions were formally published in the official statistics under the new RC-1999 starting with the year after the transition, these deaths had been originally coded under the previous SC-1988 classification, and were then roughly translated into the items of the new RC-1999 classification. Second, we found that even after all of the regions had introduced the new classification, there were still significant regional disparities for some causes of death, many of which persist up to the present day.
These observations made during our reconstruction work, together with concerns raised in previous research literature about the quality of cause-of-death statistics in Russia, provided us with the starting points for our study. We realized that there was a need to identify and systematize the problems in the reporting of the underlying causes of death at a subnational level in Russia.
We turn now to the results directly obtained using the methods described in the "Methods" section. The heatmap presents the entire range of the V r,c values in a transparent and observable form (Fig. 1). When looking at the heatmap, we can clearly see horizontal patterns that indicate that the causes of death vary greatly across the regions. We can also see some vertical patterns that show that certain regions have cause-specific mortality structures that deviate from the average more than those of other regions.
Heatmap on inter-regional variability in causes of death. Each row corresponds to a particular cause of death and each column represents a region. The cells are colored according to the values of V r,c
Table 4 provides the fragment of the regression results of the model (3). The coefficients b r indicate to what extent an average of the scores V r,c calculated for region r deviates from the omitted category "Kaluga." Similarly, the coefficients d c indicate to what extent the average of scores V r,c for a certain cause of death c varies from the average for the omitted category "trachea, bronchus, and lung cancer." We verified the robustness of the estimations with a sensitivity analysis that used different regional samples and changed the measure of inequality (relative root-mean square error instead of mean relative absolute error). The sensitivity analysis confirmed that our results have a high level of sustainability.
Table 4 Estimates of the regression coefficients of the OLS model (3)
Among the 69 causes of death that were assigned dummy variables, 45 causes showed a statistically significant (p < 0.05) deviation from the reference level, and 38 causes showed a deviation of p < 0.01. For 25 of these causes the deviation predicted by the model was higher than 20 %. The highest regression coefficients d c were found for dummy variables corresponding to AIDS (+71.4 %), senility (+70.9 %), mental and behavioral disorders (+63.1 %), atherosclerosis (+53.8 %), hypertensive diseases (+51.4 %), pulmonary heart and circulation diseases (+41.8 %), chronic obstructive pulmonary diseases (+41.8 %), and alcoholic liver disease (+40.8 %).
The highest levels of consistency (the lowest regression coefficients d c ) across regions were found for causes that represent different groups of cancers (from +0.7 % for stomach cancer to +11.8 % for the item "cancers of other digestive organs"). The other causes that deviated relatively little from the reference category were nontraumatic intracranial hemorrhage (+4.3 %) and transport accidents (+4.4 %).
Four panels of Fig. 2 present examples of distributions of causes of death that were very similar in terms of their contributions to overall mortality (period and regional average \( \overline{{S}_{\bullet, c,\bullet }} \)); however, the respective coefficients d c returned by the regression model (3) for these causes differed substantially.
Examples of distributions of cause-specific shares of the all-cause SDR across 52 regions
The distributions presented were dissimilar with respect to their kurtosis and skewness. In particular, the fourth panel illustrates three causes of death that were almost equal by average \( \overline{{S}_{\bullet, c,\bullet }} \) (2.5 % for trachea, bronchus, and lung cancers; 2.6 % for senility; and 2.9 % for myocardial infarction). But the regional distributions of \( \overline{{S}_{r,c,\bullet }} \) values were very different for these three causes. The range of variation (the difference between the maximum and the minimum regional values of \( \overline{{S}_{r,c,\bullet }} \) for trachea, bronchus, and lung cancers was only 1.4 % (with the minimum in Smolensk equal to 1.9 %, and the maximum in Altay equal to 3.3 %), while the range of variation for myocardial infarction was 5.7 % (from 1.3 % in Lipetzk to 7.0 % in Primorsky) and the range of variation for senility was 8.8 % (from 0.0 % in four regions in the sample to 8.8 % in Voronezh). All of these distributions had very different levels of skewness as well. The regional values of \( \overline{{S}_{r,c,\bullet }} \) for trachea, bronchus, and lung cancers were distributed almost symmetrically, with the modal value being approximately equal to the mean of the distribution. The distribution for myocardial infarction was highly skewed with a long right tail. Finally, the indicators \( \overline{{S}_{r,c,\bullet }} \) for the item "senility" were almost uniformly distributed across all regions.
Different approaches in the reporting of senility as the underlying cause of death undoubtedly affected regional mortality rates from the other, more specific causes. The prevalence of the other "garbage codes"Footnote 1 included in our analysis – i.e., the items "other ill-defined and unspecified causes" and "injuries with undetermined intent" – also varied considerably across regions (+34.2 and +28.5 compared with the reference category, respectively). Among the 52 regions under study, the average share of garbage codes combined was 7.0 % (period average for 2002–2012). In four of the regions the share of garbage codes was less than 3 %, while in 10 other regions the share was between 3 % and 5 %. However, in eight regions these causes contributed to overall mortality in more than 10 % of the cases, with the maximum contribution level of 15.2 % found in Ryazan.
Although the regional pattern on the heatmap is less apparent than the pattern for causes of death, some vertical structures are still clearly recognizable. The results of the least squared regression model demonstrate that in 13 regions the average of V r,c values predicted by equation (2) were statistically different from the value in the reference (Kaluga), at the p < 0.05 level; and that in seven regions the values were statistically different at the p < 0.01 level. For nine regions, the average deviation was higher than 10 %, and was thus higher than in Kaluga; the top scores were found for Dagestan (+32.6 %), the city of Moscow (+29.8 %), and the city of Saint Petersburg (+19.9 %). In eight regions the coefficients b r were negative, but none of those coefficients was statistically significant.
It is worth noting that there is no apparent spatial regularity in the distribution of the V r,c scores across the regions. For instance, the city of Moscow, which had an estimated regression coefficient of b r = 29.8; is surrounded by the Moscow Oblast, which had the same coefficient equal to 1.5.
The V r,c scores we used for the analysis of spatial variability showed average (for the period) levels of deviation, but they did not indicate the sign (positive or negative) of the deviation or how the magnitude of the deviation changed over time. To find out whether the patterns of deviations were stable in the regions over an observation period, we inspected the regional time series. We found a number of regional cause-specific series that were unexpectedly distorted during the period 2002–2012. These abrupt and/or unpredictably large changes in mortality levels from particular causes over time may indicate a modification of coding practices, whereby some number of deaths that would have previously been coded to a certain item started to be coded to another item (Fig. 3).
Examples of rapid and contrasting changes in regional cause-specific shares of all-cause SDR (both sexes combined). The trend for Russia as a whole is provided for comparison
Interestingly, the breaks in the regional time series occurred at different points in time, and the directions of these changes were sometimes even reversed in different regions. It therefore seems unlikely that regions introduced new coding practices in order to meet some baseline standards. Unpredictably large shifts in cause-specific series were not common; they took place in only a few regions and cannot be visually detected at the national level. However, when considering the small fluctuations in mortality trends by cause that can be observed at the national level, it is important to be aware that some of the changes may reflect changes in coding practices at the subnational level.
The most significant and numerous changes in regional trends for causes of death were found for AIDS, senility, mental and behavioral disorders, and atherosclerosis. The highest levels of stability over time were observed for different groups of cancers, nontraumatic intracranial hemorrhage, and transport accidents. Hence, there is an evident intersection between the causes of death with high levels of spatial variability and the causes of death for which the regional trends show a high degree of volatility. Similarly, the causes with the smallest degrees of variation across regions showed the highest levels of stability over time.
Like the Soviet system on which it is based, the current Russian system for producing information on causes of death is decentralized. The extent of this decentralization has increased substantially since the country made the transition to a new system of cause-of-death coding in 1999. Before the transition, the network of regional Statistics Offices had been responsible for coding the underlying cause of death; but since 1999, this task has been delegated to the individual medical practitioners. This shift coincided with the transition to the ICD-10. The Statistics Service coders had to code the underlying cause of death in accordance with the Soviet Abridged Classification, which offered them only 184 diagnostic items to choose from. By contrast, medical practitioners now have to assign the cause using the complete ICD-10 classification, which contains over 10,000 nosological items.
According to some experts, this change led to a deterioration of the Russian system of coding and gathering information on the causes of death, in part because no unified training in cause-of-death coding for medical workers was provided [47]. Moreover, medical professionals were not even given any centralized instructions for filling in medical death certificates and coding the causes of death in accordance with ICD rules [48]. This lack of preparation has led to specific difficulties with and discrepancies in coding practices across subnational entities and over time.
The present study has identified several problems with the cause-specific mortality statistics across the Russian territories.
We have found that while certain causes of death (e.g., cancers, transport accidents) have roughly comparable cause-specific shares across the regional mortality structures, there is a much greater degree of inconsistency in the prevalence of other cause-specific shares across the regions. For some causes, the magnitude of these inconsistencies is too large, and is therefore more likely to be artificial than to be indicative of natural variation across regions. Thus, it is possible that the regional differences in mortality from these causes reflect variation in coding practices, rather than real differences in the prevalence of diseases.
The lowest levels of consistency among the causes of death we investigated were found for AIDS. However, the high degree of variability of AIDS diagnoses cannot be regarded as a problem of coding accuracy only. AIDS was a new cause of death at the start of our study period, and the number of people who were dying from this disease was clearly increasing as the period progressed. Over time, our understanding of and ability to detect the disease have improved, and coding practices have adapted accordingly. Mortality from AIDS has been rising rapidly in Russia over the last decade (Fig. 4). One piece of evidence that supports the claim that there is "natural" wide variation in AIDS mortality across different regions of Russia is the finding that there is a strong positive correlation between the registered prevalence of HIV in a given region [49] and the share of AIDS in the regional all-cause SDR (the correlation was 0.88 in 2012).
AIDS, standardized death rates per 100,000 (Russia, both sexes)
As is the case for other communicable diseases, AIDS has spread unevenly across the population. Some regions could be a nidus of infection, while others have had much lower incidence levels. Thus, the large degrees of spatial and temporal variation in the contributions of AIDS to overall mortality can be explained. However, some portion of the variation in the prevalence of AIDS mortality across Russian regions may have also been caused by discrepancies in the cause-of-death coding practices. In many countries, deaths from AIDS are systemically miscoded under tuberculosis, endocrine disorders, Kaposi's sarcoma, meningitis, encephalitis, certain garbage codes, and other causes of death [50, 51]. C. Murray and co-authors estimated that the real number of deaths from AIDS in Russia in 2013 was 16,138 (95 % uncertainty interval 11,963 to 22,526) [51], or 52 % higher than the number that was officially reported by the Russian State Statistics Service.
In 2010, E. Tzybikova examined 6249 deaths that occurred among patients with newly diagnosed tuberculosis in 80 Russian regions. She found that in 61 regions some deaths from AIDS were mistakenly coded as tuberculosis. While tuberculosis was chosen as an underlying cause of death, AIDS was listed as an associated cause, which violates the ICD instructions for sequencing the causes of death [52]. The total number of such cases found by this study was 1,004. When we look at the 6784 deaths from AIDS recorded in the official statistics in 2010, it appears that a very significant fraction of the deaths were misclassified. Unfortunately, this study did not provide a detailed explanation of the study design, and did not investigate the regional peculiarities in the misclassification of AIDS and tuberculosis. However, the finding that there were incidents of misclassification in 61 of the 80 regions studied may indicate that the extent of the misclassification of deaths from AIDS also differs across regions.
Other groups of infectious diseases that we studied, such as "tuberculosis" and "other infectious diseases," had medium levels of inconsistency compared to other causes of death. As was shown above, regional variation in mortality from tuberculosis can be affected by the misclassification of AIDS. Difficulties in certifying deaths with AIDS/tuberculosis co-infection are common, especially in countries with a high burden of HIV [50, 53]. Nevertheless, it should be noted that during the period of observation the prevalence of tuberculosis in the mortality structure of Russian regions (\( \overline{{S}_{\bullet, c,t}} \) is equal to 1.3 %) was several times higher than mortality from AIDS (\( \overline{{S}_{\bullet, c,t}} \) is equal to 0.2 %). Hence, the miscoding of these two causes distorts the mortality statistics for AIDS much more significantly than for tuberculosis.
While high levels of spatial and temporal heterogeneity are normal in the transmission of infectious diseases, having to rely solely on the data reported by official statistics makes it difficult to determine whether high degrees of variation in mortality from infectious causes reflect real differences in the prevalence of disease, or are indicative of differences in coding practices as well. But for causes of death from non-communicable diseases, it seems rather unlikely that very high levels of within-country variation are natural. Our finding that some non-communicable diseases had much higher levels of spatial variation than some communicable diseases can serve as indirect proof that the level of variation we found for some non-communicable diseases is too high and cannot be accurate.
Very low levels of consistency were found for some groups of causes from the ICD chapter "diseases of the circulatory system." For some of these causes, the level of consistency would have been greater if we had assigned them to broader groups of items. For instance, within the group of ischemic heart diseases the ratio between the inter-regional maximum and the inter-regional minimum values of V r,c calculated according to equation (2) amounts to 13.1 for "atherosclerotic heart disease," 5.6 for "myocardial infarction," and 4.3 for "other forms of ischemic heart diseases." But if we combine all of these items into one group, "ischemic heart disease," this ratio would be only 3.1. Similar results can also be obtained for the group of "cerebrovascular diseases." These lower levels of inconsistency at higher levels of aggregation suggest that conflation often occurs when the possible causes of death are medically similar. Analyzing cause-specific mortality at higher levels of aggregation can reduce biases.
Coding discrepancies can undermine cause-specific analysis more significantly for causes that cannot be meaningfully grouped together with other items. Categories that represent complete ICD chapters, such as "diseases of the nervous system," "endocrine, nutritional, and metabolic disorders," and "mental and behavioral disorders" had very high levels of spatial and temporal inconsistencies in our analysis. Even more biases in the analysis of cause-specific mortality are caused by spatial and temporal differences in the use of garbage codes from the ICD-10 chapter XVIII, "symptoms, signs, and abnormal clinical and laboratory findings;" or groups of causes, such as "injuries of undetermined intent." The propensity to assign garbage codes as underlying causes of death varied significantly across Russian regions. As garbage codes constitute a high share of the causes of death recorded in the Russian mortality structure, regional and period discordances can heavily affect the comparability of mortality indicators for other specific groups of causes of death that are misclassified with garbage codes.
In terms of spatial variations, a few regions can be pinpointed as having the cause-specific mortality structures that deviate the most starkly from the inter-regional average: the cities of Moscow and Saint Petersburg, which are constituent federal units; and the Republic of Dagestan, a Muslim region located in the North Caucasus. We offer several hypotheses for why these particular regions had the highest scores in our analysis.
First, these three regions have the lowest overall mortality levels of the 52 regions in our sample. Lower mortality levels are generally indicative of certain mortality structures. In particular, lower mortality levels usually correspond with a higher share of deaths from neoplasms in relation to other causes of death. Accordingly, it is quite apparent on the heatmap that there are substantial differences between Moscow and Saint Petersburg on the one hand and the other Russian regions on the other in terms of the share of deaths from neoplasms relative to overall mortality. But the deviating pattern for Dagestan is mainly attributable to the relative shares of other causes of death. It is important to note that the Republic of Dagestan differs considerably from the other regions in our sample, as it is the only national republic of North Caucasus selected for this analysis, and the Muslim regions of North Caucasus have much lower mortality levels from alcohol-related causes than the rest of Russia. In addition, in these regions there are long-term concerns about the understatement of mortality at infant and old ages due to the underreporting of deaths, and about the overstatement of age [54, 55].
Second, the populations of the cities of Moscow and Saint Petersburg are entirely urban. Dagestan, by contrast, is the only region in the sample in which the urban population is still smaller than the rural population. Therefore, it is possible that the significant differences in the mortality structures between these three regions and the other regions of Russia are at least partly attributable to the differences between urban and rural populations.
The other possible explanation is a registration effect. A death in Russia can be registered either at the location of the deceased's permanent residence or at the location of death. This may result in certain biases in mortality statistics at the regional level, which can be especially large for Moscow and Saint Petersburg. First, there are a number of large federal medical centers in these two cities that specialize in the treatment of specific diseases and especially of cancers. In addition to residents of Moscow and Saint Petersburg, residents of other regions may be treated in these centers. Among all deaths from cancers in Moscow in 1990–1994, 4.8 % of the men and 5.6 % of the women who died were non-residents [56]. Additionally, Moscow and Saint Petersburg have huge migration inflows. The cause-specific mortality structures of these cities may therefore be affected by the selectivity in the health status of arriving migrants. Arkhangelsky and co-authors found that the cause-specific mortality structures of residents and non-residents in Moscow are very different [57]. Non-residents are, for example, more likely than residents to die from external causes, infectious diseases, and ill-defined conditions.
The results obtained in our study suggest that a complex series of actions will be needed to standardize regional approaches to cause-of-death coding and to improve the comparability of cause-specific mortality data within Russia. These actions should focus on strengthening the legal and regulatory framework for mortality statistics, improving the quality of human resources, and ensuring the full implementation of ICD standards. A national "gold standard" of training on death certification should be developed for medical practitioners. To increase the likelihood that medical workers will adhere to a uniform set of coding principles, the training procedures should be standardized to the greatest possible extent. Ideally, an automated, centralized coding and/or training software application would be designed and implemented across the country. The regular monitoring of the comparability of cause-specific mortality data reported by regions is also essential. In our study we took the average region/cause deviations for an 11-year period; thus, only the long-term deviations from the inter-regional average level were highlighted. Surprisingly, the number of such long-term deviations was found to be quite large. This finding suggests that regions can follow different coding practices for a long period without these discrepancies being discovered by the responsible federal authorities. Additional checks must be carried out in cases in which mortality from a certain cause in a certain region obviously deviates disproportionately from the average level, and the origins of these kinds of deviations should be thoroughly investigated.
We also suggest producing an aggregated list of causes of death that can be used in analyses of regional mortality patterns with a minimal risk of inter-regional incomparability and biases. Such a list should be regarded exclusively as a stopgap measure. Developing and implementing a national plan for strengthening the quality of cause-of-death statistics is essential, and should still be seen as the highest priority. But as making substantial improvements takes time, in the interim the aggregated list can be useful for analyzing cause-specific mortality in Russia at the regional level.
Our study has several limitations. The first arises from the indirect character of the method proposed. As we analyzed the official cause-specific mortality data as they are, we can make only indirect assessments of the quality and validity of these data. Although we can observe spatial and temporal variations in cause-specific shares, we cannot be certain whether they are caused by real mortality differences or by discrepancies in coding practices. While we can be reasonably sure that such discrepancies are present when the regional deviations in the causes of death are especially large, we cannot judge the less obvious cases. Further research is needed to determine why there are problems in the data.
The second limitation follows from the grouping of causes of death in the Russian Abridged Classification. It is impossible to extract from these data certain groups of ill-defined cancers and ill-defined cardiovascular diseases, which are also regarded as so-called garbage codes [58–60]. In most cases, such garbage codes are combined in the RC-1999 with some well-defined codes under the heading "other and unspecified." For instance, the group "cancers of other and independent (primary) multiple sites" in the RC-1999 includes, in addition to the codes for ill-defined cancers (C76, C80, C97), codes that correspond to neoplasms with specific localization, and that cannot be referred to as garbage codes, such as "cancer of eye and adnexa (C69)" and "cancers of thyroid and other endocrine glands (C73-C75)." Because these codes are ill-defined, we could not compare their prevalence across the regions, yet this is an important criterion for evaluation the quality of cause-of-death coding [2].
Third, our study was based on death counts obtained at the regional level. However, while the coding procedure in Russia is performed in a completely decentralized manner at the level of medical practitioners, there may also be some important discrepancies within the regions themselves.
The systematic analysis we performed showed that there is a high degree of variance in the coding practices for some causes of death across Russian regions. We found that the mortality statistics for some causes are more reflective of the coding practices than of the real epidemiological situation. These problems of comparability can affect the validity and the generalizability of cause-specific mortality statistics. These possible biases should be taken into account when performing mortality analyses. There is an urgent need to improve the uniformity and the stability of coding practices at the subnational level in Russia, as doing so would strengthen the accuracy and the quality of mortality statistics.
The term "garbage coding" was introduced by C. Murray and A. Lopez in 1996, and refers to the practice of assigning deaths to causes that do not meet the basic standard of generating useful information for public health analysis [58]. The prevalence of "garbage codes" in all-cause mortality structure is commonly used as one of the indicators for estimating the validity and reliability of mortality statistics [2, 59]. The overuse of garbage codes distorts mortality statistics from the other ("non-garbage") causes of death [60].
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This study was conducted in the framework of the project "From disparities in mortality trends to future health challenges (DIMOCHA)" funded by Deutsche Forschungsgemeinschaft (DFG) (Germany) (JA 2302/1-1) and Agence nationale de la recherche (ANR) (France) (ANR-12-FRAL-0003-01). This study was also supported by the MODICOD project of the AXA Research Fund and the Fund "Dynasty" (Russian Federation).
Max Planck Institute for Demographic Research, Konrad-Zuse-Strasse 1, 18057, Rostock, Germany
Inna Danilova
, Vladimir M. Shkolnikov
& Dmitri A. Jdanov
National Research University Higher School of Economics, Myasnitskaya St. 20, 101000, Moscow, Russia
New Economic School, Novaya St. 100, Skolkovo, 143026, Moscow, Russia
Vladimir M. Shkolnikov
Institut national d'études démographiques, Blvd. Davout 133, 75020, Paris, France
France Meslé
& Jacques Vallin
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Correspondence to Inna Danilova.
The authors declare they have no competing interests.
ID conducted the calculations and aided in interpreting the results and drafting the manuscript; VMS guided the analysis, developed the framework and the methods of the study, and aided in interpreting the results and drafting the manuscript; DAJ aided in developing the methods of the study and interpreting the results; and FM and JV aided in interpreting the results and drafting the manuscript. All of the authors read and approved the manuscript.
Mortality statistics | CommonCrawl |
Sophus Lie
Marius Sophus Lie (/liː/ LEE; Norwegian: [liː]; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Sophus Lie
Born(1842-12-17)17 December 1842
Nordfjordeid, Norway
Died18 February 1899(1899-02-18) (aged 56)
Kristiania, Norway
NationalityNorwegian
Alma materUniversity of Christiania
Known forOne-parameter group
Differential invariant
Contact transformation
Infinitesimal transformation
W-curve
Carathéodory–Jacobi–Lie theorem
Lie algebra
Lie bracket
Lie group
Lie product formula
Lie sphere geometry
Lie theory
Lie transform
Lie's theorem
Lie's third theorem
Lie–Kolchin theorem
See full list
AwardsLobachevsky Medal (1897)
ForMemRS (1895)
Scientific career
FieldsMathematics
InstitutionsUniversity of Christiania
University of Leipzig
Doctoral advisorCarl Anton Bjerknes
Cato Maximilian Guldberg
Doctoral studentsHans Blichfeldt
Lucjan Emil Böttcher
Gerhard Kowalewski
Kazimierz Żorawski
Élie Cartan
Elling Holst
Edgar Odell Lovett
Life and career
Marius Sophus Lie was born on 17 December 1842 in the small town of Nordfjordeid. He was the youngest of six children born to Lutheran pastor Johann Herman Lie and his wife, who came from a well-known Trondheim family.[1]
He had his primary education in the south-eastern coast of Moss, before attending high school at Oslo (known then as Christiania). After graduating from high school, his ambition towards a military career was dashed when the army rejected him due to poor eyesight. Then he enrolled at the University of Christiania.
Sophus Lie's first mathematical work, Repräsentation der Imaginären der Plangeometrie, was published in 1869 by the Academy of Sciences in Christiania and also by Crelle's Journal. That same year he received a scholarship and travelled to Berlin, where he stayed from September to February 1870. There, he met Felix Klein and they became close friends. When he left Berlin, Lie travelled to Paris, where he was joined by Klein two months later. There, they met Camille Jordan and Gaston Darboux. But on 19 July 1870 the Franco-Prussian War began and Klein (who was Prussian) had to leave France very quickly. Lie left for Fontainebleau where he was arrested, suspected of being a German spy, garnering him fame in Norway. He was released from prison after a month, thanks to the intervention of Darboux.[2]
Lie obtained his PhD at the University of Christiania (in present-day Oslo) in 1871 with a thesis entitled Over en Classe geometriske Transformationer (On a Class of Geometric Transformations).[3] It would be described by Darboux as "one of the most handsome discoveries of modern Geometry". The next year, the Norwegian Parliament established an extraordinary professorship for him. That same year, Lie visited Klein, who was then at Erlangen and working on the Erlangen program.
In 1872, Lie spent eight years together with Peter Ludwig Mejdell Sylow, editing and publishing the mathematical works of their countryman, Niels Henrik Abel.
At the end of 1872, Sophus Lie proposed to Anna Birch, then eighteen years old, and they were married in 1874. The couple had three children: Marie (b. 1877), Dagny (b. 1880) and Herman (b. 1884).
From 1876, he co-edited the journal Archiv for Mathematik og Naturvidenskab, together with the physician Jacob Worm-Müller, and the biologist Georg Ossian Sars.
In 1884, Friedrich Engel arrived at Christiania to help him, with the support of Klein and Adolph Mayer (who were both professors at Leipzig by then). Engel would help Lie to write his most important treatise, Theorie der Transformationsgruppen, published in Leipzig in three volumes from 1888 to 1893. Decades later, Engel would also be one of the two editors of Lie's collected works.
In 1886, Lie became a professor at Leipzig, replacing Klein, who had moved to Göttingen. In November 1889, Lie suffered a mental breakdown and had to be hospitalized until June 1890. Subsequently he returned to his post, but over the years his anaemia progressed to the point where he returned to his homeland. In 1898 he tendered his resignation in May, and left for home in September the same year. He died the following year in 1899 at the age of 56, due to pernicious anemia, a disease caused by impaired absorption of vitamin B12.
He was made Honorary Member of the London Mathematical Society in 1878, Member of the French Academy of Sciences in 1892, Foreign Member of the Royal Society of London in 1895 and foreign associate of the National Academy of Sciences of the United States of America in 1895.
• 1888 copy of "Theorie der Transformationsgruppen," volume I
• Title page to "Theorie der Transformationsgruppen"
• Preface to "Theorie der Transformationsgruppen"
Legacy
Lie's principal tool, and one of his greatest achievements, was the discovery that continuous transformation groups (now called, after him, Lie groups) could be better understood by "linearizing" them, and studying the corresponding generating vector fields (the so-called infinitesimal generators). The generators are subject to a linearized version of the group law, now called the commutator bracket, and have the structure of what is today called a Lie algebra.[4][5]
Hermann Weyl used Lie's work on group theory in his papers from 1922 and 1923, and Lie groups today play a role in quantum mechanics.[5] However, the subject of Lie groups as it is studied today is vastly different from what the research by Sophus Lie was about and "among the 19th century masters, Lie's work is in detail certainly the least known today".[6]
Sophus Lie was an eager proponent in the establishment of the Abel Prize. Inspired by the Nansen fund named after Fridtjof Nansen, and the lack of a prize for mathematics in the Nobel Prize. He gathered support for the establishment of an award for outstanding work in pure mathematics.[7]
Lie advised many doctoral students who went on to become successful mathematicians. Élie Cartan became widely regarded as one of the greatest mathematicians of the 20th century. Kazimierz Żorawski's work was proved to be of importance to a variety of fields. Hans Frederick Blichfeldt made contributions to various fields of mathematics.
Books
• Lie, Sophus (1888), Theorie der Transformationsgruppen I (in German), Leipzig: B. G. Teubner. Written with the help of Friedrich Engel. English translation available: Edited and translated from the German and with a foreword by Joël Merker, see ISBN 978-3-662-46210-2 and arXiv:1003.3202
• Lie, Sophus (1890), Theorie der Transformationsgruppen II (in German), Leipzig: B. G. Teubner. Written with the help of Friedrich Engel.
• Lie, Sophus (1891), Vorlesungen über differentialgleichungen mit bekannten infinitesimalen transformationen (in German), Leipzig: B. G. Teubner. Written with the help of Georg Scheffers.[8]
• Lie, Sophus (1893), Vorlesungen über continuierliche Gruppen (in German), Leipzig: B. G. Teubner. Written with the help of Georg Scheffers.[9]
• Lie, Sophus (1893), Theorie der Transformationsgruppen III (in German), Leipzig: B. G. Teubner. Written with the help of Friedrich Engel.
• Lie, Sophus (1896), Geometrie der Berührungstransformationen (in German), Leipzig: B. G. Teubner. Written with the help of Georg Scheffers.[10]
• Lie, Sophus, Engel, Friedrich; Heegaard, Poul (eds.), Gesammelte Abhandlungen, Leipzig: Teubner; 7 vols., 1922–1960{{citation}}: CS1 maint: postscript (link)[11][12]
See also
• Lie derivative
• List of simple Lie groups
• List of things named after Sophus Lie
Notes
1. James, Ioan (2002). Remarkable Mathematicians. Cambridge University Press. p. 201. ISBN 978-0-521-52094-2.
2. Darboux, Gaston (1899). "Sophus Lie". Bull. Amer. Math. Soc. 5 (7): 367–370. doi:10.1090/s0002-9904-1899-00628-1.
3. Lie, Sophus (1871). Over en classe geometriske Transformationer (PhD). University of Christiania.
4. Helgason, Sigurdur (1994), "Sophus Lie, the Mathematician" (PDF), Proceedings of the Sophus Lie Memorial Conference, Oslo, August, 1992, Oslo: Scandinavian University Press, pp. 3–21.
5. Gale, Thomson. "Marius Sophus Lie Biography". World of Mathematics. Retrieved 23 January 2009.
6. Hermann, Robert, ed. (1975), Sophus Lie's 1880 transformation group paper, Lie groups: History, frontiers and applications, vol. 1, Math Sci Press, p. iii, ISBN 0-915692-10-4
7. "The History of the Abel Prize". www.abelprize.no. Retrieved 4 February 2021.
8. Lovett, E. O. (1898). "Review: Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen". Bull. Amer. Math. Soc. 4 (4): 155–167. doi:10.1090/s0002-9904-1898-00476-7.
9. Brooks, J. M. (1895). "Review: Vorlesungen über continuerliche Gruppen mit geometrischen und anderen Anwendungen". Bull. Amer. Math. Soc. 1 (10): 241–248. doi:10.1090/s0002-9904-1895-00283-9.
10. Lovett, E. O. (1897). "Review: Geometrie der Berührungstransformationen". Bull. Amer. Math. Soc. 3 (9): 321–350. doi:10.1090/s0002-9904-1897-00430-x.
11. Schilling, O. F. G. (1939). "Book Review: Sophus Lie's Gesammelte Abhandlungen. Geometrische Abhandlungen, Volumes I & II". Bulletin of the American Mathematical Society. 45 (7): 513–514. doi:10.1090/S0002-9904-1939-07032-8. ISSN 0002-9904.
12. Carmichael, R. D. (1930). "Book Review: vol. IV of Sophus Lie's Gesammelte Abhandlungen (Samlede Avhandlinger, Norwegian edition published by Aschehoug)". Bulletin of the American Mathematical Society. 36 (5): 337–338. doi:10.1090/S0002-9904-1930-04950-2. ISSN 0002-9904. (with links to 1923 review of Vol. III, 1925 review of Vol. V, & 1928 review of Vol. VI)
References
• Fritzsche, Bernd (1999), "Sophus Lie: A Sketch of his Life and Work", Journal of Lie Theory, vol. 9, no. 1, pp. 1–38, ISSN 0949-5932, MR 1680023, Zbl 0927.01029, retrieved 2 December 2010
• Freudenthal, Hans (1970–1980), "Lie, Marius Sophus", Dictionary of Scientific Biography, Charles Scribner's Sons
• Stubhaug, Arild (2002), The mathematician Sophus Lie: It was the audacity of my thinking, Springer-Verlag, ISBN 3-540-42137-8
• Yaglom, Isaak Moiseevich (1988), Grant, Hardy; Shenitzer, Abe (eds.), Felix Klein and Sophus Lie: Evolution of the idea of symmetry in the nineteenth century, Birkhäuser, ISBN 3-7643-3316-2
External links
• Chisholm, Hugh, ed. (1911). "Lie, Marius Sophus" . Encyclopædia Britannica (11th ed.). Cambridge University Press.
• O'Connor, John J.; Robertson, Edmund F. (February 2000), "Sophus Lie", MacTutor History of Mathematics Archive, University of St Andrews
• Works by Sophus Lie at Project Gutenberg
• Works by or about Sophus Lie at Internet Archive
• "The foundations of the theory of infinite continuous transformation groups – I" An English translation of a key paper by Lie (Part I)
• "The foundations of the theory of infinite continuous transformation groups – II" An English translation of a key paper by Lie (Part II)
• "On complexes – in particular, line and sphere complexes – with applications to the theory of partial differential equations" An English translation of a key paper by Lie
• "Foundations of an invariant theory of contact transformations" An English translation of a key paper by Lie
• "The infinitesimal contact transformations of mechanics" An English translation of a key paper by Lie
• U. Amaldi, "On the principal results obtained in the theory of continuous groups since the death of Sophus Lie (1898–1907)" English translation of a survey paper that followed his death
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| Wikipedia |
\begin{document}
\begin{abstract}
A linear ordering is called context-free if it is the lexicographic ordering of some context-free language
and is called scattered if it has no dense subordering. Each scattered ordering has an associated ordinal,
called its rank. It is known that scattered context-free (regular, resp.) orderings have rank less than $\omega^\omega$
($\omega$, resp).
A language is called a one-counter language if it can be recognized by a pushdown automaton having a singleton
stack alphabet (essentially working as a counter that can hold a nonnegative integer and can be tested
against zero). The class of one-counter languages lies strictly between the classes of the regular
and the context-free languages.
In this paper we confirm the conjecture of Dietrich Kuske from 2012
that scattered one-counter languges have rank less than $\omega^2$. \end{abstract}
\title{Scattered one-counter languges have rank less than $\omega^2$} \section{Introduction} If an alphabet $\Sigma$ is equipped by a linear order $<$, this order can be extended to the lexicographic ordering $<_\ell$ on $\Sigma^*$ as $u<_\ell v$ if and only if either $u$ is a proper prefix of $v$ or $u=xay$ and $v=xbz$ for some $x,y,z\in\Sigma^*$ and letters $a<b$. So any language $L\subseteq \Sigma^*$ can be viewed as a linear ordering $(L,<_\ell)$. Since $\{a,b\}^*$ contains the dense ordering $(aa+bb)^*ab$ and every countable linear ordering can be embedded into any countably infinite dense ordering, every countable linear ordering is isomorphic to one of the form $(L,<_\ell)$ for some language $L\subseteq\{a,b\}^*$.
This way, countable order types can be represented by languages over some alphabet (by a prefix-free encoding of the alphabet by binary strings, one can restrict the alphabet to the binary one). A very natural choice is to use regular or context-free languages as these language classes are well-studied. A linear ordering (or an order type) is called \emph{regular} or \emph{context-free} if it is isomorphic to the linear ordering (or, is the order type) of some language of the appropriate class. It is known~\cite{DBLP:journals/fuin/BloomE10} that an ordinal is regular if and only if it is less than $\omega^\omega$ and is context-free if and only if it is less than $\omega^{\omega^\omega}$. Also, the Hausdorff rank~\cite{rosenstein} of any scattered regular (context-free, resp.) ordering is less than $\omega$ ($\omega^\omega$, resp)~\cite{ITA_1980__14_2_131_0,10.1007/978-3-642-29344-3_25}.
It is known~\cite{GelleIvanTCS} that the order type of a well-ordered language generated by a prefix grammar (i.e. in which each nonterminal generates a prefix-free language) is computable, thus the isomorphism problem of context-free ordinals is decidable if the ordinals in question are given as the lexicograpic ordering of \emph{prefix} grammars. Also, the isomorphism problem of regular orderings is decidable as well~\cite{DBLP:journals/ita/Thomas86,BLOOM200555}, even in polynomial time~\cite{LOHREY201371}. On the other hand, it is undecidable for a context-free grammar whether it generates a dense language, hence the isomorphism problem of context-free orderings in general is undecidable~\cite{ESIK2011107}. It is unknown whether the isomorphism problem of scattered context-free orderings is decidable -- a partial result in this direction is that if the rank of such an ordering is at most one (that is, the order type is a finite sum of the terms $\omega$, $-\omega$ and $1$), then the order type is effectively computable from a context-free grammar generating the language~\cite{GelleIvanGandalf,sofsem2020}. Also, it is also decidable whether a context-free grammar generates a scattered language of rank at most one.
It is a very plausible scenario though that the isomorphism problem of scattered context-free orderings is undecidable in general -- the rank $1$ is quite low compared to the upper bound $\omega^\omega$ of the rank of these orderings, and there is no known structural characterization of scattered context-free orderings. Clearly, among the well-orderings, exactly the ordinals smaller than $\omega^{\omega^\omega}$ are context-free but for scattered orderings the main obstacle is the lack of a finite ``normal form'' -- as every $\omega$-indexed sum of the terms $\omega$ and $-\omega$ is scattered of rank two, and these order types are pairwise different, there are already uncountably many scattered orderings of rank two and thus only a really small fraction of them can possibly be context-free.
The class of the one-counter languages lies strictly between the classes of regular and context-free languages: these are the ones that can be recognized by a pushdown automaton having only one stack symbol. In~\cite{kuske}, a family of well-ordered languages $L_n\subseteq\{a,b,c\}^*$ was given for each integer $n\geq 0$ so that the order type of $L_n$ is $\omega^{\omega\times n}$ (thus its rank is $\omega\times n$) and Kuske formulated two conjectures: i) the order type of well-ordered one-counter languages is strictly less than $\omega^{\omega^2}$ and more generally, ii) the rank of scattered one-counter languages is strictly less than $\omega^2$. Of course the second conjecture implies the first.
In this paper we prove the second conjecture of~\cite{kuske}: $\omega^2$ is a strict upper bound for the rank of scattered one-counter languages. The contents of the paper contain new results only: instead of reproving the results of~\cite{GelleIvanGandalf} and the subsequent, more general~\cite{sofsem2020} (these papers already contain full proofs and examples as well to their respective results), we push the boundaries of the knowledge of scattered context-free orderings by applying some of the tools we developed in the earlier papers to the class of one-counter languages. It turns out that it is enough to study restricted one-counter languages to prove the conjecture, and for this, a crucial step is to reason about the cycles in a generalized sequential machine -- so at the end, we can again use some graph-theoretic methods. \section{Notation}
We assume the reader has some background with formal language theory and linear orderings
(the textbooks \cite{Hopcroft+Ullman/79/Introduction,rosenstein} being excellent resources for that),
but we list the notions we use in the paper to settle the notation.
\subsection{Linear orderings}
A (strict) \emph{linear ordering} is a pair $(A,<)$ with $A$ being a set, the \emph{domain} of the ordering and
$<$ being a binary relation over $A$ which is \emph{irreflexive}: $x\not<x$, \emph{transitive}: $x<y,y<z~\Rightarrow x<z$ and \emph{total}: for any $x,y\in A$, exactly one of $x<y$, $y<x$ or $x=y$ holds. In particular,
the empty set equipped with the empty ordering relation is also a linear ordering. To ease notation, we
sometimes write $I=(I,<)$ and denote the linear ordering $(I,<)$ simply by its domain $I$ when there is
no chance of confusion.
The linear ordering $(A,<_A)$ is a \emph{subordering} of the linear ordering $(B,<_B)$ if $A\subseteq B$ and
$<_A$ is the restriction of $<_B$ to $A$. The linear ordering $(A,<_A)$ can be \emph{embedded} into $(B,<_B)$
if there is a mapping $h:A\to B$ that \emph{preserves order} (if $x<y$, then $h(x)<h(y)$) -- such mappings
are called \emph{embeddings} of $A$ into $B$. Clearly, embeddings are injective ($h(x)=h(y)$ implies $x=y$);
if an embedding is also onto $B$ (also said ``surjective'': for each $x\in B$ there is some $x'\in A$ with
$h(x')=x$), then $h$ is called an $\emph{(order) isomorphism}$ between $A$ and $B$. If there exists an
isomorphism between two linear orderings, then we call them \emph{isomorphic}, denoted $A\simeq B$.
Isomorphism of linear orderings is an equivalence relation on any set of linear orderings, an \emph{order type}
is an equivalence class of isomorphism. Of course, for any integer $n\geq 0$, the linear orderings with
$n$-element domains are isomorphic, we denote their order type also by $n$. The order type of the nonnegative
integers $0<1<\ldots$ is denoted $\omega$ while the order types of the integers and the rationals (equipped
with their standard ordering relations) are denoted $\zeta$ and $\eta$ respectively.
The order type of a linear ordering $(A,<)$ is denoted $o(A,<)$. Since embeddability of
linear orderings is preserved under isomorphism, this notion can be lifted to order types and so we write
$o_1\leq o_2$ if the linear orderings of order type $o_1$ can be embedded into the linear orderings of order type
$o_2$ and $o_1<o_2$ if $o_1\leq o_2$ but not vice versa. Note that the relation $<$ is not necessarily a linear
ordering on a set of order types, e.g. the intervals $(0,1)$ and $[0,1]$ of real numbers can be embedded into
each other and they have distinct order types.
When $I=(I,<)$ is a linear ordering and for each $i\in I$, $A_i=(A_i,<_i)$ is a linear ordering, then the
\emph{(generalized) sum} of the $A_i$s (with respect to $I$) is the linear ordering $\mathop\sum\limits_{i\in I}A_i$
with domain $\mathop\bigcup\limits_{i\in I}A_i\times\{i\}$ and ordering relation $(x,i)<(y,j)$ if and only if
$i<j$ or ($i=j$ and $x<_iy$), this ordering called the \emph{anti-lexicographic ordering} of the domain. As a special case, $(A_1,<_1)+(A_2,<_2)$ denotes $\mathop\sum\limits_{i\in\{1,2\}}A_i$.
When in such a generalized sum all the $A_i$s are the same linear ordering $A$, we write $A\times I$ for
$\mathop\sum\limits_{i\in I}A$.
When $I\simeq I'$ and for each $i\in I$, $A_i\simeq A'_i$, then $\mathop\sum\limits_{i\in I}A_i~\simeq~\mathop\sum\limits_{i\in I'}A'_i$ so the sum and product operations extend naturally to
order types, e.g. $\omega\times 2=\omega+\omega$ is the order type of the linear ordering we get by
placing two copies of the natural numbers next to each other, while $\omega\times\omega$ is the order type
of the set consisting of pairs of natural numbers, equipped with the anti-lexicographic order.
A linear ordering $(A,<)$ is a \emph{well-ordering} if it does not contain an infinite descending chain
$\ldots<x_3<x_2<x_1$, is \emph{quasi-dense} if the rationals, equipped with their standard ordering, can
be embedded into $A$, and is \emph{scattered} if it is not quasi-dense. A \emph{dense} ordering is a
linear ordering $(A,<)$ having at least two elements such that whenever $x<y$, then there exists some $z$
with $x<z<y$. These notions are preserved under isomorphism, so they can be lifted naturally to order types,
e.g. the finite order types and $\omega$ are well-ordered, $\zeta$ is scattered but not well-ordered,
and $\eta$ is dense. Each dense ordering is quasi-dense but the converse does not hold, e.g. $2\times\eta$ (the
ordering we get from the rationals by replacing each rational by two elements) is quasi-dense but not dense.
(Note that on the other hand, $\eta\times 2=\eta$ holds).
When $(A,<)$ is a linear ordering with order type $o$, then the order type of $(A,<')$ where $x<'y$ if and only
if $y<x$, is denoted by $-o$, e.g. $-\omega$ is the order type of the negative integers.
Note that we use here a minus sign instead of the more common notation $o^*$, the reason being to avoid
confusion with the Kleene star operation.
The order types of well-orderings are called \emph{ordinals}, e.g. $0,1,42,\omega,\omega+3,\omega\times\omega+\omega$
are ordinals. Since any well-ordered sum or well-orderings is also well-ordered, finite products and sums of
ordinals are ordinals as well. Any set of ordinals is well-ordered by the relation $<$ (embeddability in one
direction), e.g. $0<1<42<\omega<\omega+2<\omega\times\omega$. Moreover, for any set $X$ of ordinals, their
supremum (with respect to this relation $<$) $\bigvee X$ exists and is an ordinal as well, e.g.
$\bigvee\{0,1,\ldots\}=\omega$. Each ordinal $\alpha$ is either a \emph{successor ordinal}, in which case
$\alpha=\beta+1$ for some ordinal $\beta$, or is a \emph{limit ordinal}, in which case $\alpha=\mathop\bigvee\limits_{\beta<\alpha}\beta$. For example, $42$, $\omega+2$ and $\omega\times\omega+3$
are successor ordinals while $0$, $\omega$ and $\omega\times\omega+\omega$ are limit ordinals.
As any set of ordinals is well-ordered by $<$, one can use transfinite induction on them, by showing that
if a property $P$ holds for some ordinal $\alpha$ then it also holds for $\alpha+1$, and whenever $\alpha$
is a limit ordinal and $P$ holds for each $\beta<\alpha$, then $P$ holds for $\alpha$ as well, that proves
$P$ holds for all the ordinals. Frequently, the case when $\alpha=0$ is treated separately.
On ordinals, not only sums and products but exponentiation is also defined: when $\alpha$ and $\beta$
are ordinals, then the ordinal $\alpha^\beta$ is defined as
\begin{itemize}
\item $1$ if $\beta=0$,
\item $\alpha^\gamma\times\alpha$ if $\beta=\gamma+1$,
\item $\mathop\bigvee\limits_{\gamma<\beta}\alpha^\gamma$ if $\beta$ is a nonzero limit ordinal.
\end{itemize}
The exponentiation notation is ``right-associative'', i.e. $\alpha^{\beta^\gamma}$ stands for $\alpha^{(\beta^\gamma)}$. We omit the parentheses that are redundant applying that $+$ and $\times$ are associative,
and using the convention that exponentiation takes precedence over product, which in turn takes precedence over sum.
Hausdorff associated an ordinal rank to each scattered ordering (see e.g.~\cite{rosenstein}), but we use a slightly modified variant (not affecting the main result as this variant differs
from the original one by at most one) introduced in~\cite{10.1007/978-3-642-29344-3_25} as follows. For each ordinal
$\alpha$ we define a class $H_\alpha$ of linear orderings:
\begin{itemize}
\item $H_0$ contains all the finite linear orderings;
\item $H_\alpha$ for $\alpha>0$ is the least class of linear orderings closed under finite sum and isomorphism
which contains all the sums of the form $\mathop\sum\limits_{i\in\zeta}A_i$, where for each integer $i$, the
linear ordering $A_i$ belongs to $H_{\beta_i}$ for some ordinal $\beta_i<\alpha$.
\end{itemize}
By Hausdorff's theorem, a countable linear ordering $A$ is scattered if and only if some class $H_\alpha$ contains it:
the least such $\alpha$ is called the \emph{rank} of the ordering (or of the order type as the value factors through isomorphism) and is denoted $\mathrm{rank}(A)$ (or $\mathrm{rank}(o)$ for the order type $o=o(A)$.)
We note here that the original definition of Hausdorff includes only the empty ordering and the singletons into $H_0$
and does not require the classes $H_\alpha$ to be closed under finite sum. Since a finite sum of orderings can always be
written as a zeta-sum of the same orderings and infinitely many zeros, and a zeta-sum of finite linear orderings is also a
zeta-sum of empty and singleton orderings, this slight change can introduce only a difference of one between the rank,
e.g. $\omega+\omega$ has rank one in our rank notion but has rank two in the original one. Since $\alpha<o$ for a limit ordinal
$o$ and an ordinal $\alpha$ if and only if $\alpha+1<o$, and $o=\omega^2$ is a limit ordinal, the main theorem holds for the
original notion of rank as well.
The reader is encouraged to verify that for any (countable) ordinal $\alpha$,
the rank of $\omega^\alpha$ is $\alpha$.
\subsection{Formal languages}
An \emph{alphabet} is a finite nonempty set of symbols, which are also called \emph{letters} of the alphabet.
We assume each alphabet comes with a fixed total ordering on its letters.
For a nonempty set $\Sigma$, $\Sigma^*$ denotes the free monoid freely generated by $\Sigma$,
that is, the set of (finite) words $a_1\ldots a_n$, $n\geq 0$, $a_i\in\Sigma$ over $\Sigma$. The length
of a word $w=a_1\ldots a_n$ is $|w|=n$. For $n=0$, we get the empty word which is denoted by $\varepsilon$.
(We assume the symbol $\varepsilon$ itself is not an element of any alphabet). In this monoid, the product
operation is $a_1\ldots a_n\cdot b_1\ldots b_k=a_1\ldots a_nb_1\ldots b_k$ and the symbol $\cdot$ is often
discarded when it does not ruin readability.
When $u\in\Sigma^*$ is a word and $a\in\Sigma$ is a letter, then $|u|_a$ denotes the number of occurrences
of $a$ in $u$, formally $|\varepsilon|_a=0$, $|ua|_a=|u|_a+1$ and $|ub|_a=|u|_a$ for each $b\neq a$.
A \emph{language} over the alphabet $\Sigma$ is any subset
$L$ of $\Sigma^*$. Product of two languages $K$ and $L$ is defined as $K\cdot L=\{uv:u\in K,v\in L\}$,
again by omitting the symbol $\cdot$ when there is no danger of confusion.
When $\Sigma$ is a totally ordered set, we use two partial orderings on $\Sigma^*$: the \emph{prefix ordering} $\leq_p$
($u\leq_p v$ if and only if $v=uu'$ for some $u'\in\Sigma^*$), with $<_p$ denoting the strict variant of $\leq_p$,
and the \emph{strict ordering} $<s$ ($u<_sv$ if and only if $u=u_1au_2$ and $v=u_1bu_3$ for some words $u_1,u_2,u_3\in\Sigma^*$ and
letters $a<b$). Their union is the \emph{lexicographic ordering} $\leq_\ell$ of $\Sigma^*$ which is a total ordering
and whose strict variant is denoted $<_\ell$. This way, each language $L\subseteq\Sigma^*$ can be viewed as a linear ordering
set $(L,\leq_\ell)$; let $o(L)$ denote the order type of the language $L$. As an example, for the binary alphabet $\{0,1\}$
with $0<1$ we have $o(0^*)$ is $\omega$, $o(0^*1)$ is $-\omega$
as $\ldots<_{\ell} 001<_{\ell} 01<_{\ell} 1$, and $o((00+11)^*01)=\eta$.
We say that the language $L$ is scattered, well-ordered, etc. if so is the linear ordering $(L,<_\ell)$.
For a language $L\subseteq \Sigma^*$, we let ${\mathbf{Pref}}(L)$ stand for the set $\{u\in\Sigma^*:~u\leq_pv\hbox{ for some }v\in L\}$ of
the prefices of the members of $L$. Similarly, let $\mathbf{Suf}(L)$ stand for the set of the suffices of the members of $L$
(which is formally the reversal of the prefix language of the reversal of $L$, say).
When $L$ is a language and $n\geq 0$ is an integer, then $L^n$ is the language defined inductively
as $L^0=\{\varepsilon\}$ and $L^{n+1}=L^nL$, and $L^*$ ($L^+$, resp.) denotes the language
$\mathop\bigcup\limits_{n\geq 0}L^n$ ($\mathop\bigcup\limits_{n\geq 1}L^n$, resp.) When $L=\{u\}$ is a singleton
language, we might use $u^*$ and $uK$ for $\{u\}^*$ and $\{u\}K$. The class of \emph{regular} languages
over some alphabet $\Sigma$ is the least class which contains the empty language $\emptyset$, all the
singleton languages $\{a\}$ with $a\in\Sigma$ and which is closed under finite union, product and star.
When $u\in\Sigma^*$ is a word and $L\subseteq\Sigma^*$ is a language, then $u^{-1}L$ denotes the
language $\{v\in\Sigma^*:uv\in L\}$. It is known that if $L$ is regular, then so is $u^{-1}L$ for any word $u$.
For each word $u$ there is a shortest prefix $v$ of $u$ so that $u\in v^*$, this word $v$ is called the \emph{primitive root}
$\mathrm{root}(u)$ of $u$. The word $u$ is called \emph{primitive} if $u=\mathrm{root}(u)$.
An \emph{$\omega$-word} over $\Sigma$ is a sequence $a_1a_2\ldots$ of letters. We let $\Sigma^\omega$
denote the set of all $\omega$-words. Then, $\Sigma^\omega$ is linearly ordered by the (appropriate modification of the) relation $<_s$ and $\Sigma^*\cup\Sigma^\omega$ is linearly ordered by $<_\ell$. Of course we can
define the product $u\cdot v\in\Sigma^\omega$ with $u\in\Sigma^*$ and $v\in\Sigma^\omega$ as expected,
as well as the word $u^\omega=uu\ldots\in\Sigma^\omega$ for each $u\in\Sigma^+$.
\subsection{Transducers and (restricted) one-counter languages}
Let $D_1\subseteq\{0,1\}^*$ be the language of proper bracketings where
$0$ plays the role of the opening bracket while $1$ plays
the closing bracket. That is, a word $u\in\{0,1\}^*$ belongs to $D_1$ if and only if $|u|_0=|u|_1$ and for each
prefix $v$ of $u$, $|v|_0\geq |v|_1$.
A (nondeterministic) \emph{regular transducer} for the purposes of this paper is a tuple $M=(Q,\Sigma,\Delta,q_0,F,\mu)$
where $Q$ is the finite set of states, $q_0\in Q$ is the initial state, $F\subseteq Q$ is the set of final states,
$\Sigma$ is the \emph{output} alphabet, $\Delta\subseteq Q\times \{0,1\}\times Q$ is the transition relation and for each
$(p,a,q)\in\Delta$, $\mu(p,a,q)$, also denoted $R_{p,a,q}$ is a nonempty regular language over $\Sigma$.
For each word $w\in\{0,1\}^*$ and states $p,q\in Q$ we associate a (regular) language
$L(M,w,p,q)$ inductively as follows:
first, let $L(M,\varepsilon,p,q)=\begin{cases}\varepsilon&\hbox{if }p=q\\\emptyset&\hbox{otherwise.}\end{cases}$
Then, for each nonempty word $w=ua$, let
$L(M,ua,p,q)=\mathop\bigcup\limits_{(r,a,q)\in\Delta}L(M,u,p,r)\cdot R_{r,a,q}$.
We define $L(M,w)=\mathop\bigcup\limits_{q\in F}L(M,w,q_0,q)$ and
$L(M)=\mathop\bigcup\limits_{u\in D_1}L(M,u)$. Observe that we only allow the binary alphabet
as input, moreover, the transducer is by definition only applied to the language $D_1$ of proper
bracketings -- we make these restrictions to ease notation and to maintain readability of the paper.
A language $L\subseteq\Sigma^*$ is called a \emph{restricted one-counter language} if $L=L(M)$ for some regular transducer $M$.
As an example, consider the transducer given on Figure~\ref{fig-trans-cban},
with $q_0$ being its initial and $q_f$ being its only final state. Clearly, only words of the form $w=0^*1^+$ can have a nonempty
image $L(M,w)$ under $M$, so as $0^*1^+~\cap~D_1=\{0^n1^n:n\geq 1\}$, $L(M)=\mathop\bigcup\limits_{n\geq 1}L(M,0^n1^n)=\mathop\bigcup\limits_{n\geq 1}
c^n(b^*a)^n$, so this language $L=L(M)$ is a restricted one-counter language. In~\cite{kuske} it has been shown that $o(L)=\omega^\omega$
and $o(L^k)=\omega^{\omega\times k}$. In particular, for each $k\geq 0$, $L^k$ is a scattered language
of rank $\omega\times k$. (Note that $L^*$ is not scattered by e.g. Proposition~\ref{prop-iterate-vstar}
so $L^*$ is \emph{not} an example of a scattered language of rank $\omega^2$, though it's a one-counter
language, see below.)
\begin{center}\begin{figure}
\caption{Transducer for $c^n(b^*a)^n$}
\label{fig-trans-cban}
\end{figure}
\end{center}
A \emph{one-counter language} is usually defined via the means of pushdown automata operating with a single stack symbol.
The characterization from~\cite{Berstel79transductionsand}, see also
\cite{DBLP:journals/jcss/Latteux83} suits our purposes better:
the class of one-counter languages is the least language class
which contains the restricted one-counter languages and is closed under concatenation, union and Kleene iteration. \subsection{Linear and semilinear sets}
Let $\mathbb{N}_0$ stand for the set of nonnegative integers.
We call a set $X\subseteq\mathbb{N}_0^k$ \emph{periodic} if it has the form
$X=\{N+M\cdot t:t\geq 0\}$ for some vectors $N,M\in\mathbb{N}_0^k$;
\emph{linear} if it has the form $X=\{N_0+N_1\cdot t_1+N_2\cdot t_2+\ldots+N_n\cdot t_n:
t_1,\ldots,t_n\geq 0\}$ for some integer $n\geq 0$ and vectors $N_0,\ldots,N_n\in\mathbb{N}_0^k$;
\emph{semilinear} if it is a finite union of linear sets
and \emph{ultimately periodic} if it is a finite union of periodic sets.
(Observe that a singleton set is also periodic, by choosing the vector $M$
in the definition to be the null vector, thus finite sets are ultimately periodic.)
It is known~\cite{Matos94periodicsets} that a subset of $\mathbb{N}_0$
is ultimately periodic if and only if it is semilinear.
Moreover, by Parikh's theorem we know that the Parikh image $\Psi(L)=\{(|u|_0,|u|_1):u\in L\}$ of any context-free (thus, any regular) language $L\subseteq\{0,1\}^*$ is semilinear (the theorem holds
for arbitrary alphabets).
\section{Some order-theoretic properties of scattered languages and operations}
In this section we list several statements connecting the rank of scattered languages with language-theoretic
operations.
The reason why we use the modified rank variant instead of the original one is the following couple
of handy statements: \begin{proposition}[\cite{10.1007/978-3-642-29344-3_25}]
\label{prop-rank-ops}
Some useful properties of the version of the Hausdorff rank that we use that hold for scattered languages $K$ and $L$:
\begin{itemize}
\item $\mathrm{rank}(L)=\mathrm{rank}({\mathbf{Pref}}(L))$ (in particular, $\mathbf{Pref}(L)$ is also scattered whenever
$L$ is)
\item $\mathrm{rank}(K\cup L)=\mathrm{max}\bigl(\mathrm{rank}(K),\mathrm{rank}(L)\bigr)$
\item $\mathrm{rank}(KL)\leq \mathrm{rank}(L)+\mathrm{rank}(K)$
\item more generally, if $K$ is scattered of rank $\alpha$ and for each $w\in K$, $L_w$ is a scattered language with rank at most $\beta$,
then $\mathop\bigcup\limits_{w\in K}wL_w$ is scattered of rank at most $\beta+\alpha$.
\end{itemize} \end{proposition}
We make heavy use of the following simple propositions later:
\begin{proposition}
\label{prop-iterate-vstar}
Assume $L\subseteq\Sigma^*$ is a language such that $L^+$ is scattered. Then $L\subseteq v^*$ for some word $v\in\Sigma^*$ (and consequently, so is $L^+$).
\end{proposition}
\begin{proof}
Assume $u,v\in L$ are nonempty words with $\mathrm{root}(u)\neq\mathrm{root}(v)$.
Then, by Lyndon's theorem (see e.g.~\cite{10.5555/267846}, Theorem 2.2), $uv\neq vu$, say $uv<_svu$ (having the same length, they cannot be in the $<_p$ relation,
so it's either $uv<_svu$ or the other way around). Then the language $\{uvuv,vuvu\}^*uvvu$ forms a dense subset in $L^+$.
Thus, if $L^+$ is scattered, then the nonempty members of $L$ share a common primitive root $v$, and hence $L\subseteq v^*$.
\end{proof} Languages having a specific form will play crucial role in our proofs: \begin{definition} A \emph{prefix chain} is a language $L$ whose words are linearly ordered by the relation $<_p$. A language $L$ \emph{prefix free} if its words are pairwise incomparable with respect to the relation $<_p$ (and consequently, if and only if it is linearly ordered by the relation $<_s$). \end{definition} Observe that finite prefix chains have finite order types and thus have rank $0$, while infinite prefix chains have order type $\omega$ and thus have rank $1$. Also, for each infinite prefix chain $C$ there exists a unique $\omega$-word $w_C$ such that $C\subseteq\mathbf{Pref}(w_C)$. We call $w_C$ the \emph{limit} of $C$.
\begin{proposition}
\label{prop-dense-language-has-a-prefixfree-sublanguage}
If $L\subseteq\Sigma^*$ is a dense language,
then it has a prefix-free dense subset $K\subseteq L$.
\end{proposition}
\begin{proof}
Let $P\subseteq L$ be the language containing all the words which are members of some infinite prefix chain of
$L$, that is, $P=\mathop\bigcup\limits_{C\subseteq L\hbox{ is an infinite prefix chain}}C$. Now we have two cases:
{\textbf{Case 1.}} If $P$ is not dense, then there exist two elements $u,v\in P$ such that $u<_\ell v$ but there is no $w\in P$
with $u<_\ell w<_\ell v$. Then, the sublanguage $L'=\{x\in L ~:~ u <_\ell x <_\ell v \}$ of $L$ is still dense
and has no member in $P$. In $L'$ there can be elements which are in the prefix relation, but all the
$<_p$-chains are finite within $L'$ (since if $L'$ contained an infinite $<_p$ chain, its elements would be in $P$). So let $K\subseteq L'$ be the language containing the $<_p$-maximal elements of $L'$. Of course, $K$ is prefix-free.
We now show that $K$ is dense: let $u_1$, $u_2$ be members of $K$ with $u_1<_\ell u_2$. Since $K$ is prefix-free,
it has to be the case that $u_1<_su_2$. Now let $u'_2$ be the shortest prefix of $u_2$ with $u_1<_su'_2\leq_p u_2$
and $u'_2\in L'$. Since $L'$ itself is dense and $u_1,u'_2$ belong to $L'$, there is some word $w\in L'$ with
$u_1<_\ell w<_\ell u'_2$. With $u_1$ being a $<_p$-maximal element of $L'$, it has to be the case $u_1<_sw$
and as $w<_pu'_2$ would contradict the minimality of $u'_2$, it also has to be $w<_su'_2$. Hence, $u_1<_sw<_su_2$
as well and by the assumption of $P$, there has to be a $<_p$-maximal element $w'$ of $L'$ with $w<_p L'$
(otherwise there would be an infinite prefix chain present in $P$). Hence this $w'$ also belongs to $K$ and so
$K$ is dense.
{\textbf{Case 2.}} If $P$ is dense, we define a word $x_u\in P$ inductively for each word $u\in \{0,2\}^*\{\varepsilon,1\}$ such that
$u <_p v$ implies $x_u <_p x_v$ and $u <_s v$ implies $x_u <_s x_v$. This way we embed the dense
language $\{0,2\}^*\{1\}$ into $P$, proving the statement.
First observe that for each $x\in P$, there has to be an infinite number of $\omega$-words $w$
such that $x\in{\mathbf{Pref}}(w)$ and ${\mathbf{Pref}}(w)\cap P$ is infinite (that is, there have to be infinitely many
different prefix chains containing $w$ within $L$), for if there were some $x\in P$ with only a finite number
of such $\omega$-words, say $\{w_1,\ldots,w_k\}$,
then by choosing one of them, say $w_1$, there would be a length $N$ such that
if $u\in{\mathbf{Pref}}(w_1)$ with $|u|\geq N$, then $u\notin{\mathbf{Pref}}(w_i)$ for $i>1$ (as $\omega$-words are linearly ordered
by $<_s$). Hence, if $u$ and $v$ were
long enough members of ${\mathbf{Pref}}(w_1)$, then only a finite number of elements of $P$ would fit between them
(each of them being prefixes of the same $w_1$) and $P$ wouldn't be a dense set.
So, moving back to the construction, for the base step, we choose an arbitrary word from $P$, for $x_\varepsilon$.
Having defined $x_u\in P$ with $u\in\{0,2\}^*$, we define $x_{u0}$, $x_{u1}$ and $x_{u2}$ as follows.
Since there are infinitely many infinite prefix chains in $P$ containing $x_u$, we can choose three different
$\omega$-words, $w_1$, $w_2$ and $w_3$ with $x_u$ being a prefix of each of them and with $w_1<_s w_2<_sw_3$
and of course with $\mathbf{Pref}(w_i)\cap P$ being infinite (i.e. three $\omega$-words corresponding to
three different maximal prefix chains within $P$, each containing $x_u$).
Since the three $\omega$-words differ, long enough prefices of $w_i$ are not prefices of the other two words,
and since each $w_i$ is a limit of an infinite prefix chain, we can choose long enough prefices of each $w_i$
which are in $P$ and not prefices of the other two $\omega$-words. We define $x_{u0}$, $x_{u1}$ and $x_{u2}$
to be this prefix of $w_1$, $w_2$ and $w_3$ respectively. Then of course, $x_u<_p x_{u0}, x_{u1}, x_{u2}$
as well as $x_{u0}<_sx_{u1}<_sx_{u2}$ are satisfied.
Thus, the words of the form $u_{x1}$ form a dense subset of $P$.
\end{proof}
\begin{proposition}
\label{prop-sub-of-scattered-is-scattered}
If $L\subseteq\Sigma^*$ is a scattered language and $uK\subseteq{\mathbf{Pref}}(L)$ for some word $u\in\Sigma^*$ and
language $K\subseteq\Sigma^*$, then $K$ is scattered as well.
\end{proposition}
\begin{proof}
Assume $K$ is quasi-dense with $uK\subseteq\mathbf{Pref}(L)$ and let $X\subseteq K$ be a dense subset of $K$.
Then, $uX$ is still dense and $uX\subseteq\mathbf{Pref}(L)$ which would imply $\mathbf{Pref}(L)$ being
quasi-dense and thus by Proposition~\ref{prop-rank-ops}, $L$ would have to be quasi-dense as well,
contradicting the assumptions of the proposition.
\end{proof}
\begin{corollary}
\label{cor-product-members-are-scattered}
If $L=L_1L_2$ is a nonempty scattered language, then so are $L_1$ and $L_2$.
\end{corollary}
\begin{proof}
Apply Proposition~\ref{prop-sub-of-scattered-is-scattered} with $K=L_1$ and $u=\varepsilon$ for
$L_1$ and with $K=L_2$ and an arbitrary $u\in L_1$ for $L_2$.
\end{proof}
\section{The main result} We are ready to state the main result of the paper. After that, in this we sketch a birds-eye view of its proof, which is fleshed out in the remaining Sections. \begin{theorem} \label{thm-main}
The rank of any scattered one-counter language is smaller than $\omega^2$. \end{theorem} First, observe that it suffices to prove the statement for restricted one-counter languages. Since a language is one-counter if and only if it can be constructed from restricted one-counter languages by a finite number of union, product and star applications, we can use induction on the required number of those applications where the induction steps are \begin{itemize}
\item if $K^*$ is a scattered one-counter language for the one-counter language $K$,
then by Proposition~\ref{prop-iterate-vstar},
$K^*\subseteq v^*$ for some word $v$ and hence has rank at most $1$,
\item while if $K=L_1L_2$ is a scattered one-counter language for the one-counter languages
$L_1$ and $L_2$, then by Corollary~\ref{cor-product-members-are-scattered} both $L_1$ and
$L_2$ are scattered. Applying the induction hypothesis we get that the ranks of both $L_1$
and $L_2$ are smaller than $\omega^2$, thus applying Proposition~\ref{prop-rank-ops},
the rank of $L$ is at most $\mathrm{rank}(L_2)+\mathrm{rank}(L_1)$ that is still smaller than
$\omega^2$ if so are the two summands,
\item and if $K=L_1\cup L_2$ is a scattered one-counter language for the one-counter languages
$L_1$ and $L_2$, then again, both of $L_1$ and $L_2$ are scattered as well, thus applying
the induction hypothesis and $\mathrm{rank}(K)\leq\mathrm{max}\{\mathrm{rank}(L_1),\mathrm{rank}(L_2)\}$
of Proposition~\ref{prop-rank-ops} we get that the rank of $K$ is again smaller than $\omega^2$. \end{itemize} Hence it is enough to show that any scattered restricted one-counter language has rank smaller than $\omega^2$.
We prove this in the following way: \begin{itemize}
\item We start from a transducer $M$ with $L(M)=L$ being a scattered language.
\item First we assume that $M$ has the ``feasible cycle property'', stating that whenever
there is a cycle in the graph of $M$, then there exists a member $u$ of $D_1$ and a run of $M$
over $u$ which visits this cycle.
\item Then we prove that, by studying the possible cycles in $M$ (which will be categorized to ``$0$-cycles'',
``positive cycles'' and ``negative cycles'' based on the sign of the difference of the $0$s and $1$s in them)
that for these transducers, the rank of $L$ has to be smaller than $\omega^2$ (using induction on the height
of the connected components of the transducer).
\item Finally, we prove that for each transducer $M$ there exists a transducer $M'$ satisfying the
feasible cycles property with $L(M)=L(M')$. \end{itemize}
\section{Transducers with feasible paths only - handling nonnegative cycles} Let $M=(Q,\Sigma,\Delta,q_0,F,\mu)$ be a transducer. Without loss of generality, we can assume that $q_0$ is a source state with no incoming transitions, $F=\{q_f\}$ is a singleton and $q_f$ is a sink state with no outgoing transitions. For each transition $(p,a,q)\in\Delta$, let $R_{p,a,q}$ stand for the (regular, nonempty) output language $\mu(p,a,q)$ and by extension, if $(p,a,q)$ is not a transition in $M$, then let $R_{p,a,q}$ be $\emptyset$, let $R_{p,\varepsilon,q}=\{\varepsilon\}$ and for each word $u\in\{0,1\}^*$ and letter $a\in\{0,1\}$, let $R_{p,ua,q}$ be $\mathop\bigcup\limits_{r\in Q}R_{p,u,r}R_{r,a,q}$. Then, each language $R_{p,u,q}$ is regular and thus has a finite rank if it is scattered. Also, let $R_{p,q}$ stand for $\mathop\bigcup\limits_{u\in\{0,1\}^*}R_{p,u,q}$.
We also do the similar construction for the input part: let us define the $\leadsto$ relation as follows: $p\mathop{\leadsto}\limits^\varepsilon q$ if and only if $p=q$, $p\mathop{\leadsto}\limits^aq$ for a letter $a\in\{0,1\}^*$ if and only if $(p,a,q)\in\Delta$ and $p\mathop{\leadsto}\limits^{ua}q$ for $u\in\{0,1\}^*$ and $a\in\{0,1\}$ if and only if there exists a state $r$ with $p\mathop{\leadsto}\limits^ur\mathop{\leadsto}\limits^aq$. (That is, the ``reachability by an input word'' relation, without taking the output into account.) Let $p\leadsto q$ denote that $p\mathop{\leadsto}^uq$ holds for some word $u$. Then $(Q,\leadsto)$ can be seen as a directed transitive graph. Let a \emph{component} of $M$ be a strongly connected component of this graph (that is, $p$ and $q$ are in the same component of $M$ if $p\leadsto q$ and $q\leadsto p$).
For two states $p,q\in Q$, let us denote by $\mathbf{In}(p,q)=\{u\in\{0,1\}^*:p\mathop{\leadsto}\limits^uq\}$ the language of those input words that can take $p$ to $q$.
A key property of transducers in this section is that of having feasible cycles only: \begin{definition}
The transducer $M$ has the \emph{feasible cycles property}, also said \emph{has feasible cycles only}
if the following conditions hold: \begin{enumerate}
\item
Whenever $q\in Q$ is a state and $u\in\{0,1\}^*$ is an input word so that $q\mathop{\leadsto}\limits^u q$
(also called a cycle on $q$),
then there exist words $v,w\in\{0,1\}^*$ such that $vuw\in D_1$ and $q_0\mathop\leadsto\limits^vq\mathop\leadsto\limits^wq_f$.
\item
Whenever $q\in Q$ is a state and $u\in\{0,1\}^*$ is an input word so that $q\mathop{\leadsto}\limits^uq$
and $|u|_0>|u|_1$ (also called a positive cycle on $q$), then there exist a word $v$ and for each
integer $t\geq 0$ a word $w_t$ so that each $vu^tw_t$ is in $D_1$ and
$q_0\mathop\leadsto\limits^vq\mathop\leadsto\limits^{w_t}q_f$.
\item
Whenever $q\in Q$ is a state for which for each integer $N\geq 0$ there exist words $u_N$ and $v_N$
with $q_0\mathop\leadsto\limits^{u_N}q\mathop\leadsto\limits^{v_N}q_f$, $u_Nv_N\in D_1$
and $|u_N|_0-|u_N|_1\geq N$, then for each word $u\in\{0,1\}^*$ with $q\mathop\leadsto\limits^{u}q$,
there exists some integer $N\geq 0$
and word $v$ with $u_Nuv\in D_1$ and $q\mathop\leadsto\limits^{v}q_f$.
\end{enumerate} \end{definition}
We begin studying such transducers by classifying its possible cycles. First we show that if in a component there exists some \emph{positive cycle}, that is, a state $p$ and a word $u$ with $|u|_0>|u|_1$ and $p\mathop{\leadsto}\limits^up$, then all the cycles on this state can only generate a language of order type at most $\omega$ (and thus of rank at most $1$). \begin{proposition}
Assume $M$ has feasible cycles only, $L(M)$ is scattered
and there exists a state $p$ of $M$ and some word $u\in\mathbf{In}(p,p)$
with $|u|_0>|u|_1$.
Then there exists a primitive word $r_p\in\Sigma^*$ such that $R_{p,p}\subseteq r_p^*$. \end{proposition} \label{prop-positive-cycles-imply-rank-1-inner-loops} \begin{proof}
Observe that since $q_0$ is a source and $q_f$ is a sink, $p\notin\{q_0,q_f\}$.
Let us consider the output language $R_{p,u,p}$. Then for each $t\geq 1$ we have
$R_{p,u,p}^t\subseteq R_{p,u^t,p}$. Moreover, as $M$ has feasible cycles only,
there exist input words
$v,w_t\in\{0,1\}$ such that $q_0\mathop{\leadsto}\limits^v p\mathop{\leadsto}\limits^w_t q_f$, and $vu^tw_t\in D_1$, thus in particular, $vu^t\in\mathbf{Pref}(D_1)$.
Hence, for an arbitrary output word $v'\in R_{q_0,v,p}$ and integer $t\geq 1$,
we get $v'R_{p,u^t,p}R_{p,w_t,q_f}\subseteq L(M)$ and (as $R_{p,u,p}^t\subseteq R_{p,u^t,p}$)
thus $v'R_{p,u,p}^t\subseteq\mathbf{Pref}(L(M))$. Hence $v'R_{p,u,p}^*\subseteq\mathbf{Pref}(L(M))$, the latter
being a scattered language by Proposition~\ref{prop-sub-of-scattered-is-scattered}, $R_{p,u,p}^*$ is scattered
as well, which implies $R_{p,u,p}^*\subseteq r_u^*$ for some primitive output word $r_u\in\Sigma^*$ by
Proposition~\ref{prop-iterate-vstar}.
Now let $x\in R_{p,p}$, say $x\in R_{p,u_1,p}$ for some word $u_1\in\{0,1\}^*$, be arbitrary.
Then $u_2=u^{|u_1|+1}u_1$ is also a member of $\mathbf{In}(p,p)$ (as both $u$ and $u_1$ can lead from $p$ to $p$)
and has more $0$s than $1$s (as $|u|_0>|u|_1$, it is sure that $|u_1|+1$ copies of $u$ has more than $|u_1|$
more $0$s than $1$s, hence $u^{|u_1|+1}u_1$ still has more $0$s than $1$s).
Hence,
$v'(R_{p,u,p}^{|u_1|+1}R_{p,u,p}^*R_{p,u_1,p})^*\subseteq\mathbf{Pref}(L(M))$ which implies
$R_{p,u,p}^{|u_1|+1}R_{p,u,p}^*R_{p,u_1,p}$ being a subset of $r_p^*$ for some primitive word $r_p\in\Sigma^*$.
But then, by $R_{p,u,p}^*\subseteq r_u^*$ for the also primitive word $r_u$,
it has to be the case $r_p=r_u$ and so
$R_{p,u_1,p}\subseteq r_u^*$ as well, thus as $x\in R_{p,p}$ was arbitrary we indeed get $R_{p,p}\subseteq r_p^*$
for the $r_p=r_u$ defined above. \end{proof} \begin{corollary} If $M$ has feasible cycles only and $C$ is a component of $M$ such that for some $p\in C$ there exists an
$u\in\{0,1\}^*$ with $p\mathop{\leadsto}\limits^up$ and $|u|_0>|u|_1$, then for each state $q\in C$ there exists a primitive word $r_q\in\Sigma^*$ such that $R_{q,q}\subseteq r_q^*$. \end{corollary} \begin{proof}
Let $p\mathop{\leadsto}\limits^up$, and $q$ be in the same component as $p$, say $q\mathop{\leadsto}\limits^{u_1} p
\mathop{\leadsto}\limits^{u_2}q$. Then for the word $u'=u_1u^{|u_1u_2|+1}u_2$ we have $q\mathop{\leadsto}\limits^{u'}q$ and $|u'|_0>|u'|_1$: applying Proposition~\ref{prop-positive-cycles-imply-rank-1-inner-loops} on $q$ and $u'$ we get the corollary. \end{proof}
Now we turn our attention towards ,,$0$-cycles'', input words $u$ with $|u|_0=|u|_1$ and states $p$ with $p\mathop{\leadsto}\limits^up$. \begin{proposition} \label{prop-zero-cycles-imply-rank-1} If $M$ has feasible cycles only, $L(M)$ is scattered, $p$ is a state of $M$, and there exists some word $u\in\{0,1\}^+$ with
$|u|_0=|u|_1$ and $p\mathop{\leadsto}\limits^u p$, then there exists a primitive output word $r_p$, depending only on $p$, such that whenever $|v|_0=|v|_1$ and $p\mathop{\leadsto}\limits^vp$ for a word $v\in\{0,1\}^*$, then $R_{p,v,p}\subseteq r_p^*$. \end{proposition} Note that if there is also a positive cycle on $p$, its $r_p$ from Proposition~\ref{prop-positive-cycles-imply-rank-1-inner-loops} has to be the same as $r_p$ of Proposition~\ref{prop-zero-cycles-imply-rank-1} so using $r_p$ for both Propositions' primitive roots is not ambiguous. \begin{proof}
Analogously to the proof of Proposition~\ref{prop-positive-cycles-imply-rank-1-inner-loops},
assume $M$, $p$ and $u$ satisfy the conditions of the Proposition. Then, there exist words $u_1$ and $u_2$
with $q_0\mathop{\leadsto}\limits^{u_1}p\mathop{\leadsto}\limits^up\mathop{\leadsto}\limits^{u_2}q_f$
and $u_1uu_2\in D_1$, implying $u_1u^tu_2\in D_1$ for each $t\geq 0$.
Thus if $x\in R_{q_0,u_1,p}$ is arbitrary, we get
$xR_{p,u,p}^t\subseteq xR_{p,u^t,p}\subseteq\mathbf{Pref}(L(M))$, hence $xR_{p,u,p}^*\subseteq\mathbf{Pref}(L(M))$
and thus $R_{p,u,p}\subseteq r_p^*$ for some primitive word $r_p\in\Sigma^*$ as $L(M)$ is scattered.
Now if $v\in\{0,1\}^*$ is also some word with $|v|_0=|v|_1$ and $p\mathop{\leadsto}\limits^vp$, then
there exist some words $v_1,v_2\in\{0,1\}^*$ such that $q_0\mathop{\leadsto}\limits^{v_1}p\mathop{\leadsto}\limits^{v}p\mathop{\leadsto}\limits^{v_2}q_f$.
Now depending on whether $|v_1|_0-|v_1|_1$ or $|u_1|_0-|u_1|_1$ is greater, for some $w\in\{u,v\}$
we have $w_1uw_2$ and $w_1vw_2$ both being in $D_1$, hence both $w_1u$ and $w_1v$ are in $\mathbf{Pref}(D_1)$
and thus by picking an arbitrary $x\in R_{q_0,w_1,p}$ we get that $x(R_{p,u,p}\cup R_{p,v,p})^*\subseteq\mathbf{Pref}(D_1)$, implying $R_{p,u,p}\cup R_{p,v,p}$ being contained in $r_v^*$ for some
primitive word $r_v\in\Sigma^*$ but as we already know that $R_{p,u,p}\subseteq r_p^*$, it has to be the case
that $r_v=r_p$, proving the statement of the Proposition. \end{proof} There is a third option where cycles in $M$ can only output a language of rank at most one: \begin{proposition}
\label{prop-infinite-nx-imply-rank-1}
Assume $M$ has feasible cycles only, $L(M)$ is scattered and for some state $p$ of $M$ and output
word $x'\in\Sigma^*$
it holds that for any integer $N\geq 0$ there exists some words $u_N$ and $v_N$ with
$q_0\mathop{\leadsto}\limits^{u_N}p$, $p\mathop\leadsto\limits^{v_N}q_f$,
$x'\in R_{q_0,u_N,p}$ and $|u_N|_0\geq |u_N|_1+N$.
Then $R_{p,p}\subseteq r_p^*$ for some primitive word $r_p$ (and consequently, for each state $q$
belonging to the same component of $M$ as $p$, there is a primitive word $r_q$ with $R_{q,q}\subseteq r_q^*$). \end{proposition} \begin{proof}
Let $p\in Q$ and $x'\in\Sigma^*$ satisfy the condition of the Proposition.
Assume $u'_1$ and $v'_1$ are both members of $R_{p,p}$.
We prove that $x'\{u'_1,v'_1\}^*\subseteq\mathbf{Pref}(L(M))$ which, as $L(M)$ is scattered, implies that $\{u'_1,v'_1\}\subseteq r_p^*$ for some primitive word $r_p\in\Sigma^*$ which implies $u'_1$ and $v'_1$
have the same primitive root. As $u'_1$ and $v'_1$ were chosen arbitrarily from $R_{p,p}$, all the members
of $R_{p,p}$ have to have this same primitive root $r_p$, proving the statement.
So let $u_1,v_1\in\{0,1\}^*$ be input words with $u'_1\in R_{p,u_1,p}$ and $v'_1\in R_{p,v_1,p}$
and let $y'\in\{u'_1,v'_1\}^*$ be arbitrary. Then, $y'\in R_{p,y,p}$ for some $y\in\{u_1,v_1\}^*$.
Since $M$ has feasible cycles only, there is an integer $N\geq 0$ and word $v$ with $u_Nyv\in D_1$,
$q_0\mathop\leadsto\limits^{u_N}p\mathop\leadsto\limits^vq_f$.
Thus, as $x'\in R_{q_0,u_N,p}$, we get $x'y'\in\mathbf{Pref}(L(M))$ and so, as $y'$ was an arbitrary
member of $\{u'_1,v'_1\}^*$, we get that $x'\{u'_1,v'_1\}^*\subseteq\mathbf{Pref}(L(M))$ indeed holds,
proving the claim. \end{proof} \section{Transducers with feasible cycles only -- the finishing move}
In this section we show that if $M$ is a transducer with feasible cycles only such that $L(M)$ is scattered, then the rank of $L(M)$ is smaller than $\omega^2$. To this end, for such a transducer $M=(Q,\Sigma,\Delta,q_0,\{q_f\},\mu)$, let us call a transition $\delta=(p,a,q)\in\Delta$ an \emph{intercomponent} transition if $p$ and $q$ belong to different components of $M$. As $q_0$ is a source and $q_f$ is a sink, transitions involving these two states are always intercomponent transitions. Now let us define for each intercomponent transition $\delta=(p,a,q)$ the language $L(\delta)$ as \[\mathop\bigcup\left(R_{q_0,u,p}R_{p,a,q}:~u\in\{0,1\}^*:~\exists v\in\{0,1\}^*~uav\in D_1,q_0\mathop\leadsto\limits^up,q\mathop\leadsto\limits^vq_f\right)\]
that is, the language containing all possible output words that are associated with a run in $M$ that starts in $q_0$ and ends in $q$, using the transition $\delta$ as its last step.
We will show that each such $L(\delta)$ is scattered with rank smaller than $\omega^2$. As $L(M)\subseteq \mathop\bigcup\limits_{\delta=(q,a,q_f)}L(\delta)$ (not necessarily being the same as the latter language contains not only all the images of the words $u\in D_1$ with $q_0\mathop\leadsto\limits^uq_f$ but also the images of those which are not in $D_1$ but in $\mathbf{Pref}(D_1)$), and each transition arriving to $q_f$ is an intercomponent one, thus by Proposition~\ref{prop-rank-ops} and that suborderings cannot have a larger rank we get $\mathrm{rank}(L(M))\leq\max\{\mathrm{rank}(L(\delta)):~\delta=(q,a,q_f)\}<\omega^2$, proving the main result of the section.
For two intercomponent transitions $\delta=(p,a,q)$ and $\delta'=(p',a',q')$ let us write $\delta<\delta'$ if $q\leadsto p'$. Then this relation $<$ is a strict partial order on the finite set of intercomponent transitions (as should $q'\leadsto p$ also hold, then the states $p,q,p'$ and $q'$ all belong to the same component of $M$ and thus neither $\delta$ nor $\delta'$ would be intercomponent). We will apply induction with respect to $<$.
For an intercomponent transition $\delta'=(p',a',q')$ with $q'$ being in the same component $C$ as $p$, and a word $u\in\{0,1\}^*$ with $ua'\in\mathbf{Pref}(D_1)$ and $q_0\mathop\leadsto\limits^up'$, let us define the input language $\mathbf{In}(u,\delta',\delta)$ as the set of those words $v\in\{0,1\}^*$ such that $q'\mathop\leadsto\limits^vp$ and $ua'va\in\mathbf{Pref}(D_1)$, that is: if a computation path enters $q'$ via $\delta'$ after reading the word $ua'$, then it can read $v$ with being still inside $C$, end in $p$, then leave $C$ via $\delta$ so that the input word $ua'va$ read so far is still in $\mathbf{Pref}(D_1)$.
We organize the proof of the induction step into a separate proposition. \begin{proposition}
\label{prop-ldelta-inductive}
Assume $M=(Q,\Sigma,\Delta,q_0,\{q_f\},\mu)$ is a transducer with feasible cycles only and $L(M)$ is scattered.
Let $\delta=(p,a,q)$ be an intercomponent transition of $M$ and assume for each intercomponent transition
$\delta'<\delta$, $L(\delta')$ is scattered of rank smaller than $\omega^2$.
Then $L(\delta)$ is also scattered and also has rank smaller than $\omega^2$. \end{proposition} \begin{proof}
The case when $p=q_0$ is clear, with $q_0$ being a source state, $L(\delta)$ is then simply either the
regular language $R_{q_0,a,q}$ (if $a=0$) or $\emptyset$ (if $a=1$ as no word in $D_1$ can start with
the letter $1$). As scattered regular languages always have a finite rank, the claim is proved for this
case.
Now assume $p$ belongs to some component $C\neq \{q_0\}$. Let $\Delta'\subseteq\Delta$ be the
set of intercomponent transitions entering $C$, that is, of the form $(p',a',q')$ with $p'\notin C$ and
$q'\in C$. Then,
\[L(\delta)~=~\mathop\bigcup\limits_{(p',a',q')\in\Delta'}\mathop\bigcup\limits_{u\in\{0,1\}^*:q_0\mathop\leadsto\limits^up',ua'\in\mathbf{Pref}(D_1)}\mathop\bigcup\limits_{v\in\mathbf{In}(u,\delta',\delta)}R_{q_0,u,p'}R_{p',a',q'}
R_{q',v,p}R_{p,a,q}.\]
(The reason: any valid computation path over some word within $\mathbf{Pref}(D_1)$
that leaves $C$ via $\delta$, has to enter $C$ first, via some intercomponent
transition $\delta'$, after reading in some input word $u$, then taking $\delta'$, after take some route
within $C$ ending in $p$, reading in some input word $v$ during this phase,
and then finally taking $\delta$ as well, making sure that the word $ua'va$ read in so far still belongs
to $\mathbf{Pref}(D_1)$.)
Of course the $R$ languages there are each regular and thus each of them (and their product
as well) has a finite rank but this does not entail the result as there are infinite unions there. However, the very
first union is finite as there are only a finite number of transitions, so if we can show that the
language
\[\mathop\bigcup\limits_{u\in\{0,1\}^*:q_0\mathop\leadsto\limits^up',ua'\in\mathbf{Pref}(D_1)}\mathop\bigcup\limits_{v\in\mathbf{In}(u,\delta',\delta)}R_{q_0,u,p'}R_{p',a',q'} R_{q',v,p}R_{p,a,q}\]
is scattered of rank at most $\omega^2$ for each $(p',a',q')\in\Delta'$, then the statement is proved
(applying Proposition~\ref{prop-rank-ops}).
To this end, let us rewrite the above union as follows: for each output word $x\in\Sigma^*$,
let $U_x\subseteq\{0,1\}^*$ contain those words $u\in\{0,1\}^*$ with $ua'\in\mathbf{Pref}(D_1)$
for which there exists some $v'\in\{0,1\}^*$ such that $q_0\mathop\leadsto\limits^up'\mathop\leadsto\limits^{a'}q'
\mathop\leadsto\limits^{v'}q_f$, $ua'v'\in D_1$
and $x\in R_{q_0,u,p'}$. (Of course $U_x$ might be empty if there is no suitable $u$ at all.)
Then, we can write the above union as
\[\mathop\bigcup\limits_{x\in \mathop\bigcup\limits_{u'a'\in\mathbf{Pref}(D_1)}R_{q_0,u',p'}}\left(\mathop\bigcup\limits_{u\in U_x}\mathop\bigcup\limits_{v\in\mathbf{In}(u,\delta',\delta)}xR_{p',a',q'}R_{q',v,p}R_{p,a,q}\right).\]
The reason why this can help us is the last part of Proposition~\ref{prop-rank-ops}:
substituting $w=x$, $K=\mathop\bigcup\limits_{u'a'\in\mathbf{Pref}(D_1)}R_{q_0,u',p'}$ and $L_w=\mathop\bigcup\limits_{u\in U_x}\mathop\bigcup\limits_{v\in\mathbf{In}(u,\delta',\delta)}xR_{p',a',q'}R_{q',v,p}R_{p,a,q}$
we exactly have a language of the form $\mathop\bigcup\limits_{w\in K}wL_w$ here.
Regarding $R=\mathop\bigcup\limits_{u'a'\in\mathbf{Pref}(D_1)}R_{q_0,u',p'}$,
we have $RR_{p',a',q'}=L(\delta')$ which is scattered and has rank smaller than $\omega^2$ by the assumption
of the Proposition. Thus as $R$ is a subset of $\mathbf{Pref}(L(\delta'))$, it's also scattered and has
a rank $\alpha$ which is smaller than $\omega^2$.
So if we manage to show that all the languages of the form $L_w$ above are scattered
and have some rank smaller than some $\beta$, then by Proposition~\ref{prop-rank-ops} we get that
the whole union $\mathop\bigcup\limits_{w\in K}L_w$ is scattered of rank at most $\beta+\alpha$.
Now if $\beta$ is smaller than $\omega^2$ (that is, it has the form $\omega\times k+n$ for some integers
$k$ and $n$) and so is $\alpha$, then their sum still is smaller than $\omega^2$ and the Proposition is proved.
So let us fix a word $x\in\{0,1\}^*$ belonging to $\mathop\bigcup\limits_{u'a'\in\mathbf{Pref}(D_1)}R_{q_0,u',p'}$
and consider the language
\begin{equation}
\label{eq-distributive-juggling}
\mathop\bigcup\limits_{u\in U_x}\mathop\bigcup\limits_{v\in\mathbf{In}(u,\delta',\delta)}R_{p',a',q'}R_{q',v,p}R_{p,a,q}~=~R_{p',a',q'}
\Bigl(\mathop\bigcup\limits_{u\in U_x}\mathop\bigcup\limits_{v\in\mathbf{In}(u,\delta',\delta)}R_{q',v,p}\Bigr)R_{p,a,q}.
\end{equation}
That's a product of three languages, with the outermost two being regular, scattered nonempty languages, hence
having a finite rank. Thus, if the languages of the form
\begin{equation}
\label{eq-onlytwounionsleft}
\mathop\bigcup\limits_{u\in U_x}\mathop\bigcup\limits_{v\in\mathbf{In}(u,\delta',\delta)}R_{q',v,p}
\end{equation}
can be shown to have rank $\beta$ smaller than $\omega^2$, then by Proposition~\ref{prop-rank-ops},
the product of these three languages will have a rank smaller than $n+\beta+k$ for some integers $n$
and $k$, which is still smaller than $\omega^2$ if so is $\beta$.
Now we rewrite again the union above to a more managable form. Observe that if $u_1$ and $u_2$
are in $\mathbf{Pref}(D_1)$ with $|u_1|_0-|u_1|_1=|u_2|_0-|u_2|_1$, then for any word $v$,
$u_1a'va\in\mathbf{Pref}(D_1)$ if and only if $u_2a'va\in\mathbf{Pref}(D_1)$ (the set of possible
suffixes depends only on the current number of still opened parentheses, assuming the prefix so far
is valid at all). So let $N_x$ stand for the set $\{|ua'|_0-|ua'|_1:u\in U_x\}$ of nonnegative integers
and for each integer $n$, let $\mathbf{In}(n,\delta',\delta)\subseteq\{0,1\}^*$ stand for the set
$\mathop\bigcup\limits_{u\in U_x,|ua'|_0-|ua'|_1=n}\mathbf{In}(u,\delta',\delta)$, that is, the
set of those input words $v$ which can lead $M$ from $q'$ to $p$ and then use $\delta$
while ``closing at most $n$ opening parentheses'',
i.e. if $|ua'|_0-|ua'|_1=n\geq 0$ for a word $ua'\in\mathbf{Pref}(D_1)$, then $ua'va$ is still in $\mathbf{Pref}(D_1)$.
We can rewrite Equation~\ref{eq-onlytwounionsleft} as
\begin{equation}
\label{eq-againthreeunions}
\mathop\bigcup\limits_{n\in N_x}\mathop\bigcup\limits_{v\in\mathbf{In}(n,\delta',\delta)}R_{q',v,p}.
\end{equation}
Now the part $\mathbf{In}(n,\delta',\delta)$ contains those words $v$ from $\{0,1\}^*$ which can be
appended after a word in $\mathbf{Pref}(D_1)$ still having $n$ opened parentheses so that the resulting
word is still in $\mathbf{Pref}(D_1)$, moreover, $v$ can lead from $q'$ to $p$. This is still an infinite
union from which we aim to create a finite one.
Call a state $r\in C$ \emph{loopable} if $R_{r,r}\subseteq u_r^*$ for some primitive word $u_r$.
By Propositions~\ref{prop-positive-cycles-imply-rank-1-inner-loops} and
\ref{prop-infinite-nx-imply-rank-1} we have that
\begin{itemize}
\item if there is some word $u\in\Sigma^*$ with $|u|_0>|u|_1$ and a state $r'\in C$ with
$r'\mathop\leadsto\limits^ur'$, then all the states of $C$ are loopable;
\item if $N_x$ is infinite, then all the states of $C$ are loopable.
\end{itemize}
Now let us fix a word $v=a_1\ldots a_k\in\mathbf{In}(n,\delta',\delta)$.
Then $R_{q',v,p}$ is a subset of the union of the languages of the form
\begin{equation}
\label{eqn-allowed-product-basic}
R_{q_1,a_1,q_2}R_{q_2,a_2,q_3}\ldots R_{q_k,a_k,q_{k+1}}
\end{equation}
with the union ranging over all the possible sequences $q'=q_1,q_2,\ldots,q_k,q_{k+1}=p$ within $C$.
For each product of the form \ref{eqn-allowed-product-basic}, there exists at least one, possibly
more, product of the form
\begin{equation}
\label{eqn-allowed-product-advanced}
R=R_{q'_1,v_1,q'_2}R_{q'_2,v_2,q'_3}\ldots R_{q'_\ell,v_\ell,q'_{\ell+1}}
\end{equation}
with $\ell\geq 0$, $q'=q'_1,q'_2,\ldots,q'_\ell,q'_{\ell+1}=p$ being a state sequence within $C$,
each $v_i\in\{0,1\}^+$ being a word with $v=v_1\ldots v_\ell$ such that whenever $|v_i|>1$,
then $q'_i=q'_{i+1}$ and either $q'_i$ is a loopable state, or $|v_i|_0=|v_i|_1$ (or both),
moreover, $R_{q_1,a_1,q_2}\ldots R_{q_k,a_k,q_{k+1}}\subseteq R$,
with one possible such product being the decomposition~\ref{eqn-allowed-product-basic} itself.
Let us choose one such product $R_1\ldots R_\ell$ satisfying the above conditions for~\ref{eqn-allowed-product-advanced} which minimizes $\ell$.
Now we bound $\ell$ in terms of $|C|$ and $n$.
Observe that if for a product of the form~\ref{eqn-allowed-product-advanced}
there exist $i<j$ with $q'_i=q'_{j+1}$ either being a loopable state, or with
$v_i\ldots v_j$ having the same number of $0$s and $1$s, then the factor sequence
$R_{q'_i,v_i,q_{i+1}}\ldots R_{q'_j,v_j,q'_{j+1}}$ can be replaced to its superset $R_{q'_i,v_i\ldots v_j,q'_{j+1}}$,
lowering the number of factors and still producing a product formed satisfying the condition, so
in a shortest product of the form \ref{eqn-allowed-product-advanced} no loopable states get repeated
and there is no $0$-cycle either spanning over more factors.
We do now a case analysis.
{\textbf{Case 1.}} If all the states are loopable, this means $\ell$ is at most $2|C|$ as no $q'_i$ can
be the same as $q'_j$ for any $i<j$ unless $j=i+1$. Hence the longest possible shortest sequence can have
$q'_1=q'_2$, then $q'_3=q'_4$, and so on, for $q'_{2|C|-1}$ and $q'_{2|C|}$ to finish the sequence (enumerating
each state of $C$ in some order, spelling each state twice).
{\textbf{Case 2.}} If not all the states are loopable, then $N_x$ is finite and there are absolutely no
words $u$ and states $r\in C$ with $r\mathop\leadsto\limits^ur$ and $|u|_0>|u|_1$ (due to Propositions
~\ref{prop-positive-cycles-imply-rank-1-inner-loops} and~\ref{prop-infinite-nx-imply-rank-1}).
Then, each factor of the form $R_{r,v,r}$ with $|v|>1$ has $|v|_0=|v|_1$ and all other factors have
the form $R_{q_i,a_i,q_{i+1}}$ for some $i$. Now assume $\ell>(n+1+|C|)\cdot|C|$. Then there is a state
$r$ which appears at least $n+2+|C|$ times in the sequence $q'_1,\ldots,q'_{\ell+1}$.
By minimality of $\ell$, if $q'_i=q'_{j+1}$ for some $i<j$, then for the word $v[i,j]=v_iv_{i+1}\ldots v_j$
we have $|v[i,j]|_0<|v[i,j]|_1$ (since if it were the other way around, we would have a positive cycle
and if they were equal, we could collapse this interval into $R_{q'_i,v[i,j],q'_{j+1}}$). Hence
if $r$ appers at least $n+2+|C|$ times in the sequence, with $q'_i$ being its first appearance and $q'_{j+1}$
being the last one, then for the word $v[i,j]$ we have $|v[i,j]|_0+n+1+|C|<|v[i,j]|_1$. Since we started
from a word $v\in\mathbf{In}(n,\delta',\delta)$, it has to be the case that $|v[1,j]|_0+n\geq |v[1,j]|_1$
as otherwise $v$ could not be in any set $\mathbf{In}(u,\delta',\delta)$ with $|ua'|_0-|ua'|_1=n$ as
the word $ua'v[1,j]$, which is a prefix of $ua'va\in\mathbf{Pref}(D_1)$ would contain more $1$s than $0$s
which cannot happen. Hence, from $|v[i,j]|_0+n+1+|C|<|v[i,j]|_1$ and $|v[1,j]|_0+n\geq |v[1,j]|_1$
we get $|v[1,i-1]|_0> |v[1,i-1]|_1+|C|$, that is, in the prefix $v[1,i-1]$ there have to be much more $0$s
than $1$s to be able to handle all the $1$s arriving later with still remaining in $\mathbf{Pref}(D_1)$.
However, whenever $|v_k|_0\neq |v_k|_1$, then (since we are within Case 2) $v_k$ has to be one of the symbols
$0$ or $1$, so the difference between $|v[1,k]|_0-|v[1,k]|_1$ and $|v[1,k+1]|_0-|v[1,k+1]|_1$ is at most $1$
for each index $k$ (and starts from $0$ as $v[1,0]$ can be seen as the empty word).
Now this means that if we reach up to $|C|+1$ with this difference, then there are indices $i_1<i_2<\ldots<i_{|C|+1}$
in the sequence such that for each $k=1,\ldots,|C|+1$, $i_k$ is the first index satisfying $|v[1,i_k]|_0-|v[1,1_{k+1}]|_1=k$. Hence, as this sequence has $|C|+1$ elements, there has to be a state
which gets repeated but if (say) $q'_{i_k}=q'_{i_m}$ for $k<m$, then for the word $w=v[i_k,i_{m-1}]$ we would get
$q'_{i_k}\mathop\leadsto\limits^w q'_{i_k}$ with $|w|_0-|w|_1>0$, contradicting to the assumption we are
in Case 2.
Hence, in this case $\ell$ is at most $(n+1+|C|)\cdot|C|$. Note that this quantity does not depend on $v$
anymore, only on $n$ (which depends on the word $x$) and on $|C|$ (which depends on $\delta$).
Thus we now know that each product of the form~\ref{eqn-allowed-product-basic} is a subset of a product
of the form
$R_1\ldots R_\ell$ with each $\ell$ being at most $(n+1+|C|)\cdot |C|$ and each $R_i$ being
either an $R_{q'_i,a'_i,q'_{i+1}}$ for some states $q'_i$ and $q'_{i+1}$ of $C$ and letter $a'_i\in\{0,1\}$,
or an $R_{q'_i,v_i,q'_i}$ with either $q'_i$ being a loopable state, or $|v_i|_0=|v_i|_1$, in both cases
being a scattered language of rank at most $1$ (as in both cases, these languages are subsets of
$w^*$ for some appropriate primitive word $w$ due to Propositions~\ref{prop-positive-cycles-imply-rank-1-inner-loops} and \ref{prop-zero-cycles-imply-rank-1}).
The other languages $R_{q'_i,a'_i,q'_{i+1}}$ are all regular languages, defined by the transducer and there
is only a finite number of them. Hence, there is an absolute constant integer $N\geq 0$ depending only on
$M$ such that each language
$R_{q'_i,v_i,q'_i}$ is scattered of at most $N$, thus by Proposition~\ref{prop-rank-ops} the rank of
each such product $R_1\ldots R_\ell$ is bounded by either $2\cdot|C|\cdot N$ (in Case 1) or by
$(n+1+|C|)\cdot |C|\cdot N$ (in Case 2). As there are only $|\Delta|$ ``elementary'' languages and at most $|C|$
languages of either the form $R_{q'_i,q'_i}$ (when $q'_i$ is loopable) or $\mathop\bigcup\limits_{|u|_0=|v|_0}R_{q'_i,u,q'_i}$ (when $v_i$ is a $0$-cycle), if we let
$R'$ denote the union of all the languages of the form $R_{q'_i,a'_i,q'_{i+1}}$, $R_{q'_i,q'_i}$ for $q'_i$ loopable
and $\mathop\bigcup\limits_{|u|_0=|v|_0}R_{q'_i,u,q'_i}$, we get that $R'$ is a finite union of languages
of finite rank, and also for each $n\in N_x$, $\mathop\bigcup\limits_{v\in\mathbf{In}(n,\delta',\delta)}R_{q',v,p}$ is a subset of $R'^\ell$ for the
power $\ell$ computable from $M$ and $n$, hence these languages are products of scattered languages of
finite rank, thus they are also scattered of finite rank as well.
When $N_x$ is finite, this makes the language of Equation~\ref{eq-againthreeunions} to be a subset of a
finite union of scattered languages, each having a finite rank and we are done proving that the languages
of the form Equation~\ref{eq-againthreeunions} are always scattered and have a finite rank.
Now, $N_x$ in Equation~\ref{eq-againthreeunions} might be infinite but in that case all the states are
loopable and $\ell\leq 2\cdot |C|$ does not depend on $n$. Hence in that case the whole union itself
is a subset of the language $R'^{2\cdot |C|}$ which is again a scattered language of finite rank as so is $R'$.
In summary, we proved that the languages in Equation~\ref{eq-againthreeunions} are scattered and have a finite rank, which in turn implies the languages of Equation~\ref{eq-onlytwounionsleft} are also scattered and have a finite
rank, hence their rank (which was called $\beta$ just above Equation~\ref{eq-distributive-juggling})
is indeed smaller than $\omega^2$ (smaller than $\omega$ actually), finishing the proof of the Proposition. \end{proof} \begin{corollary}
\label{cor-feasible-are-small}
Suppose $M$ is a transducer having feasible cycles only and $L(M)$ is scattered.
Then the rank of $L(M)$ is smaller than $\omega^2$. \end{corollary} \begin{proof}
Applying Proposition~\ref{prop-ldelta-inductive} as the inductive step, we get that for each intercomponent
transition $\delta$, the language $L(\delta)$ is scattered and has a rank smaller than $\omega^2$.
As $L(M)$ itself is a subset of the finite union $\mathop\bigcup\limits_{\delta=(q,a,q_f)\in\Delta}L(\delta)$,
the claim is proved applying Proposition~\ref{prop-rank-ops}. \end{proof}
\section{Making all cycles feasible}
In this section we show that for every transducer $M$ there is a transducer $M'$ having feasible
cycles only with $L(M)=L(M')$, assuming $L(M)\neq\emptyset$.
Together with Corollary~\ref{cor-feasible-are-small} it implies that
scattered restricted one-counter languages have rank smaller than $\omega^2$, which in turn implies
Theorem~\ref{thm-main}.
Let us define the (net) \emph{opening depth} of a word $w\in\{0,1\}^*$ as
$\mathrm{open}(w)=|w|_0-|w|_1$. Clearly, a word $w$ belongs to $\mathbf{Pref}(D_1)$
if and only if $\mathrm{open}(w')\geq 0$ for each prefix $w'$ of $w$, and
to $D_1$ if additionally, $\mathrm{open}(w)=0$. As an extension, we define
$\mathrm{open}':\mathbb{N}_0^2\to\mathbb{N}_0$ as $(n,m)\mapsto n-m$.
Then clearly, $\mathrm{open}(w)=\mathrm{open}'(\Psi(w))$ for each word $w\in\{0,1\}^*$
(recall that $\Psi(w)=(|w|_0,|w|_1)$ is the Parikh image of $w$)
and the image under $\mathrm{open}'$ of a linear set $\{(n_0,m_0)+(n_1,m_1)\cdot t_1+\ldots+(n_k,m_k)\cdot t_k:t_1,\ldots,t_k\geq 0\}\subseteq\mathbb{N}_0^2$ is the linear (thus ultimately periodic)
set $\left\{(n_0-m_0)+\mathop\sum\limits_{i=1}^k(n_i-m_i)\cdot t_i:t_1\ldots,t_k\geq 0\right\}\subseteq\mathbb{N}_0$. Hence, $\mathrm{open}(L)$ is an ultimately periodic
set for any context-free language $L\subseteq\{0,1\}^*$, in particular, for $D_1$,
$\mathbf{Pref}(D_1)$, $\mathbf{Suf}(D_1)$, their intersections with regular languages,
and finite unions and products of such languages.
Similarly, let us define the \emph{closing depth} of a word $w\in\{0,1\}^*$ as
$\mathrm{close}(w)=|w|_1-|w|_0$. Then, a word $w$ belongs to $\mathbf{Suf}(D_1)$
if and only if $\mathrm{close}(w')\geq 0$ for each suffix $w'$ of $w$, and
belongs to $D_1$ if and only if additionally $\mathrm{close}(w)=0$.
Again, we define $\mathrm{close'}(n,m)=m-n$. We get also that for any context-free
language $L\subseteq\{0,1\}^*$, $\mathrm{close}(L)\subseteq\mathbb{N}_0$ is ultimately periodic.
Given a transducer $M=(Q,\Sigma,\Delta,q_0,F,\mu)$,
we associate to each state $q\in Q$ the following sets $N_{-}(q), N_{+}(q)$ and $N(q)\subseteq\mathbb{N}_0$
of integers:
\begin{itemize}
\item $n\in N_{-}(q)$ if and only if there exists some $u\in\mathbf{Pref}(D_1)$ with $\mathrm{open}(u)=n$
and $q_0\mathop\leadsto\limits^uq$
\item $n\in N_{+}(q)$ if and only if there exists some $v\in\mathbf{Suf}(D_1)$ with $\mathrm{close}(v)=n$
and $q\mathop\leadsto\limits^vq_f$ for some $q_f\in F$
\item and $N(q)=N_{-}(q)\cap N_{+}(q)$.
\end{itemize}
Then, e.g. $n\in N(q)$ if and only if there exists at least some successful computation path in $M$
reading in some word $uv\in D_1$ for which after reading $u$ in, the path is in $q$ and there are
exactly $n$ open parentheses at that instant.
\begin{proposition}
\label{prop-nq-ultimately-periodic}
For each state $q$ of a transducer $M$, the sets $N(q)$, $N_{-}(q)$ and $N_{+}(q)$ are ultimately periodic.
\end{proposition}
\begin{proof}
As $N_{-}(q)=\mathrm{open}(\{u\in\mathbf{Pref}(D_1):q\in q_0u\})$ and this
language is the intersection of the context-free language $\mathbf{Pref}(D_1)$
and the regular language $\{u\in\{0,1\}^*:q_0\mathop\leadsto\limits^uq\}$, thus is context-free
as well, we have that $N_{-}(q)$ is ultimately periodic.
Similarly, $N_{+}(q)$ is ultimately periodic as well. As the intersection of
finitely many ultimately periodic sets is ultimately periodic~\cite{Matos94periodicsets},
so is $N(q)$.
\end{proof}
For an example for a transducer (without the output function as that does not play a role in the
sets $N(q)$) and the sets $N(q)$ see Figure~\ref{fig-nq}. The reader is encouraged to verify some of
these sets, e.g. for $N_+(q_1)$ we have that the words accepted from $q_1$ are the members of the
language $(000+01)^*0(1(11)^*+11)~\cap~\mathbf{Suf}(D_1)$ on which if we apply the $\mathrm{close}$ function we get the
nonnegative numbers belonging to the set $\{-3t_1-1+1+2t_2:t_1,t_2\geq 0\}~\cup~\{-3t_1-1+2:t_1\geq 0\}$,
that is, $\{2t_2-3t_1:t_1,t_2\geq 0,2t_2\geq 3t_1\}~\cup~\{1\}$ which in turn is simply $\mathbb{N}_0$, or
$\{t:t\geq 0\}$ as each nonnegative integer $k$ can be written as either $k=2\cdot t_2-3\cdot 0$ if $k$ is even
and as $k=2t_1-3\cdot 1$ if $k$ is odd.
\begin{figure}
\caption{The sets $N_-(q)$, $N_+(q)$ and $N(q)$, denoted by $-$, $+$ and $\cap$ respectively.}
\label{fig-nq}
\end{figure}
\begin{proposition}
\label{prop-period-and-tau-exist}
For any transducer $M$, there exists some integer $P>1$, called a \emph{period} of $M$
and for each state $q$ of $M$,
some subset $\tau(q)$ of $\{0,\ldots,2P-1\}$, called the \emph{type} of $q$
such that
\[N(q)=\bigl(\tau(q)\cap\{0,\ldots,P-1\}\bigr)~\cup~\{n\in\mathbb{N}:~n\geq P,n\equiv r~\mathrm{mod}~P\hbox{ for some }r\geq P,r\in\tau(q)\}.\]
\end{proposition}
\begin{proof}
By Proposition~\ref{prop-nq-ultimately-periodic}, each set $N(q)$ is ultimately periodic,
that is, a finite union of sets of the form $\{r+p\cdot t:t\geq 0\}$ for some constants
$r,p\geq 0$ (called the remainder and the period -- the case $p=0$ defines a singleton set).
Let $P$ be the least integer which is a multiple of each nonzero period and larger than
all the remainders and is also at least two.
We claim that $X(q)=\{n:0\leq n\leq 2P-1\}\cap N(q)$ is a good choice for the type of $q$.
To this end, let $\widehat{X}(q)$ stand for the (ultimately periodic) set
\[\bigl(X(q)\cap\{0,\ldots,P-1\}\bigr)~\cup~\mathop\bigcup\limits_{r\in X(q),r\geq P}\{n\geq P:n\equiv r~\mathrm{mod}~P\}.\]
So we have to show that $N(q)=\widehat{X}(q)$.
First, observe that $\widehat{X}(q)\cap\{0,\ldots,P-1\}~=~N(q)\cap\{0,\ldots,P-1\}$ by the definition of
$X(q)$ so we have to show that for any integer $n\geq P$, $n\in\widehat{X}(q)$ if and only if $n\in N(q)$.
Let us write $N(q)=\mathop\bigcup\limits_{i\in[k]}\{r_i+p_i\cdot t:t\geq 0\}$
And indeed, for $n\geq P$ (and thus $n\geq r_i,p_i$ for each $i\in[k]$) we have
\begin{align*}
n\in\widehat{X}(q) &\Leftrightarrow n\equiv r~\mathrm{mod}~P\hbox{ for some }r\in X(q),r\geq P\\
&\Leftrightarrow n\equiv r~\mathrm{mod}~P\hbox{ for some }r\in N(q),P\leq r<2P\\
&\Leftrightarrow n\equiv r_i+p_i\cdot t~\mathrm{mod}~P\hbox{ for some }i\in[k], 0\leq t\\
&\Leftrightarrow n\equiv r_i+p_i\cdot t~\mathrm{mod}~P\hbox{ for some }i\in[k], 0\leq t<P/p_i\\
&\Leftrightarrow n\equiv r_i~\mathrm{mod}~p_i,n\geq r_i\hbox{ for some }i\in[k]\\
&\Leftrightarrow n\in N(q).
\end{align*}
\end{proof}
{\textbf{Now we create a transucer $M'$ from $M$ by creating copies of each state.}}
The states of $M'$ will be triples of the form $(q,n,\sigma)$ with $q\in Q$, $n\in\tau(q)$ and $\sigma\in\{\equiv,\uparrow,\downarrow\}$.
Let $P$ be a period of $M$. From the state $q$ of $M$, we will create states $(q,n,\equiv)$ for each $P\leq n\in\tau(q)$
and two states, $(q,n,\uparrow)$ and $(q,n,\downarrow)$ for each $n\in\tau(q)$ with $n<P$.
Observe that since $q_0\mathop\leadsto\limits^wq_f$ for some $w\in D_1$ (otherwise $L(M)$ is empty)
and $q_f\in F$,
ˇwe have $0\in\tau(q_0)$.
In $M'$, let $(q_0,0,\uparrow)$ be the initial state.
Also, if $q_f\in F$, then we can assume that there exists some word $w\in D_1$ with
$q_0\mathop\leadsto\limits^wq_f$ (otherwise we can remove $q_f$ from $F$, the resulting transducer will
be equivalent with $M$), and so $0\in N(q_f)$ as well. So let $\{(q_f,0,\downarrow):q_f\in F\}$ be the
(nonempty) set of accepting states in $M'$.
We define the transitions of $M'$ as follows: let $((p,n,\sigma_1),a,(q,m,\sigma_2))\in\Delta'$
if and only if $(p,a,q)\in\Delta$ and one of the following conditions holds:
\begin{enumerate}
\item[i)] $n+1=m<P$, $\sigma_1=\sigma_2$ and $a=0$
\item[ii)] $n-1=m$, $m<P$, $\sigma_2\in\{\sigma_1,\downarrow\}$ and $a=1$
\item[iii)] $n+1\equiv m~\mathrm{mod}~P$, $m\geq P$, $n\geq P-1$, $a=0$, $\sigma_2=\equiv$
and $\sigma_1\neq\downarrow$
\item[iv)] $n-1\equiv m~\mathrm{mod}~P$, $n\geq P$, $m\geq P-1$, $a=1$, $\sigma_1=\equiv$
and $\sigma_2\neq\uparrow$.
\end{enumerate}
Moreover, for $((p,n),a,(q,m))\in\Delta'$, let $\mu'((p,n),a,(q,m))=\mu(p,a,q)$.
Finally, if there is any non-accessible or non-coaccessible state in $M'$, then let us drop it.
Figure~\ref{fig-mprime} shows a part of the transducer $M'$ constructed from the transducer $M$ of Figure~\ref{fig-nq}
with some states missing and without the output function, to maintain readability of the transition diagram.
The idea is that when $M'$ reads some input word, then for a while it uses states labeled by $\uparrow$, then
if for the currently read prefix the opening depth reaches $P$, then from that point it uses states labeled by
$\equiv$, then, after reading in the longest prefix with opening depth at least $P$ it switches to states
labeled by $\downarrow$. In the $\uparrow$ and $\downarrow$ states, the exact opening depth is maintained while
in the $\equiv$ states it's maintained only up to modulo $P$. (During the switch from an $\equiv$ state to a $\downarrow$
state, nondeterminism is used to guess the end of the longest prefix and this guess is then checked against by the
$\downarrow$ states.) Finally, if the depth of the word never reaches $P$, then the transducer switches at some point
from an $\uparrow$-state to a $\downarrow$ state by a transition of type ii). Most of these latter
transitions are missing intentionally from the diagram of $M'$ of Figure~\ref{fig-mprime}.
\begin{figure}
\caption{The automaton $M'$.}
\label{fig-mprime}
\end{figure}
\begin{proposition}
\label{prop-consistent-runs}
For each word $u=a_1\ldots a_n\in D_1$ and run $q_0\mathop{\longrightarrow}\limits^{a_1/R_1}q_1\mathop{\longrightarrow}\limits^{a_2/R_2}\ldots\mathop{\longrightarrow}\limits^{a_n/R_n}q_n$ in $M$ with $q_n\in F$ there is a run
$(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{a_1/R_1}(q_1,t_1,\sigma_1)\mathop{\longrightarrow}\limits^{a_2/R_2}\ldots\mathop{\longrightarrow}\limits^{a_n/R_n}(q_n,t_n,\sigma_n)$ in $M'$ with $(q_n,t_n,\sigma_n)\in F\times\{0\}\times\{\downarrow\}$ in $M'$.
\end{proposition}
\begin{proof}
Let $u=a_1\ldots a_n\in D_1$ be a word and $q_0\mathop{\longrightarrow}\limits^{a_1/R_1}q_1\mathop{\longrightarrow}\limits^{a_2/R_2}\ldots\mathop{\longrightarrow}\limits^{a_n/R_n}q_n$ be a run in $M$ with $q_n\in F$.
There are two cases: either $\mathrm{open}(v)<P$ for each prefix $v$ of $u$, or $\mathrm{open}(v)\geq P$ for
at least one prefix $v$ of $u$. We construct an accepting run
$(q_0,0,\uparrow)\mathop{\longrightarrow}\limits^{a_1/R_1}(q_1,t_1,\sigma_1)\mathop{\longrightarrow}\limits^{a_2/R_2}\ldots\mathop{\longrightarrow}\limits^{a_n/R_n}(q_n,t_n,\sigma_n)$
of $M'$ in both cases.
\begin{enumerate}
\item If $\mathrm{open}(v)<P$ for each prefix $v$ of $u$, then let us define $t_i=\mathrm{open}(v)$ for each
$0\leq i\leq n$, $\sigma_i=\uparrow$ for each $0\leq i<n$ and $\sigma_n=\downarrow$.
Then, the first $n-1$ transitions are of type i) and type ii) depending on $a_i$, with $\sigma_1=\sigma_2=\uparrow$, and the last transition is of type ii) with $\sigma_2=\downarrow$,
since by $u\in D_1$ we get $a_n=1$. Thus this is indeed an accepting run in $M'$.
\item If $\mathrm{open}(v)\geq P$ for at least one prefix $v$ of $u$, then let $i_\uparrow\geq 0$
be the largest index
so that for each $j\leq i_\uparrow$, $\mathrm{open}(a_1\ldots a_j)<P$ and let $i_\downarrow$ be
the smallest index so that for each $j\geq i_\downarrow$, $\mathrm{open}(a_1\ldots a_j)<P$.
These indices exist since $\mathrm{open}(a_1)=1<P$ and $\mathrm{open}(a_1\ldots a_n)=0<P$,
moreover, $i_\uparrow<i_\downarrow$ since there exists some $i$ with $\mathrm{open}(a_1\ldots a_i)\geq P$
and all of these $i$s have to fall strictly between $i_\uparrow$ and $i_\downarrow$.
Now let us define
\begin{align*}
t_i&=\begin{cases}
\mathrm{open}(a_1\ldots a_i)&\hbox{if }i\leq i_\uparrow\hbox{ or }i\geq i_\downarrow\\
(\mathrm{open}(a_1\ldots a_i)~\mathrm{mod}~P)+P&\hbox{otherwise}
\end{cases}&
\sigma_i&=\begin{cases}
\uparrow&\hbox{if }i\leq i_\uparrow\\
\equiv&\hbox{if }i_\uparrow<i<i_\downarrow\\
\downarrow&\hbox{if }i_\downarrow\leq i.
\end{cases}
\end{align*}
We claim that for each $0\leq i<n$, $((q_i,t_i,\sigma_i),a_{i+1},(q_{i+1},t_{i+1},\sigma_{i+1}))$ is
a transition in $M'$. Indeed: $(q_i,a_{i+1},q_{i+1})$ is a transition of $M$ and
\begin{itemize}
\item if $i<i_\uparrow$ and $a_{i+1}=0$, then $t_i=\mathrm{open}(a_1\ldots a_i)$, $t_{i+1}=\mathrm{open}(a_1\ldots a_{i+1})=t_i+1<P$ and $\sigma_1=\sigma_2=\uparrow$,
thus then the triple is a type i) transition
\item if $i<i_\uparrow$ and $a_{i+1}=1$, then $t_i=\mathrm{open}(a_1\ldots a_i)$, $t_{i+1}=\mathrm{open}(a_1\ldots a_{i+1})=t_i-1$, $t_i<P$ and $\sigma_1=\sigma_2=\uparrow$,
thus then the triple is a type ii) transition
\item if $i=i_\uparrow$, then (by the maximality of $i_\uparrow$) $a_{i+1}=0$,
$\mathrm{open}(a_1\ldots a_i)=t_i=P-1$, $\mathrm{open}(a_1\ldots a_{i+1})=t_{i+1}=P$
(as $(P~\mathrm{mod}~P)+P=0+P=P$), $\sigma_1=\uparrow$, $\sigma_2=\equiv$ and the triple is
a type iii) transition
\item if $i_\uparrow<i<i_\downarrow-1$ and $a_{i+1}=0$, then $\sigma_i=\sigma_{i+1}=\equiv$,
$t_i=(\mathrm{open}(a_1\ldots a_i)~\mathrm{mod}~P)+P\geq P$, $t_{i+1}=((\mathrm{open}(a_1\ldots a_i)+1)~\mathrm{mod}~P)+P\geq P$ and the triple is a type iii) transition
\item if $i_\uparrow<i<i_\downarrow-1$ and $a_{i+1}=1$, then $\sigma_i=\sigma_{i+1}=\equiv$,
$t_i=(\mathrm{open}(a_1\ldots a_i)~\mathrm{mod}~P)+P\geq P$, $t_{i+1}=((\mathrm{open}(a_1\ldots a_i)-1)~\mathrm{mod}~P)+P\geq P$ and the triple is a type iv) transition
\item if $i=i_\downarrow-1$, then (by the minimality of $i_\downarrow$)
$t_i=\mathrm{open}(a_1\ldots a_i)=P$, $a_{i+1}=1$, $t_{i+1}=\mathrm{open}(a_1\ldots a_{i+1})=P-1$,
$\sigma_i=\equiv$, $\sigma_2=\downarrow$ and the triple is a type iv) transition
\item if $i_\downarrow\leq i$ and $a_{i+1}=0$, then $t_i=\mathrm{open}(a_1\ldots a_i)$,
$t_{i+1}=\mathrm{open}(a_1\ldots a_{i+1})=t_i+1<P$ and $\sigma_1=\sigma_2=\downarrow$,
thus then the triple is a type i) transition
\item if $i_\downarrow\leq i$ and $a_{i+1}=1$, then $t_i=\mathrm{open}(a_1\ldots a_i)$,
$t_{i+1}=\mathrm{open}(a_1\ldots a_{i+1})=t_i-1$, $t_i<P$ and $\sigma_1=\sigma_2=\downarrow$,
thus then the triple is a type ii) transition
\end{itemize}
\end{enumerate}
\end{proof} Let us call the run $r'$ of $M'$ constructed from a run $r$ of $M$ in the proof of Proposition~\ref{prop-consistent-runs} the \emph{canonical lifted run} of $r$. \begin{corollary}
\label{cor-pm-is-pmprime}
$L(M)=L(M')$ for the transducers $M$ and $M'$ of Proposition~\ref{prop-consistent-runs}.
\end{corollary} \begin{proof}
From Proposition~\ref{prop-consistent-runs} we have $L(M)\subseteq L(M')$. For the other direction, $L(M')\subseteq L(M)$ also clearly holds since the mapping $(q,n,\sigma)\mapsto q$ for each $q\in Q$, $n\in\tau(q)$, $\sigma\in\{\uparrow,\downarrow,\equiv\}$ transforms an accepting run in $M'$ into an accepting run in $M$, with the same labels on the transitions. \end{proof}
We will show that the created $M'$ has feasible cycles only. To this end, we first prove a batch of statements regarding the states of $M'$. \begin{proposition}
\label{prop-props-of-states-of-mprime}
The following all hold for the transducer $M'$ we constructed from $M$:
\begin{enumerate}
\item For any state $(q,n,\uparrow)$ of $M'$, (thus $0\leq n<P$), whenever
$u\in\mathbf{Pref}(D_1)$ is a word with $(q_0,0,\uparrow)\mathop\leadsto\limits^u(q,n,\sigma)$,
then $\mathrm{open}(u)=n$, moreover, at least one such word exists.
\item For any $(q,n,\downarrow)$ of $M'$ (thus $0\leq n<P$), whenever
$v\in\mathbf{Suf}(D_1)$ is a word with $(q,n,\sigma)\mathop\leadsto\limits^v(q_f,0,\downarrow)$ for
some $q_f\in F$, then $\mathrm{close}(v)=n$, moreover, at least one such word exists.
\item For any state $(q,n,\equiv)$ of $M'$ (and thus $P\leq n<2P$),
whenever $u\in\mathbf{Pref}(D_1)$ is a word with $(q_0,0,\uparrow)\mathop\leadsto\limits^u(q,n,\equiv)$,
then $\mathrm{open}(u)\equiv n~\mathrm{mod}~P$.
\item For any state $(q,n,\equiv)$ of $M'$ (and thus $P\leq n<2P$),
whenever $v\in\mathbf{Suf}(D_1)$ is a word with $(q,n,\equiv)\mathop\leadsto\limits^v(q_f,0,\downarrow)$
for some $q_f\in F$, then $\mathrm{close}(v)\equiv n~\mathrm{mod}~P$.
\item For any state $(q,n,\equiv)$ of $M'$ and integer $N\geq P$ with $N\equiv n~\mathrm{mod}~P$,
there exists a word $uv\in D_1$
such that $\mathrm{open}(u)=\mathrm{close}(v)=N$ and $(q_0,0,\uparrow)\mathop\leadsto\limits^u(q,n,\equiv)\mathop\leadsto\limits^v(q_f,0,\downarrow)$ for
some $q_f\in F$. \end{enumerate} \end{proposition} \begin{proof} First observe that each transition either increases the second coordinate modulo $P$ when it's reading a $0$, or decreases the second coordinate when it's reading a $1$. Thus in particular, \begin{itemize}
\item whenever $(q_0,0,\uparrow)\mathop\leadsto\limits^u(q,n,\sigma)$ for some word $u$ and
state $(q,n,\sigma)$ of $M'$, it holds that $\mathrm{open}(u)\equiv n~\mathrm{mod}~P$,
\item and whenever $(q,n,\sigma)\mathop\leadsto\limits^v(q_f,0,\downarrow)$ for some word $v$
and state $(q,n,\sigma)$ of $M'$ and $q_f\in F$, it holds that $\mathrm{close}(v)\equiv n~\mathrm{mod}~P$. \end{itemize} These already prove Items $3$ and $4$ above. For $1$ and $2$, observe additionally that \begin{itemize}
\item whenever $(q_0,0,\uparrow)\mathop\leadsto\limits^u(q,n,\uparrow)$ for some state $(q,n,\uparrow)$
of $M'$, then for each prefix $u'$ of $u$ we have $\mathrm{open}(u')<P$,
\item and whenever $(q,n,\downarrow)\mathop\leadsto\limits^v(q_f,0,\downarrow)$ for some state
$(q,n,\downarrow)$ of $M'$ and $q_f\in F$, then for each suffix $v'$ of $v$ we have $\mathrm{close}(v')<P$. \end{itemize} Indeed, for the first item to reach $(q,n,\uparrow)$ as there is no transition leading from a non-$\uparrow$
state to an $\uparrow$-state, we have to stay within the set of $\uparrow$-states during the whole run reading $u$ starting from $(q_0,0,\uparrow)$. Then we can only use transitions of type i) and ii) and it is easy to see by induction on $|u|$ that the first point holds, showing Item 1. A similar reasoning applies to the second bullet point as once we are in a $\downarrow$-component, we cannot leave that, thus again we can only use transitions of type i) and ii) starting from such a state, showing Item 3.
The parts ``moreover, at least such one such word exists'' parts are clear as during the construction we explicitly remove all those states which are either not accessible or not coaccessible.
Now let us turn to Item 5 and let $(q,n,\equiv)$ be a state of $M'$ and let $N\geq P$ be an integer with $N\equiv n~\mathrm{mod}~P$. Since by construction, we have the state $(q,n,\equiv)$ with $P\leq n<2P$ in $M'$ because $n\in\tau(q)$, and $n\geq P$, we get that $N\in N(q)$. Thus, there exist words $u$ and $v$ with $uv\in D_1$ such that $q_0\mathop\leadsto\limits^uq\mathop\leadsto\limits^vq_f$ for some $q_f\in F$ and $\mathrm{open}(u)=\mathrm{close}(v)=N$.
Considering a run $r$ of $M$ on $uv$ which is in $q$ after reading in the $u$ prefix of the input, let us see the canonical lifted run $r'$ of $r$ in $M'$. It has to be the case that in this run, we have $(q_0,0,\uparrow)\mathop\leadsto\limits^u(q,m,\sigma)\mathop\leadsto\limits^v(q_f,0,\downarrow)$ for some $\sigma\in\{\uparrow,\downarrow,\equiv\}$ and value $m\equiv N~\mathrm{mod}~P$. But as $\mathrm{open}(u)=N\geq P$, $\sigma$ cannot be $\uparrow$ due to Item 1; as $\mathrm{close}(v)=N\geq P$, $\sigma$ cannot be $\downarrow$ due to Item 2; thus, $\sigma=\equiv$ and hence $m=n$ (as that's the only possible value between $P$ and $2P-1$ inclusive for which we have $m\equiv N~\mathrm{mod}~P$), proving Item 5. \end{proof} Proposition~\ref{prop-props-of-states-of-mprime} has some interesting corollaries related to having feasible cycles: \begin{corollary} \label{cor-cycles-of-m-prime} The following all hold for the cycles present in $M'$: \begin{enumerate}
\item Whenever $(q,n,\sigma)\mathop\leadsto\limits^u(q,n,\sigma)$ is a cycle in $M'$ with $u\in\{0,1\}^+$,
then any run in $M'$ corresponding the cycle either visits states only within a $\downarrow$-component,
or within a $\uparrow$-component, or within a $\equiv$-component.
\item Cycles within $\uparrow$- and $\downarrow$-components are always $0$-cycles.
\item For any cycle $(q,n,\sigma)\mathop\leadsto\limits^w(q,n,\sigma)$ with $\sigma\in\{\uparrow,\downarrow\}$,
and word $uv\in D_1$ with $q_0\mathop\leadsto\limits^uq\mathop\leadsto\limits^vq_f$ it holds that $uwv\in D_1$
as well. \end{enumerate} \end{corollary} \begin{proof}
The first item is clear since a cycle can be present inside a component of $M'$ and we cannot reach an
$\uparrow$-state from either a $\equiv$- or a $\downarrow$-state, nor an $\equiv$-state from a $\downarrow$-state
so all the components of $M'$ are homogeneous with respect to $\sigma$.
Assume $(q,n,\uparrow)\mathop\leadsto\limits^u(q,n,\uparrow)$ for some $q\in Q$ and $0\leq n<P$
and let $x\in\{0,1\}^*$ be a word with $(q_0,0,\uparrow)\mathop\leadsto\limits^x(q,n,\uparrow)$.
Then
by Proposition~\ref{prop-props-of-states-of-mprime}, $\mathrm{open}(x)=n$. Now as
$(q_0,0,\uparrow)\mathop\leadsto\limits^{xu}(q,n,\uparrow)$ also holds, we also know $\mathrm{open}(xu)=n$ as well,
yielding $|u|_0=|u|_1$.
For $\downarrow$-states having $0$-cycles only the proof is analogous.
For the third point, if $(q,n,\sigma)\mathop\leadsto\limits^w(q,n,\sigma)$ with $\sigma\in\{\uparrow,\downarrow\}$,
then in particular, during this cycle the run stays within the same $\sigma$-component all the time and the
second coordinate always tracks the number of currently opened parentheses. Hence it cannot happen that
for any prefix $w'$ of $w$ to have $\mathrm{close}(w')>n$ as then there would be an undefined transition
from some state $(q,0,\sigma)$ with an input symbol $1$ during the run. Thus in that case, $uw\in\mathbf{Pref}(D_1)$ as well for any word $u$ with $(q_0,0,\uparrow)\mathop\leadsto^{u}(q,n,\sigma)$ (and at least one such word
exists by Proposition~\ref{prop-props-of-states-of-mprime}) and from the second point, $\mathrm{open}(w)=0$
so for any word $v$ with $(q,n,\sigma)\mathop\leadsto\limits^{v}(q_f,0,\downarrow)$ (and at least one
such word exists) we get both $uv\in D_1$ and $uwv\in D_1$ as well. \end{proof} Now we show the main result of this section: \begin{proposition}
The transducer $M'$ has feasible cycles only. \end{proposition} \begin{proof}
For the first requirement of having feasible cycles only,
let $(q,n,\sigma)$ be a state of $M'$ and $u\in\{0,1\}^+$ such that $(q,n,\sigma)\mathop\leadsto\limits^u(q,n,\sigma)$.
\begin{itemize}
\item If $\sigma\in\{\uparrow,\downarrow\}$, then Corollary~\ref{cor-cycles-of-m-prime} and Proposition~\ref{prop-props-of-states-of-mprime} show that the cycle can be extended into a run.
\item If $\sigma=\equiv$, then let $N$ be the maximum value of $\mathrm{close}(u')$ ranging over
the prefixes $u'$ of $u$. By Proposition~\ref{prop-props-of-states-of-mprime}, there exists some word
$x\in\mathbf{Pref}(D_1)$ with $\mathrm{open}(x)\geq N+P$, $\mathrm{open}(x)\equiv n~\mathrm{mod}~P$
and $(q_0,0,\uparrow)\mathop\leadsto\limits^x(q,n,\sigma)$. As $\mathrm{open}(x)$ is large enough,
$xu$ is then still in $\mathbf{Pref}(D_1)$ and $\mathrm{open}(xu)\geq P$, moreover, by
Proposition~\ref{prop-props-of-states-of-mprime}, $\mathrm{open}(xu)\equiv n~\mathrm{mod}~P$.
Again by Proposition~\ref{prop-props-of-states-of-mprime}, there exists then a word $v$ such
that $v\in\mathbf{Suf}(D_1)$, $\mathrm{close}(v)=\mathrm{open}(xu)$ and $(q,n,\equiv)\mathop\leadsto\limits^{v}(q_f,0,\downarrow)$ showing the claim.
\end{itemize}
For the second requirement, observe that positive cycles can be present only in $\equiv$-components by
Corollary~\ref{cor-cycles-of-m-prime}. Again, if $u$ is a positive cycle from the state
$(q,n,\equiv)$, then we can construct a word $v$ with $(q_0,0,\uparrow)\mathop\leadsto\limits^{v}(q,n,\equiv)$ such that $\mathrm{open}(v)$ is large enough
to make sure $vu$ is still in $\mathbf{Pref}(D_1)$. Then for each integer $t\geq 1$, the word $vu^t$
still belongs to $\mathbf{Pref}(D_1)$ and by Proposition~\ref{prop-props-of-states-of-mprime}, to each such
word there exists a suitable $w_t$ with $vu^tw_t\in D_1$ and $(q,n,\equiv)\mathop\leadsto\limits^{w_t}(q_f,0,\downarrow)$.
For the last requirement, observe that the statement again requires for the state $(q,n,\sigma)$
input words with arbitrary large $\mathrm{open}$ value to be completable from $(q,n,\sigma)$.
By Proposition~\ref{prop-props-of-states-of-mprime}, this leaves only the possibility $\sigma=\equiv$.
But repeating our previous argument, for any such cycle $u$ of $(q,n,\sigma)$ we indeed can pick
some word $u_N\in\mathbf{Pref}(D_1)$ having a large enough opening value, leading to $(q,n,\sigma)$
from the initial state, making sure that $u_Nu$ is still in $\mathbf{Pref}(D_1)$ and its opening is
still at least $P$, hence there exists (again by Proposition~\ref{prop-props-of-states-of-mprime}) some
suitable word $v$ with $u_Nuv\in D_1$, $(q,n,\sigma)\mathop\leadsto\limits^v(q_f,0,\downarrow)$. \end{proof}
So we proved that each to and every transducer there exists an equivalent one which also satisfies the feasible cycles property, which (along with Proposition~\ref{prop-ldelta-inductive} show that for any scattered restricted one-counter language has a rank smaller than $\omega^2$, which, applying the induction argument of Section 4, proves Theorem~\ref{thm-main}.
\section{Conclusion}
We confirmed the conjecture of~\cite{kuske} that scattered one-counter languages always have a rank strictly smaller than $\omega^2$, thus in particular, well-ordered one-counter languages always have an order type smaller than $\omega^{\omega^2}$. In the proof we used some upper bounds on the rank -- it would be an interesting question to turn this into an algorithm which computes the exact rank of the language. Also, since scattered order types lack a Cantor-like normal form, it is not clear whether the order type of a scattered one-counter language is presentable by some expression involving, say, $\omega$, $-\omega$, $1$, finite products, sums and powers and if so, whether such a presentation is computable, or from the descriptive complexity point of view, whether representing such an expression by a transducer can be more succint than storing the expression itself. Also, it is still not known whether the order isomorphism problem of two scattered context-free languages is decidable (for the general case of arbitrary context-free languages it is known to be undecidable), and not even for one-counter languages. For the case of regular languages the order isomorphism is known to be decidable, so to extend decidability the class of restricted one-counter languages might be a good choice.
\section{Acknowledgements.} This research was supported by project TKP2021-NVA-09. Project no. TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.
The author wishes to thank two anonymous referees, sending valuable feedbacks to a much earlier version of the manuscript, their inputs made it possible to improve the presentation of the result, very appreciated.
Also, thanks to Kitti Gelle for digitizing Figure~\ref{fig-mprime}.
{}
\end{document} | arXiv |
On the structure of graded transitive Lie algebras
Gerhard F. Post
Discrete Mathematics and Mathematical Programming
Research output: Book/Report › Report › Other research output
We study finite-dimensional Lie algebras ${\mathfrak L}$ of polynomial vector fields in $n$ variables that contain the vector fields $\dfrac{\partial}{\partial x_i} \; (i=1,\ldots, n)$ and $x_1\dfrac{\partial}{\partial x_1}+ \dots + x_n\dfrac{\partial}{\partial x_n}$. We show that the maximal ones always contain a semi-simple subalgebra $\bar{{\mathfrak g}}$, such that $\dfrac{\partial}{\partial x_i}\in \bar{{\mathfrak g}} \; (i=1,\ldots, m)$ for an $m$ with $1 \leq m \leq n$. Moreover a maximal algebra has no trivial $\bar{{\mathfrak g}}$-module in the space spanned by $\dfrac{\partial}{\partial x_i} (i=m+1,\ldots, n)$. The possible algebras $\bar{{\mathfrak g}}$ are described in detail, as well as all $\bar{{\mathfrak g}}$-modules that constitute such maximal ${\mathfrak L}$. All maximal ${\mathfrak L}$ are described explicitly for $n\leq 3$.
University of Twente, Department of Applied Mathematics
Department of Applied Mathematics, University of Twente
MSC-17B05
EWI-3377
Post, G. F. (2000). On the structure of graded transitive Lie algebras. University of Twente, Department of Applied Mathematics.
Post, Gerhard F. / On the structure of graded transitive Lie algebras. Enschede : University of Twente, Department of Applied Mathematics, 2000.
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Post, GF 2000, On the structure of graded transitive Lie algebras. University of Twente, Department of Applied Mathematics, Enschede.
On the structure of graded transitive Lie algebras. / Post, Gerhard F.
Enschede : University of Twente, Department of Applied Mathematics, 2000.
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AB - We study finite-dimensional Lie algebras ${\mathfrak L}$ of polynomial vector fields in $n$ variables that contain the vector fields $\dfrac{\partial}{\partial x_i} \; (i=1,\ldots, n)$ and $x_1\dfrac{\partial}{\partial x_1}+ \dots + x_n\dfrac{\partial}{\partial x_n}$. We show that the maximal ones always contain a semi-simple subalgebra $\bar{{\mathfrak g}}$, such that $\dfrac{\partial}{\partial x_i}\in \bar{{\mathfrak g}} \; (i=1,\ldots, m)$ for an $m$ with $1 \leq m \leq n$. Moreover a maximal algebra has no trivial $\bar{{\mathfrak g}}$-module in the space spanned by $\dfrac{\partial}{\partial x_i} (i=m+1,\ldots, n)$. The possible algebras $\bar{{\mathfrak g}}$ are described in detail, as well as all $\bar{{\mathfrak g}}$-modules that constitute such maximal ${\mathfrak L}$. All maximal ${\mathfrak L}$ are described explicitly for $n\leq 3$.
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BT - On the structure of graded transitive Lie algebras
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Post GF. On the structure of graded transitive Lie algebras. Enschede: University of Twente, Department of Applied Mathematics, 2000. | CommonCrawl |
Hostname: page-component-7ccbd9845f-jxkh9 Total loading time: 0.267 Render date: 2023-02-01T22:28:25.298Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true
>Compositio Mathematica
>Volume 147 Issue 2
>Uniqueness of Morava K-theory
Compositio Mathematica
Uniqueness of Morava K-theory
Part of: Homology and cohomology theories Categories with structure Operations and obstructions
Published online by Cambridge University Press: 27 September 2010
Vigleik Angeltveit
Vigleik Angeltveit*
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA (email: [email protected])
HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.
We show that there is an essentially unique S-algebra structure on the Morava K-theory spectrum K(n), while K(n) has uncountably many MU or -algebra structures. Here is the K(n)-localized Johnson–Wilson spectrum. To prove this we set up a spectral sequence computing the homotopy groups of the moduli space of A∞ structures on a spectrum, and use the theory of S-algebra k-invariants for connectiveS-algebras found in the work of Dugger and Shipley [Postnikov extensions of ring spectra, Algebr. Geom. Topol. 6 (2006), 1785–1829 (electronic)] to show that all the uniqueness obstructions are hit by differentials.
S-algebraMorava K-theorymoduli space
MSC classification
Secondary: 55N22: Bordism and cobordism theories, formal group laws 55S35: Obstruction theory 18D50: Operads
Compositio Mathematica , Volume 147 , Issue 2 , March 2011 , pp. 633 - 648
DOI: https://doi.org/10.1112/S0010437X10005026[Opens in a new window]
Copyright © Foundation Compositio Mathematica 2010
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Vigleik Angeltveit (a1)
DOI: https://doi.org/10.1112/S0010437X10005026 | CommonCrawl |
# Overview of CUDA and its applications in signal processing
CUDA (Compute Unified Device Architecture) is a parallel computing platform and programming model developed by NVIDIA for general computing on GPUs. It allows developers to use GPUs for general-purpose computing, including signal processing tasks. In signal processing, CUDA can be used to perform real-time Fast Fourier Transform (FFT) calculations, which are essential for tasks such as audio and video processing, radio communications, and radar systems.
## Exercise
1. What are the key features of CUDA?
- CUDA supports multiple programming languages, including C, C++, and Fortran.
- CUDA provides a large number of registers and shared memory, enabling efficient computation.
- CUDA enables developers to write kernels, which are functions executed by multiple threads in parallel.
- CUDA provides a runtime API for managing memory and launching kernels.
- CUDA supports a wide range of GPUs from NVIDIA, making it a versatile platform for signal processing.
# Real-time processing and its challenges in signal processing
Real-time processing is a critical aspect of signal processing, as it requires efficient computation and minimal latency. In real-time signal processing, data is continuously acquired and processed, with results delivered in a timely manner. Achieving real-time processing can be challenging due to factors such as data throughput, computation time, and hardware limitations.
## Exercise
2. What are some challenges of real-time signal processing?
- Data throughput: Ensuring that the data is processed quickly enough to meet real-time requirements.
- Computation time: Minimizing the time taken to compute the results.
- Hardware limitations: Dealing with the limitations of the hardware, such as memory bandwidth and processing power.
# FFT algorithm basics
The Fast Fourier Transform (FFT) is an algorithm that computes the discrete Fourier transform of a sequence, or its inverse. It is widely used in signal processing for tasks such as filtering, spectral analysis, and data compression. The Cooley-Tukey algorithm is a popular implementation of the FFT.
## Exercise
3. What is the discrete Fourier transform?
The discrete Fourier transform (DFT) is a mathematical transformation that converts a sequence of values in the time domain into its frequency domain representation. It is defined as:
$$X_k = \sum_{n=0}^{N-1} x_n e^{-j\frac{2\pi kn}{N}}$$
where $X_k$ is the $k$th frequency component, $x_n$ is the $n$th time domain value, $N$ is the sequence length, and $j$ is the imaginary unit.
# Cooley-Tukey algorithm for FFT
The Cooley-Tukey algorithm is a divide-and-conquer approach to computing the FFT. It recursively splits the input sequence into smaller FFTs, which are then combined to form the final result. This approach allows for efficient computation on parallel architectures, such as GPUs.
## Exercise
4. What are the key steps of the Cooley-Tukey algorithm?
1. Divide the input sequence into smaller FFTs.
2. Compute the FFTs in parallel.
3. Combine the results to form the final FFT.
# Parallelization of FFT using CUDA
Parallelizing the FFT algorithm using CUDA involves dividing the input sequence into smaller FFTs and executing them in parallel on the GPU. This can be achieved by launching multiple threads and kernels, each responsible for a portion of the FFT computation.
## Exercise
5. How can the FFT be parallelized using CUDA?
- Divide the input sequence into smaller FFTs.
- Launch multiple threads and kernels to compute each FFT in parallel.
- Combine the results to form the final FFT.
# CUDA programming basics
To implement a real-time FFT system using CUDA, you need to be familiar with CUDA programming basics, such as kernel functions, memory management, and synchronization.
## Exercise
6. What are some basic concepts in CUDA programming?
- Kernel functions: Functions executed by multiple threads in parallel on the GPU.
- Memory management: Allocating and managing memory on the GPU.
- Synchronization: Ensuring that threads and kernels complete their tasks before proceeding.
# Designing and optimizing FFT kernels
Designing and optimizing FFT kernels for CUDA involves selecting efficient algorithms, data layouts, and thread configurations. This can be done by considering factors such as memory access patterns, thread coalescing, and register usage.
## Exercise
7. What are some key factors to consider when designing and optimizing FFT kernels?
- Memory access patterns: Ensuring that data is read and written efficiently.
- Thread coalescing: Ensuring that threads access memory in a coalesced manner.
- Register usage: Minimizing the use of registers to free up resources for other threads.
# Implementing a real-time FFT system using CUDA
Implementing a real-time FFT system using CUDA involves designing and optimizing FFT kernels, managing memory efficiently, and handling input and output data. This can be achieved by following best practices and guidelines for heterogeneous computing.
## Exercise
8. What are the key steps in implementing a real-time FFT system using CUDA?
1. Design and optimize FFT kernels.
2. Manage memory efficiently.
3. Handle input and output data.
# Handling input and output data in real-time systems
Handling input and output data in real-time systems is critical for efficient computation and minimal latency. This can be achieved by using efficient data structures, managing memory efficiently, and ensuring that data is read and written in a coalesced manner.
## Exercise
9. What are some key aspects to consider when handling input and output data in real-time systems?
- Efficient data structures: Using data structures that minimize memory access and computation time.
- Memory management: Allocating and managing memory efficiently.
- Thread coalescing: Ensuring that threads access memory in a coalesced manner.
# Benchmarking and profiling real-time FFT systems
Benchmarking and profiling real-time FFT systems is essential for evaluating their performance and identifying areas for optimization. This can be done using tools such as NVIDIA's CUDA Profiler and Nsight.
## Exercise
10. What are some key aspects to consider when benchmarking and profiling real-time FFT systems?
- Performance metrics: Measuring relevant metrics such as throughput, latency, and energy consumption.
- Optimization opportunities: Identifying areas for improvement in the algorithm, data structures, and memory management.
- Hardware limitations: Considering the limitations of the hardware and how they affect the performance of the system.
# Applications of real-time FFT in signal processing
Real-time FFT systems have a wide range of applications in signal processing, including audio and video processing, radio communications, and radar systems. These applications require efficient computation and minimal latency to provide real-time results.
## Exercise
11. What are some applications of real-time FFT in signal processing?
- Audio and video processing: Real-time FFT can be used for tasks such as audio equalization, noise reduction, and video compression.
- Radio communications: Real-time FFT can be used for tasks such as frequency modulation and demodulation, signal detection, and spectrum analysis.
- Radar systems: Real-time FFT can be used for tasks such as target detection, tracking, and navigation.
# Conclusion
In conclusion, CUDA provides a powerful platform for real-time FFT computation in signal processing. By understanding the basics of CUDA programming, designing and optimizing FFT kernels, and handling input and output data efficiently, it is possible to develop high-performance real-time FFT systems. These systems can be applied to a wide range of signal processing applications, such as audio and video processing, radio communications, and radar systems. | Textbooks |
Deep person re-identification in UAV images
Aleksei Grigorev ORCID: orcid.org/0000-0002-7317-79611,
Zhihong Tian2,
Seungmin Rho3,
Jianxin Xiong4,
Shaohui Liu1 &
Feng Jiang1
EURASIP Journal on Advances in Signal Processing volume 2019, Article number: 54 (2019) Cite this article
The person re-identification is one of the most significant problems in computer vision and surveillance systems. The recent success of deep convolutional neural networks in image classification has inspired researchers to investigate the application of deep learning to the person re-identification. However, the huge amount of research on this problem considers classical settings, where pedestrians are captured by static surveillance cameras, although there is a growing demand for analyzing images and videos taken by drones. In this paper, we aim at filling this gap and provide insights on the person re-identification from drones. To our knowledge, it is the first attempt to tackle this problem under such constraints. We present the person re-identification dataset, named DRone HIT (DRHIT01), which is collected by using a drone. It contains 101 unique pedestrians, which are annotated with their identities. Each pedestrian has about 500 images. We propose to use a combination of triplet and large-margin Gaussian mixture (L-GM) loss to tackle the drone-based person re-identification problem. The proposed network equipped with multi-branch design, channel group learning, and combination of loss functions is evaluated on the DRHIT01 dataset. Besides, transfer learning from the most popular person re-identification datasets is evaluated. Experiment results demonstrate the importance of transfer learning and show that the proposed model outperforms the classic deep learning approach.
Recently, person re-identification (re-id) problem has attracted the attention of the computer vision community due to its significant role in modern surveillance systems. Moreover, the impressive performance of deep convolutional neural networks (CNN) in the image classification task made deep CNN one of the most significant tools for computer vision. It has caused the performance push and has inspired researchers to collect and release more complicated re-id datasets. In short words, person re-id is about how to successfully find a person identity in a database, where the database may contain only one image of that person. It is important to carefully design the network, which will be able to learn optimal features for re-id task. However, re-id has been mostly studied in default constraints, where images or videos were collected by static CCTV cameras. But such cameras lack mobility and typically requires a big amount of time to set up and connect to the surveillance system. The current development of quadrocopters and their high availability make them a desirable choice for creating a surveillance system in terms of mobility and price. In this paper, we study the drone-based person re-identification problem.
Accompanied by deep learning, research on person re-id has already achieved impressive performance on the most popular re-id benchmark datasets. However, the existed datasets are composed of images captured from static CCTV cameras, although those cameras are part of the existed surveillance systems, and the datasets were collected under real-life conditions. It actually does not cover all use-cases, under which person re-identification may be required. For example, one may want to use drones to perform crowd analysis, such as object counting, object detection, and person re-id. Moreover, static CCTV cameras have their own disadvantages. For instance, it is impossible to move it quickly and set up anywhere. They require much more expertise to connect to the existed surveillance system, from mount cameras to set up working places for security operators. Quadrocopter drones do not have such disadvantages. Moreover, there is a rich choice of drones in terms of hardware and prices. But, there is a lack of research on person re-identification on images or videos collected by drones. One of the significant reason is the complexity of collecting and annotation such datasets. It requires a large amount of time to process and annotate images by a hand. In this work, we aim at filling this gap. We present the new person Re-ID dataset that was collected around the university campus by a drone. The paper contribution can be summarized as follows:
The drone-based person re-identification dataset, which is collected by using an unmanned aerial vehicle (UAV). The dataset is different from the existing re-id datasets in terms of angle of view, light conditions, etc.
We analyze the effectiveness of the transfer learning for person re-identification problem and demonstrate that a fine-tuning model from more suitable dataset outperforms fine-tuning model from other datasets.
We propose to use a combination of L-GM and triplet loss functions to tackle re-id problem. Experimental result shows that the proposed combination of loss functions outperforms the network trained with triplet loss only.
Due to the ability of deep convolution neural networks to learn the optimal features for person re-identification task[1], it has demonstrated the impressive performance in this problem. Pioneer work [1] has shown that the deep learning methods outperform the classic re-id approaches and do not require a user to create the hand-crafted features. The person re-identification problem has attracted the wide attention of the computer vision community. Although a huge amount of work [2–9, 9–12] has been done to improve the performance of the convolution neural networks in person re-id, it still remains a challenging task, due to dramatic variations in human body poses, light conditions, background clutter, etc. In classical settings, pedestrians usually captured by many cameras that mounted in the various places which leads to the different body pose, huge variation in illumination, different image sizes, occlusions, etc. Such problems force the computer vision community not only to research and propose novel methods to tackle the person re-id problem but to collect and release the new re-id datasets to simulate the different real-world scenarios as well. Thus, the large amount of research is focused not only on extracting a better image representation but creating new re-id datasets as well. Recent studies in deep re-id are focused on several directions, such as combining the global and local features [2–4], generative adversarial network (GAN)-based methods [11, 12], metric learning [5–9], and video re-id [9, 10].
Some recent research was focused on the design of the loss function [5–9]. Since the study [5], triplet loss became the most widely used loss function in person re-identification. It is designed to narrow the distance between the positive sample pairs and push the negative sample pairs away within the batch. Quadruplet loss [7] is the improved version of the triplet loss, which considers not only the relative distance between the positive and negative samples but the absolute distance between them.
Methods [11, 12] based on GAN models are aiming at increasing dataset size. The re-id datasets usually lack cross-view paired training data and do not have rich pose variations. The study [12] proposed to use GAN models to synthesize new training data with different poses and extra information. It helped to increase network generalization and boosted performance. Another study [11] used GAN to bridge the gap between different domains. They collected and released a new large-scale dataset and used GAN to transfer an image from a source to the target by the coping style of the target dataset. It increased the size of target dataset and boosted the final performance.
Because of the growing demand for the application of the re-id in real-world situations, the video re-id methods are aiming at performing person re-identification in a video. The study [9] proposed a new loss function to overcome the disadvantages of the softmax loss, and the new network architecture to perform detection and re-identification in one step. Zheng et al.'s [10] study proposed a new large-scale video dataset for person re-id. This study attempted to determine how object detection can affect re-id performance.
The studies [2, 3] proved that combining global and local features can boost performance of the model. Extracting and matching local features are significant issues as well. Many CNN-based approaches design two-branches networks, where each branch independently or jointly learns global and local features. Then, local features are matched by employing pre-defined or learned matches strategies. Methods based on pre-defined maths strategy split image into fixed parts or provide extra information about the image for such partition. For instance, study [2] applied the region proposal network to extract body regions and feed it to the network. Then, the micro and macro body features are aligned across images. The learned matches strategies [3] force network to learn how to better align the local features across images. For instance, study [3] proposed to perform automatic part alignment during the learning. Li et al. [4] forced network to pay attention to specific parts of images by using attention mechanics. In the inference stage, the local branch is discarded and the only global branch is used to extract the feature from image.
Re-id dataset review
Recently, the computer vision community has spent a huge amount of efforts to collect and release different re-id datasets. However, some studies [13, 14] highlighted that the current amount of datasets is still far from satisfactory. Because a lot of existed re-id datasets does not cover real-world use cases, for instance, study [13] points out that the huge amount of research ignores the temporal aspect of the re-id problem; existed algorithms are usually evaluated on academic re-id datasets [15], where pedestrians' images are already extracted, while the real surveillance system generates the gallery candidates on the fly. The new datasets aim to be as close as possible to real-word re-identification scenario, but this is still far from satisfactory, especially because of annotation complexity. We briefly describe the most popular re-id datasets.
The first effort to create a re-id dataset goes back to 2009. The ViPeR [16] dataset was collected by two cameras, each of which captured one image per person. It also provides the viewpoint angle of each image. It contains 632 identities and 1264 images, each image has a size of 128×48. The 3DPes [17] dataset was collected by 8 non-overlapped outdoor cameras. It has 192 identities, 1011 images, which have different size. In video sequences, only the bounding boxed of the first appearing frame of each identity is provided. The PRID [18] dataset was collected by 2 cameras. It has 385 trajectories from camera A and 749 trajectories from camera B. Among them, only 200 people appear in both cameras. It contains 24,541 images with a size of 128×64. The CUHK01 dataset contains two images for every identity from each camera. It contains 971 identities and 3884 images with a size of 160×60.
CUHK03 [1] is an extended version of the CUHK01. Besides the camera pair in CUHK01, it has four more camera pair settings. It has 1816 identities and 7264 images with a size of 160×60. The CUHK03 dataset was the first attempt to collect enough data for deep learning. It provides the bounding boxes detected by using deformable part models (DPM) and manually labeling. It was collected by 5 camera pairs and contains 1467 identities and 13,164 images with different image size. Market1501 [19] contains a large number of identities, and each identity has several images from disjoin cameras. This dataset also includes 2793 false alarms from DPM as distracters to mimic the real scenario. Moreover, 500K distracters were integrated to make the dataset large scale. It contains 1501 identities and 32,217 images with a size of 128×64.
The MARS [20] dataset is an extension version of the Market1501 [19]. It is the first large-scale video-based person re-id dataset. All bounding boxes and tracklets are generated automatically. It contains distracters, and each identity may have more than one tracklets. It has 1261 identities and 1,191,003 images with a size of 125×128. The PRW [10] dataset is an extension of the Markert1501 dataset as well. It was the first attempt to create a dataset that can be used to evaluate person re-identification in the wild, while we need not only to perform re-identification of a person but his detection as well. The dataset contains full frames with annotations. Therefore, one can evaluate the effect of different person detectors. It contains 932 identities and 34,304 images with different size. Person identities were labeled by hand.
The DukeMTMC [21] dataset is a large-scale heavily labeled multi-target multi-camera tracking dataset. It was collected by 8 cameras and also contains a lot of extra information, such as full frames, frame level, ground truth, and calibration information. It has 1812 identities and 36,441 images with different image size. The person search dataset [9] provides full frame access and a large number of labeled bounding boxes. It tries to mimic the real scenario of a person search. Therefore, to test this dataset, a reliable person detector is needed. To make the dataset more difficult, the gallery part includes frames from hand-held camera and movies. It contains the low-resolution and occlusion subset as well. It has 11,934 identities and 34,574 images with different size.
Collected datasets by drones
Numerous benchmark datasets have contributed to the evolution of computer vision, such as Caltech [22], KITTI [23], CityPersons [24], COCOPersons [25], CrowdHuman [26], and the EuroCity Persons [27]. These datasets were collected to evaluate human detection systems in different real-world scenarios. They usually were collected by static or moving CCTV cameras and include a huge amount of samples to fully utilize advantages of deep convolutional networks.
Drones equipped with cameras become highly in demand in a wide range of applications, such as fast delivery, aerial photography, surveillance, and agricultural. Due to wireless networks [28–30], they can be controlled remotely. Traditional person detection datasets are not usually optimal for dealing with sequences or image captured by drones, because they were collected by using a fixed camera angle, scale, and view. Objects in images captured by drones typically are different in terms of scale, size, and view angle (Fig. 1). It was mentioned [31–34] that research towards images captured by drones is limited by the lack of publicly available datasets.
Examples of images taken by UAV
Some recent efforts [31–34] have been devoted to collect datasets with a drone focusing on object detection or tracking. Although [31–33] datasets are still limited in size and covered scenarios, because of the difficulties in data collection and annotation, they provide rich insights about a drone's data processing. The study [31] proposes an aerial video benchmark dataset (UAV123) for low-altitude drone target tracking which contains 123 video sequences. It provides the evaluation of the different state-of-the-art trackers on data collected by the drone. They used UAV to follow different objects at altitudes varying between 5 and 25 m. The dataset also contains low-quality video sequences to make the tracking even more challenging. The study [32] presents drone-based object counting approach. The authors collected large-scale car parking dataset, which contains almost 90,000 cars in drone-based high-resolution images. It was captured from 4 different parking lots. The images were collected with the drone view at approximate 40-m height. Authors [33] employ drones to understand human trajectories in crowded scenes. They collected a dataset which contains images and videos of different types of targets that are moving and interacting in a real-world university campus. The dataset contains about 19,000 targets, such as pedestrians, bicyclists, cars, skateboarders, and golf carts. It contains information about targets' interactions as well.
Methodology and experimental settings
We use a standard remote-operated quadrocopter to collect data around the university campus. The drone was flying at an altitude of about 25 m; it was equipped with an HD camera with a video resolution of 1920×1080 pixels at 30 fps. The several video sequences were recorded by a drone. Each video sequence contains about 5000 frames. We use the deep convolutional neural network to detect pedestrians on captured videos. The special annotation software was implemented to make the annotation process more easy. We use it to label and extract identities by hand. A total of 101 unique pedestrians' identities are extracted, where each person has about 459 images (Fig. 2).
Examples of pedestrians' images extracted by the object detector in the DRHIT01 dataset
Given the video sequence, the next step is to extract and label pedestrians. We choose the Faster R-CNN [35] with ResNet50 [36] backbone as an object detector and use the Detectron [37] as the main framework for network training and inference. RoiAlign [38] is used as the ROI extraction method. RPN has sizes of (32, 64, 128, 256, 512). We train the detector on eyesky dataset [39], which provides bounding boxes and annotations for persons and pedestrians. Before training, the dataset is converted to the COCO dataset [25] format, and each image is scaled to 800×648 size. We train a model for 360,000 iterations, with the base learning rate 0.01, which decays after 240,000 and 320,000 iterations by 0.1. It takes 2 days to train a model on a workstation with NVIDIA GTX1080TI. The same settings are used for the inference; we also remove detections with a confidence lower than 80%.
Large-margin Gaussian mixture loss
L-GM loss [40] was proposed as a better alternative to the softmax cross-entropy loss for deep convolutional neural networks in classification tasks. The proposed loss function assumes that the features of the training set come from a Gaussian mixture distribution. L-GM loss combines a likelihood regularization and a classification margin. According to the author's experiment results, it shows a better performance than softmax loss in classification tasks.
Different from the softmax loss, the authors assumed that the extracted deep feature x on the training set comes from Gaussian mixture distribution:
$$ p(x) = \sum_{k=1}^{K} {N}(x;\mu_{k},\Sigma_{k}) p(k), $$
where \(\sum _{k}\) and μk are the covariance and mean of class k in the feature space and p(k) is the prior probability of the class k.
Thus, the conditional probability distribution of a feature xi given its class label zi∈[1,K] can be written as:
$$ p(x_{i}|z_{i}) = \mathcal{N}(x_{i};\mu_{z_{i}},\Sigma_{z_{i}}) $$
and posterior probability distribution is:
$$ p(z_{i}|x_{i}) = \frac{\mathcal{N}(x_{i};\mu_{z_{i}},\Sigma_{z_{i}})p(z_{i})}{\sum_{k=1}^{K}\mathcal{N}(x_{i};\mu_{k},\Sigma_{k}) p(k)} $$
Then, a classification loss Lcls can be expressed as the cross-entropy between the posterior probability distribution and the one-hot class label:
$$ \begin{aligned} L_{\text{cls}} &= -\frac{1}{N}\sum_{i=1}^{N}\sum_{k=1}^{K}\amalg(z_{i}=k)\log p(k|x_{i})\\ &= -\frac{1}{N}\sum_{i=1}^{N}\log\frac{N(x_{i};\mu_{i},\Sigma_{z_{i}})p(z_{i})}{\sum_{k=1}^{K}N(x_{i};\mu_{k},\Sigma_{k})p(k)}, \end{aligned} $$
where ∐ is the indicator function, which equal 1 of zi equals k, or 0 otherwise. To make sure the training samples fit the assumed distribution, the authors introduced a likelihood regularization term. The likelihood for complete data set can be written as:
$$ p(X,Z|\mu,\Sigma)=\prod_{i=1}^{N}\prod_{k=1}^{K}1(z_{i} = k)N(x_{i};\mu_{z_{i}},\Sigma_{z_{i}})p(z_{i}) $$
The likelihood regularization term is defined as the negative likelihood:
$$ \log p(X,Z|\mu, \Sigma) = -\sum_{i=1}^{N}(log N(x_{i};\mu_{z_{i}},\Sigma_{z_{i}}) + \log p(z_{i})) $$
And the likelihood regularization Llkd can be expressed as:
$$ L_{\text{lkd}}=-\sum_{i=1}^{N}\log N(x_{i};\mu_{z_{i}},\Sigma_{z_{i}}) $$
The proposed GM loss LGM is defined as:
$$ \mathcal{L}_{\text{GM}} = \mathcal{L}_{\text{cls}} + \lambda \mathcal{L}_{\text{lkd}}, $$
where λ is a non-negative weighting coefficient.
The contribution of xi to the classification loss is:
$$\begin{array}{@{}rcl@{}} L_{\text{cls}} = -\log \frac{p(z_{i})|\sum_{z_{i}}|^{-\frac{1}{2}}e^{-d_{z_{i}}}}{\sum_{k}p(k)|\sum_{k}|^{-\frac{1}{2}}e^{-d_{k}}} \end{array} $$
$$\begin{array}{@{}rcl@{}} d_{k} = (x_{i} - \mu_{k})^{T}\sum_{k}^{-1}(x_{i}-\mu_{k})/2 \end{array} $$
The classification loss Lcls with margin can be formulated as follows:
$$\begin{array}{@{}rcl@{}} L_{\text{cls},i}^{m} = -\log \frac{p(z_{i})|\sum_{z_{i}}|^{-\frac{1}{2}}e^{-d_{z_{i}} - m}}{\sum_{k}p(k)|\sum_{k}|^{-\frac{1}{2}}e^{-d_{k}} - 1 (k = z_{i})m} \end{array} $$
The problem settings in person re-identification can be abstracted as follows. Given a photo of the person of interest, which is often called the query or probe, and a collection of images, which is called the gallery, an algorithm is required to rank the gallery images according to their similarity with the query photo.
Cumulative Matching Characteristics (CMC) curves are the most popular evaluation metrics for such problem settings. Consider a simple single-gallery-shot setting, where each gallery identity has only one instance. For each query, an algorithm should rank all the gallery samples according to their distances to the query from small to large, and the CMC top-k accuracy is
$${\kern-16.5pt} {\begin{aligned} \text{cmc}_{k} \,=\, \left\{\!\!\begin{array}{ll} 1 & \text{if the query identity is contained in the top-\textit{k} ranked gallery samples,} \\ 0 & \text{otherwise}, \end{array}\right. \end{aligned}} $$
which is a shifted step function. The final CMC curve is computed by averaging the shifted step functions over all the queries.
Mean average precision (mAP) is the average of the precision value across all queries' average precision. Because the target can appear in multiple cameras, which means the model cannot be represented by rank-1 rate only, by using the mAP, the algorithm can evaluate the performance from rank-1 to rank-n. In order to calculate mAP, we need to perform the following steps:
Calculate the precision. For some query, we return the arranged set of gallery images, where we consider only the first n images. Then, we calculate the precision by tacking into account how many query images contain in n (we define it as T). Thus, P(n)=T/n
Calculate the average precision. For the first K query, remember sequences of arranges results set M. Calculate the average precision; thus, \(AP_{k}=\sum (P(I)/M)\), where i∈{i1,i2,…,iM}.
Calculate the mean of the average precision for all queries. Thus, \(mAP = \sum _{K}({AP}_{K}/N)\).
Transfer learning
Transfer learning [41–43] is a common approach to handle lack of training data in a dataset. It is widely believed that networks trained on the ImageNet dataset [44] are able to learn general features from it; then, this network can be fine-tuned on other datasets for a specific task such as face recognition [45, 46], classification [47–49], detection [50, 51], and visual tracking [52–54]. Therefore, it makes transfer learning an essential approach, especially for the small datasets. The performance of the person re-id suffers from the many challenging issues, such as pose and viewpoint changes, complex scenes, and different illumination. Different re-id datasets usually exploit the different real-world scenarios, and the existed domain gaps between datasets influence the re-id performance. For instance, the model trained on one dataset does not produce good results when tested on other datasets [11]. Such domain gap forces us to carefully choose the datasets we want to fine-tune from.
Due to a large amount of data, the ImageNet dataset is usually used to train network to learn general features. Then, the pre-trained network is fine-tuned on a specific computer vision dataset. The fine-tuning typically demonstrates the higher performance, than a network trained from random initialization [41]. The pre-trained networks are available for the computer vision community and help to decrease the amount of work. To demonstrate the effectiveness of transfer learning, we employ a two-stage training procedure. In the first stage, we use the ResNet-50 pre-trained on the ImageNet dataset and train it on the most popular re-id datasets, such as Market1501, CUHK03, and CUHK-SYSU. Then, the trained network is fine-tuned on the DRDIT01 dataset.
We follow the recent advances in person re-id and use the proposed channel group learning [55, 56] and multi-branch loss, which demonstrated that network can be trained more efficiently with a combination of different loss functions. The group learning aims to exploit discriminative information about an image from different channel groups. Two-branches approaches [3, 55] are aiming at combining the global and local information, because the network trained only on global features focus on certain parts and ignore the local details, while the network trained only on the local features cannot effectively exploit all the local information and usually does not take the global context into account. Training the network with only one loss function usually cannot overcome such drawbacks, and all the image information remains unexplored. Another drawback of the local features is the misalignment. The local feature may not correspond to the local body region due to inaccurate person detection, pose variation, etc. To tackle such problems, some studies propose to use dynamic programming to align features [3] or use attention mechanism [57] to force the network to pay attention to specific parts of the image. However, such methods increase the complexity of the networks and their training time. The studies [55, 56] propose to use channel grouping design to handle the local feature misalignment problem without increasing network complexity.
To employ a group learning approach, we follow [55, 56] and use ResNet-50 to extract the global feature vector after the global average pooling (GAP) (Fig. 3). Then, the vector is split into Nc channel groups, where each channel is the partial global feature of the input image. After it, the features are fed to the 1×1 convolutional layers to transform them into 128-d feature vectors. The last fully connected layer is used to perform prediction. In the inference stage, the partition part is discarded and only the global feature vector is extracted to perform re-identification.
The proposed framework. We feed the input image to a base network (ResNet-50) and extract the global feature vector after global average pooling (GAP). The global branch contains two fully connected layers, which transform the feature vector into 128-d embedding. The embedding is feed to the triplet loss. The local branch divides the feature vector into Nc channel groups. Each channel group is feed to the shared 1×1 convolutional layer. Then, each feature vector is feed to its own fully connected layer. The output of the fully connected layer is feed to the L-GM loss function
In the next sections, we perform several experiments to demonstrate the effectiveness of the proposed method and transfer learning. In our experiments, we use the most popular re-id datasets such as CUHK03, Market-1501, and CUHK-SYSU for transfer learning (Fig. 4). We use ResNet-50 as the backbone. To increase the spatial resolution of the feature maps before global average pooling, we modify the stride of the last convolution block from 2 to 1. After the GAP, we have global and local branches. The global branch contains two fully connected layers. The first layer transforms inputs to the 2048-d feature vectors. It is followed by the batch normalization [58], ReLU [59], and dropout [60] layers. Then, the feature vectors are feed to the second fully connected layer which outputs 128-d embeddings. The local branch aims to learn multiple channel group features. First, it uniformly splits the input feature vector into Nc channel groups; then, the 1×1 convolutional layer is used to transform features into 256-d vectors. This layer is followed by batch normalization and ReLU layers. Finally, the last fully connected layer produces the 128-d embeddings. Then, the embeddings produced by the first and the second branches are feed to the triplet and L-GM loss correspondingly.
The flow diagram of experiments with transfer learning. The baseline does not have transfer learning part and network directly fine-tuned on the DRHIT01 dataset
First, we train the network only with the global branch and triplet loss with different margin on CUHK03, Market1501, and CUHK-SYSU datasets. The authors demonstrated [5] that the model achieved the best performance with triplet loss with margin value in the interval [0.1, 0.3]. The same interval is used in experiments. We follow [5] and use the Adam [61] optimizer with the following parameters: e=10−3,B1=0.9,B2=0.999. The network is trained for 25,000 iterations, where the initial learning rate is set as 0.0001 and begin decay after 15,000 iterations:
$$ \mathrm{e}(t) = \left\{\begin{array}{ll} e_{0} & \text{if}\ t < t_{0} \\ e_{0}^{0.001^{\frac{t-t_{0}}{t_{1}-t_{0}}}} & \text{if}\ t_{0} < t < t_{1} \end{array}\right. $$
where t is a current iteration, t0 is the 15,000 iterations, and t1 is the 25,000 iterations. Before training, ResNet-50 is initialized by the weights trained on the ImageNet dataset. In addition, we directly fine-tune ImageNet pre-trained network on the DRHIT01 dataset with the same settings.
Then, we use the weights from trained networks to initialize ResNet-50 and fine-tune it on the DRHIT01 dataset. In this experiment, we use the same settings as before, except the learning rate begin decay after 8000 iterations. The network is trained for 16,000 iterations. In the next experiment, we use the same settings, but train ResNet-50 with two branches which include triplet and L-GM loss functions, where L-GM loss for the second branch can be formulated as follows:
$$ L_{lgm} = \sum_{i=1}^{N_{c}}{L_{i}} $$
and the total loss is:
$$ L = L_{tri} + k{L_{lgm}}, $$
where Ltri is the triplet loss and k=0.25. For each experiment, we set the mini-batch size to 128 which contains 32 persons with 4 images each. Each image of size H×W is resized to \(1 \frac {1}{8}(H\times W)\). Before feeding the image to the network, the random crop of size H×W is taken, where H=256 and W=128. The extra data augmentation includes the random horizontal flip.
In the inference stage, the extra branches are removed. We feed an image to the network and extract the global feature vector which is produced by the global average pooling and use L2 metric to calculate the distance matrix between query and gallery images. Based on the calculated matrix, we compute the rank-1 and mAp scores and report it in Table 1. We do not employ any re-ranking and test-time augmentation techniques. The PyTorch deep learning framework is used to implement a proposed approach.
Table 1 For dataset evaluation, we use the ResNet-50 with triplet loss function and the proposed model with a combination of different loss functions
According to Table 1, CUHK-SYSU is the most suitable dataset to fine-tune from. We carry out several experiments with a different margin for triplet loss function and demonstrate that triplet loss with margin 0.2 achieves the best results. Although the Market1501 and CUHK-SYSU datasets contain almost the same amount of images, they have a different number of unique identities: 1261 and 11,934 respectively. The CUHK-SYSU dataset is richer in terms of the image backgrounds, occlusion, light conditions, etc. Such difference is critical for the transfer learning, and according to Table 1 for the same margin, the fine-tuning on the DRHIT01 datasets produces significantly different results. The fine-tuning from CUHK-SYSU outperforms the fine-tuning from Market1501 by 4.9%. In addition, the fine-tuning from ImageNet on the DRHIT01 dataset shows the worst performance among others. Although the features learned from ImageNet are general, it is still important to train the network on a more domain-specific dataset to force it to learn domain-specific features.
In this paper, we present the new person re-identification benchmark. The dataset consists of 101 unique pedestrians collected by a drone. All pedestrians are extracted by employing the object detector and manually annotated. The dataset is used to evaluate and study the drone-based person re-identification. We build a network with multi-branch design, group channel learning and combination of different loss functions, which can effectively tackle the re-id problem. The network contains the local and global branches which learn local and global image features correspondingly. The channel group learning is used to extract discriminative features of each channel group from the global feature vector. Different from most existed studies, the large Gaussian mixture loss is used to perform local feature classification. The proposed network outperforms the baseline deep learning approach. We study the transfer learning mechanism and demonstrate that the dataset from which the network is fine-tuned significantly affects the final performance. The fine-tuning network on the existed re-id datasets forces it to learn the domain-specific features. We hope the proposed dataset will contribute to the computer vision community and attract its attention to the drone-based computer vision problems. In the future, we expect to extend the current dataset to include more sequence captured in different weather conditions.
Please contact the author for data requests.
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There is no funding support for this research.
School of Computer Science and Technology, Harbin Institute of Technology, Harbin, 150001, China
Aleksei Grigorev
, Shaohui Liu
& Feng Jiang
Cyberspace Institute of Advanced Technology, Guangzhou University, Guangzhou, 510006, China
Zhihong Tian
College of Software and Convergence Technology, Sejong University, Seoul, Republic of Korea
Seungmin Rho
School of Computer, Beijing Institute of Technology, Beiing, 100081, China
Jianxin Xiong
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All authors read and approved the final manuscript.
Correspondence to Zhihong Tian.
Grigorev, A., Tian, Z., Rho, S. et al. Deep person re-identification in UAV images. EURASIP J. Adv. Signal Process. 2019, 54 (2019) doi:10.1186/s13634-019-0647-z
Received: 26 February 2019
DRHIT01
Triplet loss
Visual human motion understanding in the Wild | CommonCrawl |
Women through the glass ceiling: gender asymmetries in Wikipedia
Claudia Wagner ORCID: orcid.org/0000-0002-0640-82211,2,
Eduardo Graells-Garrido3,
David Garcia4 &
Filippo Menczer5
EPJ Data Science volume 5, Article number: 5 (2016) Cite this article
187 Altmetric
Contributing to the writing of history has never been as easy as it is today thanks to Wikipedia, a community-created encyclopedia that aims to document the world's knowledge from a neutral point of view. Though everyone can participate it is well known that the editor community has a narrow diversity, with a majority of white male editors. While this participatory gender gap has been studied extensively in the literature, this work sets out to assess potential gender inequalities in Wikipedia articles along different dimensions: notability, topical focus, linguistic bias, structural properties, and meta-data presentation.
We find that (i) women in Wikipedia are more notable than men, which we interpret as the outcome of a subtle glass ceiling effect; (ii) family-, gender-, and relationship-related topics are more present in biographies about women; (iii) linguistic bias manifests in Wikipedia since abstract terms tend to be used to describe positive aspects in the biographies of men and negative aspects in the biographies of women; and (iv) there are structural differences in terms of meta-data and hyperlinks, which have consequences for information-seeking activities. While some differences are expected, due to historical and social contexts, other differences are attributable to Wikipedia editors. The implications of such differences are discussed having Wikipedia contribution policies in mind. We hope that the present work will contribute to increased awareness about, first, gender issues in the content of Wikipedia, and second, the different levels on which gender biases can manifest on the Web.
Wikipedia aims to provide a platform to freely share the sum of all human knowledge. It represents an influential source of information on the Web, containing encyclopedic information about notable people from different countries, epochs, and disciplines. It is also a community-created effort driven by a self-selected set of editors. In theory, by following its guidelines about verifiability, notability, and neutral point of view, Wikipedia should be an unbiased source of knowledge. In practice, the community of Wikipedians is not diverse, but predominately white and male [1–3], and women are not being treated as equals in the community [1]. In our previous work we found that gender asymmetries exist in Wikipedia content [4, 5]. Here we extend our prior work and provide an in-depth analysis of who makes it into Wikipedia and how these people are presented.
Objectives: This work sets out to assess potential gender inequalities in Wikipedia articles along different dimensions. Concretely, we aim to address the following research questions: (i) Are men and women who are depicted in Wikipedia equally notable - i.e., do Wikipedians use the same thresholds for women and men when deciding who should be depicted on Wikipedia? (ii) Are any topical aspects overrepresented in articles about men or women? (iii) Does linguistic bias manifest in Wikipedia? (iv) Do articles about men and women have similar structural properties, i.e., similar meta-data, and network properties in the hyperlink network?
Approach: We define gender inequality as a systematic asymmetry [6] in the way that the two genders are treated and presented. To assess the extent to which Wikipedia suffers from potential gender bias, we compare biographies about men and women in Wikipedia along the following dimensions: external and internal global notability, topical and linguistic presentation, structural position, and meta-data presentation.
Contributions and findings: Our results show that:
Women in Wikipedia are on average slightly more notable than their male counterparts. Furthermore, the gap between the number of men and women is larger for 'local heroes' (people who are only depicted in few language editions) than for 'superstars' (people who are present in almost all language editions). These effects can be explained by interpreting Wikipedia's entry barrier as a subtle glass ceiling. While it is obvious that very notable people should be included in Wikipedia, the decision is questionable for people who are less notable. We find that bias and inequality manifest themselves in the presence of such uncertainty, as the Wikipedia editor community must make more subjective decisions about inclusion.
There are differences in the topical focus of biographical content, where gender-, family-, and relationship-related topics are more dominant in the stand-alone overviews of biographies about women in the English Wikipedia.
Linguistic bias becomes evident when looking at the abstractness and positivity of language. Abstract terms tend to be used to describe positive aspects in biographies of men, and negative aspects in biographies of women.
There are structural differences in terms of meta-data and hyperlinks, which have consequences for information-seeking activities.
The contributions of this work are twofold: (i) we present a computational method for assessing gender bias in Wikipedia along multiple dimensions and (ii) we apply this method to the English Wikipedia and share empirical insights on the observed gender inequalities. The methods presented in this paper can be used to assess, monitor and evaluate these issues in Wikipedia on an ongoing basis. We translate our findings into potential actions for the Wikipedia editor community to reduce gender bias in the future.
To study gender bias in Wikipedia, we consider the following data sources:
The DBpedia 2014 dataset [7].Footnote 1
Inferred gender for Wikipedia biographies by [8].Footnote 2
DBpedia [7] is a structured version of Wikipedia that provides meta-data for articles; normalized article Uniform Resource Identifiers (URIs) that allow to interlink articles about the same entity in different language editions; normalized links between articles (taking care of redirections); and a categorization of articles into a shallow ontology, which includes a Person category. This information is available for 125 Wikipedia editions.
To obtain gender meta-data for biographies in the English Wikipedia edition we match article URIs with the dataset by Bamman and Smith [8], which contains inferred gender for biographies based on the number of grammatically gendered words (e.g., he, she, him, her, etc.). Note that only male and female genders are considered in this dataset. The gender meta-data in other language editions are obtained from Wikidata by exploiting the links between DBpedia and Wikidata. Wikidata reports more genders (e.g., transgender male and transgender female). However, those genders have a very small presence, and thus we only focus on male and female.
Table 1 shows the biography statistics of the 20 largest Wikipedia editions in terms of entities available with meta-data in DBpedia. The English edition contains the largest number of biographies with gender information (893,380), while the Basque edition (eu) contains the lowest number of biographies (3,449). In terms of representation of women, 15.5% of biographies in the English edition are about women. The smallest fraction of women can be found in the German edition (13.2%), while the maximum fraction is found in the Korean edition (22.6%). Since the English language edition has the largest number of articles covering personalities from multiple editions and all language editions share in average 97% of people with the English language editions, we focus our analysis on the English edition.
Table 1 The largest 20 language editions of Wikipedia
We split this dataset in Pre-1900 and Post-1900. The Pre-1900 sample contains all people born before 1900, while the Post-1900 sample consists of people born in or after 1900.
To assess the extent to which gender bias manifests in Wikipedia, we compare Wikipedia articles about men and women along the following dimensions:
Global notability of people according to external and internal proxy measures.
Topical focus and linguistic bias of biography articles.
Structural properties of articles, including meta-data and network-theoretic position of people in the Wikipedia article link network.
Global notability
Let us first compare how difficult it is for men and women to make it into Wikipedia. Do Wikipedians use the same notabilfity threshold for men and women when deciding who should be included? Or does the so called glass-ceiling effect make it more difficult for women to be recognized for their achievements? Recall that the glass-ceiling effect refers to the situation in which women cannot reach higher positions because an 'invisible barrier' (namely, gender bias) prevents them from doing so.
We hypothesize that if the entry point of Wikipedia functions as a glass ceiling, fewer women will be included in Wikipedia, but those women will be more notable than their male counterparts on average. Especially if we compare the number of male and female 'local heroes' (people with low levels of notability, without worldwide fame), we expect to see a larger gender gap (i.e., fewer women than men) than for worldwide 'superstars,' because fewer female 'local heroes' will be able to make it into Wikipedia.
To address the question of whether a glass-ceiling effect exists in Wikipedia, we study the population of men and women who are depicted in Wikipedia and analyze their global notability from an internal and external perspective.
Assessing the notability of people is a difficult task. Fortunately, Wikipedia and search engines like Google allow us to gauge public interest in different people and from different locations over time. Such signals can be employed as proxies for the notability of people. These proxy measures are noisy and may also be biased, since they reflect the interests of Google users or Wikipedia editors, which in turn are influenced by many factors. Nevertheless, both signals that we explore let us compare the public interest in men and women. While our analysis allows us to quantify the existence of a glass-ceiling effect, it does not permit an assessment of its origin. It could be that Wikipedians unconsciously apply different thresholds for men and women or that Wikipedia only reflects the glass ceiling of our society and other media, which only document the life of women who have higher capacities and abilities than men which are covered.
Concretely, we use the following external and internal proxy measures:
Number of language editions: The number of Wikipedia language editions that contain an article about a person is used as an internal proxy measure for that person's global notability. The idea is that people who only show up in a few language editions are less relevant from a global perspective than those who show up in more language editions. The DBpedia dataset provides a mapping for articles between different language editions, enabling us to count the number of editions in which a biography appears. In particular, we consider the biographies that appear in at least one of the top 20 languages of DBpedia, and count how often they show up in any other language editions.
To explore whether the number of editions is influenced by gender, we fit a negative binomial (NB) regression model. The number of editions in which a person is depicted is used as dependent variable, while gender is used as independent variable. We include the profession of a person (obtained through the DBpedia ontology classes) as well as the decade in which the person was born (obtained from the DBpedia date of birth meta-data) as control variables. The NB model is appropriate since we consider overdispersed count data.
Google search volume: The Google trendFootnote 3 data gauge the interest of Google users between 2004 and 2015. Google trend data serve as an external proxy for the public interest toward a person, or information need about that person, and can be measured in different countries and at different points in time.
For a random sample of around 5,000 people born after 1900 and before 2000 we collected Google trend data using the full name of the person as input. Google trends shows how often search terms are entered in Google relative to the total search volume in a region or globally. Using full names as search terms will of course introduce noise since several people may share the same name. However, a similar level of noise can be expected for men and women.
We count the number of countries and the number of months between January 2004 and October 2015 (from a worldwide perspective) that reveal a relative search volume above a threshold chosen by Google. The Google threshold is relative to the total number of searches in the region and month under consideration. To explore whether the number of countries and number of months in which we observe search volume above the threshold is influenced by gender, we fit two negative binomial regression models that both use gender as the independent variable. We also used a linear regression model and obtained similar results, but a loss of power.
Topical and linguistic bias
After the investigation of potential differences in entry barriers, let us focus on the lexical presentation of those who made it into Wikipedia. Language use is reportedly different when speaking about different genders [9]. For example, the Finkbeiner test [10] suggests that an article about a woman often emphasizes the fact that she is a woman, mentions her husband and his job, her children and childcare arrangements, how she nurtures her underlings, how she is taken aback by the competitiveness in her field, and how she is such a role model for other women. Historian Gillian Thomas investigated the role of women in Encyclopaedia Britannica, finding that as contributors, women were relegated to matters of 'social and purely feminine affairs' and as subjects, women were often little more than addenda to male biographies (e.g., Marie Curie as the wife of Pierre Curie) [11].
Beside topical bias, previous research also suggests that linguistic biases may manifest when people describe other people that are part of their in- or out-group [12]. Linguistic bias is a systematic asymmetry in language patterns as a function of the social group of the persons described, and is often subtle and therefore unnoticed. The Linguistic Intergroup Bias (LIB) theory [13] suggests that for members of our in-group, we tend to describe positive actions and attributes using more abstract language, and their undesirable behaviors and attributes more concretely. In other words, we generalize their success but not their failures. Note that verbs are usually used to make more concrete statements (e.g., 'he failed in this play'), while adjectives are often used in abstract statement (e.g., 'he is a bad actor'). Conversely, when an out-group individual does or is something desirable, we tend to describe them with more concrete language (we do not generalize their success), whereas their undesirable attributes are encoded more abstractly (we generalize them). Maass et al. point out that LIB may serve as a device that signals to others both our status with respect to an in- or out-group, as well as our expectations for their behavior and attributes [13]. Our expectations are of course not only determined by our group-membership but also by the society in which we live. For example, in some situations or domains not only men but also women may expect other women to be inferior to men.
While it is well known that topical and linguistic biases exist, it is unknown to what extent these biases manifest in Wikipedia. To investigate this question we compare the overview of biographies about men and women in the English Wikipedia. The overview (also known as lead section) is the first section of an article. According to Wikipedia, it 'should stand on its own as a concise overview of the article's topic. It should define the topic, establish context, explain why the topic is notable, and summarize the most important points.'Footnote 4 We focus on the lead section for two reasons. On one hand, the first part of the article is potentially read by most people who look at the article. On the other hand, Wikipedia editors need to focus on what they consider most important about the person, and biases are likely to play a role in this selection process.
Topical bias: To unveil topical biases in Wikipedia content, we analyze the following three topics that could be over-represented in articles about women according to what is suggested by Thomas's observations in Britannica and the Finkbeiner test:
The gender topic contains words that emphasize that someone is a man or woman (i.e., man, women, mr, mrs, lady, gentleman) as well as sexual identity (e.g., gay, lesbian).
The relationship topic consists of words about romantic relationships (e.g., married, divorced, couple, husband, wife).
The family topic aggregates words about family relations (e.g., kids, children, mother, grandmother).
To associate words with these topics (plus an unrelated category, other), we follow an open vocabulary approach [14]. Because we want to include concepts that may comprise more than one word, we consider n-grams with \(n \leq2\). We then analyze the association between the top 200 n-grams for each gender and the four topics (gender, relationship, family, or other). To rank the n-grams for men and women we use Pointwise Mutual Information [15]. PMI measures the relationship between the joint appearance of two outcomes (X and Y) and their independent appearances. It is defined as:
$$\operatorname{PMI}(X, Y) = \log\frac{P(X, Y)}{P(X) P(Y)}, $$
where, in our case, X is a gender and Y is an n-gram. The value of \(P(X)\) can be estimated from the proportions of biographies about men and women, and the other probabilities can be estimated from n-gram frequencies. PMI is zero if X is independent of Y, it is greater than 0 if X is positively associated with Y, and it is smaller than 0 if X is negatively associated with Y. We exclude words that appear in biographies from one gender only, because such words have undefined PMI for the other gender, and thus the comparison is not meaningful. We are interested in words/n-grams that may appear in any gender, and which presumably could be independent of gender. Finally, we compare the proportion of topics that are present in the top 200 n-grams that we associated with men and women using chi-square tests. In the absence of topical asymmetries, one would expect to observe only minor differences in the proportions of topics for men and women.
Linguistic bias: To measure linguistic bias, we use a lexicon-based approach and syntactic annotations to detect abstract and subjective language as proposed by Otterbacher [12]. The level of abstraction of language can be detected through the syntactic class of terms, where adjectives are the most abstract class, as for example comparing 'is violent' with 'hurt the victims' [16].
To test for the existence of linguistic biases in Wikipedia, we quantify the tendency of expressing positive and negative aspects of biographies with adjectives, as a measure of the degree of abstraction of positive and negative content. We quantify the tendency to use abstract language in each class as the ratio of adjectives among positive and negative words. To do so, we detect positive and negative terms taken from the Subjectivity Lexicon [17]. For each term that in the lexicon, we check if it is an adjective or not based on part-of-speech tags [18].
After processing the text, we count for each biography the numbers of positive \(W_{+}\) and negative \(W_{-}\) words, and from those the numbers of positive adjectives \(A_{+}\) and negative adjectives \(A_{-}\). We combine these counts into ratios of abstract positivity and negativity computed as \(r_{+}=A_{+}/W_{+}\) and \(r_{-}=A_{-}/W_{-}\). This way, we quantify the tendency to generalize positive and negative aspects of the biographies, with the purpose of testing if this generalization depends on the gender of the person being described.
The presence of gender stereotypes and sexism and the Linguistic Intergroup Bias (LIB) theory suggest that abstract terms would be more likely to be used to describe positive aspects in the biographies of men than in biographies of women. Similarly, abstract language would be more likely to describe negative aspects in the biographies of women in comparison to biographies of men. We test this hypothesis first through a chi-square test on the aggregated ratios of adjectives over positive and negative words in all biographies of each gender. To test if the bias appears at the individual level, we then focus on biographies with at least 250 words and one evaluative term, testing if the measured \(r_{+}\) and \(r_{-}\) depends on gender while controlling for professions and the century in which a person was born.
Structural properties
Structural properties impact how visible and reachable articles about notable men and women are, since users and algorithms rely on this information when navigating Wikipedia or when assessing the relevance of content within a certain context. For instance, search result rankings are often informed by centrality measures such as PageRank. Furthermore, search results show meta-data when the query is related to notable personalities (using, e.g., the Google Knowledge Graph [19]). These examples show that gender inequalities that manifest in the structure of Wikipedia may have important implications since they impact the information consumption process.
Meta-data: To provide structured meta-data, DBpedia processes content from the infoboxes in Wikipedia articles. The infoboxes are tables with specific attributes that depend on the main activity associated with the person portrayed in the article. For instance, anyone has attributes like date/place of birth, but philosophers have 'Main Ideas' in their attributes, and soccer players have 'Current Team' as an attribute. To explore asymmetries between attribute distributions according to gender, we first identify all meta-data attributes present in the dataset. Then, for each attribute we count the number of biographies that contain it. Finally, we compare the relative proportions of attribute presence between genders using chi-square tests, considering the male proportion as baseline, and discuss which differences go beyond what can be explained by professional areas.
Hyperlink network: We build a network of biographies using the hyperlink structure among Wikipedia articles about people in the English language edition. Concretely, we use the structured links between the canonical URLs of articles provided by DBpedia, where redirects are resolved. On this network we perform two different analyses: first, we explore to what extent the connectivity between people is influenced by gender, and second, we investigate the relation between the centrality of people and their gender. To this end, we compute the PageRank of articles about people. PageRank is a widely used measure of network centrality [20, 21]. To explore potential asymmetries in network centrality, we sort the list of biographies according to their PageRank values in descending order. We estimate the fraction of biographies that are about women at different ranks k. In the absence of any kinds of inequality, whether endogenous or exogenous to Wikipedia, one would expect the fraction of women to be around the overall proportion of women biographies, irrespective of k.
To discern whether the observed asymmetries with respect to gender go beyond what we would expect to observe by chance, we compare our empirical results with those obtained from baseline graphs that are constructed as follows:
Random. We shuffle the edges in the original network. For each edge \((u,v)\), we select two random nodes \((i,j)\) and replace \((u,v)\) with \((i,j)\). The resulting network is a random graph with neither the heterogeneous degree distribution nor the clustered structure that the Wikipedia graph reveals [22].
Degree sequence. We generate a graph that preserves both in-degree and out-degree sequences (and therefore both distributions) by shuffling the structure of the original network. For a random pair of edges \(((u,v), (i,j)) \) rewire to \(((u,j), (i,v))\). We repeat this shuffling as many times as there are edges. Note that although the in- and out-degree of each node are unchanged, the degree correlations and the clustering are lost.
Small world. We generate an undirected small world graph using the model by Watts and Strogatz [23]. This model interpolates a random graph and a lattice in a way that preserves two properties of small world networks: average path length and clustering coefficient. After building the graph, we randomly assign a gender to each node, maintaining the proportions from the observed network.
We provide implementations of our methods, as well as data-gathering tools, in a public repository available at github.com/clauwag/WikipediaGenderInequality.
In this section we present the results of our empirical study about gender inequalities in Wikipedia.
Inequalities in global notability thresholds
Let us first test our hypothesis that the Wikipedia entry point functions as a glass ceiling, making it more difficult for women to be included. If this is the case, women who made it into Wikipedia should be more notable than men. We measure notability using the internal and external proxies based on language editions and search volume, respectively. We filtered biographies that did not have a birth date in their meta-data, as well as those with birth date previous to year 0, and those with birth date greater than year 2015. Consequently, in this analysis we consider \(N= 590\mbox{,}741\) biographies (with 14.7% women). In addition to examining all biographies at once, we split the dataset in two parts to account for the fact that the visibility of women and presumably also their access to resources has changed drastically over time. We thus consider biographies of people born before 1900 (\(N_{b}= 134\mbox{,}306\), with 7.8% women) and biographies of people born after that year (\(N_{a}= 456\mbox{,}435\), with 16.8% women).
Number of language editions
We measure the ratio between men and women as a function of the number of language editions in which they are depicted. If the Wikipedia entry indeed functions as a glass ceiling, we expect to see a larger gender gap for 'local heroes' than for 'superstars,' because fewer female local heroes would be able to overcome the glass ceiling. The exclusion of less notable women would also imply that, on average, women in Wikipedia should be more notable than their male counterparts. On the contrary, the inclusion of less notable men would decrease the average notability of men in Wikipedia.
Figure 1 shows that since 1900, the gap between men and women is indeed larger for people with low or medium level of global notability than for the 'global superstars,' compared to a baseline. If we focus on strictly local heroes (people who only appear in one language edition), the men to women ratio is larger than expected by chance. In the population of people born since 1900, men are 5.62 times more likely than women to be included in Wikipedia if they are only included in one language edition. By random chance (estimated by reshuffling the gender) we would expect a ratio of 4.94 for those people. This means that the population is 15.1% women versus the expected 16.8%, i.e., women are around 10% less likely to be included than we would expect by chance. This difference is important because almost half of our population (45% of men and 40% of women) belongs to the group of strictly local heroes. For global superstars, the gap tends to be smaller than expected.
Men-women ratio. Ratio of men to women included in N language editions before 1900 (left) and since 1900 (right), as a function of N. The gender gap since 1900 is larger for people with low or medium global notability than for the global superstars. The empirical ratio is smoothed with locally weighted regression (solid lines) and compared with a baseline obtained by random shuffling of genders (dashed lines).
We also find a higher than expected gap for strictly local heroes born before 1900. The men/women ratio is 13.28 versus an 11.73 baseline. In this case women are about 11% less likely to be included as local heroes than we would expect by chance. Again, a large portion of our population belongs to this group (44% of men and 39% of women). The main difference between the two populations in Figure 1 is that the gender gap for people born before 1900 does not decrease systematically with increasing notability.
A possible explanation for the high men-to-women ratio for local heroes is that the entry barrier into Wikipedia is higher for women than for men. Note that people can also create articles about themselves in Wikipedia; men are on average more self-absorbed than women [24], and thus may be more likely to create articles about themselves. Another possible explanation is that more information may be available online about less notable men than about less notable women. Since Wikipedia editors rely on secondary information sources, their decisions also reflect the biases that exist in other media.
To further quantify the glass-ceiling effect while controlling for other factors that may potentially explain our results (e.g., profession and age), we use a negative binomial regression model and explore the effect of gender on the number of language editions including a person. We performed three different regressions: one for people born before 1900 (\(N_{b}\)), one for people born since 1900 (\(N_{a}\)), and one for the entire dataset (N). The coefficients that are reported in Table 2 can be interpreted as follows: if all other factors in the corresponding model were held constant, an increase of one unit in the factor (e.g., from male to female, from Person to Scientist, etc.) would increase the logarithm of the number of editions by the fitted coefficient β. The Incidence Rate Ratio (IRR) of each factor is obtained by exponentiating its coefficient.
Table 2 Notability via number of language editions
The regression from the full dataset (last column in Table 2) reveals that being female makes a biography increase its edition count by an IRR of 1.13, all other parameters equal. This effect is significant (\(p < 0.001\)), indicating that women in Wikipedia are 13% more notable than their male counterparts. If we only look at people born since 1900, we see that women are 12% more notable than men, while limiting our dataset to people born before 1900 indicates that women are 4% less notable than men. For people in Wikipedia born before 1900, being a female decreases the chances of notability, as one would predict based on the historical exclusion of women [25]. Conversely, for people in Wikipedia born since 1900, being female increases the chances of notability. Due to the noted relation between being historic and global notability (see Figure 2), we cannot claim a glass-ceiling effect for inclusion in Wikipedia of women born prior to 1900.
Notability by year of birth. The mean number of language editions in which men and women are included as a function of their birth year. The global importance decreases with birth years, suggesting that less historic people are covered by Wikipedia in a more local way. This can be explained in part by the availability of information about these people, but also by the collective process whereby the editors of each language edition describe their own local heroes. Women are slightly more notable than men among people born after 1600, while before 1600 it is the other way around.
We also observe interesting differences for professions. For example, being a philosopher has the strongest positive effect on being of global importance (\(\mathrm{IRR} = 4.8\), \(p < 0.001\)), while being a journalist has the strongest negative effect on global importance (\(\mathrm{IRR} = 0.37\), \(p < 0.001\)). This indicates that people with certain professions are more likely to be recognized globally if they contributed something, while others are more likely to be recognized locally. While we do observe interesting differences among professions, further analysis is necessary to investigate whether professional differences in notability are confounded by the average birth decade. For instance, a quarter of the top 100 historical figures are philosophers [26], while journalists are more likely to have become famous in recent years.
The model further indicates that the decade when a person was born is negatively associated with notability (\(\mathrm{IRR} = 0.99\), \(p < 0.001\)); the more historic a person is, the more notable they are from a global perspective. This is expected: people from older centuries appear on Wikipedia because their ideas and actions have transcended time (through secondary sources). Conversely, people of recent fame can be notable in terms of availability of secondary sources, but not necessarily because their ideas will remain valuable in time. Interestingly, we find that the birth decade factor has a different effect when we look at people pre-1900 and post-1900. For people born before 1900, as with the global dataset, being historic is associated with notability (\(\mathrm{IRR}_{b} = 0.98\), \(p < 0.001\)). When we consider people born since 1900 we find that Wikipedia developed a 'recency bias'; people in this group are slightly more notable if they were born more recently (\(\mathrm{IRR} = 1.01\), \(p = 0.008\)). A possible explanation is that younger people may benefit from the greater availability of digital information about them or generated by them, making them more likely to be recognized by Wikipedia editors.
Google search trends
Let us next compare the external notability proxy (based on geographic and temporal search interest) of a random sample of men and women in Wikipedia born since 1900. Table 3 shows that women in Wikipedia are slightly more of interest to the world according to Google's relative search volume statistics. Both coefficients are significantly positive: on average, women are of interest in more regions (\(\mathrm{IRR} = 1.555\)) and during more months (\(\mathrm{IRR} = 1.322\)). The mean number of regions with search volume above the Google threshold is 2.10 for women, 1.56 for men; the median is zero for both. The mean number of months during which we observe a global search volume above the Google threshold is 34 for women, 30 for men. The median number of months is one for women and zero for men.
Table 3 Notability via Google trend data
While our results suggest that the gender of a person that made it into Wikipedia is significantly related to the number of regions and months in which this person is of interest, we cannot exclude other confounders. For example, women included in Wikipedia tend to be born in recent years (see Figure 3) and people born in recent years may have received more attention on Google between 2004 and 2015. Controlling for year of birth and profession was not possible due to the technical challenges of collecting large amounts of Google trend data. Focusing on sub-samples of people who are born in the same year and share the same profession may allow to address these confounding factors future research.
Distribution of biographies in time. The number of men and women in Wikipedia that are born in a certain year. The number increases with birth year. The fraction of notable women increases as well.
Topical and linguistic asymmetries
Language is one of the primary media through which stereotypes are conveyed. We next explore differences in the words and word sequences that are frequently used when writing about men or women to uncover topical and linguistic biases.
Topical bias
Following the notability analysis, we must consider time as a confounding factor. We therefore consider two groups of biographies: those with birth date prior to 1900, and those with birth date from 1900 onwards. We estimated the PMI of each word and bi-gram in our vocabulary for each gender. Since the PMI give more weight to words with very small frequencies, we considered only n-grams that appear in at least 1% of men's or women's biography overviews. Our findings for each dataset are summarized as follows:
Pre-1900: the three words most strongly associated with females are her husband, women's, and actress. The three most strongly associated with males are served, elected, and politician.
1900-onwards: the three words most strongly associated with females are actress, women's, and female. The three most strongly associated with males are played, league, and football.
Figure 4 shows the n-grams that are strongly associated with each gender. The bi-grams that are strongly associated with women born before 1900 relate frequently to categories such as gender, family, and relationships. Words associated with men mainly relate to other categories, such as politics and sports. Table 4 shows the proportion of the top 200 n-grams that fall into each category, for both genders in both periods. The categories gender, relationship, and family are more prominent for women than men. However, the distributions of those categories are different in the two periods under consideration. The distribution is significantly different across genders only pre-1900, according to a chi-square test (\(\chi^{2} = 14.33\), \(p < 0.01\)). In prior work we have shown that the differences are significant if time is not considered [5] and that similar results hold for five other language editions [4].
Topical bias. Word clouds for biographies of women (top) and men (bottom), with birth date before 1900 (left) and since 1900 (right). Spaces in bi-grams are replaced with an underscore. Font size is proportional to PMI with each gender. Colors depict the four categories: gender in orange, family in green, relationship in violet, and other in blue. Beside professional and topical areas, words in the gender, relationship, and family categories are more dominant in articles about women born before 1900. Gender-specific differences are much less pronounced in articles about people born since 1900.
Table 4 Topical bias
Linguistic bias
Table 5 shows the ratios of abstract terms among positive and negative terms when aggregating all the text in the summaries of the biographies of men and women separately. One-tailed chi-square tests suggest that linguistic biases appear along the predicted directions: more abstract terms are used for positive aspects of men's biographies and for negative aspects of women's biographies. Effect sizes, measured by Cohen's w, are very small, in line with the typically small effects in other studies in psycholinguistics. When measuring relative changes, we find that adjectives are almost 9% more likely to be used to describe positive aspects of men's biographies, while 1.62% more likely to describe negative aspects in women's biographies.
Table 5 Linguistic bias
We apply linear regression in two models, one with \(r_{+}\) as dependent variable and another one with \(r_{-}\), expressed as a linear combination of gender, class, and century of birth. We focus on all biographies with valid birth dates and at least 250 words in their summary. Our results indicate that women's biographies tend to have fewer abstract terms for positive aspects and more abstract terms for negative aspects, as predicted by the LIB (see Table 6). This effect is robust to the inclusion of control variables like profession and century of birth. We repeated the analysis using a logit transformation of \(r_{-}\) and \(r_{+}\), as well as with beta regression, finding the same results.
Structural inequalities
Structured information in Wikipedia serves many purposes, from providing input data to search engines, to feeding knowledge databases. Thus, inequalities in structure have an influence that goes beyond Wikipedia, regardless of being a reflection of society or history, or being inherent to Wikipedia contributors.
Meta-data
In total, the DBpedia dataset contains 340 attributes extracted from infobox templates. Of those attributes, 33 display statistically significant differences. Only 14 of them are present in at least 1% of the male or female biographies. These attributes are shown in Table 7. As in previous sections, we have estimated the significance of their differences for people born before and since 1900. An analysis of the entire dataset without considering time is presented in our previous work [5].
Table 7 Meta-data asymmetries
Due to the number of available attributes, the portion of biographies that contains each of them is small. Thus, instead of considering p-value correction, we discuss the statistically significant gender differences manifested in the meta-data to qualitatively assess whether they have significance in our context:
Attributes activeYearsEndDate, activeYearsStartYear, careerStation, numberOfMatches, position, team, and years are more frequently used to describe men. All of these attributes are related to sports, therefore the differences can be explained by the prominence of men in sports-related DBpedia classes (e.g., Athlete, SportsManager and Coach [5]). Differences in activeYearsStartYear are only significant at the entire dataset level, and differences in activeYearsEndDate are only significant before the 20th century. The other attributes are mostly significantly different in recent times.
Attributes deathDate and deathYear are more frequently used for men born before 1900. A possible explanation is that the life of women was less well documented than the life of men in the past, and therefore it is more likely that the death date or birth date is unknown for women.
Attribute birthName is more frequently used for women in recent times. Its value refer mostly to the original name of artists, and women have considerable presence in this class [5]. A likely explanation is that married women change their surnames to those of their husbands in some cultures.
Attributes occupation and title are more frequently used to describe women in recent times, and seem to serve the same purpose but through different mechanisms. On one hand, title is a text description of a person's occupation (the most common values found are Actor and Actress). On the other hand, occupation is a DBpedia resource URI (e.g., http://dbpedia.org/resource/Actress). These attributes are present in the infoboxes of art-related biographies. Conversely, the infoboxes of sport-related biographies do not contain these attributes because their templates are different and contain other attributes (like the aforementioned careerStation and position). Thus the meta-data of athletes, who are mostly men, do not contain such attributes.
The homepage attribute is more frequently used for women in recent times. Our manual inspection showed that biographies from the Artist class tend to have homepages, which explains why the attribute is used more frequently for women.
The spouse attribute is more frequently used for women in recent times. This attribute indicates whether the portrayed person was married or not, and with whom. In some cases, it contains the resource URI of the spouse, while in other cases, it contains the name (i.e., when the spouse does not have a Wikipedia article), or the resource URI of the article of 'divorced status.' This difference is consistent with our results about topical gender difference, where terms related to relationships show a stronger association with women than men.
All differences found have large effect sizes (Cohen's \(w > 0.5\)).
We constructed the empirical network from the inter-article links among 893,380 biographical articles in the English Wikipedia. After removing 192,674 singleton nodes (of which 15.3% were female), the resulting graph had n= 700,706 nodes (of which 15.6% were female) and 4,153,978 edges. All baseline graphs have the same number of nodes n and approximately the same mean degree \(k \approx4\) as the empirical network. The small world baseline has a parameter \(\beta= 0.34\) representing the probability of rewiring each edge. Its value was set using the Brent root finding method in such a way as to recover the clustering coefficient of the original network.
Figure 5 shows the top 30 men and women according to their PageRank. The top-ranked women are slightly less central than men, and the centrality of women decreases faster than that of men with decreasing rank. The top-ranked biographies are similar to those found in previous work [26, 27].
Top 30 biographies sorted by PageRank. Women are slightly less central than men and their centrality decreases faster with decreasing rank.
In addition to the full hyperlink network, we created two sub-networks: one only contains people born before 1900 and the other only contains people born since 1900. For each empirical network, we created several null models and compared the proportion of links within and across genders using a chi-square test. Table 8 indicates that in both empirically observed Wikipedia graphs, women biographies have more links to other women articles than one would expect by chance. A possible explanation for this asymmetry stems from the reported interests of female editors, who frequently edit biographies about women in Wikipedia [28].
Table 8 Hyperlink network asymmetries
The effect of structural differences on visibility can be analyzed in terms of how many women are ranked among the top biographies by centrality scores. Figure 6 displays the fraction of women in subsets of top-ranked biographies. For people born before 1900, the fraction of women in the top k biographies is below the expected ratio of 7.8% up to \(k \approx 10^{3}\), and above when lower-ranked biographies are considered. For people born since 1900, the fraction or women is below the expected ratio of 16.8% for the entire range of k. This indicates that the empirically observed structure of the Wikipedia hyperlink network puts women at a disadvantage when it comes to ranking algorithms, especially for women born since 1900. For people born before 1900, as k increases, the relative fractions of women among the top k biographies in the baseline networks converge to the expected ratios faster than in the empirical networks. This implies an asymmetry that cannot simply be explained by heterogeneities in the structure of the networks, since our baseline graphs preserve several characteristics of the empirical network, including the broad distribution of node degrees. Therefore one must conclude that there exists a bias in the generation of links by Wikipedia editors, favoring articles about men.
Fraction of women in top k biographies by PageRank. The relative fraction of women among the top k biographies in the empirical and baseline networks must converge to the expected ratios (dashed lines) as k increases. A fraction below the expected ratio and a slower convergence suggest that the empirically observed structure of the hyperlink network puts women (especially women born since 1900) at a disadvantage when it comes to ranking algorithms.
In previous work we found that notable women and men from three different reference lists have equal probability of being represented in Wikipedia [4]. While this result is encouraging, external reference lists may also be biased. For example, if women that show up in these reference lists are more notable than their male counterparts, then equality in coverage does not imply the absence of gender bias. However, assessing the notability of people is a difficult task. In this work we propose to use Wikipedia edits in different language editions and search engines like Google to estimate the public interest in a person at different times and in different regions. Wikipedia view statistics could be used to extend or replace this internal proxy measure of notability in the future, especially if automated cross-language article creation tools become widely used.
Our analysis of the global notability of men and women in Wikipedia reveals that women are slightly more notable than men using internal and external proxy measures for notability. In parts we controlled for confounding factors such as professions (e.g., philosophers have high global notability and most of them are men) and year of birth (historic people are more notable and until recently our history was dominated by men) and obtained the same results: women in Wikipedia are on average slightly more notable than similar men. Further, the men-to-women ratio is higher than expected for local heroes (i.e. people who only show up in 1 language edition) and lower for superstars. These findings suggest the existence of a subtle glass-ceiling effect that makes it more difficult for women to be included in Wikipedia than for men.
At least three plausible explanations exist that describe why the glass-ceiling effect may be present in Wikipedia: (1) the narrow diversity of editors may foster the glass-ceiling effect since it is well known that individuals generally favor people from their in-group over people from their out-group [29, 30]; (2) men are potentially more likely to create an article about themselves since previous research suggests that men are on average more self-absorbed than women [24]; (3) the external materials on which Wikipedia editors rely may introduce this bias, since the life of women or certain ethnic minorities may be less well documented and less visible on the Web. We leave the question of identifying what causes this effect for future research.
One way to mitigate the glass-ceiling effect is by relaxing notability guidelines for women, in order to include women who are locally notable, and for whom secondary sources might be hard to find. We acknowledge that this is not easy, because relaxing notability guidelines can open the door for original research, which is not allowed in Wikipedia. However, a well-defined affirmative strategy would allow for the proportion of women in Wikipedia to grow and make women easier to find, alleviating several asymmetries found.
The topical and linguistic asymmetries that we found highlight that editors need to pay attention to the ways women are portrayed in Wikipedia. Critics may rightly say that by relying on secondary sources, Wikipedia just reflects the biases found in them. However, editors are expected to write in their own words 'while substantially retaining the meaning of the source material'Footnote 5 and thus, the differences found in terms of language are caused explicitly by them. Efforts to mitigate linguistic bias could include a revision of the neutral point of view (NPOV) guidelinesFootnote 6 to explicitly address gender bias. A simple example would be the Finkbeiner test: does the article mention the person's gender? Is it needed?
Even though the structural inequalities that we found suggest that editors (especially those who edit articles about women) do a great job in interlinking articles about women, the visibility of women is still lower than expected when link-based ranking algorithms such as PageRank are applied. The low visibility of women cannot simply be explained by heterogeneities in the structure of the networks, since our baseline graphs preserve several characteristics of the empirical network, including the broad distribution of node degrees. Therefore one must conclude that there exists a bias in the generation of links by Wikipedia editors, favoring articles about men. Since the majority of biographies are about men and men tend to link more to men than to women (see Figure 6 in [31] for preliminary comparison of ranking algorithms), future research should focus on developing search and ranking algorithms that account for potential discrimination of minority groups due to homophily, i.e., the tendency of nodes to link to similar nodes.
Wikipedia should provide tools to help editors, for instance, by considering already existing manuals of gender-neutral language [32], or by indicating missing links between articles. For example, if an article about a woman links to the article about her husband, the husband should also link back. Internal Wikipedia discussions that started after we published our preliminary studies on gender inequalities in the content of Wikipedia [4, 5] suggest such actions.Footnote 7 However they are not yet internal policies.
Gender inequalities in traditional media: Feminists often claim that news is not just mostly about men, but overwhelmingly seen through the eyes of men. Analysis of longitudinal data from the Global Media Monitoring Project (GMMP) spanning over 15 years indicates that the role of women as producers and subjects of news has seen a steady improvement, but the relative visibility of women compared to men has been stuck at 1:3 [33]. Gender inequalities are also manifested in films used for education purposes, as revealed by the application of the Bechdel test to teaching content [34].
Gender inequalities in Wikipedia: Our work is not the first to recognize the importance of understanding gender biases in Wikipedia [4, 5, 27, 31, 35, 36].
Reagle and Lauren [35] compare the coverage and article length of thousands of biographical subjects from six reference sources (e.g., The Atlantic's 100 most influential figures in American history, TIME Magazine's list of 2008's most influential people) in the English-language Wikipedia and the online Encyclopedia Britannica. The authors do not find gender-specific differences in the coverage and article length in Wikipedia, but Wikipedia's missing articles are disproportionately female relative to those of Britannica. Wagner et al. [4] also analyzed the coverage of notable people in Wikipedia based on three external reference lists (Pantheon [37], Freebase [38] and Human Accomplishment [39]) and found no significant difference in the proportional coverage of men and women in six different language edition of Wikipedia.
Bamman and Smith [8] present a method to learn biographical structures from text and observe that in the English Wikipedia, the biographies of women disproportionately focus on marriage and divorce compared to those of men, in line with our findings on the lexical dimension. Similar results are found by Graells-Garrido et al. [5] where the most important n-grams and LIWC categories of men and women are compared. Similar topical biases are found in six different language editions (German, English, French, Italian, Spanish and Russian) [4].
Recent research shows that most important historical figures across Wikipedia language editions are born in western countries after the 17th century, and are male [31]. The authors use different link-based ranking algorithms and focus on the top 100 figures in each language edition. Their results show that very few women are among the top 100 figures - 5.2 on average across language editions. Since the authors do not use external reference lists, it remains unclear how many women we would expect to see among the top 100 figures.
In terms of network structure, we built a biography network [27] in which we estimated PageRank, a measure of node centrality based on network connectivity [20, 21]. In similar contexts, PageRank has been used to provide an approximation of historical importance [26, 27] and to study the bias leading to the gender gap [26].
Previous research has also explored gender inequalities in the editor community of Wikipedia and potential reasons [1–3]. The importance of this issue has been acknowledged among Wikipedians, for example through the initiation of the 'Countering Systemic Bias' WikiProjectFootnote 8 in 2004.
Though previous research identified gender bias on a topical and structural level in Wikipedia, the present work goes beyond previous efforts by (i) providing an in-depth analysis of the content and structure of the English Wikipedia, (ii) analyzing external and internal signals of global notability of men and women that are depicted in Wikipedia, and (iii) exploring to what extent linguistic biases manifest in the content of Wikipedia.
In this paper we studied various aspects of gender bias in the content of Wikipedia biographies. This is an important issue since the usage of Wikipedia is growing, and with that, its importance as a central knowledge repository that is used around the globe, including for educational purposes.
Our empirical results uncover significant gender differences at various levels that cannot only be attributed to the fact that Wikipedia is mirroring the off-line world and its biases. For instance, the lexical, linguistic and structural differences must be attributed to Wikipedia editors, since they are expected to use their own words and interlink articles manually. We believe that the differences in the notability of men and women that are present in Wikipedia can in part be explained by how the life of men and women is documented in our society [11]. Since Wikipedia editors do rely on this biased information for informing their decisions (e.g., who is notable enough to be depicted in Wikipedia? What are the most important facts about this person?), it is not surprising that the content they produce reflects these pre-existing biases. However, it is also well known from social psychology that human-beings generally favor people in their in-group over people in their out-group [29, 30] and our results show that Wikipedia editors reveal a linguistic in-group/out-group bias [13].
The extent to which this bias also impacts the selection (or article creation) process of notable people remains however unclear. Interestingly, we find that women that are depicted in Wikipedia tend to be more notable than men from a global perspective, which can be seen as an indication of gender-specific entry barriers.
Our empirical results are limited to the English Wikipedia, which is biased towards western cultures [40]. However, in previous work [4] we found that similar structural, topical and coverage biases exist across six different language editions. We leave a more detailed exploration of gender bias across all language editions for future work. Our methods can be applied in other contexts given an ad-hoc manual coding of associated keywords to each gender.
In summary, the contributions of this work are twofold: (i) we presented a computational method for assessing gender bias in Wikipedia along multiple dimensions and (ii) we applied this method to the English Wikipedia and shared empirical insights on observed gender inequalities. The methods presented in this work can be used to assess, monitor and evaluate these issues in Wikipedia on an ongoing basis. We translate our findings into some potential actions for the Wikipedia editor community to reduce gender biases in the future. We hope our work will contribute to increased awareness about gender biases online, and about the different ways these biases can manifest themselves. We propose that Wikipedia may wish to consider revising its guidelines, both to account for the low visibility of women and to encourage a less biased use of language.
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GESIS - Leibniz Institute for the Social Sciences, Unter Sachsenhausen 5-8, Cologne, Germany
Claudia Wagner
University of Koblenz-Landau, Koblenz, Germany
Telefónica I+D, Av. Manuel Montt 1404, Third Floor, Santiago, Chile
Eduardo Graells-Garrido
ETH Zurich, Weinbergstrasse 56/58, Zurich, 8092, Switzerland
Center for Complex Networks and Systems Research, School of Informatics and Computing, Indiana University, 919 East Tenth St, Bloomington, IN, 47408, USA
Filippo Menczer
Correspondence to Claudia Wagner.
All authors contributed to the research design and writing of the paper. Claudia Wagner was mainly responsible for the internal and external notability study and the topical analysis. Eduardo Graells-Garrido was collecting and preparing the data. Further he was working on the internal notability study, the network and topic analyses. David Garcia focused on the linguistic bias exploration. Filippo Menczer was mainly responsible for the network analysis.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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DOI: https://doi.org/10.1140/epjds/s13688-016-0066-4
historical relevance
lexical bias
Advances in data-driven computational social sciences | CommonCrawl |
Discovering model structure for partially linear models
Xin He1 &
Junhui Wang2
Annals of the Institute of Statistical Mathematics volume 72, pages45–63(2020)Cite this article
Partially linear models (PLMs) have been widely used in statistical modeling, where prior knowledge is often required on which variables have linear or nonlinear effects in the PLMs. In this paper, we propose a model-free structure selection method for the PLMs, which aims to discover the model structure in the PLMs through automatically identifying variables that have linear or nonlinear effects on the response. The proposed method is formulated in a framework of gradient learning, equipped with a flexible reproducing kernel Hilbert space. The resultant optimization task is solved by an efficient proximal gradient descent algorithm. More importantly, the asymptotic estimation and selection consistencies of the proposed method are established without specifying any explicit model assumption, which assure that the true model structure in the PLMs can be correctly identified with high probability. The effectiveness of the proposed method is also supported by a variety of simulated and real-life examples.
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This research is supported in part by HK GRF-11302615, HK GRF-11331016, and City SRG-7004865. The authors would like to thank the associate editor and two anonymous referees for their constructive suggestions. The authors would also like to thank Dr. Heng Lian (City University of Hong Kong) for sharing his code on the DPLM method.
School of Statistics and Management, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai, 200433, China
Xin He
Department of Mathematics, City University of Hong Kong, 83 Tat Chee Ave, Kowloon Tong, 999077, Hong Kong, China
Junhui Wang
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Correspondence to Xin He.
Below is the link to the electronic supplementary material.
Supplementary material 1 (pdf 230 KB)
Appendix: technical proofs
Proof of Theorem 1
For some constant \({a_1}\), denote
$$\begin{aligned} \mathcal {C}&=\Big \{ \mathcal{E}(\widehat{\mathbf{g}},\widehat{\mathbf{H}})-2\sigma ^2_s \ge a_{1} \left( \log \frac{4}{\delta _n}\right) ^{1/2} \\&\qquad \big ( n^{-1/4} + n^{-1/2} \lambda _0^{-1} + n^{-1/2}\lambda _1^{-2} +s^{p+6} +\lambda _0 +\lambda _1 \big ) \Big \}. \end{aligned}$$
Then it suffices to bound \(P(\mathcal{C})\). First,
$$\begin{aligned} P(\mathcal {C})&= P \left( \mathcal {C}\cap \{ |y| \le n^{1/8}~\text{ and }~ U_n \le M_0\} \right) \\&\quad +P \left( \mathcal {C}\cap \{ |y| \le n^{1/8}~\text{ and }~ U_n \le M_0\}^{C} \right) \\&\le P\left( |y|> n^{1/8}\right) + P\left( |y| \le n^{1/8}~\text{ and }~U_n> M_0 \right) \\&\quad +P\Big (\mathcal {C}\cap \{ |y| \le n^{1/8}~\text{ and }~ U_n \le M_0\}\Big )=P_1+P_2+P_3, \end{aligned}$$
where \(U_n=\frac{1}{n(n-1)}\sum _{i,j=1}^n(y_i-y_j)^2\), and \(M_0 = 4A^2 + 2\sigma ^2+1\) with A being the upper bound of \(f^*(\mathbf{x})\) on \(\mathcal{X}\) and \(\sigma ^2=\mathrm{Var}(\epsilon )\). Next, we bound \(P_1, P_2,\) and \(P_3\) separately. To bound \(P_1\), we have \(P_1\le {E(|y|)}n^{-1/8}\) by the Markov's inequality, where E(|y|) is a bounded quantity. To bound \(P_2\), note that \(E(U_n)=E(E(U_n|\mathbf{x}_i,\mathbf{x}_j))=E((f^*(\mathbf{x}_i)-f^*(\mathbf{x}_j))^2)+E((\epsilon _i-\epsilon _j)^2) \le 4A^2 + 2\sigma ^2\). And thus by Bernstein's inequality for U-statistics (Hoeffding 1963), we have that
$$\begin{aligned} P_2&\le P\left( U_n> M_0 \big | |y|\le n^{1/8}\right) \\&\le P\left( U_n -E(U_n) >1 \big | |y|\le n^{1/8}\right) \le \exp \left( -\frac{n^{{1}/{2}}}{16}\right) . \end{aligned}$$
To bound \(P_3\), within the set \(\{ |y| \le n^{\frac{1}{8}}~\text{ and }~ U_n{\tiny {\tiny }} \le M_0\}\), equlaity (1) and by Lemma 3 in the supplementary file, we have with probability at least \(1-\delta _n\) that
$$\begin{aligned} 0&\le \mathcal{E}(\widehat{\mathbf{g}},\widehat{\mathbf{H}}) - 2\sigma ^2_s \le a_1 \left( \log \frac{4}{\delta _n}\right) ^{1/2} \\&\quad \left( n^{-{1}/{4}}+ M_0^2n^{-{1}/{2}}\lambda _0^{-1} + M_0^2n^{-{1}/{2}}\lambda _1^{-2} +s^{p+6} + \lambda _0 +\lambda _1 \right) , \end{aligned}$$
which implies \(P_3\le \delta _n\), and thus \(P(\mathcal {C})\le \delta _n+O(n^{-{1}/{8}})\) for some constant \(a_1\). Specially, when \(\lambda _0= n^{-{1}/{4}}\), \(\lambda _1=n^{-{1}/{4(p+2)}}\) and \(s=n^{-{1}/{4(p+6)(p+2+2\theta )}}\), there exists a constant \(c_5\) such that with probability at least \(1-\delta _n\)
$$\begin{aligned} 0\le \mathcal{E}(\widehat{\mathbf{g}},\widehat{\mathbf{H}}) -2\sigma ^2_s \le c_5 \left( \log \frac{4}{\delta _n}\right) ^{{1}/{2}}n^{-\frac{1}{4(p+2+2\theta )}}. \end{aligned}$$
Next, we establish the estimation consistency. By Assumptions 1 and 2 and equlaity (1), for some constant \(a_2\) there holds
$$\begin{aligned} 0\le \mathcal{E}({\mathbf{g}}^*,{\mathbf{H}}^*) - 2 \sigma _s^2&\le \iint w(\mathbf{x},\mathbf{u})c_0^2\Vert \mathbf{x}-\mathbf{u}\Vert _2^6\rho _{\mathbf{x}}\rho _{\mathbf{u}}\\&\le c_0^2c_4s^{p+6}\int e^{-\mathbf{t}^\mathrm{T}\mathbf{t}}{\mathbf{t}}^\mathrm{T}{\mathbf{t}} \mathrm{d}\mathbf{t}\le a_2s^{p+6}, \end{aligned}$$
where \(a_2=c_0^2c_4\int e^{-\mathbf{t}^\mathrm{T}\mathbf{t}}{\mathbf{t}}^\mathrm{T}{\mathbf{t}} \mathrm{d}\mathbf{t}\), \(\mathbf{t}=(\mathbf{u}-\mathbf{x})/s\) and \(\int e^{-\mathbf{t}^\mathrm{T}\mathbf{t}}{\mathbf{t}}^\mathrm{T}{\mathbf{t}} \mathrm{d}\mathbf{t}\) is a bounded quantity. Specially, with \(s=n^{-{1}/{4(p+6)(p+2+2\theta )}}\), we have \(\mathcal{E}({\mathbf{g}}^*,{\mathbf{H}}^*) - 2 \sigma _s^2\le a_2n^{-{1}/{4(p+2+2\theta )}}\). Therefore, for some constant \(c_6\), triangle inequality implies that
$$\begin{aligned} |\mathcal{E}(\widehat{\mathbf{g}},\widehat{\mathbf{H}}) - \mathcal{E}({\mathbf{g}}^*,{\mathbf{H}}^*)|&\le |\mathcal{E}(\widehat{\mathbf{g}},\widehat{\mathbf{H}})-2\sigma ^2_s| + |\mathcal{E}({\mathbf{g}}^*,{\mathbf{H}}^*)-2\sigma ^2_s|\\&\le c_6 \left( \log \frac{4}{\delta _n}\right) ^{{1}/{2}}n^{-\frac{1}{4(p+2+2\theta )}}. \end{aligned}$$
This completes the Proof of Theorem 1. \(\square \)
Proof of Theorem 2:
First we show that for any \(l \in \mathcal{L}^*\), \(\Vert \widehat{H}_{ll'}\Vert _{L_{\rho _{\mathbf{x}}}^2}=0\) for any \(l'\in \mathcal{S}\). Note that \(\Vert \widehat{c}_{ll'}\Vert _2=0\) implies that \(\Vert \widehat{H}_{ll'}\Vert _{L_{\rho _{\mathbf{x}}}^2}=0\) based on the representer theorem for the RKHS, and thus it suffices to show \( \Vert \widehat{\mathbf{c}}_{ll'}\Vert _2=0\) for any \(l \in \mathcal{L}^*\) and \(l'\in \mathcal{S}\).
Suppose \(\Vert \widehat{\mathbf{c}}_{ll'}\Vert _2>0\) for some \(l \in \mathcal{L}^*\) and \(l' \in \mathcal{S}\). The derivative of (5) with respect to \({\mathbf{c}}_{ll'}\) yields that
$$\begin{aligned} \sum _{i,j=1}^n x_{ijl}x_{jil'} A_1 ({\mathbf{x}}_i,y_i,{\mathbf{x}}_j,y_j){\mathbf{K}}_{{\mathbf{x}}_i} = \lambda _1 A_2(\widehat{\mathbf{c}}_{ll'}), \end{aligned}$$
$$\begin{aligned} A_1({\mathbf{x}}_i,y_i,{\mathbf{x}}_j,y_j)= & {} \frac{1}{n(n-1)}w_{ij} ( y_i -y_j - \widehat{\mathbf{g}}({\mathbf{x}}_i)^\mathrm{T} ({\mathbf{x}}_i - {\mathbf{x}}_j)\\&+ \frac{1}{2} ({\mathbf{x}}_i - {\mathbf{x}}_j)^\mathrm{T} \widehat{\mathbf{H}}({\mathbf{x}}_i)({\mathbf{x}}_i - {\mathbf{x}}_j) ), \end{aligned}$$
\(A_2(\widehat{\mathbf{c}}_{ll'})=\frac{ \pi _{ll'} {\mathbf{K}} \widehat{\mathbf{c}}_{ll'} }{\left( \widehat{\mathbf{c}}_{ll'}^\mathrm{T} {\mathbf{K}} \widehat{\mathbf{c}}_{ll'} \right) ^{{1}/{2}}}\), and \(x_{ijl}=x_{il}-x_{jl}.\) For the right-hand side of (8), its norm divided by \(n^{{1}/{2}}\) is \(n^{-{1}/{2}}\lambda _1\Vert A_2(\widehat{\mathbf{c}}_{ll'})\Vert _2 \ge n^{-{1}/{2}} \lambda _{1}\pi _{ll'} \psi _{min}\psi _{max}^{-{1}/{2}}\), which diverges to infinity by Assumption 5. For the left-hand side of (8), by Assumption 1, \(x_{ijl}, x_{jil'}\), and every elements of \({\mathbf{K}}_{\mathbf{x}}\) are bounded. Denote \(A_{\mathcal{Z}^n}(\widehat{{\varvec{\alpha }}},\widehat{\mathbf{c}})=\sum _{i,j=1}^n A_1 ({\mathbf{x}}_i,y_i,{\mathbf{x}}_j,y_j)\), we will show that \(|A_{\mathcal{Z}^n}(\widehat{{\varvec{\alpha }}},\widehat{\mathbf{c}})|\) is bounded as well.
For some constant \(a_{3}\) and \(\delta _n\in (0,1)\), denote
$$\begin{aligned} \mathcal {D}&=\Big \{ |A_{\mathcal{Z}^n}(\widehat{{\varvec{\alpha }}},\widehat{\mathbf{c}})| > a_{3}\left( \log {\frac{2}{\delta _n}}\right) ^{{1}/{2}} \\&\quad \left( n^{-{1}/{8(p+2+2\theta )}} + n^{-{3}/{8}} +n^{-{1}/{2}}\lambda _0^{-{1}/{2}}+n^{-{1}/{2}}\lambda _1^{-1} \right) \Big \}, \end{aligned}$$
and thus it suffices to bound \(P(\mathcal{D})\). First, we have
$$\begin{aligned} P(\mathcal {D})&=P\left( \mathcal {D}\cap \{|y|\le n^{{1}/{8}} ~\text{ and }~U_n\le M_0\} \right) \\&\quad + P\left( \mathcal {D}\cap \{|y|\le n^{{1}/{8}} ~\text{ and }~U_n\le M_0\}^{C} \right) \\&\le P\left( |y|> n^{{1}/{8}} \right) + P\left( |y|\le n^{{1}/{8}} ~\text{ and }~U_n> M_0 \right) \\&\quad + P\left( \mathcal {D}\cap \{|y|\le n^{{1}/{8}} ~\text{ and }~U_n\le M_0\}\right) \le P_1+P_2+P_4, \end{aligned}$$
where \(U_n\) and \(M_0\) are defined as in Theorem 1. Note that \(P_1+P_2=O(n^{-1/8})\) as in the Proof of Theorem 1. To bound \(P_4\), by Cauchy–Schwarz inequality, we conclude that
$$\begin{aligned} E(A_{\mathcal{Z}^n}(\widehat{{\varvec{\alpha }}}, \widehat{\mathbf{c}} ))&\le \Big (\iint w({\mathbf{x}},{\mathbf{u}}) \Big (f^*(\mathbf{x}) - f^*(\mathbf{u}) -\widehat{\mathbf{g}}(\mathbf{x})^\mathrm{T}({\mathbf{x}}-{\mathbf{u}}) \\&\quad + \frac{1}{2} ({\mathbf{x}}-{\mathbf{u}})^\mathrm{T}\widehat{\mathbf{H}}(\mathbf{x})({\mathbf{x}}-{\mathbf{u}}) \Big )^2 \mathrm{d} \rho _{\mathbf{x}}\mathrm{d} \rho _{\mathbf{u}} \Big )^{{1}/{2}}=\left( \mathcal{E}(\widehat{\mathbf{g}},\widehat{\mathbf{H}})-2\sigma ^2_s\right) ^{{1}/{2}}. \end{aligned}$$
Within the set \(\{|y|\le n^{{1}/{8}} ~\text{ and }~U_n\le M_0\}\), following similar proofs of Lemma 1 and Proposition 2, we have for some constant \(a_{3}\), with probability at least \(1-{\delta _n}\) there holds
$$\begin{aligned} |A_{\mathcal{Z}^n}(\widehat{{\varvec{\alpha }}}, \widehat{\mathbf{c}})| \le a_{3} \left( \log \frac{2}{\delta _n}\right) ^{{1}/{2}}\left( n^{-{1}/{{8(p+2+2\theta )}}} + n^{-{3}/{8}} +n^{-{1}/{2}}\lambda _0^{-1}+n^{-{1}/{2}}\lambda _1^{-1} \right) , \end{aligned}$$
which implies \(P_4\le \delta _n\), and thus we have \(P(\mathcal {D})\le {\delta _n} + O(n^{-{1}/{8}})\). Combining the above results, the norm of the left-hand side of (8) divided by \(n^{{1}/{2}}\) converges to zero in probability, which contradicts with the fact that the right-hand side of (8) diverges to infinity when \(n\rightarrow \infty \). Therefore, for any \(l \in \mathcal{L}^*\) and \(l'\in \mathcal{S}\), \(\Vert \widehat{\mathbf{c}}_{ll'}\Vert _2\equiv 0\), implying \(\Vert \widehat{H}_{ll'}\Vert _{L_{\rho _{\mathbf{x}}}^2}=0\) for any \(l \in \mathcal{L}^*\) and \(l'\in \mathcal{S}\), and thus there holds \(\widehat{\mathcal{L}} \subset \mathcal{L}^*\).
Next, we show \(\Vert \widehat{H}_{ll'}\Vert _{L^2_{\rho _{\mathbf{x}}}}^2\ne 0\) for any \(l\in \mathcal{N}^* \) and some \(l'\in \mathcal{S}\). By Lemma 4, for the set \(\mathcal{X}_s=\{ \mathbf{x}\in \mathcal{X}: d(\mathbf{x},\partial \mathcal{X})>s, p(\mathbf{x})>s+c_1s^{\theta }\}\) and some constant as in the supplementary file, there holds
$$\begin{aligned} \int \nolimits _{\mathcal{X}_{s}} \Vert \widehat{\mathbf{H}}(\mathbf{x}) - {\mathbf{H}}^*(\mathbf{x})\Vert _F^2 \mathrm{d} \rho _{\mathbf{x}} \le \frac{b_{6}}{s^{p+5 }} (s^{6+p} + \mathcal{E}\left( \widehat{\mathbf{g}}, \widehat{\mathbf{H}})- 2\sigma _s^2\right) , \end{aligned}$$
which converges to zero in probability by Theorem 1. Suppose that there exist some \(l\in \mathcal{N}^*\) such that \(\Vert \widehat{H}_{ll'}\Vert ^2_{L_{\rho _X}^2}=0\) for any \(l'\in \mathcal{S}\), then
$$\begin{aligned} \int \nolimits _{\mathcal{X}_{s}} ({H}^*_{ll'}(\mathbf{x}))^2 \mathrm{d} \rho _{\mathbf{x}}\le \int \nolimits _{\mathcal{X}_{s}} \Vert \widehat{\mathbf{H}}(\mathbf{x}) - {\mathbf{H}}^*(\mathbf{x})\Vert _F^2 \mathrm{d} \rho _{\mathbf{x}}. \end{aligned}$$
However, Assumption 4 implies that for some \(l'\in \mathcal{S}\), \(\int \nolimits _{\mathcal{X}_{s}} ({H}^*_{ll'}(\mathbf{x}))^2 \mathrm{d} \rho _{\mathbf{x}}>\int \nolimits _{\mathcal{X}\backslash \mathcal{X}_{t}} ({H}^*_{ll'}(\mathbf{x}))^2 \mathrm{d} \rho _{\mathbf{x}}\) when s is sufficiently small, which is a positive constant, and thus leads to contradiction. Therefore, \(\Vert \widehat{H}_{ll'}\Vert _{L^2_{\rho _{\mathbf{x}}}}^2\ne 0\) for any \(l\in \mathcal{N}^* \) and some \(l'\in \mathcal{S}\), and thus there holds \(\widehat{\mathcal{N}}\subset \mathcal{N}^*\).
Finally, since \(\mathcal{S}=\mathcal{L}^*\cup \mathcal{N}^*=\widehat{L}\cup \widehat{\mathcal{N}}\) and \(\mathcal{L}^*\cap \mathcal{N}^*=\widehat{L}\cap \widehat{\mathcal{N}}=\varnothing \), combining with the above results we have \(P(\widehat{\mathcal{L}} = \mathcal{L}^*)\rightarrow 1\) and \(P(\widehat{\mathcal{N}}=\mathcal{N}^*)\rightarrow 1\) when n diverges. Moreover, we have \(P(\widehat{\mathcal{L}}=\mathcal{L}^*,\widehat{\mathcal{N}}=\mathcal{N}^*) \ge 1 - P(\widehat{\mathcal{L}} \ne \mathcal{L}^*) - P(\widehat{\mathcal{N}}\ne \mathcal{N}^*) \rightarrow 1\) as \(n \rightarrow \infty \). This completes the Proof of Theorem 2. \(\square \)
He, X., Wang, J. Discovering model structure for partially linear models. Ann Inst Stat Math 72, 45–63 (2020) doi:10.1007/s10463-018-0682-9
Revised: 04 June 2018
Issue Date: February 2020
Gradient learning
Partially linear models
Proximal gradient descent
Reproducing kernel Hilbert space (RKHS) | CommonCrawl |
\begin{document}
\title[Constructing model categories with prescribed fibrant objects] {Constructing model categories with prescribed fibrant objects} \author[{A. E.} {Stanculescu}]{{Alexandru E.} {Stanculescu}} \address{\newline Department of Mathematics and Statistics, \newline Masaryk University, Kotl\'{a}{\v{r}}sk{\'{a}} 2,\newline 611 37 Brno, Czech Republic} \email{[email protected]} \thanks{Supported by the project CZ.1.07/2.3.00/20.0003 of the Operational Programme Education for Competitiveness of the Ministry of Education, Youth and Sports of the Czech Republic \newline \indent}
\begin{abstract} We present a weak form of a recognition principle for Quillen model categories due to J.H. Smith. We use it to put a model category structure on the category of small categories enriched over a suitable monoidal simplicial model category. The proof uses a part of the model structure on small simplicial categories due to J. Bergner. We give an application of the weak form of Smith's result to left Bousfield localizations of categories of monoids in a suitable monoidal model category. \end{abstract}
\maketitle
There are nowadays several recognition principles that allow one to put a Quillen model category structure on a given category. For the purposes of this work we divide them into those that make use of the small object agument and those that don't. A recognition principle that makes use of the small object argument is the following theorem of J.H. Smith.
\begin{theorem}\cite[Theorem 1.7]{Bek} Let $\mathcal{E}$ be a locally presentable category, {\rm W} a full accessible subcategory of $\mathrm{Mor}(\mathcal{E})$, and $I$ a set of morphisms of $\mathcal{E}$. Suppose they satisfy:
$c0:$ $\mathrm{W}$ has the two out of three property.
$c1:$ $\mathrm{inj}(I)\subset{\rm W}$.
$c2:$ The class ${\rm cof}(I)\cap {\rm W}$ is closed under transfinite composition and under pushout.
Then setting weak equivalences:={\rm W}, cofibrations:=$\mathrm{cof}(I)$ and fibrations:=$\mathrm{inj}(\mathrm{cof}(I)\cap {\rm W})$, one obtains a cofibrantly generated model structure on $\mathcal{E}$. \end{theorem} We can say that $(a)$ in practice, it is condition $c2$ above that is often the most difficult to check and $(b)$ the result gives \emph{no} description of the fibrations of the resulting model structure. Another recognition principle that makes use of the small object argument is a result of D.M. Kan \cite[Theorem 11.3.1]{Hi}, \cite[Theorem 2.1.19]{Ho}. We can say that Kan's result gives a \emph{full} description of the fibrations of the resulting model structure. In this paper we
(1) advertise (see Proposition 1.3) an abstraction of a technique due to D.-C. Cisinski \cite[Proof of Th\'{e}or\`{e}me 1.3.22]{Ci} and A. Joyal (unpublished, but present in his proof, circa 1996, of the model structure for quasi-categories) that addresses both $(a)$ and $(b)$ above, in the sense that it makes $c2$ easier to check and it gives a \emph{partial} description of the fibrations of the resulting model structure---namely the fibrant objects and the fibrations between them are described---provided that other assumptions hold, and
(2) give an application of this technique to the homotopy theory of categories enriched over a suitable monoidal simplicial model category (see Theorem 2.3) and to left Bousfield localizations of categories of monoids in a suitable monoidal model category (see Theorem 4.5).
The paper is organized as follows. In Section 1 we detail the above mentioned technique. The two out of six property of a class of maps of Dwyer et al. \cite{DHKS} plays an important role. In Section 2 we prove that the category of small categories enriched over a monoidal simplicial model category that satisfies some assumptions, admits a certain model category structure. Our proof uses one result of the non-formal part of the proof of the analogous model structure for categories enriched over the category of simplicial sets, due to J. Bergner \cite{Be}. We modify one of the steps in Bergner's proof; this modification is a key point in our approach and it enables us to apply the technique from Section 1. We also fix (see Remark 2.8), in an appropiate way, a mistake in \cite{St}. The idea to use the model structure for categories enriched over the category of simplicial sets is due to G. Tabuada \cite{Ta}. In Section 3 we extend a result of R. Fritsch and D.M. Latch \cite[Proposition 5.2]{FL} to enriched categories; this is needed in the proof of the main result of Section 2. The section is self contained. Motivated by considerations from \cite{Ho2}, we apply in Section 4 the technique from Section 1 to the study of left Bousfield localizations of categories of monoids. Precisely, let $L${\bf M} be a left Bousfield localization of a monoidal model category {\bf M}. We consider the problem of putting a model category structure on the category of monoids in {\bf M}, somehow related to $L${\bf M}.
\section{Constructing model categories with prescribed fibrant objects}
We recall from \cite{DHKS} the following definitions. Let $\mathcal{E}$ be an arbitrary category and {\rm W} a class of maps of $\mathcal{E}$. {\rm W} is said to satisfy the \emph{two out of six property} if for every three maps $r,s,t$ of $\mathcal{E}$ for which the two compositions $sr$ and $ts$ are defined and are in {\rm W}, the four maps $r,s,t$ and $tsr$ are in {\rm W}. {\rm W} is said to satisfy the \emph{weak invertibility property} if every map $s$ of $\mathcal{E}$ for which there exist maps $r$ and $t$ such that the compositions $sr$ and $ts$ exist and are in {\rm W}, is itself in {\rm W}. The two out of six property implies the two out of three property. The converse holds in the presence of the weak invertibility property.
The terminal object of a category, when it exists, is denoted by $1$.
Let $\mathcal{E}$ be a locally presentable category and $J$ a set of maps of $\mathcal{E}$. Then the pair $({\rm cof}(J),{\rm inj}(J))$ is a weak factorization system on $\mathcal{E}$ \cite[Proposition 1.3]{Bek}. We call a map of $\mathcal{E}$ that belongs to ${\rm inj}(J)$ a \emph{naive fibration}, and say that an object $X$ of $\mathcal{E}$ is \emph{naively fibrant} if $X\rightarrow 1$ is a naive fibration. We denote the class of naive fibrations between naively fibrant objects by ${\rm inj}_{0}(J)$. \begin{lem} {\rm (D.-C. Cisinski, A. Joyal)} Let $\mathcal{E}$ be a locally presentable category, $(\mathcal{A},\mathcal{B})$ a weak factorisation system on $\mathcal{E}$, {\rm W} a class of maps of $\mathcal{E}$ satisfying the two out of six property and $J$ a set of maps of $\mathcal{E}$.
$(1)$ Suppose that $cell(J)\subset {\rm W}$. Then a map that has the left lifting property with respect to maps in ${\rm inj}_{0}(J)$ belongs to {\rm W}.
$(2)$ Suppose that $cell(J)\subset {\rm W}$ and that ${\rm inj}_{0}(J)\cap{\rm W}\subset\mathcal{B}$. Then a map in $\mathcal{A}$ belongs to {\rm W} if and only if it has the left lifting property with respect to the maps in ${\rm inj}_{0}(J)$. In particular, $\mathcal{A}\cap {\rm W}$ is closed under pushouts and transfinite compositions.
$(3)$ Suppose that $cell(J)\subset \mathcal{A}\cap{\rm W}$ and that ${\rm inj}_{0}(J)\cap{\rm W}\subset\mathcal{B}$. Then an object $X$ of $\mathcal{E}$ is naively fibrant if and only if the map $X\rightarrow 1$ is in $\mathrm{inj}(\mathcal{A}\cap {\rm W})$. Also, a map between naively fibrant objects is in $\mathrm{inj}(\mathcal{A}\cap {\rm W})$ if and only if it is a naive fibration. \end{lem} \begin{proof} $(1)$ Let $i:A\rightarrow B$ be a map which has the left lifting property with respect to the naive fibrations between naively fibrant objects. Factorize (see, for example, \cite[Proposition 1.3]{Bek}) the map $B\rightarrow 1$ as $B\rightarrow \bar{B}\rightarrow 1$, where $B\rightarrow \bar{B}$ is in $cell(J)$ and $\bar{B}$ is naively fibrant. Next, factorize the composite map $A\rightarrow \bar{B}$ as a map $A\rightarrow \bar{A}$ in $cell(J)$ followed by a naive fibration $\bar{A} \rightarrow \bar{B}$. The resulting commutative diagram \[ \xymatrix{ A \ar[r] \ar[d]_{i} & \bar{A} \ar[d]\\ B \ar[r] & \bar{B}\\ }
\] has then a diagonal filler, and so the hypothesis and the two out of six property of {\rm W} imply that $i$ is in {\rm W}.
$(2)$ Let \[
\xymatrix{ A \ar[r]^{u} \ar[d]_{i} & X \ar[d]^{p}\\ B \ar[r]^{v} & Y\\
}
\] be a commutative diagram with $i$ in $\mathcal{A}\cap {\rm W}$ and $p$ in ${\rm inj}_{0}(J)$. Factorize $v$ as a map $B\rightarrow \bar{B}$ in $cell(J)$ followed by a naive fibration $\bar{B}\rightarrow Y$. Next, factorize the canonical map $A\rightarrow \bar{B}\underset{Y}\times X$ as a map $A\rightarrow \bar{A}$ in $cell(J)$ followed by a naive fibration $\bar{A}\rightarrow \bar{B}\underset{Y}\times X$. It suffices to show that the square \[
\xymatrix{ A \ar[r] \ar[d]_{i} & \bar{A} \ar[d]\\ B \ar[r] & \bar{B}\\ }
\] has a diagonal filler. The map $\bar{A}\rightarrow \bar{B}$ is a naive fibration between naively fibrant objects. It also belongs to {\rm W} by the two out of three property, and so by hypothesis it is in $\mathcal{B}$. Therefore the diagonal filler exists. The converse follows from $(1)$. Thus, in order to detect if an element of $\mathcal{A}$ is in {\rm W} one can use the left lifting property with respect to a class of maps, namely ${\rm inj}_{0}(J)$. In particular, $\mathcal{A}\cap {\rm W}$ is closed under pushouts and transfinite compositions.
$(3)$ This is straightforward from $(2)$. \end{proof} \begin{remark} {\rm One can make variations in Lemma 1.1. For example, the path object argument devised by Quillen shows that the conclusion of $(1)$ remains valid if instead of $cell(J)\subset {\rm W}$ one requires that} $\mathcal{E}$ has a functorial naively fibrant replacement functor and every naively fibrant object has a naive path object. {\rm This new requirement implies that $cell(J)\subset {\rm W}$.} \end{remark} The following result makes the connection between Smith's Theorem and Lemma 1.1. \begin{proposition} Let $\mathcal{E}$ be a locally presentable category, {\rm W} a full accessible subcategory of $\mathrm{Mor}(\mathcal{E})$ and $I$ and $J$ be two sets of morphisms of $\mathcal{E}$. Let us call a map of $\mathcal{E}$ that belongs to ${\rm inj}(J)$ a \emph{naive fibration}, and an object $X$ of $\mathcal{E}$ \emph{naively fibrant} if $X\rightarrow 1$ is a naive fibration. Suppose the following conditions are satisfied:
$c0:$ $\mathrm{W}$ has the two out of three property.
$c1:$ $\mathrm{inj}(I)\subset{\rm W}$.
$nc0:$ {\rm W} has the weak invertibility property.
$nc1:$ $cell(J)\subset {\rm cof}(I)\cap {\rm W}$.
$nc2:$ A map between naively fibrant objects that is both a naive fibration and in {\rm W} is in {\rm inj}$(I)$.
Then the triple $({\rm W}, {\rm cof}(I), {\rm inj}({\rm cof}(I)\cap {\rm W}))$ is a model structure on $\mathcal{E}$. Moreover, an object of $\mathcal{E}$ is fibrant if and only if it is naively fibrant, and the fibrations between fibrant objects are the naive fibrations. \end{proposition} \begin{proof} We shall use Theorem 0.1. All the assumptions of this result hold, except possibly condition $c2$. To check that $c2$ holds we apply the last part of Lemma 1.1(2) to the weak factorization system $(\mathcal{A},\mathcal{B})=({\rm cof}(I),{\rm inj}(I))$. It follows that the triple $({\rm W}, {\rm cof}(I), {\rm inj}({\rm cof}(I)\cap {\rm W}))$ is a model structure. The characterization of fibrant objects and of the fibrations between fibrant objects is then a consequence of Lemma 1.1(3) applied to the weak factorization system $({\rm cof}(I),{\rm inj}(I))$. \end{proof} The following result is a variation of Proposition 1.3, essentially due to A.K. Bousfield \cite[Proof of Theorem 9.3]{Bo}. We leave the proof to the interested reader. \begin{proposition} Let $\mathcal{E}$ be a category that is closed under limits and colimits and let {\rm W} be a class of maps of $\mathcal{E}$ that has the two out of three property. If $I$ and $J$ are two sets of morphisms of $\mathcal{E}$ such that
$(1)$ both $I$ and $J$ permit the small object argument \cite[Definition 10.5.15]{Hi},
$(2)$ ${\rm inj}(I)\subset{\rm W}$,
$(3)$ $cell(J)\subset {\rm cof}(I)\cap {\rm W}$,
$(4)$ ${\rm inj}_{0}(J)\cap{\rm W}\subset {\rm inj}(I)$, and
$(5)$ the class {\rm W} is stable under pullback along maps in ${\rm inj}_{0}(J)$,
then the triple $({\rm W}, {\rm cof}(I), {\rm inj}({\rm cof}(I)\cap {\rm W}))$ is a right proper model structure on $\mathcal{E}$. Moreover, an object of $\mathcal{E}$ is fibrant if and only if it is naively fibrant, and the fibrations between fibrant objects are the naive fibrations. \end{proposition} Here is an application of Lemma 1.1. Let $\mathcal{E}$ be a locally presentable closed category with initial object $\emptyset$. We denote by $\otimes$ the monoidal product of $\mathcal{E}$ and for two objects $X,Y$ of $\mathcal{E}$ we write $Y^{X}$ for their internal hom. In the language of Lemma 1.1 we have \begin{proposition} Let {\rm W} be a class of maps of $\mathcal{E}$ having the two out of six property and let $I$ and $J$ be two sets of maps of $\mathcal{E}$. Suppose that the domains of the elements of $I$ are in {\rm cof}$(I)$, that $cell(J)\subset {\rm cof}(I)\cap {\rm W}$ and that a map between naively fibrant objects which is both a naive fibration and in {\rm W} is in {\rm inj}$(I)$. Then the following are equivalent:
$(a)$ for any maps $A\rightarrow B$ and $K\rightarrow L$ of {\rm cof}$(I)$, the canonical map $$A\otimes L\underset{A\otimes K}\cup B\otimes K\rightarrow B\otimes L$$ is in {\rm cof}$(I)$, which is in {\rm W} if either one of the given maps is in {\rm W};
$(b)$ for any maps $A\rightarrow B$ and $K\rightarrow L$ of {\rm cof}$(I)$, the canonical map $$A\otimes L\underset{A\otimes K}\cup B\otimes K\rightarrow B\otimes L$$ is in {\rm cof}$(I)$ \emph{and} for every element $A\rightarrow B$ of $I$ and every naive fibration $X\rightarrow Y$ between naively fibrant objects, the canonical map $$X^{B}\rightarrow Y^{B}\times_{Y^{A}}X^{A}$$ is a naive fibration between naively fibrant objects. \end{proposition} \begin{proof} The fact that the domains of the elements of $I$ are in {\rm cof}$(I)$ means that for every element $A\rightarrow B$ of $I$, the map $\emptyset \to A$ is in {\rm cof}$(I)$ (and therefore so is $\emptyset \to B$).
We prove $(a)\Rightarrow (b)$. Let $A\rightarrow B$ be an element of $I$, $X\rightarrow Y$ a naive fibration between naively fibrant objects and $C\rightarrow D$ an element of $J$. A commutative diagram \[ \xymatrix{ C \ar[rr] \ar[d] & & X^{B} \ar[d]\\ D \ar[rr] & & Y^{B}\times_{Y^{A}}X^{A}\\
}
\] has a diagonal filler if and only if its adjoint transpose \[
\xymatrix{ C\otimes B\underset{C\otimes A}\cup D\otimes A \ar[rr] \ar[d] & & X \ar[d]\\ D\otimes B \ar[rr] & & Y\\
}
\] has one. The latter is true by Lemma 1.1(2) applied to the weak factorization system $({\rm cof}(I),{\rm inj}(I))$. It follows that $X^{B}\rightarrow Y^{B}\times_{Y^{A}}X^{A}$ is a naive fibration. A similar adjunction argument shows that $X^{A}\rightarrow Y^{A}$ and $X^{B}\rightarrow Y^{B}$ are naive fibrations between naively fibrant objects, therefore $Y^{B}\times_{Y^{A}}X^{A}$ is naively fibrant.
We prove $(b)\Rightarrow (a)$. Suppose first that $A\rightarrow B$ is an element of $I$ and let $K\rightarrow L$ be a fixed map in ${\rm cof}(I)\cap {\rm W}$. Then the canonical map $$A\otimes L\underset{A\otimes K}\cup B\otimes K\rightarrow B\otimes L$$ is in {\rm W} by Lemma 1.1(2) applied to the weak factorization system $({\rm cof}(I),{\rm inj}(I))$ and an adjunction argument. Thus, it suffices to show that the class of maps $A'\rightarrow B'$ of {\rm cof}$(I)$ such that $$A'\otimes L\underset{A'\otimes K}\cup B'\otimes K\rightarrow B'\otimes L$$ is in {\rm cof}$(I)\cap {\rm W}$ is closed under pushout, transfinite composition and retracts. This is the case since by Lemma 1.1(2) applied to the weak factorization system $({\rm cof}(I),{\rm inj}(I))$ the elements of ${\rm cof}(I)$ which are in {\rm W} can be detected by the left lifting property with respect to a class of maps. \end{proof}
\section{Application: categories enriched over monoidal simplicial model categories}
We denote by {\bf S} the category of simplicial sets, regarded as having the standard model structure (due to Quillen). We let {\bf Cat} be the category of small categories. We say that an arrow $f:C\rightarrow D$ of {\bf Cat} is an \emph{isofibration} if for any $x\in Ob(C)$ and any isomorphism $v:y'\rightarrow f(x)$ in $D$, there exists an isomorphism $u:x'\rightarrow x$ in $C$ such that $f(u)=v$. The class of isofibrations is invariant under isomorphisms in the sense that given a commutative diagram in {\bf Cat} \[ \xymatrix{ {A}\ar[r] \ar[d]_{f} &{B} \ar[d]^{g}\\ {C} \ar[r] &{D} } \] in which the horizontal arrows are isomorphisms, the map $f$ is an isofibration if and only if $g$ is so.
\subsection{Monoidal simplicial model categories}
Let {\bf M} be a monoidal model category with cofibrant unit. We recall \cite[Definition 4.2.20]{Ho} that {\bf M} is said to be a \emph{monoidal} {\bf S}-\emph{model category} if it is given a Quillen pair $F:{\bf S}\rightleftarrows {\bf M}:G$ such that $F$ is strong monoidal. Since $F$ is strong monoidal, $G$ becomes a monoidal functor.
\subsection{Classes of {\bf M}-functors and the main result}
Let {\bf M} be a monoidal model category with cofibrant unit $e$. We denote by {\bf M}-{\bf Cat} the category of small {\bf M}-categories. If $S$ is a set, we denote by {\bf M}-{\bf Cat}$(S)$ (resp. {\bf M}-{\bf Graph}$(S)$) the category of small {\bf M}-categories (resp. {\bf M}-graphs) with fixed set of objects $S$. When $S$ is a one element set $\{\ast\}$, {\bf M}-{\bf Cat}$(\{\ast\})$ is the category $Mon({\bf M})$ of monoids in {\bf M}. There is a free-forgetful adjunction $$F_{S}:{\bf M}\text{-}{\bf Graph}(S)\rightleftarrows {\bf M}\text{-}{\bf Cat}(S):U_{S}$$ We denote by $\varepsilon^{S}$ the counit of this adjunction. Every function $f:S\rightarrow T$ induces an adjoint pair $$f_{!}:{\bf M}\text{-}{\bf Cat}(S)\rightleftarrows {\bf M}\text{-}{\bf Cat}(T):f^{\ast}$$ If $\mathcal{K}$ is a class of maps of {\bf M}, an {\bf M}-functor $f:\mathcal{A}\rightarrow \mathcal{B}$ is said to be \emph{locally in} $\mathcal{K}$ if for each pair $x,y\in \mathcal{A}$ of objects, the map $f_{x,y}:\mathcal{A}(x,y)\rightarrow \mathcal{B}(f(x),f(y))$ is in $\mathcal{K}$.
We have a functor $[\_]_{{\bf M}}:{\bf M}\text{-}{\bf Cat} \rightarrow {\bf Cat}$ obtained by change of base along the symmetric monoidal composite functor \[
\xymatrix{
{\bf M} \ar[r] & Ho({\bf M})
\ar[rr]^{Ho({\bf M})(e,-)} & & Set\\ } \] \begin{definition} Let $f:\mathcal{A}\rightarrow \mathcal{B}$ be a morphism in {\bf M}\text{-}{\bf Cat}.
1. The morphism $f$ is a $\mathrm{DK}$-\emph{equivalence} if $f$ is locally a weak equivalence of {\bf M} and $[f]_{{\bf M}}:[\mathcal{A}]_{{\bf M}}\rightarrow [\mathcal{B}]_{{\bf M}}$ is essentially surjective.
2. The morphism $f$ is a $\mathrm{DK}$-\emph{fibration} if $f$ is locally a fibration of {\bf M} and $[f]_{{\bf M}}$ is an isofibration.
3. The morphism $f$ is called a \emph{trivial fibration} if it is both a DK-equivalence and a DK-fibration.
4. The morphism $f$ is called a \emph{cofibration} if it has the left lifting property with respect to the trivial fibrations. \end{definition} It follows from Definition 2.1 that $(a)$ an {\bf M}-functor $f$ is a DK-equivalence if and only if $Ho(f)$ is an equivalence of $Ho({\bf M})$-categories, and $(b)$ an {\bf M}-functor is a trivial fibration if and only if it is surjective on objects and locally a trivial fibration of {\bf M}. In particular, the class of DK-equivalences has the two out of three and weak invertibility properties.
We denote by $\mathcal{I}$ the {\bf M}-category with a single object $\ast$ and $\mathcal{I}(\ast,\ast)=e$. For an object $X$ of {\bf M} we denote by $2_{X}$ the {\bf M}-category with two objects 0 and 1 and with $2_{X}(0,0)=2_{X}(1,1)=e$, $2_{X}(0,1)=X$ and $2_{X}(1,0)=\emptyset$. When {\bf M} is cofibrantly generated, an {\bf M}-functor is a trivial fibration if and only if it has the right lifting property with respect to the saturated class generated by $\{\emptyset\rightarrow \mathcal{I}\}\cup \{2_{X}\overset{2_{i}} \rightarrow 2_{Y}$, $i$ generating cofibration of {\bf M}\}, where $\emptyset$ denotes the initial object of {\bf M}\text{-}{\bf Cat}. We have the following fundamental result of J. Bergner. \begin{theorem} \cite{Be} The category {\bf S}\text{-}{\bf Cat} of simplicial categories admits a cofibrantly generated model structure in which the weak equivalences are the DK-equivalences and the fibrations are the DK-fibrations. A generating set of trivial cofibrations consists of
(B1) $\{2_{X}\overset{2_{j}} \longrightarrow 2_{Y}\}$, where $j$ is a horn inclusion, and
(B2) inclusions $\mathcal{I} \overset{\delta_{y}} \rightarrow \mathcal{H}$, where $\{\mathcal{H}\}$ is a set of representatives for the isomorphism classes of simplicial categories on two objects which have countably many simplices in each function complex. Furthermore, each such $\mathcal{H}$ is required to be cofibrant and weakly contractible in ${\bf S}\text{-}{\bf Cat}(\{x,y\})$. Here $\{x,y\}$ is the set with elements $x$ and $y$ and $\delta_{y}$ omits $y$. \end{theorem} Recall from \cite[Definition 3.3]{SS1} the monoid axiom. The main result of this section is \begin{theorem} Let {\bf M} be a cofibrantly generated monoidal {\bf S}-model category having cofibrant unit and which satisfies the monoid axiom. Suppose furthermore that {\bf M} is locally presentable and that a transfinite composition of weak equivalences of {\bf M} is a weak equivalence.
Then {\bf M}\text{-}{\bf Cat} admits a cofibrantly generated model category structure in which the weak equivalences are the DK-equivalences, the cofibrations are the elements of {\rm cof}$(\{\emptyset\rightarrow \mathcal{I}\}\cup \{2_{X}\overset{2_{i}} \rightarrow 2_{Y}$, $i$ generating cofibration of {\bf M}\}{\rm )}, the fibrant objects are the locally fibrant {\bf M}-categories and the fibrations between fibrant objects are the DK-fibrations.
If the model structure on {\bf M} is right proper, then so is the one on {\bf M}\text{-}{\bf Cat}. \end{theorem} \begin{proof} We shall apply Theorem 0.1 via Proposition 1.3. We take $\mathcal{E}$ to be {\bf M}\text{-}{\bf Cat} and {\rm W} to be the class of DK-equivalences. The fact that {\bf M}\text{-}{\bf Cat} is locally presentable can be seen in a few ways, one is presented in \cite{KL}. The fact that the class of DK-equivalences is accessible follows essentially from the fact that the classes of weak equivalences of {\bf M} and of essentially surjective functors are accessible. We take $I$ to be the set $\{\emptyset\rightarrow \mathcal{I}\}\cup \{2_{X} \overset{2_{i}} \rightarrow 2_{Y}$, $i$ generating cofibration of {\bf M}\}. Let $$F:{\bf S}\rightleftarrows {\bf M}:G$$ be the Quillen pair guaranteed by the definition. $(F,G)$ induces adjoint pairs $$F':{\bf S}\text{-}{\bf Cat} \rightleftarrows {\bf M}\text{-}{\bf Cat}:G'$$ and $$F':{\bf S}\text{-}{\bf Cat}(S) \rightleftarrows {\bf M}\text{-}{\bf Cat}(S):G'$$ for every set $S$. The first $G'$ functor preserves trivial fibrations and the {\bf M}-functors which are locally a fibration. The latter adjoint pair is a Quillen pair. Finally, we take $J$ to be the set $F'(B2)\cup \{2_{X}\overset{2_{i}} \rightarrow 2_{Y}$, $i$ generating trivial cofibration of {\bf M}\}, where $B2$ is as in Theorem 2.2.
\emph{Step 1}. Conditions $c0, c1$ and $nc0$ from Proposition 1.3 were dealt with above.
\emph{Step 2}. Since every map $\delta_{y}$ belonging to the set B2 from Theorem 2.2 has a retraction, one readily checks that an {\bf M}-category is naively fibrant if and only if it is locally fibrant. We claim that if an {\bf M}-functor between locally fibrant {\bf M}-categories is a naive fibration, then it is a DK-fibration. To see this, let first {\bf M}\text{-}{\bf Cat}$_{f}$ be the full subcategory of {\bf M}\text{-}{\bf Cat} consisting of the locally fibrant {\bf M}-categories. By \cite[Proposition 8.5.16]{Hi} we have a natural isomorphism of functors $$\eta:[\_]_{{\bf S}}G'\cong [\_]_{{\bf M}}: {\bf M}\text{-}{\bf Cat}_{f}\rightarrow {\bf Cat}$$ such that for all $\mathcal{A} \in {\bf M}\text{-}{\bf Cat}_{f}$, $\eta_{\mathcal{A}}$ is the identity on objects: indeed, for each pair $x,y\in \mathcal{A}$ of objects we have natural isomorphisms $$[\mathcal{A}]_{{\bf M}}(x,y)\cong Ho({\bf M})(e,\mathcal{A}(x,y))\cong Ho({\bf M})(F1,\mathcal{A}(x,y))\cong Ho({\bf S})(1,G\mathcal{A}(x,y))\cong [G'\mathcal{A}]_{{\bf S}}(x,y)$$ Second, we use the following relaxed version of \cite[Proposition 2.3]{Be}. Let $f$ be a simplicial functor between categories enriched in Kan complexes such that $f$ is locally a Kan fibration. If $f$ has the right lifting property with respect to every element of the set B2, then $f$ is a DK-fibration. (This is the only fact from \cite{Be} that we need.) These facts, together with the observation that the class of isofibrations is invariant in {\bf Cat} under isomorphisms, imply the claim. It is now clear that condition $nc2$ from Proposition 1.3 holds.
\emph{Step 3}. We check condition $nc1$ from Proposition 1.3. Let $j:X\rightarrow Y$ be a trivial cofibration of {\bf M}. We show that for every {\bf M}-category $\mathcal{A}$, in the pushout diagram \[
\xymatrix{ 2_{X} \ar[r]^{2_{j}} \ar[d] & 2_{Y} \ar[d]\\ \mathcal{A} \ar[r] & \mathcal{B}\\ }
\] the map $\mathcal{A}\rightarrow \mathcal{B}$ is a DK-equivalence. Let $S=Ob(\mathcal{A})$. This pushout can be calculated as the pushout \[
\xymatrix{ F_{S}U_{S}\mathcal{A} \ar[r] \ar[d]^{\varepsilon^{S}_{\mathcal{A}}}
& F_{S}\mathcal{X} \ar[d]\\ \mathcal{A} \ar[r] & \mathcal{B}\\ }
\] where $U_{S}\mathcal{A}\rightarrow \mathcal{X}$ is a certain map of {\bf M}-graphs with fixed set of objects $S$. But then the map $\mathcal{A} \rightarrow \mathcal{B}$ is known to be locally a weak equivalence of {\bf M}, see \cite[Proof of Proposition 6.3(1)]{SS2}.
We now claim that if $\delta_{y}:\mathcal{I}\rightarrow \mathcal{H}$ is a map belonging to the set B2 from Theorem 2.2 and $\mathcal{A}$ is any {\bf M}-category, then in the pushout diagram \[ \xymatrix{ F'\mathcal{I} \ar[r]^{a} \ar[d]_{F'\delta_{y}} & \mathcal{A} \ar[d]\\ F'\mathcal{H} \ar[r] & \mathcal{B}\\ }
\] the map $\mathcal{A}\rightarrow \mathcal{B}$ is a DK-equivalence. We factorize the map $\delta_{y}$ as $\mathcal{I} \overset{\delta_{y}'} \longrightarrow \mathcal{H}' \rightarrow \mathcal{H}$, where the simplicial category $\mathcal{H}'$ has $\{x\}$ as set of objects and $\mathcal{H}'(x,x)=\mathcal{H}(x,x)$, and then we take consecutive pushouts: \[ \xymatrix{ F'\mathcal{I} \ar[r]^{a} \ar[d]_{F'\delta_{y}'} & \mathcal{A} \ar[d]^{j}\\ F'\mathcal{H}' \ar[d] \ar[r] & \mathcal{A'} \ar[d]\\ F'\mathcal{H} \ar[r] & \mathcal{B}\\ }
\] The map $j$ can be obtained from the pushout diagram in ${\bf M}\text{-}{\bf Cat}(Ob(\mathcal{A}))$ \[
\xymatrix{ a_{!}F'\mathcal{I} \ar[r] \ar[d]_{a_{!}F'\delta_{y}'} & \mathcal{A} \ar[d]^{j}\\ a_{!}F'\mathcal{H}' \ar[r] & \mathcal{A}'\\ }
\] where $a_{!}:Mon({\bf M})\rightarrow {\bf M}\text{-}{\bf Cat}(Ob(\mathcal{A}))$. By Lemma 2.4 the map $\delta_{y}'$ is a trivial cofibration in the category of simplicial monoids, therefore $F'\delta_{y}'$ is a trivial cofibration in the category of monoids in {\bf M}. Since $a_{!}$ is a left Quillen functor, $j$ is a trivial cofibration in ${\bf M}\text{-}{\bf Cat}(Ob(\mathcal{A}))$.
The map $F'\mathcal{H}'\rightarrow F'\mathcal{H}$ is a full and faithful inclusion, so by Proposition 3.1 the map $\mathcal{A}'\rightarrow \mathcal{B}$ is a full and faithful inclusion. Therefore the map $\mathcal{A}\rightarrow \mathcal{B}$ is locally a weak equivalence of {\bf M}. Applying the functor $[\_]_{{\bf M}}$ to the diagram \[ \xymatrix{ F'\mathcal{I} \ar[r]^{a} \ar[d]_{F'\delta_{y}} & \mathcal{A} \ar[d]\\ F'\mathcal{H} \ar[r] & \mathcal{B}\\ }
\] and taking into account that $F'$ preserves DK-equivalences and that $Ob(\mathcal{B})=Ob(\mathcal{A})\cup \{\ast\}$, it follows that $\mathcal{A}\rightarrow \mathcal{B}$ is a DK-equivalence as well. The claim is proved.
So far we have shown that the pushout of a map from $J$ along any {\bf M}-functor is in ${\rm cof}(I)\cap {\rm W}$. Since a transfinite composition of weak equivalences of {\bf M} is a weak equivalence, we readily obtain that $cell(J)\subset {\rm cof}(I)\cap {\rm W}$. Thus, condition $nc1$ is checked.
Now, putting all the three steps together we obtain the desired model structure on {\bf M}\text{-}{\bf Cat}.
\emph{Step 4}. Suppose that {\bf M} is right proper. Using the explicit construction of pullbacks in {\bf M}\text{-}{\bf Cat}, the description of the fibrations between fibrant objects and \cite[Lemma 9.4]{Bo}, we conclude that the model structure on {\bf M}\text{-}{\bf Cat} is right proper. \end{proof} \begin{lem} Let $\mathcal{A}$ be a cofibrant simplicial category. Then for each $a\in Ob(\mathcal{A})$ the simplicial monoid $a^{\ast}\mathcal{A}=\mathcal{A}(a,a)$ is cofibrant. \end{lem} \begin{proof} Let $S=Ob(\mathcal{A})$. $\mathcal{A}$ is cofibrant in {\bf S}\text{-}{\bf Cat} if and only if it is cofibrant as an object of ${\bf S}\text{-}{\bf Cat}(S)$. The cofibrant objects of ${\bf S}\text{-}{\bf Cat}(S)$ are characterized in \cite[7.6]{DK}: they are the retracts of free simplicial categories. Therefore it suffices to prove that if $\mathcal{A}$ is a free simplicial category then $a^{\ast}\mathcal{A}$ is a free simplicial category for all $a\in S$. There is a full and faithful functor $\varphi:{\bf S}\text{-}{\bf Cat}\rightarrow {\bf Cat}^{\Delta^{op}}$ given by $Ob(\varphi(\mathcal{A})_{n})=Ob(\mathcal{A})$ for all $n\geq 0$ and $\varphi(\mathcal{A})_{n}(a,a')=\mathcal{A}(a,a')_{n}$. Recall \cite[4.5]{DK} that $\mathcal{A}$ is a free simplicial category if and only if ($i$) for all $n\geq 0$ the category $\varphi(\mathcal{A})_{n}$ is a free category on a graph $G_{n}$, and ($ii$) for all epimorphisms $\alpha:[m]\rightarrow [n]$ of $\Delta$, $\alpha^{\ast}:\varphi(\mathcal{A})_{n}\rightarrow \varphi(\mathcal{A})_{m}$ maps $G_{n}$ into $G_{m}$.
Let $a\in S$. The category $\varphi(a^{\ast}\mathcal{A})_{n}$ is a full subcategory of $\varphi(\mathcal{A})_{n}$ with object set $\{a\}$, hence it is free as well. A set $G^{a^{\ast}\mathcal{A}}_{n}$ of generators can be described as follows. An element of $G^{a^{\ast}\mathcal{A}}_{n}$ is a path from $a$ to $a$ in $\varphi(\mathcal{A})_{n}$ such that every arrow in the path belongs to $G_{n}$ and there is at most one arrow in the path with source and target $a$. The fact that $\varphi(a^{\ast}\mathcal{A})_{n}$ is indeed freely generated by $G^{a^{\ast}\mathcal{A}}_{n}$ follows from Lemma 2.5 and its proof. Since every epimorphism $\alpha:[m]\rightarrow [n]$ of $\Delta$ has a section, $\alpha^{\ast}$ maps $G^{a^{\ast}\mathcal{A}}_{n}$ into $G^{a^{\ast}\mathcal{A}}_{m}$. \end{proof} \begin{lem} A full subcategory of a free category is free. \end{lem} \begin{proof} Let $F(G)$ be a free category generated by a graph $G=(G_{1}\rightrightarrows G_{0})$. An arrow $f$ of $F(G)$ is a generator if and only if $f$ is \emph{indecomposable} ($f$ is not a unit and $f=vu$ implies $v$ or $u$ is a unit). Let $C$ be a full subcategory of $F(G)$ with $Ob(C)=C_{0}\subset G_{0}$. If $x,y\in C_{0}$, let us say that a path $(x_{1},f_{1},...,f_{n-1},x_{n})\colon x\to y$ in the graph $G$ is $C_{0}$-\emph{free} if $target(f_{i})\notin C_{0}$ for $1\leq i< n$. Let $G_{1}'$ be the set of $C_{0}$-free paths. It is easy to see that every arrow of $C$ can be uniquely written as a finite composition of $C_{0}$-free paths, so that $C$ is freely generated by the graph $(G_{1}'\rightrightarrows C_{0})$. \end{proof} \begin{remark} The class of cofibrations of the model category constructed in Theorem 2.3 can be given an explicit description \cite[Section 4.2]{St1}. \end{remark} \begin{remark} We noticed during the proof of Theorem 2.3 that our result is almost independent on Theorem 2.2, only a relaxed version of \cite[Proposition 2.3]{Be} being needed. In particular, taking ${\bf M}={\bf S}$ in Theorem 2.3 results in a weaker version of Bergner's result. However, using the fact that {\bf S} has a monoidal fibrant replacement functor that preserves fibrations, the full Theorem 2.2 can be recovered. \end{remark} \begin{remark} One can change the assumptions of Theorem 2.3 and the recognition principle used in its proof to obtain a similar outcome. For example, let {\bf M} be a cofibrantly generated monoidal {\bf S}-model category having cofibrant unit and which satisfies the monoid axiom. Suppose furthermore that
$(a)$ a transfinite composition of weak equivalences of {\bf M} is a weak equivalence,
$(b)$ {\bf M} satisfies the technical condition of \cite[Theorem 2.1]{Ho1}, and
$(c)$ in the Quillen pair $F:{\bf S}\rightleftarrows {\bf M}:G$ guaranteed by the definition, the functor $G$ preserves weak equivalences.
Then \cite[Theorem 11.3.1]{Hi} can be used to show that {\bf M}\text{-}{\bf Cat} admits a cofibrantly generated model category structure in which the weak equivalences are the DK-equivalences and the fibrations are the DK-fibrations. The proof proceeds in th same way as the proof of Theorem 2.3, Step 3 remains unchanged but Step 2 requires suitable modifications. Condition $(b)$ can be relaxed, it was stated in this form in order to include examples such as compactly generated spaces \cite{Ho1}. \end{remark}
\section{Pushouts along full and faithful functors} A result of R. Fritsch and D.M. Latch \cite[Proposition 5.2]{FL} says that the pushout of a full and faithful functor is full and faithful. The purpose of this section is to extend this result to categories enriched over a monoidal category.
Let $(\mathcal{V},\otimes,I)$ be a cocomplete closed category. We denote by $\mathcal{V}$\text{-}{\bf Cat} the category of small $\mathcal{V}$-categories and by $\mathcal{V}$\text{-}{\bf Graph} that of small $\mathcal{V}$-graphs. A $\mathcal{V}$-functor, or a map of $\mathcal{V}$-graphs, that is locally an isomorphism (Section 2.2) is said to be \emph{full and faithful}. If $S$ is a set, we denote by $\mathcal{V}$\text{-}{\bf Cat}$(S)$ (resp. $\mathcal{V}$\text{-}{\bf Graph}$(S)$) the category of small $\mathcal{V}$-categories (resp. $\mathcal{V}$-graphs) with fixed set of objects $S$. The category $\mathcal{V}$\text{-}{\bf Graph}$(S)$ is a monoidal category with monoidal product $\square_{S}$ and unit which we denote by $\mathcal{I}_{S}$. \begin{proposition} Let $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ be three small $\mathcal{V}$-categories and let $i:\mathcal{A}\hookrightarrow \mathcal{B}$ be a full and faithful inclusion. Then in the pushout diagram of $\mathcal{V}$-categories \[ \xymatrix{ \mathcal{A} \ar[r]^{i} \ar[d]_{f} & \mathcal{B} \ar[d]^{g}\\ \mathcal{C} \ar[r]^{i'} & \mathcal{D}\\ }
\] the map $i':\mathcal{C}\rightarrow \mathcal{D}$ is a full and faithful inclusion. \end{proposition} \begin{proof} We shall construct $\mathcal{D}$ explicitly, as was done in the proof of \cite[Proposition 5.2]{FL}. On objects we put $Ob(\mathcal{D})=Ob(\mathcal{C})\sqcup (Ob(\mathcal{B})-Ob(\mathcal{A}))$ and $\mathcal{D}(p,q)=\mathcal{C}(p,q)$ if $p,q\in Ob(\mathcal{C})$. For $p\in Ob(\mathcal{C})$ and $q\in (Ob(\mathcal{B})-Ob(\mathcal{A}))$ we define $$\mathcal{D}(p,q)=\int^{x\in Ob(\mathcal{A})}\mathcal{B}(x,q)\otimes \mathcal{C}(p,f(x))$$ For $p\in (Ob(\mathcal{B})-Ob(\mathcal{A}))$ and $q\in Ob(\mathcal{C})$ we define $$\mathcal{D}(p,q)=\int^{x\in Ob(\mathcal{A})}\mathcal{C}(f(x),q)\otimes \mathcal{B}(p,x)$$ For $p,q\in (Ob(\mathcal{B})-Ob(\mathcal{A}))$ we define $\mathcal{D}(p,q)$ to be the pushout \[ \xymatrix{ \int^{x\in Ob(\mathcal{A})}\mathcal{B}(x,q)\otimes \mathcal{B}(p,x) \ar[r] \ar[d] & \int^{x\in Ob(\mathcal{A})}\int^{y\in Ob(\mathcal{A})}\mathcal{B}(x,q)\otimes \mathcal{C}(f(y),f(x))\otimes \mathcal{B}(p,y) \ar[d]\\ \mathcal{B}(p,q) \ar[r] & \mathcal{D}(p,q)\\ }
\] We shall describe a way to see that, with the above definition, $\mathcal{D}$ is indeed a $\mathcal{V}$-category.
Let $(\mathcal{B}-\mathcal{A})^{+}$ be the preorder with objects all finite subsets $S\subset Ob(\mathcal{B})-Ob(\mathcal{A})$, ordered by inclusion. For $S\in (\mathcal{B}-\mathcal{A})^{+}$, let $\mathcal{A}_{S}$ be the full sub-$\mathcal{V}$-category of $\mathcal{B}$ with objects $Ob(\mathcal{A})\cup S$. Then $\mathcal{B}=\underset{(\mathcal{B}-\mathcal{A})^{+}}\lim \mathcal{A}_{S}$. On the other hand, a filtered colimit of full and faithful inclusions of $\mathcal{V}$-categories is a full and faithful inclusion. This is because the forgetful functor from $\mathcal{V}$\text{-}{\bf Cat} to $\mathcal{V}$\text{-}{\bf Graph} preserves filtered colimits \cite[Corollary 3.4]{KL} and a filtered colimit of full and faithful inclusions of $\mathcal{V}$-graphs is a full and faithful inclusion. Therefore one can assume from the beginning that $Ob(\mathcal{B})=Ob(\mathcal{A})\cup \{q\}$, where $q\not \in Ob(\mathcal{A})$.
\emph{Case 1: $f$ is full and faithful.} In this case the pushout giving $\mathcal{D}(q,q)$ is simply $\mathcal{B}(q,q)$, all the other formulas remain unchanged. Then to show that $\mathcal{D}$ is a $\mathcal{V}$-category is straightforward.
\emph{Case 2: $f$ is the identity on objects.} The map $i$ induces an adjoint pair $$i_{!}:\mathcal{V}\text{-}{\bf Cat}(Ob(\mathcal{A}))\rightleftarrows \mathcal{V}\text{-}{\bf Cat}(Ob(\mathcal{B})):i^{\ast}$$ One has $$i_{!}\mathcal{A}(a,a') =
\begin{cases} \mathcal{A}(a,a'), & \text{if } a,a'\in Ob(\mathcal{A}),\\ \emptyset, & \text{otherwise},\\ I, & \text{if } a=a'=q,\\ \end{cases} $$ and $i$ factors as $\mathcal{A} \rightarrow i_{!}\mathcal{A} \rightarrow \mathcal{B}$, where $i_{!}\mathcal{A} \rightarrow \mathcal{B}$ is the obvious map in $\mathcal{V}\text{-}{\bf Cat}(Ob(\mathcal{B}))$. Then the original pushout can be computed using the pushout diagram \[ \xymatrix{ i_{!}\mathcal{A} \ar[r] \ar[d]_{i_{!}f} & \mathcal{B} \ar[d]\\ i_{!}\mathcal{C} \ar[r] & \mathcal{D}\\ }
\] in $\mathcal{V}\text{-}{\bf Cat}(Ob(\mathcal{B}))$. Next, we claim that $\mathcal{D}$ can be calculated as the pushout, in the category $_\mathcal{B}Mod_{\mathcal{B}}$ of $(\mathcal{B},\mathcal{B})$-bimodules in $$(\mathcal{V}\text{-}{\bf Cat}(Ob(\mathcal{B})), \square_{Ob(\mathcal{B})}, \mathcal{I}_{Ob(\mathcal{B})})$$ of the diagram \[ \xymatrix{ \mathcal{B}\square_{i_{!}\mathcal{A}}\mathcal{B} \ar[rrr]^{ \mathcal{B}\square_{i_{!}\mathcal{A}}i_{!}f\square_{i_{!}\mathcal{A}}\mathcal{B}} \ar[d] & & & \mathcal{B}\square_{i_{!}\mathcal{A}}i_{!}\mathcal{C}\square_{i_{!}\mathcal{A}}\mathcal{B} \ar[d]^{m}\\ \mathcal{B} \ar[rrr] & & & \mathcal{D}\\ }
\] For this we have to show that $\mathcal{D}$ is a monoid in $_\mathcal{B}Mod_{\mathcal{B}}$. We first show that $\mathcal{B}\square_{i_{!}\mathcal{A}}i_{!}\mathcal{C}\square_{i_{!}\mathcal{A}}\mathcal{B}$ is a monoid in $_\mathcal{B}Mod_{\mathcal{B}}$. There is a canonical isomorphism $$i_{!}\mathcal{C}\square_{i_{!}\mathcal{A}}i_{!}\mathcal{C}\cong i_{!}\mathcal{C}\square_{i_{!}\mathcal{A}}\mathcal{B} \square_{i_{!}\mathcal{A}}i_{!}\mathcal{C}$$ of $(i_{!}\mathcal{A},i_{!}\mathcal{A})$-bimodules which is best seen pointwise, using coends. This provides a multiplication for $\mathcal{B}\square_{i_{!}\mathcal{A}}i_{!}\mathcal{C}\square_{i_{!}\mathcal{A}}\mathcal{B}$ which is again best seen to be associative by working pointwise, using coends. To define a multiplication for $\mathcal{D}$ consider the cube diagrams \[
\xymatrix{ \mathcal{B}\cdot i_{!}\mathcal{A}\cdot\mathcal{B}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{A}\cdot\mathcal{B} \ar[rr] \ar[dr] \ar[dd] & & \mathcal{B}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{A}\cdot\mathcal{B} \ar[drr] \ar[dd] \\ & \mathcal{B}\cdot i_{!}\mathcal{C}\cdot\mathcal{B}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{A}\cdot\mathcal{B} \ar[rrr] \ar[dd] & & & \mathcal{D}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{A}\cdot\mathcal{B} \ar[dd]\\ \mathcal{B}\cdot i_{!}\mathcal{A}\cdot\mathcal{B}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{C}\cdot\mathcal{B} \ar[rr] \ar[dr] & & \mathcal{B}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{C}\cdot\mathcal{B} \ar[drr]\\ & \mathcal{B}\cdot i_{!}\mathcal{C}\cdot\mathcal{B}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{C}\cdot\mathcal{B}
\ar[rrr] & & & \mathcal{D}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{C}\cdot\mathcal{B}\\
}
\] and \[
\xymatrix{ \mathcal{B}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{A}\cdot\mathcal{B} \ar[rr] \ar[dr] \ar[dd] & &
\mathcal{B}\cdot_{\mathcal{B}}\mathcal{B} \ar[drr] \ar[dd] \\ & \mathcal{D}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{A}\cdot\mathcal{B}
\ar[rrr] \ar[dd] & & & \mathcal{D}\cdot_{\mathcal{B}}\mathcal{B} \ar[dd]\\
\mathcal{B}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{C}\cdot\mathcal{B}
\ar[rr] \ar[dr] & & \mathcal{B}\cdot_{\mathcal{B}}\mathcal{D} \ar[drr]\\ & \mathcal{D}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{C}\cdot\mathcal{B}
\ar[rrr] & & & \mathcal{D}\cdot_{\mathcal{B}}\mathcal{D}\\
}
\] For space considerations we have suppressed tensors (always over $i_{!}\mathcal{A}$, unless explicitly indicated) from notation. The right face of the first cube is the same as the left face of the latter cube. Let $PO_{1}$ (resp. $PO_{2}$) be the pushout of the left (resp. right) face of the first cube diagram. Let $PO_{3}$ be the pushout of the right face of the second cube diagram. We have pushout digrams \[ \xymatrix{ PO_{1} \ar[r] \ar[d] & PO_{2} \ar[r] \ar[d] & PO_{3} \ar[d]\\ \mathcal{B}\cdot i_{!}\mathcal{C}\cdot \mathcal{B}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{C}\cdot\mathcal{B} \ar[r] & \mathcal{D}\cdot_{\mathcal{B}}\mathcal{B}\cdot i_{!}\mathcal{C}\cdot\mathcal{B} \ar[r] & \mathcal{D}\cdot_{\mathcal{B}}\mathcal{D}
}
\] Using these pushouts and the fact that $\mathcal{B}\square_{i_{!}\mathcal{A}}i_{!} \mathcal{C}\square_{i_{!}\mathcal{A}}\mathcal{B}$ is a monoid one can define in a canonical way a map $\mu:\mathcal{D}\cdot_{\mathcal{B}}\mathcal{D}\rightarrow \mathcal{D}$. We omit the long verification that $\mu$ gives $\mathcal{D}$ the structure of a monoid. The map $\mu$ was constructed in such a way that $m$ becomes a morphism of monoids. The fact that $\mathcal{D}$ has the universal property of the pushout in the category $\mathcal{V}\text{-}{\bf Cat}(Ob(\mathcal{B}))$ follows from its definition.
\emph{Case 3: $f$ is arbitrary.} Let $u=Ob(f)$. We factorize $f$ as $\mathcal{A}\overset{f^{u}} \rightarrow u^{\ast}\mathcal{C} \rightarrow \mathcal{C}$, where $Ob(u^{\ast}\mathcal{C})=Ob(\mathcal{A})$, $u^{\ast}\mathcal{C}(a,a')=\mathcal{C}(fa,fa')$ and $f^{u}$ is the obvious map, and take consecutive pushouts: \[ \xymatrix{ \mathcal{A} \ar[r]^{i} \ar[d]_{f^{u}} & \mathcal{B} \ar[d]\\ u^{*}\mathcal{C} \ar[d] \ar[r] & \mathcal{A'} \ar[d]\\ \mathcal{C} \ar[r] & \mathcal{D}\\ }
\] Now apply Case 2 to $f^{u}$ and Case 1 to $u^{\ast}\mathcal{C}\rightarrow \mathcal{C}$. \end{proof}
\section{Application: left Bousfield localizations of categories of monoids} This section was motivated by the paragraph `As we mentioned above,...in general.' on page 111 of \cite{Ho2}.
\subsection{The problem} Let {\bf M} be a (suitable) monoidal model category, $L${\bf M} a left Bousfield localization of {\bf M} which is itself a monoidal model category and $Mon({\bf M})$ the category of monoids in {\bf M}. The problem is to induce on $Mon({\bf M})$ a model category structure somehow related to $L${\bf M}. As pointed out in \cite{Ho2}, such a model structure exists if, for example, $(a)$ $L${\bf M} satisfies the monoid axiom or $(b)$ $Mon({\bf M})$ has a suitable left proper model category structure. In order for $(a)$ to be fulfilled one needs to know the (generating) trivial cofibrations of $L${\bf M}. However, it often happens that one does not have an explicit description of them. For $(b)$, the category of monoids in a monoidal model category is rarely known to be left proper (it is left proper when the underlying model category has all objects cofibrant, for instance, which seems to us too restrictive to work with).
\subsection{Our solution} We shall propose below a solution to the above problem. We shall reduce the verification of the monoid axiom for $L${\bf M} to a smaller---and hopefully more tractable in practice, set of maps and we shall avoid left properness by using Theorem 0.1 via Proposition 1.3. The model category theoretical framework will be the `combinatorial' counterpart of the one of \cite[Section 8]{Ho2}.
It will be clear that the method could potentially be applied to other structures than monoids.
\subsubsection{Recollections on enriched left Bousfield localization} We recall some facts from \cite{Ba}. Let $\mathcal{V}$ be a monoidal model category and {\bf M} a model $\mathcal{V}$-category with tensor, hom and cotensor denoted by $$-\ast-:\mathcal{V}\times {\bf M}\rightarrow {\bf M}$$ $$Map(-,-):{\bf M}^{op}\times {\bf M}\rightarrow \mathcal{V}$$ $$(-)^{(-)}:\mathcal{V}^{op}\times {\bf M}\rightarrow {\bf M}$$ Let $S$ be a set of maps of {\bf M} between cofibrant objects. \begin{definition} A fibrant object $W$ of {\bf M} is $S$-\emph{local} if for every $f\in S$ the map $Map(f,W)$ is a weak equivalence of $\mathcal{V}$. A map $f$ of {\bf M} is an $S$-\emph{local equivalence} if for every $S$-local object $W$ and for some (hence any) cofibrant approximation $\tilde{f}$ to $f$, the map $Map(\tilde{f},W)$ is a weak equivalence of $\mathcal{V}$. \end{definition} In the previous definition, if the map $Map(\tilde{f},W)$ is a weak equivalence of $\mathcal{V}$, then for any other cofibrant approximation $\tilde{g}$ to $f$, the map $Map(\tilde{g},W)$ is a weak equivalence of $\mathcal{V}$ \cite[Proposition 14.6.6(1)]{Hi}. \begin{theorem} \cite{Ba} Let $\mathcal{V}$ be a combinatorial monoidal model category, {\bf M} a left proper, combinatorial model $\mathcal{V}$-category and $S$ a set of maps of {\bf M} between cofibrant objects. Suppose that $\mathcal{V}$ has a set of generating cofibrations with cofibrant domains.
Then the category {\bf M} admits a left proper, combinatorial model category structure, denoted by $L_{S}{\bf M}$, with the class of $S$-local equivalences as weak equivalences and the same cofibrations as the given ones. The fibrant objects of $L_{S}{\bf M}$ are the $S$-local objects. $L_{S}{\bf M}$ is a model $\mathcal{V}$-category.
Suppose that, moreover, {\bf M} is a monoidal model $\mathcal{V}$-category which has a set of generating cofibrations with cofibrant domains. Let us denote by $\otimes$ the monoidal product on {\bf M}. If $X\otimes f$ is an $S$-local equivalence for every $f\in S$ and every $X$ belonging to the domains and codomains of the generating cofibrations of {\bf M}, then $L_{S}{\bf M}$ is a monoidal model $\mathcal{V}$-category. \end{theorem}
\subsubsection{The $S$-extended monoid axiom} Let $\mathcal{V}$ be a monoidal model category and {\bf M} a monoidal model $\mathcal{V}$-category with monoidal product $\otimes$ and tensor, hom and cotensor denoted as in 4.2.1. If $i:K\rightarrow L$ is a map of $\mathcal{V}$ and $f:A\rightarrow B$ a map of {\bf M}, we denote by $i\ast' f$ the canonical map $$L\ast A\underset{K\ast A}\cup K\ast B\rightarrow L\ast B$$ Let $S$ be a set of maps of {\bf M} between cofibrant objects. For every $f\in S$, let $f=v_{f}u_{f}$ be a factorization of $f$ as a cofibration $u_{f}$ followed by a weak equivalence $v_{f}$; a concrete one is the mapping cylinder factorization. \begin{definition} We say that {\bf M} satisfies the $S$-\emph{extended monoid axiom} if, in the notation of \cite[Section 3]{SS1}, every map in $$(\{{\rm trivial \ cofibrations \ of\ {\bf M}}\} \cup (\{{\rm cofibrations\ of\ \mathcal{V}}\}\ast' u_{f})_{f\in S}) \otimes {\bf M}\text{-}{\rm cof_{reg}}$$ is an $S$-local equivalence. \end{definition} As usual \cite[Lemma 3.5(2)]{SS1}, if $\mathcal{V}$ and {\bf M} are cofibrantly generated and every map in $$(\{{\rm generating \ trivial \ cofibrations\ of\ {\bf M}}\}\cup (\{{\rm generating \ cofibrations \ of\ \mathcal{V}}\}\ast' u_{f})_{f\in S}) \otimes {\bf M}\text{-}{\rm cof_{reg}}$$ is an $S$-local equivalence, then the $S$-extended monoid axiom holds.
Let $Mon({\bf M})$ be the category of monoids in {\bf M} and let $$T:{\bf M} \rightleftarrows Mon({\bf M}):U$$ be the free-forgetful adjunction. \begin{definition} A monoid $M$ in {\bf M} is $TS$-\emph{local} if $U(M)$ is $S$-local. A map $f$ of monoids in {\bf M} is a $TS$-\emph{local equivalence} if $U(f)$ is an $S$-local equivalence. \end{definition} \begin{theorem} Let $\mathcal{V}$ be a combinatorial monoidal model category having a set of generating cofibrations with cofibrant domains. Let {\bf M} be a left proper, combinatorial monoidal model $\mathcal{V}$-category which has a set of generating cofibrations with cofibrant domains. Let us denote by $\otimes$ the monoidal product on {\bf M}. Let $S$ be a set of maps of {\bf M} between cofibrant objects.
Suppose that $X\otimes f$ is an $S$-local equivalence for every $f\in S$ and every $X$ belonging to the domains and codomains of the generating cofibrations of {\bf M} and that {\bf M} satisfies the $S$-extended monoid axiom.
Then the category $Mon({\bf M})$ admits a combinatorial model category structure with $TS$-local equivalences as weak equivalences and with $T(\{{\rm cofibrations\ of \ {\bf M}}\})$ as cofibrations. The fibrant objects are the $TS$-local objects. \end{theorem} \begin{proof} We shall apply Theorem 0.1 via Proposition 1.3. We take $\mathcal{E}$ to be $Mon({\bf M})$, {\rm W} to be the class of $TS$-local equivalences, $I$ to be the set $T(\{{\rm generating \ cofibrations \ of\ {\bf M}}\})$ and $J$ to be $$T(\{{\rm generating \ trivial \ cofibrations \ of \ {\bf M}}\} \cup \{{\rm generating \ cofibrations\ of\ \mathcal{V}}\ast' u_{f}\}_{f\in S})$$ Notice that a map $g$ of monoids in {\bf M} belongs to ${\rm inj}(T(\{{\rm generating \ cofibrations\ of \ {\bf M}}\}))$ if and only if $U(g)$ belongs to ${\rm inj}(\{{\rm generating \ cofibrations\ of \ {\bf M}}\})$ if and only if $U(g)$ is a trivial fibration of {\bf M}. Therefore condition $c1$ from Proposition 1.3 holds.
We claim that a monoid $M$ in {\bf M} is naively fibrant if and only if $M$ is $TS$-local. We may assume without loss of generality that $U(M)$ is fibrant. We observe that if $i$ is any map of $\mathcal{V}$ and $f\in S$, then $M$ has the right lifting property with respect to $T(i\ast' u_{f})$ if and only if $Map(u_{f},U(M))$ has the right lifting property with respect to $i$. Since $Map(v_{f},U(M))$ is a weak equivalence of $\mathcal{V}$ and $Map(u_{f},U(M))$ is a fibration of $\mathcal{V}$, the claim follows from this observation.
Let now $g$ be a map of monoids in {\bf M} between $TS$-local monoids such that $g$ is both a $TS$-local equivalence and a naive fibration. Then $U(g)$ is an $S$-local equivalence between $S$-local objects, so $U(g)$ is a weak equivalence. $U(g)$ is also a fibration, therefore condition $nc2$ from Proposition 1.3 holds.
Condition $nc1$ from Proposition 1.3 is guaranteed by the $S$-extended monoid axiom and \cite[Proof of Lemma 6.2]{SS1}. \end{proof}
{\bf Acknowledgements.} We are deeply indebted to Andr\'{e} Joyal for many useful discussions and suggestions.
\end{document} | arXiv |
\begin{document}
\thanks{The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). Marianna Chatzakou is a postdoctoral fellow of the Research Foundation – Flanders (FWO) under the postdoctoral grant No 12B1223N. Michael Ruzhansky and Aidyn Kassymov are also supported by EPSRC grant EP/R003025/2 and the MESRK grant AP19676031, respectively.\\ \indent {\it Keywords:} log-Sobolev inequality; log-Gagliardo-Nirenberg inequality; log-Caffarelli-Kohn-Nirenberg inequality; Shannon inequality; Nash inequality; homogeneous groups; stratified groups; graded groups; Lie groups}
\begin{abstract} In this note we prove the anisotropic version of the Shannon inequality. This can be conveniently realised in the setting of Folland and Stein's homogeneous groups. We give two proofs: one giving the best constant, and another one using the Kubo-Ogawa-Suguro inequality. \end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
In this paper we derive the logarithmic versions of several well-known functional inequalities. Some inequalities are obtained with best constants, or with semi-explicit constants, the information that is useful for some further applications. Our techniques allow us to derive these inequalities in rather general settings, so we will be working in the settings of general Lie groups, as well as on several classes of nilpotent Lie groups, namely, graded and homogeneous Lie groups. Since on stratified Lie groups we also have the horizontal gradient at our disposal, we will also formulate versions of some of the inequalities in the setting of stratified groups, using the horizontal gradient instead of a power of a sub-Laplacian.
In the Euclidean space, in one of the Sobolev's pioneering works, Sobolev obtained the following inequality, which at this moment is bearing his name: \begin{equation}
\|u\|_{L^{p^{*}}(\mathbb{R}^{n})}\leq C \|\nabla u\|_{L^{p}(\mathbb{R}^{n})}, \end{equation} where $1<p<n$, $p^{*}=\frac{np}{n-p}$ and $C=C(n,p)>0$ is a positive constant. The best constant of this inequality was obtained by Talenti in \cite{T76}. The Sobolev inequality is one of the most important tools in studying PDE and variational problems. Folland and Stein extended Sobolev's inequality to general stratified groups (see e.g. \cite{GV}): if $\mathbb{G}$ is a stratified group and $\Omega\subset \mathbb{G}$ is an open set, then there exists a constant $C>0$ such that we have \begin{equation}\label{Sobolev-Folland-Stein}
\|u\|_{L^{p^{*}}(\Omega)}\leq C\left(\int_{\Omega}|\nabla_{H} u|^{p}dx\right)^{\frac{1}{p}},\; 1<p<Q,\; p^{*}=\frac{Qp}{Q-p}, \end{equation} for all $u\in C_{0}^{\infty}(\Omega)$. Here $\nabla_{H}$ is the horizontal gradient and $Q$ is the homogeneous dimension of $\mathbb{G}$. Inequality \eqref{Sobolev-Folland-Stein} is called the Sobolev or Sobolev-Folland-Stein inequality. Furthermore, in relation to groups, we can mention Sobolev inequalities and embeddings on general unimodular Lie groups \cite{VSCC93}, on general locally compact unimodular groups \cite{AR20}, on general noncompact Lie groups \cite{BPTV19,BPV21}, as well as Hardy-Sobolev inequalities on general Lie groups \cite{RY19}. The Sobolev inequality on graded groups using Rockland operators was proved in \cite{FR17} and the best constant for it was obtained in \cite{RTY20}.
On the other hand, the logarithmic Sobolev inequality was shown to hold on $\mathbb{R}^n$ in the following form: \begin{equation}\label{ddlogin}
\int_{\mathbb{R}^{n}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{R}^{n})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{R}^{n})}}\right)dx\leq\frac{n}{p}\log\left(C\frac{\|\nabla u\|^{p}_{L^{p}(\mathbb{R}^{n})}}{\|u\|^{p}_{L^{p}(\mathbb{R}^{n})}}\right). \end{equation} We can refer to \cite{Wei78} for the case $p=2$, but to e.g. \cite{DD03} for some history review of cases $1\leq p<\infty$, including the discussion of best constants.
In \cite{Mer08}, the author obtained a logarithmic Gagliardo-Nirenberg inequality. In \cite {FNQ18} and \cite{KRS20} the authors proved the logarithmic Sobolev inequality and the fractional logarithmic Sobolev inequality on the Heisenberg group and on homogeneous groups, respectively. A fractional weighted version of \eqref{ddlogin} on homogeneous groups was proved in \cite{KS20}. In this paper, we prove logarithmic Sobolev inequalities on graded groups and weighted logarithmic Sobolev inequalities on general Lie groups. As applications of these inequalities we show Nash and weighted Nash inequalities on graded and general Lie groups, respectively. The log-Sobolev type inequalities with weights are also sometimes called the log-Hardy inequalities \cite{DDFT10}.
In this paper we establish Shannon's inequality on general homogeneous groups, and we can refer to its links to Shannon's entropy \cite{AO73, Isi72, Kap87} and information theory \cite{Sha48, Khi57, MPP00}.
After Shannon's seminal paper \cite{Sha48} in 1948, several versions of Shannon's inequality have appeared either in discrete, cf. \cite{Khi57,AB00,Wei78,AO73}, or in integral form, cf. \cite{Isi72,Kap87,KOS19}, on certain metric spaces. The underlying motivation is the study of inequalities concerning the entropy function, and, as such, can be regarded as the mathematical foundation of information theory; we refer to \cite{MPP00,MF93} for an overview of the topic. Characterisations of the entropy appear, in the integral form, as the gain of information with functional inequalities. The latter, in the case of a homogeneous Lie group $\mathbb{G}$, with homogeneous dimension $Q$, where $|\cdot|$ an arbitrary homogeneous quasi-norm, and $\alpha\in(1,\infty)$, reads as follows: For all $u\not=0$ we have \begin{equation}\label{EQ:Shannon}
\int_{\mathbb{G}}\frac{|u(x)|}{\|u\|_{L^{1}(\mathbb{G})}}\log\left(\frac{|u(x)|}{\|u\|_{L^{1}(\mathbb{G})}}\right)^{-1}dx\leq \frac{Q}{\alpha}\log\left(\frac{\alpha e A_{Q,\alpha}}{Q}\frac{\||\cdot|^{\alpha}u\|_{L^{1}(\mathbb{G})}}{\|u\|_{L^{1}(\mathbb{G})}}\right), \end{equation}
with an explicit value for $ A_{Q,\alpha}$ (see \eqref{EQ:Aexp1}) that is best possible. Shannon's inequality gives sufficient conditions under which the generalised entropy function, particularly in our case the left-hand side of \eqref{EQ:Shannon}, converges. Shannon's inequality can be viewed, in some sense, as the counter part of the log-Sobolev inequality as it arises as the limiting case of \eqref{ddlogin} for $p=1$, where, however, instead of the regularity of $u$ it is assumed that $|\cdot|^{\alpha}u$ is in $L^1(\mathbb{G}).$
\section{Preliminaries} In this section, we briefly recall definitions and main properties of the homogeneous groups. The comprehensive analysis on such groups has been initiated in the works of Folland and Stein \cite{FS}, but in our exposition below we follow a more recent presentation in the open access book \cite{FR16}.
\begin{defn}[\cite{FS, FR16}, Homogeneous group] A Lie group (on $\mathbb{R}^{N}$) $\mathbb{G}$ with the dilation $$D_{\lambda}(x):=(\lambda^{\nu_{1}}x_{1},\ldots,\lambda^{\nu_{N}}x_{N}),\; \nu_{1},\ldots, \nu_{n}>0,\; D_{\lambda}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N},$$ which is an automorphism of the group $\mathbb{G}$ for each $\lambda>0,$ is called a {\em homogeneous (Lie) group}. \end{defn} For simplicity, in this paper we use the notation $\lambda x$ for the dilation $D_{\lambda}(x)$. We denote \begin{equation} Q:=\nu_{1}+\ldots+\nu_{N}, \end{equation} the homogeneous dimension of a homogeneous group $\mathbb{G}$.
Let $dx$ denote the Haar measure on $\mathbb{G}$ and let $|S|$ denote the corresponding volume of a measurable set $S\subset \mathbb{G}$. Then we have \begin{equation}\label{scal}
|D_{\lambda}(S)|=\lambda^{Q}|S| \quad {\rm and}\quad \int_{\mathbb{G}}f(\lambda x) dx=\lambda^{-Q}\int_{\mathbb{G}}f(x)dx. \end{equation} We also note that from \cite[Proposition 1.6.6]{FR16}, the standard Lebesgue measure $dx$ on $\mathbb{R}^{N}$ is the Haar measure on $\mathbb{G}$. Then we have the following widely used property in this paper, see e.g. \cite[p. 19]{RS19}: Let $\mathbb{G}$ be a homogeneous Lie group with homogeneous dimension $Q$, $r>0$, and let $dx$ be a Haar measure. Then, we have $$d(rx)=r^{Q}dx.$$ \begin{defn}[{\cite[Definition 3.1.33]{FR16} or \cite[Definition 1.2.1]{RS19}}]\label{quasi-norm} For any homogeneous group $\mathbb{G}$ there exist homogeneous quasi-norms, which are continuous non-negative functions \begin{equation}
\mathbb{G}\ni x\mapsto |x|\in[0,\infty), \end{equation} with the properties
\begin{itemize}
\item[a)] $|x|=|x^{-1}|$ for all $x\in\mathbb{G}$,
\item[b)] $|\lambda x|=\lambda|x|$ for all $x\in \mathbb{G}$ and $\lambda>0$,
\item[c)] $|x|=0$ if and only if $x=0$. \end{itemize} \end{defn} Moreover, the following polarisation formula on homogeneous Lie groups will be used in our proofs, as established by Folland and Stein \cite{FS}. \begin{prop}[e.g. {\cite[Proposition 3.1.42]{FR16}}]
Let $\mathbb{G}$ be a homogeneous Lie group and $\mathfrak{S}:=\{x\in \mathbb{G}:\,|x|=1\},$ be the unit sphere with respect to the homogeneous quasi-norm $|\cdot|.$ Then there is a unique Radon measure $\sigma$ on $\mathfrak{S}$ such that for all $f\in L^{1}(\mathbb{G}),$ we have \begin{equation}\label{EQ:polar} \int_{\mathbb{G}}f(x)dx=\int_{0}^{\infty} \int_{\mathfrak{S}}f(ry)r^{Q-1}d\sigma(y)dr. \end{equation} \end{prop}
\section{Shannon inequality on homogeneous groups}
In this section we show the Shannon inequality on homogeneous Lie groups. Let us introduce the weighted Lebesgue space $$L^{p,\alpha}(\mathbb{G}):=\{u:u\in L^{p}_{loc}(\mathbb{G}),\,\,\langle x\rangle^{\alpha} u\in L^{p}(\mathbb{G})\},$$ where $\alpha>0$ and
$$\langle x \rangle:=(1+|x|^{2})^{\frac{1}{2}},\,\,\,\,\,\text{for}\,\,\,x\in \mathbb{G},$$
with $|\cdot|$ a homogeneous quasi-norm on $\mathbb{G}.$ Firstly, let us show Shannon inequality. \begin{thm}[Shannon inequality]\label{shthm1}
Let $\mathbb{G}$ be a homogeneous Lie group with homogeneous dimension $Q$ and let $|\cdot|$ be a homogeneous quasi-norm on $\mathbb{G}$. Suppose that $\alpha\in(0,\infty)$ and $u\in L^{1,\alpha}(\mathbb{G})\setminus \{0\}$. Then we have \begin{equation}\label{sh11}
\int_{\mathbb{G}}\frac{|u(x)|}{\|u\|_{L^{1}(\mathbb{G})}}\log\left(\frac{|u(x)|}{\|u\|_{L^{1}(\mathbb{G})}}\right)^{-1}dx\leq \frac{Q}{\alpha}\log\left(\frac{\alpha e A_{Q,\alpha}}{Q}\frac{\||\cdot|^{\alpha}u\|_{L^{1}(\mathbb{G})}}{\|u\|_{L^{1}(\mathbb{G})}}\right), \end{equation} where \begin{equation}\label{EQ:Aexp1}
A^{\frac{Q}{\alpha}}_{Q,\alpha}=\frac{|\mathfrak{S}|\Gamma\left({\frac{Q}{\alpha}}\right)}{\alpha}, \end{equation}
with $|\mathfrak{S}|$ the $Q-1$ dimensional surface measure of the unit quasi-sphere with respect to $|\cdot|$. Moreover, $A_{Q,\alpha}$ is the best possible constant. This constant is attained with $E_{\alpha}(x)=\exp(-A_{Q,\alpha}|x|^{\alpha})$. \end{thm} \begin{proof}
Without loss of generality, it is enough to prove inequality \eqref{sh11} for $\|u\|_{L^{1}(\mathbb{G})}=1,$ it means, it is enough to prove \begin{equation}\label{shwith1}
\int_{\mathbb{G}}|u(x)|\log\left(|u(x)|\right)^{-1}dx\leq \frac{Q}{\alpha}\log\left(\frac{\alpha e A_{Q,\alpha}}{Q}\||\cdot|^{\alpha}u\|_{L^{1}(\mathbb{G})}\right). \end{equation}
Let us denote $d\mu=|u(x)|dx$, then we have $\int_{\mathbb{G}}d\mu=1$ is a probability measure. First, let us compute the following integral using the change $r^{\alpha}=z$: \begin{equation}
\begin{split}
A^{\frac{Q}{\alpha}}_{Q,\alpha}&=\int_{\mathbb{G}}e^{-|x|^{\alpha}}dx\\&
\stackrel{\eqref{EQ:polar}}=\int_{0}^{\infty}\int_{\mathfrak{S}}e^{-r^{\alpha}}r^{Q-1}d\sigma(y)dr\\&
=|\mathfrak{S}|\int_{0}^{\infty}e^{-r^{\alpha}}r^{Q-1}dr\\&
=\frac{|\mathfrak{S}|}{\alpha}\int_{0}^{\infty}e^{-z}z^{\frac{Q}{\alpha}-1}dz\\&
=\frac{|\mathfrak{S}|\Gamma\left(\frac{Q}{\alpha}\right)}{\alpha}.
\end{split} \end{equation}
By using Jensen's inequality with polarisation and changing variables $A_{Q,\alpha}r^{\alpha}=z$, with $E_{\alpha}(x)=\exp(-A_{Q,\alpha}|x|^{\alpha})$, we compute \begin{equation}\label{sh321}
\begin{split}
\exp\left(\int_{\mathbb{G}}|u(x)|\log\left(\frac{|u(x)|}{E_{\alpha}(x)}\right)^{-1}dx\right)&=\exp\left(\int_{\mathbb{G}}\log\left(\frac{|u(x)|}{E_{\alpha}(x)}\right)^{-1}d\mu\right)\\&
\leq \int_{\mathbb{G}}\left(\frac{|u(x)|}{E_{\alpha}(x)}\right)^{-1}d\mu\\&
=\int_{\mathbb{G}}e^{-A_{Q,\alpha}|x|^{\alpha}}dx\\&
=|\mathfrak{S}|\int_{0}^{\infty}e^{-A_{Q,\alpha}r^{\alpha}}r^{Q-1}dr\\&
=\frac{|\mathfrak{S}| A^{-\frac{Q}{\alpha}}}{\alpha}\int_{0}^{\infty}e^{-z}z^{\frac{Q}{\alpha}-1}dz\\&
=\frac{|\mathfrak{S}|\Gamma\left(\frac{Q}{\alpha}\right)A^{-\frac{Q}{\alpha}}}{\alpha}\\&
=1,
\end{split} \end{equation} then, we obtain \begin{equation}\label{shin1} \begin{split}
\int_{\mathbb{G}}|u(x)|\log\left(|u(x)|\right)^{-1}dx\leq \int_{\mathbb{G}}|u(x)|\log\left(E_{\alpha}(x)\right)^{-1}dx=A_{Q,\alpha}\int_{\mathbb{G}}|x|^{\alpha}|u(x)|dx. \end{split} \end{equation} For $\lambda>0$, let us denote by $u_{\lambda}\in L^{1}(\mathbb{G})$ the function $u_{\lambda}(x)=\lambda^{Q}u(\lambda x)$. Putting $u_{\lambda}$ in \eqref{shin1} instead of $u$, we have \begin{equation}\label{shin12345} \begin{split}
\int_{\mathbb{G}}|u_{\lambda}(x)|\log\left(|u_{\lambda}(x)|\right)^{-1}dx\leq A_{Q,\alpha}\int_{\mathbb{G}}|x|^{\alpha}|u_{\lambda}(x)|dx, \end{split} \end{equation} and multiplying both sides by $\frac{\alpha}{Q}$, we have \begin{equation}\label{shin1234} \begin{split}
\frac{\alpha}{Q}\int_{\mathbb{G}}|u_{\lambda}(x)|\log\left(|u_{\lambda}(x)|\right)^{-1}dx\leq \frac{\alpha A_{Q,\alpha}}{Q}\int_{\mathbb{G}}|x|^{\alpha}|u_{\lambda}(x)|dx. \end{split} \end{equation} Then let us compute left hand side of \eqref{shin1234}, \begin{equation}\label{shinpr1}
\begin{split}
\frac{\alpha}{Q}\int_{\mathbb{G}}|u_{\lambda}(x)|\log(|u_{\lambda}(x)|)^{-1}dx&=\frac{\alpha}{Q}\int_{\mathbb{G}}\lambda^{Q}|u(\lambda x)|\log(\lambda^{Q}|u(\lambda x)|)^{-1}dx\\&
\stackrel{\eqref{scal}}=\frac{\alpha}{Q}\int_{\mathbb{G}}|u(x)|\log(\lambda^{Q}|u(x)|)^{-1}dx\\&
=\frac{\alpha}{Q}\int_{\mathbb{G}}|u(x)|\log(|u(x)|)^{-1}dx-\log \lambda^{\alpha},
\end{split} \end{equation} and the right hand side of \eqref{shin1234}, \begin{equation}\label{shinpr2}
\begin{split}
\frac{\alpha A_{Q,\alpha}}{Q} \int_{\mathbb{G}}|x|^{\alpha}|u_{\lambda}(x)|dx&
=\frac{\alpha A_{Q,\alpha}}{Q} \int_{\mathbb{G}}|x|^{\alpha}\lambda^{Q}|u(\lambda x)|dx\\&
=\frac{\alpha A_{Q,\alpha}}{Q} \int_{\mathbb{G}}\frac{\lambda^{\alpha}}{\lambda^{\alpha}}|x|^{\alpha}\lambda^{Q}|u(\lambda x)|dx\\&
=\frac{\alpha A_{Q,\alpha}}{Q} \int_{\mathbb{G}}\frac{1}{\lambda^{\alpha}}|\lambda x|^{\alpha}\lambda^{Q}|u(\lambda x)|dx\\&
\stackrel{\eqref{scal}}=\frac{\alpha A_{Q,\alpha}\lambda^{-\alpha}}{Q}\int_{\mathbb{G}}|x|^{\alpha}|u(x)|dx.
\end{split} \end{equation} Putting the last two facts in \eqref{shin1234}, we get \begin{equation}
\frac{\alpha}{Q}\int_{\mathbb{G}}u(x)\log(|u(x)|)^{-1}dx\leq \log\lambda^{\alpha}+\frac{\alpha}{Q}\lambda^{-\alpha}A_{Q,\alpha}\int_{\mathbb{G}}|x|^{\alpha}|u(x)|dx.
\end{equation}
Then by taking $\lambda^{\alpha}=\frac{\alpha A_{Q,\alpha}}{Q}\int_{\mathbb{G}}|x|^{\alpha}|u(x)|dx$ in the last fact, we have
\begin{equation}
\frac{\alpha}{Q}\int_{\mathbb{G}}|u(x)|\log(|u(x)|)^{-1}dx\leq\log\left(\frac{e\alpha A_{Q,\alpha}}{Q}\int_{\mathbb{G}}|x|^{\alpha}|u(x)|dx\right).
\end{equation}
Let us prove the best possible constant in \eqref{sh11}. It is enough to show that the function $E_{\alpha}(x)$ gives equality in \eqref{shin1}, which means that we have \begin{equation}\label{shpr3}
\begin{split}
\frac{\alpha}{Q}\int_{\mathbb{G}}E_{\alpha}(x)\log(E_{\alpha}(x))^{-1}dx&=\frac{\alpha}{Q}\int_{\mathbb{G}}E_{\alpha}(x)\log(\exp(A_{Q,\alpha})|x|^{\alpha})dx\\&
=\frac{\alpha A_{Q,\alpha}}{Q}\int_{\mathbb{G}}|x|^{\alpha}E_{\alpha}(x)dx.
\end{split} \end{equation}
By taking $E_{\alpha,\lambda}(x)=\lambda^{Q}e^{-A_{Q,\alpha}|\lambda x|^{b}}$ with $\lambda^{b}=\frac{\alpha A_{Q,\alpha}}{Q}\int_{\mathbb{G}}|x|^{\alpha}E_{\alpha}(x)dx$ in \eqref{shpr3}, and repeating same calculation as \eqref{shinpr1} and \eqref{shinpr2}, we get equality in \eqref{shwith1}. \end{proof} Let us now show another proof of the Shannon inequality. Firstly, we show the Kubo-Ogawa-Suguro inequality and as an application, we derive the Shannon inequality.
\begin{thm}[Kubo-Ogawa-Suguro inequality]
Let $\mathbb{G}$ be a homogeneous Lie group with homogeneous dimension $Q$ and a homogeneous quasi-norm $|\cdot|$ on $\mathbb{G}$. Let $\alpha\in(1,\infty)$ and $u\in L^{1,\alpha}(\mathbb{G})\setminus \{0\}$. Then we have \begin{equation}\label{KOSin}
- \int_{\mathbb{G}}|u(x)|\log\frac{|u(x)|}{\|u\|_{L^{1}(\mathbb{G})}}dx\leq Q\int_{\mathbb{G}}|u(x)|\log \left(C_{Q,\alpha}(1+|x|^{\alpha})\right)dx, \end{equation} where \begin{equation}
C_{Q,\alpha}=\left(\frac{|\mathfrak{S}|\Gamma\left(\frac{Q}{\alpha}\right)\Gamma\left(\frac{Q}{\alpha'}\right)}{\alpha\Gamma\left(Q\right)}\right)^{\frac{1}{Q}}, \end{equation}
is the best constant with $\frac{1}{\alpha}+\frac{1}{\alpha'}=1$ and $|\mathfrak{S}|$ is the $Q-1$ dimensional surface measure of the unit quasi-sphere with respect to $|\cdot|$. \end{thm} \begin{proof}
Without loss of generality, assume that $\|u\|_{L^{1}(\mathbb{G})}=1$. Then, by denoting $d\mu=|u(x)|dx$, we have $\int_{\mathbb{G}}d\mu=1$.
Let us denote by $\varphi(x)=c_{Q,\alpha}(1+|x|^{\alpha})^{-Q}$, where $c_{Q,\alpha}=\frac{\alpha\Gamma\left(Q\right)}{|\mathfrak{S}|\Gamma\left(\frac{Q}{\alpha}\right)\Gamma\left(\frac{Q}{\alpha'}\right)}$ and let us prove that $\|\varphi\|_{L^{1}(\mathbb{G})}=1$. By using the polar decomposition with the change of variables $(1+r^{\alpha})^{-Q}=t^{Q}$, we compute \begin{equation}
\begin{split}
\int_{\mathbb{G}}(1+|x|^{\alpha})^{-Q}dx&=\int_{0}^{\infty}\int_{\mathfrak{S}}(1+r^{\alpha})^{-Q}r^{Q-1}drd\sigma(y)\\&
=|\mathfrak{S}|\int_{0}^{\infty}(1+r^{\alpha})^{-Q}r^{Q-1}dr\\&
=\frac{|\mathfrak{S}|}{\alpha}\int_{0}^{1}(1-t)^{\frac{Q}{\alpha}-1}t^{\frac{Q}{\alpha'}-1}dt\\&
=\frac{|\mathfrak{S}|}{\alpha}\text{B}\left(\frac{Q}{\alpha},\frac{Q}{\alpha'}\right)\\&
=\frac{|\mathfrak{S}|\Gamma\left(\frac{Q}{\alpha}\right)\Gamma\left(\frac{Q}{\alpha'}\right)}{\alpha\Gamma\left(Q\right)},
\end{split} \end{equation}
where $\text{B}(\cdot,\cdot)$ is the Beta function. Then $\|\varphi\|_{L^{1}(\mathbb{G})}=1$. By using this last fact with Jensen's inequality, we get \begin{equation} \begin{split}
\int_{\mathbb{G}}|u(x)|\log\left(\frac{\varphi(x)}{|u(x)|}\right)dx&\leq \log\left(\int_{\mathbb{G}}\frac{\varphi(x)}{|u(x)|}d\mu\right)\\&
=\log\left(\int_{\mathbb{G}}\varphi(x)dx\right)\\&
=0. \end{split} \end{equation} It means that we have \begin{equation} \begin{split}
-\int_{\mathbb{G}}|u(x)|\log|u(x)|dx&\leq -\int_{\mathbb{G}}|u(x)|\log\varphi(x)dx\\&
=-\int_{\mathbb{G}}|u(x)|\log c_{Q,\alpha}(1+|x|^{\alpha})^{-Q}dx\\&
=Q\int_{\mathbb{G}}|u(x)|\log c^{-\frac{1}{Q}}_{Q,\alpha}(1+|x|^{\alpha})dx\\&
=Q\int_{\mathbb{G}}|u(x)|\log C_{Q,\alpha}(1+|x|^{\alpha})dx. \end{split} \end{equation} Also, in the last inequality, equality holds, if and only if \begin{equation}
u(x)=c_{Q,\alpha}(1+|x|^{\alpha})^{-Q}. \end{equation} By using Jensen's inequality, we get \begin{equation}\label{sh1}
\begin{split}
\int_{\mathbb{G}}|u(x)|\log(1+|x|^{\alpha})dx&\leq \log\left(\int_{\mathbb{G}}(1+|x|^{\alpha})d\mu\right)\\&
=\log\left(\int_{\mathbb{G}}|u(x)|(1+|x|^{\alpha})dx\right)\\&
\leq C\log\left(\int_{\mathbb{G}}\langle x\rangle^{\alpha}|u(x)|dx\right).
\end{split} \end{equation} By using \eqref{sh1}, we have that \begin{equation} \begin{split}
-\int_{\mathbb{G}}|u(x)|\log|u(x)|dx&\leq Q\int_{\mathbb{G}}|u(x)|\log C_{Q,\alpha}(1+|x|^{\alpha})dx\\&
=Q\int_{\mathbb{G}}|u(x)|\log (1+|x|^{\alpha})dx+Q\int_{\mathbb{G}}|u(x)|\log C_{Q,\alpha}dx\\&
\stackrel{\eqref{sh1}}\leq C\log\left(\int_{\mathbb{G}}\langle x\rangle^{\alpha}|u(x)|dx\right)+Q\int_{\mathbb{G}}|u(x)|\log C_{Q,\alpha}dx\\&
<\infty, \end{split} \end{equation} also implying \eqref{KOSin}. \end{proof} Let us show that Kubo-Ogawa-Suguro inequality also implies Shannon's inequality. \begin{cor}[Shannon inequality]\label{shthm2}
Let $\mathbb{G}$ be a homogeneous Lie group with homogeneous dimension $Q$ and a homogeneous quasi-norm $|\cdot|$ on $\mathbb{G}$. Let $\alpha\in(1,\infty)$ and $u\in L^{1,\alpha}(\mathbb{G})\setminus \{0\}$. Then we have \begin{equation}\label{shkos}
\int_{\mathbb{G}}\frac{|u(x)|}{\|u\|_{L^{1}(\mathbb{G})}}\log\left(\frac{|u(x)|}{\|u\|_{L^{1}(\mathbb{G})}}\right)^{-1}dx\leq \frac{Q}{\alpha}\log\left(\frac{B_{Q,\alpha}}{\|u\|_{L^{1}(\mathbb{G})}}\||\cdot|^{\alpha}u\|_{L^{1}(\mathbb{G})}\right), \end{equation} where \begin{equation}\label{EQ:Aexp}
B_{Q,\alpha}=\alpha^{\alpha}(\alpha-1)^{1-\alpha}\left(\frac{|\mathfrak{S}|\Gamma\left(\frac{Q}{\alpha}\right)\Gamma\left(\frac{Q}{\alpha'}\right)}{\alpha\Gamma\left(Q\right)}\right)^{\frac{\alpha}{Q}}, \end{equation}
with $\frac{1}{\alpha}+\frac{1}{\alpha'}=1$ and $|\mathfrak{S}|$ is the $Q-1$ dimensional surface measure of the unit quasi-sphere with respect to $|\cdot|$. \end{cor} \begin{proof}
Similarly to the previous theorem, without loss generality, we can assume that $\|u\|_{L^{1}(\mathbb{G})}=1$ for $u\in L^{1}(\mathbb{G})$. Let us denote $d\mu=|u(x)|dx,$ then we have $\int_{\mathbb{G}}d\mu=1$ is a probability measure. By combining \eqref{KOSin} and Jensen's inequality, we get \begin{equation}\label{sh3}
\begin{split}
- \int_{\mathbb{G}}|u(x)|\log\frac{|u(x)|}{\|u\|_{L^{1}(\mathbb{G})}}dx&\leq Q\int_{\mathbb{G}}|u(x)|\log \left(C_{Q,\alpha}(1+|x|^{\alpha})\right)dx\\&
= Q\int_{\mathbb{G}}\log \left(C_{Q,\alpha}(1+|x|^{\alpha})\right)d\mu\\&
\leq Q\log\left(\int_{\mathbb{G}} C_{Q,\alpha}(1+|x|^{\alpha})d\mu\right)\\&
=Q\log\left(\int_{\mathbb{G}} C_{Q,\alpha}|u(x)|(1+|x|^{\alpha})dx\right),
\end{split} \end{equation}
where $C_{Q,\alpha}=\left(\frac{|\mathfrak{S}|\Gamma\left(\frac{Q}{\alpha}\right)\Gamma\left(\frac{Q}{\alpha'}\right)}{\alpha\Gamma\left(Q\right)}\right)^{\frac{1}{Q}}.$
For $\lambda>0$, let us denote by $u_{\lambda}\in L^{1}(\mathbb{G})$ the function $u_{\lambda}(x)=\lambda^{Q}u(\lambda x)$. Then we have \begin{equation*}\label{sh4}
\begin{split}
-\int_{\mathbb{G}}|u_{\lambda}(x)|\log|u_{\lambda}(x)|dx&=-\int_{\mathbb{G}}\lambda^{Q}|u(\lambda x)|\log(\lambda^{Q}|u(\lambda x)|)dx\\&
=-\int_{\mathbb{G}}\lambda^{Q}|u(\lambda x)|\log\lambda^{Q}dx-\int_{\mathbb{G}}\lambda^{Q}|u(\lambda x)|\log|u(\lambda x)|dx\\&
= -\log\lambda^{Q}\int_{\mathbb{G}}\lambda^{Q}|u(\lambda x)|dx-\int_{\mathbb{G}}\lambda^{Q}|u(\lambda x)|\log|u(\lambda x)|dx\\&
=-\log\lambda^{Q}\int_{\mathbb{G}}|u(\lambda x)|d(\lambda x)-\int_{\mathbb{G}}|u(\lambda x)|\log|u(\lambda x)|d(\lambda x)\\&
=-Q\log\lambda-\int_{\mathbb{G}}|u(x)|\log|u(x)|dx,
\end{split} \end{equation*} and \begin{equation*}\label{sh5}
\begin{split}
Q\log \left(C_{Q,\alpha}\int_{\mathbb{G}}(1+|x|^{\alpha})|u_{\lambda}(x)|dx\right)&= Q\log \left(C_{Q,\alpha}\int_{\mathbb{G}}\lambda^{Q}(1+|x|^{\alpha})|u(\lambda x)|dx\right)\\&
=Q\log C_{Q,\alpha}+Q\log \left(\int_{\mathbb{G}}\lambda^{Q}(1+|x|^{\alpha})|u(\lambda x)|dx\right)\\&
=Q\log C_{Q,\alpha}+Q\log \left(\int_{\mathbb{G}}\lambda^{Q}(1+\frac{\lambda^{\alpha} }{\lambda^{\alpha}}|x|^{\alpha})|u(\lambda x)|dx\right)\\&
=Q\log C_{Q,\alpha}+Q\log \left(\int_{\mathbb{G}}(1+\lambda^{-\alpha} |\lambda x|^{\alpha})|u(\lambda x)|d(\lambda x)\right)\\&
=Q\log C_{Q,\alpha}+Q\log \left(\int_{\mathbb{G}}(1+\lambda^{-\alpha} |x|^{\alpha})|u(x)|dx\right)\\&
=Q\log C_{Q,\alpha}+Q\log \left(1+\lambda^{-\alpha} \| |\cdot|^{\alpha}u\|_{L^{1}(\mathbb{G})}\right).
\end{split} \end{equation*} Using these two facts in \eqref{sh3}, we get \begin{equation}
-\int_{\mathbb{G}}|u(x)|\log|u(x)|dx\leq Q\log C_{Q,\alpha}+Q\log \left(\lambda+\lambda^{1-\alpha} \| |\cdot|^{\alpha}u\|_{L^{1}(\mathbb{G})}\right). \end{equation}
By choosing $\lambda=(\alpha-1)^{\frac{1}{\alpha}}\||\cdot|^{\alpha}u\|^{\frac{1}{\alpha}}_{L^{1}(\mathbb{G})}$, we get \begin{equation*} \begin{split}
Q\log \left(\lambda+\lambda^{1-\alpha} \| |\cdot|^{\alpha}u\|_{L^{1}(\mathbb{G})}\right)&=Q\log\left((\alpha-1)^{\frac{1}{\alpha}}\||\cdot|^{\alpha}u\|^{\frac{1}{\alpha}}_{L^{1}(\mathbb{G})}+(\alpha-1)^{\frac{1-\alpha}{\alpha}}\||\cdot|^{\alpha}u\|^{\frac{1}{\alpha}}_{L^{1}(\mathbb{G})}\right)\\&
=Q\log\left(\alpha(\alpha-1)^{\frac{1}{\alpha}-1}\||\cdot|^{\alpha}u\|^{\frac{1}{\alpha}}_{L^{1}(\mathbb{G})}\right)\\&
=\frac{Q}{\alpha}\log\left(\alpha^{\alpha}(\alpha-1)^{1-\alpha}\||\cdot|^{\alpha}u\|_{L^{1}(\mathbb{G})}\right). \end{split} \end{equation*} Finally, we get \begin{equation} \begin{split}
-\int_{\mathbb{G}}|u(x)|\log|u(x)|dx&\leq \frac{Q}{\alpha}\log \left(C^{\alpha}_{Q,\alpha}\alpha^{\alpha}(\alpha-1)^{1-\alpha}\||\cdot|^{\alpha}u\|_{L^{1}(\mathbb{G})}\right)\\&
=\frac{Q}{\alpha}\log \left(B_{Q,\alpha}\||\cdot|^{\alpha}u\|_{L^{1}(\mathbb{G})}\right), \end{split} \end{equation} implying \eqref{shkos}. \end{proof}
\begin{rem}
For large $Q\gg 1$, we have that the constant $B_{Q,\alpha}$ in \eqref{shkos} coincides with the best constant $\frac{\alpha A_{Q,\alpha}}{Q}$ in \eqref{sh11}, that is, $$B_{Q,\alpha}\simeq \frac{\alpha e A_{Q,\alpha}}{Q},\,\,\,\,Q\gg 1.$$ \end{rem} \begin{proof} From Stirling approximation formula \begin{equation*}
\Gamma(Q)\simeq(2\pi)^{\frac{1}{2}}e^{-Q}Q^{Q-\frac{1}{2}},\,\,\,\,\,Q\gg1, \end{equation*} we get, \begin{equation}
\begin{split}
\frac{\alpha e A_{Q,\alpha}}{Q B_{Q,\alpha}}&\stackrel{\eqref{EQ:Aexp1},\eqref{EQ:Aexp}}=Q^{-1}\alpha^{1-\alpha}(\alpha-1)^{\alpha-1}e\left(\frac{\frac{|\mathfrak{S}|\Gamma\left({\frac{Q}{\alpha}}\right)}{\alpha}}{\frac{|\mathfrak{S}|\Gamma\left(\frac{Q}{\alpha}\right)\Gamma\left(\frac{Q}{\alpha'}\right)}{\alpha\Gamma\left(Q\right)}}\right)^{\frac{\alpha}{Q}}\\&
=Q^{-1}(\alpha')^{1-\alpha}e\left(\frac{\Gamma\left(Q\right)}{\Gamma\left(\frac{Q}{\alpha'}\right)}\right)^{\frac{\alpha}{Q}}\\&
\simeq Q^{-1}(\alpha')^{1-\alpha}e\left(\frac{e^{-Q}Q^{Q-\frac{1}{2}}}{e^{-\frac{Q}{\alpha'}}\left(\frac{Q}{\alpha'}\right)^{\frac{Q}{\alpha'}-\frac{1}{2}}}\right)^{\frac{\alpha}{Q}}\\&
=(\alpha')^{-\frac{\alpha}{2Q}}\\&
\stackrel{Q\rightarrow \infty}\rightarrow 1.
\end{split} \end{equation} \end{proof}
\end{document} | arXiv |
\begin{document}
\thispagestyle{empty}
\title{\bf E. Cartan's attempt at bridge-building between Einstein and the Cosserats -- or how translational curvature became to be known as {\em torsion}} \author{Erhard Scholz\footnote{University of Wuppertal, Faculty of Math./Natural Sciences, and Interdisciplinary Centre for History and Philosophy of Science, \quad [email protected]}} \date{09. 10. 2018 } \maketitle
\begin{abstract} \'Elie Cartan's ``g\'en\'eralisation de la notion de courbure'' (1922) arose from a creative evaluation of the geometrical structures underlying both, Einstein's theory of gravity and the Cosserat brothers generalized theory of elasticity. In both theories groups operating in the infinitesimal played a crucial role. To judge from his publications in 1922--24, Cartan developed his concept of generalized spaces with the dual context of general relativity and non-standard elasticity in mind. In this context it seemed natural to express the translational curvature of his new spaces by a rotational quantity (via a kind of Grassmann dualization). So Cartan called his translational curvature "torsion" and coupled it to a hypothetical rotational momentum of matter several years before spin was encountered in quantum mechanics. \end{abstract}
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\section*{Introduction}
In a series of notes in the {\em Comptes Rendus} of the Paris Academy of Sciences submitted between February and April 1922 \'Elie Cartan sketched the basic ideas of a new type of geometry which was centrally based on employing the method of differential forms as far as possible \citep{Cartan:1922a,Cartan:1922[57],Cartan:1922[58],Cartan:1922[59],Cartan:1922[60],Cartan:1922[61]}. The notes were an outgrowth of his investigations of Einstein's gravity theory under the perspective of his own ideas in differential geometry. He completed this first round of publications by applying his methods to the problem of space as it had recently been re-formulated by Hermann Weyl in the light of relativity. A detailed presentation of the ideas followed during the next years.\footnote{See \citep{Chorlay:Cartan,Nabonnand:2016Cartan}.}
One aspect of Cartan's peculiar approach to differential geometry consisted in formulating the curvature concept of Riemannian differential geometry in terms of differential forms with values in the inhomogeneous Euclidean group operating in the infinitesimal neighbourhoods of any point. But the core of his new geometry lay elsewhere; it generalized the concept of curvature in two respects. The first generalization consisted in adding a translational component to the connection and, correspondingly, to the curvature. For the latter he chose the somehow surprising name ``torsion''. The earliest public presentation of his idea was given in his note {\em Sur une g\'en\'eralisation de la notion de courbure de Riemann et les espaces \`a torsion} of February, 22nd, \citep{Cartan:1922[58]}.
The second, perhaps even more consequential, generalization lay in his proposal for allowing different types of groups operating in the infinitesimal neighbourhoods, rather than just concentrating on the group of Euclidean motions (respectively their Lorentzian counterpart, the Poincar\'e group). This made it possible to study various types of geometries arising from the conformal, the affine, the projective groups, or even more general Lie groups, with their respective pairing of inhomogeneous/homogeneous constituents.
By this move Cartan reshaped the Kleinian program of structuring different types of geometry according to their automorphism groups in the context of differential geometry. In his note of March 13th, {\em Sur les espaces g\'en\'eralis\'ees et la th\'eorie de la Relativit\'e} \citep{Cartan:1922[59]} the idea was first stated in some generality. The double aspect of infinitesimalizing the Kleinian view of geometry and of taking into account a translational component of connection and curvature was crucial for Cartan's {\em espaces g\'en\'ealis\'ees} which later became to known as {\em Cartan spaces}.\footnote{For a modern presentation see \citep{Sharpe:DiffGeo}.}
The following paper concentrates on the first aspect of Cartan's generalization of differential geometry and the peculiar contexts which lay at the base of the, prima facie paradoxical, terminology of {\em torsion} for the {\em translational} component of the curvature. In the paper in which Cartan announced this new concept he described it in quite intuitive terms. He expressed the difference of his approach to classical (Euclidean) geometry similarly to what had been done by Levi-Civita and Weyl. That is, he considered the change a vector would undergo, if it is transported along an infinitesimal closed path according to the rules established by the generalized connection: \begin{quote} En d\'efinitive, \`a tout contour ferm\'e infiniment petit de l'espace donn\'e sont associ\'ees une translation et une rotation infiniment petites (. . . ) qui manifestent la divergence entre cet espace et l'espace Euclidien \citep[p. 594]{Cartan:1922[58]}.\footnote{{\em Definitely, to any infinitesimally closed curve of the space an infinitesimal translation and a rotation are associated (\ldots); they express the divergence between this space and Euclidean space.} -- Translations in emphasized letters by ES; other translations in quotes with source indicated.} \end{quote} The mentioned infinitesimal translation and rotation expresses the curvature properties of the space. Cartan immediately identified the well known case of Riemannian geometry with its Levi-Civita connection as the situation in which the translational component of the curvature vanishes.
A little later in the note he came back to the difference to the more classical geometries again and introduced a new terminology for the translational curvature mentioned above: \begin{quote} Dans les cas g\'en\'eral o\`u il y a une translation associ\'ee \`a tout contour ferm\'e infiniment petit, on peut dire que l'espace donn\'e se diff\'erencie de l'espace euclidien de deux mani\`eres: 1$^{\circ}$ par une {\em courbure} au sens de Riemann, qui se traduit par la rotation; $2^{\circ}$ par une {\em torsion}, qui se traduit par la translation \citep[594f.]{Cartan:1922[58]}.\footnote{ {\em To any closed infinitesimal loop there is generally an associated translation; in this case one can say that the given space differs from Euclidean space in two respects: 1. by a \underline{curvature} in the sense of Riemann, which is expressed by the rotation; 2. by a \underline{torsion} which is expressed by the translation} (emphasis in the original). } \end{quote} But why did he call the translational curvature ``torsion''?
A first clue follows immediately; but at first glance it enhances the riddle and introduces an even wider {\em quid pro quo}: \label{quid pro quo} \begin{quote} La rotation peut \^etre repr\'esent\'ee par un vecteur d'origine $A$ et la translation par un couple (ibid.).\footnote{\em The rotation can be represented by a vector of origin $A$ and the translation by a couple.} \end{quote} Now everything has been turned upside down, rotations were expressed by vectors, translations by couples.
The last word of the sentence indicates that a mechanical context stood behind this move. In fact, Cartan indicated that one can study the equilibrium of an elastic medium in terms of his connection and curvature. This led him to formulate a geometrical picture of the constellation of forces: \begin{quote} On a ainsi une image g\'eom\'etrique d'un milieu mat\'eriel continu en \'equilibre, mais dans le cas o\`u ces forces se manifesteraient sur chaque \'el\'ement de surface, non seulement par une force unique (tension ou pression), mais par un couple (torsion) \citep[594]{Cartan:1922[58]}.\footnote{\em One thus has a geometrical picture of a continuous material medium in equilibrium, but in the case where the forces express themselves not only by a single force (tension or pressure) but also by a couple (torsion).} \end{quote}
By {\em couple} Cartan referred to the traditional (18th and 19th century) expression for a rotational momentum (torque) by a pair of forces of the same norm, acting along different parallel lines in opposing orientations. This might superficially explain the rephrasing of translational curvature as ``torsion''. But for the unprepared reader it still remains a riddle why Cartan identified the infinitesimal {\em rotations} with forces (vectors) and infinitesimal {\em translations} with rotational momenta (couples). From a purely geometrical point of view this identification would not appear particularly plausible. But at the end of the note Cartan gave a hint for the motivation of such an interchange. He indicated that \begin{quote}
\ldots les consid\'erations pr\'ec\'edentes (\ldots) du point de vue m\'ecanique, s'apparentent aux beaux travaux de MM. E. et F. Cosserat sur, l'action Euclidienne \ldots''. (ibid.)\footnote{\em \ldots from the mecanical point of view (\ldots) the preceding considerations are similar to the beautiful works of the Messieurs E. and F. Cosserat on the Euclidean action \ldots } \end{quote} In addition he mentioned another link, namely to to H. Weyl's studies of the problem of space; but this does not lead us further for our question.\footnote{For Weyl's space problem see, among others, \citep{Bernard:2018Paris,Scholz:2016Weyl/Cartan}; for the elasticity of the Cosserats \citep{Brocato/Chatzis:Cosserat,Pommaret:1997,Hehl/Obukhov:2007}.}
If we want to understand the background of Cartan's choice of terminology for the translational curvature we have to reconstruct the historical context of the { unconventional theory of elastic media} of the brothers \'Eug\`ene and Fran\c{c}ois Cosserat, which Cartan referred to. On the other hand, the geometrical picture of the elastic medium Cartan had in mind arose from his way of reading Einstein's gravity theory in a mathematical analogy to elasticity. In order to understand Cartan's intentions epressed in the note \citep{Cartan:1922[58]} we have to follow the traces of a ``threefold knot'' tied by Cartan between the {\em mathematical methods} developed for his new type of geometry, {\em Einstein's theory} of gravity, and the {\em generalized theory of elasticity} of the Cosserats.
We therefore start this paper with a short description of Cartan's mathematical arsenal used for constituting his generalized geometries (section 1), continue with a r\'esum\'e of Cosserat elasticity and its historical context (section 2) before we shed a glance at Cartan's reading of Einstein gravity (section 3). This allows us to reconstruct how Cartan linked these three components in an intriguing interplay between his geometrical picture of a Cosserat type elasticity theory and a (speculative) generalization of Einstein gravity by torsion (section 4). We then look back at his practice of organizing the three-sided interplay between mathematics/geometry, elasticity theory, and gravity (section 5), and give some indications of its repercussions on the work of physicists in the second half of the 20th century (section 6).
\section{\small A short outline of basic ideas of Cartan geometry \label{section Cartan geometry}}
The usual differential geometric description of a {\em metric} $ds^2$ (Euclidean, Minkowski or (pseudo-)Riemannian) uses the differentials $dx_i$ of the coordinates of a point $x=(x_1, \ldots, x_n)$
\[ ds^2 = \sum_{i=1}^n g_{ij} \ dx_i^2 dx_j^2 \] This corresponds to a choice of a coordinate basis in the tangent spaces (infinitesimal neighbourhood) of any point \ldots \\
\hspace*{4cm}{\includegraphics[angle=0,scale=0.2]{Cartan-1} } \\ and an expression of the metric with regard to this basis.
Cartan, in contrast, preferred to describe the metric in terms of {differential forms $\omega_1, \ldots \omega_n$} which diagonalize the metric: \\[-0.5em] \begin{equation} ds^2 = \sum_{i=1}^n \epsilon_i \, \omega_i^2, \qquad \epsilon_i = \pm 1 \label{metric} \end{equation}
The $\epsilon_i$ (used by Cartan himself) account for different signatures of the metric, most importantly Euclidean/Riemannian and Minkowski/Lorentzian.\footnote{\citep[p. 150, eq. (9)]{Cartan:1922a}}
This form can be arrived at by linear algebraic considerations in each infinitesimal neighbourhood. In his papers of 1922ff. Cartan emphasized that geometrically the diagonalization indicates a choice of point-dependent (``mobile'') {\em orthonormal reference systems}.\footnote{The paper \citep{Cartan:1922a} was written in 1921 and published only in the following year. Cartan remarked that the ``germs'' of his new geometry can be found at the beginning and the end of this paper. Before it was published, Cartan announced the basic ideas of his new geometry in several {\em Comptes Rendus} notes, \citep{Cartan:1922[57],Cartan:1922[58],Cartan:1922[59],Cartan:1922[60],Cartan:1922[61],Cartan:1922[62]}. Technical details followed in his long m\'emoire {\em Les vari\'et\'es a connexion affine et la th\'eorie de la relativit\'e g\'en\'eralis\'ee} \citep{Cartan:1923/24,Cartan:1925} and subsequent publications.}
Cartan called them Euclidean reference systems, ``syst\`eme de r\'eference euclidien'' (ibid., p. 151) or ``tri\`edre trirectangulaire'' \citep{Cartan:1922[58]} etc. and denoted them
by { $e_1, e_2, \ldots, e_n$} (orthonormal basis ONB, or frame). \\ \hspace*{2cm} \includegraphics[angle=0,scale=0.4]{Cartan-2neu.jpg} \\ If we introduce the analogous symbols for the dual basis (at evey point), \begin{equation} \{e^1, \ldots e^n\} \quad \mbox{dual basis of 1-forms to ONB} \quad \{e_1, \ldots, e_n\}, , \label{ONB-dual} \end{equation} Cartan's $\omega_i$ turn out to be nothing but these, $\omega_i = e^i$. In a coherent use of lower and upper indexes one therefore better writes Cartan's component forms as $\omega^i$. In fact Cartan often, although not always, used upper and lower indices like in the tensor calculus,\footnote{Upper indices ones for vector like, and lower ones for differential form like transformation behaviour under change of coordinates or reference systems.}
e.g. $\omega^i_{\; k}$ in place of $\omega_{ik}$. He also applied the Einstein summation convention abbreviating, e.g., $ \sum_k \omega_{ik}\omega_{kj}$ by $ \omega^i_{\; k}\omega^k_{\; j}$ etc.
If one moves between infinitesimally close points $x, x'$ the reference systems undergo an infinitesimal rotation given by a system of coefficients $(\omega_{ij})$ depending on the start point $x$ and $\delta x= x'-x$: \\[0.8em] \hspace*{2cm} \includegraphics[angle=0,scale=0.5]{Cartan-3neu.jpg} \\ Cartan realized that the coefficients of {$\omega_{i j}$} can be understood as a system of differential forms (antisymmetric in the indices $i,j$). They encode the {\em rotational connection} of the space.
By analogy Cartan interpreted the {$\omega_i$} as assigning to any $\delta x$ a translational shift of the reference system identical to $\delta x$:\\ \hspace*{2cm} \includegraphics[angle=0,scale=0.12]{Cartan-4.jpg} \\ This was a new idea which paved the way for the first of Cartan's two innovations mentioned above. In addition to the role of the $\omega_i$ for representing the metric in diagonal form (and for specifying a ``tri\`edre trirectangulaire'') he used them for assigning a translation with components $\omega_i(\delta x)$ to any infinitesimal shift $\delta x$ from the point $x$ to an infinitesimally close one $x'$ \citep[p. 152]{Cartan:1922a}. This was a first step towards turning the $\omega_i$ into a {\em translational connection} which complements the rotational one of the reference systems:\\ \hspace*{2cm} \includegraphics[angle=0,scale=0.12]{Cartan-5.jpg} \\
An important feature of Cartan's approach was that both parts of connection, the rotational and the translational one, were given component-wise by (real valued) differential forms. Present day readers may prefer to read them more collectively as two differential forms, one with values in the Lie algebra of the rotational group $\overline{\omega}=(\omega^i_{\;j})$ and one with values in the translations $\omega=(\omega^i)$. Moreover, a present reader might like to see an explicit expression for the covariant derivative $\nabla$ of Cartan's connection $\widetilde{\omega}=(\omega, \overline{\omega})$, which would generalize the Levi-Civita connection of the metric (\ref{metric}).\footnote{For a lucid modern presentation see \citep{Sternberg:Curvature}; for more technicalities \citep[appendix]{Gasperini:2017}.} Cartan emphasized calculations which could be expressed in the calculus of differential forms, rather than rewriting the bulk of Ricci's and Levi-Civita's covariant tensor calculus in his symbolism.
For calculating the analogue of exterior differentials of the connection forms Cartan had to take the rotational coefficients into account. If both generalized exterior differentials are zero, \begin{eqnarray} d \omega_i + \epsilon_i \sum_k \omega_{ik} \omega_k &=& 0, \label{flatness condition} \\ \nonumber d \omega_{ij} + \sum_k \epsilon_k \omega_{ik}\omega_{kj} &=& 0, \end{eqnarray} so Cartan noticed, the space is {\em Euclidean} (flat).\footnote{\citep[pp, 145, 148]{Cartan:1922a}}
But in general this need not be the case, and one encounters an {\em espace g\'en\'eralis\'e} (generalized space).
If one then lets a point $M$ traverse an infinitesimal loop starting and ending in $A$
\begin{quote} \ldots {\em on ne retrouvera pas dans l'espace euclidien le tri\`edre initial} mais il faudra, pour l'obtenir, effectuer un d\'eplacemant compl\'ementaire dont les composants sont bien d\'efinies par rapport au tri\`edre initial \citep[p. 594, emphasis in the original]{Cartan:1922[58]}.\footnote{\em \ldots \underline{one does not find the initial three-frame in the Euclidean space} ({\em meant is the tangent space in modern terms, ES}), in order to arrive at it one rather has to apply a complementary displacement the components of which are well defined with regard to the original three-frame.}
\end{quote} Cartan noted explicitly that this {\em d\'eplacement complementaire} is {\em independent} of the choice of reference systems. Today it is called the {\em Cartan curvature} of the space.
The translational and rotational {\em d\'eplacement complementaire} $\Omega^i$ and $ \Omega^i_{\;j}$, i.e. the deviations from zero of the above given expressions are the 2-forms:
\begin{eqnarray} \Omega^i &=& d \omega^i + \sum_k \omega^i_{\; k} \omega^k =A^i_{\;j k} \, [\omega^i \omega^j ] \qquad \mbox{( {``torsion''})} \label{equ structure}\\
\Omega^i_{\;j} &=&d \omega^i_{\;j} + \sum_k \omega^i_{\; k}\omega^k_j = A^i_{\;jkl}\, [\omega^k \omega^l ] \qquad \mbox{( {``courbature''}),} \label{equation courbature}
\end{eqnarray} where the square brackets denote alternating products. They were adapted by Cartan from Grassmann.\footnote{Later authors, more precisely E. K\"ahler, introduced the now common symbolism $[a b] = a \wedge b$.} Cartan called the equations (\ref{flatness condition}) and (\ref{equ structure}) ``les \'equations de structure'' (structural equationis) of the generalized space \citep[p. 368]{Cartan:1923/24}. For vanishing torsion the $A^i_{\;jkl}$ characterize the Riemannian curvature in Cartan's symbolism \citep[p. 154]{Cartan:1922a}. But it may also happen that the rotational curvature vanishes, while the torsion is non-trivial.
A curve with a tangent vector field which is parallel in the sense of the Cartan connection is called an {\em autoparallel}, while a curve of extremal length is called a {\em geodesic}. In Riemannian geometry both concepts agree, but in Cartan geometry (modeled on the Euclidean or pseudo-Euclidean group) they usually fall apart. But this need not be necessarily so. Cartan gave a simple example of a structure in dimension $n=3$ with vanishing rotational curvature and non-trivial torsion, in which geodesics and autoparallels coincide \citep[p. 595]{Cartan:1922[58]}.\footnote{In the context of the studies of (generalized) Cosserat media this structure attracts attention until today as an example with intriguing geometrical properties \citep{Lazar/Hehl,Hehl/Obukhov:2007}.}
Cartan generalized this approach to allow for more general groups than the orthogonal ones, operating in the infinitesimal neighbourhoods. At the moment we need not follow this generalization in more details; but in general, the metric lost its central place and and the ``tri\`edre trirectangulaires'' had to be replaced by more general ``r\' eperes''. Cartan called the arising spaces {\em espaces non-holonomes} (non-holonomous spaces).\footnote{For the historical background of this terminology see \citep{Nabonnand:Cartan_2009}.} During the 1920s he studied such spaces of increasingly complex type with the following groups:
\begin{itemize} \item The Poincar\'e group in papers on the geometrical foundation of general relativity \citep{Cartan:1922[59],Cartan:1923PoS}, \citep{Cartan:1923/24}. For torsion $\Omega ^i =0$ such a Cartan space reduces to a Lorentzian manifold. Cartan could use this reduction for treating Einstein's theory in his own geometric terms. \item The inhomogeneous similarity group. For torsion $ =0$, this case reduces to Weylian manifolds \citep{Cartan:1923PoS}. \item The conformal group \cite{Cartan:1922[60]}. \item The projective group \cite{Cartan:1924[70]}. \end{itemize}
In this way, Cartan developed a wide conceptual frame for studying different types of differential geometries, Riemannian, Lorentzian, Weylian, affine, conformal, projective. All were enriched by the possibility to allow for the new phenomenon of torsion, and all arose from Cartan's unified method of adapting the Kleinian viewpoint to infinitesimal geometry. But if we want to understand his first papers of the year 1922 and the immediately following ones, we have to know a bit the ``beaux travaux de M. E. et F. Cosserat'' \citep{Cartan:1922[58]}.
\section{\small Generalized elasticity theory \label{section elasticity}} In the early 19th century a group of mainly French authors developed the foundations of the linear elasticity theory of solid bodies. A.J. Fresnel (1821) and C.L. Navier (1827) derived their theories on the basis of a molecular theory of matter with central forces acting between the discrete units of matter. When A.-L. Cauchy jumped in between 1823 and 1828, he first approached the question from the point of view of a continuum theory of matter and derived his influential representation of the linear relationship between the {\em strain} matrix characterizing the deformation of the material and the {\em stress} matrix (both later understood as tensors) from a phenomenological {\em Ansatz}.\footnote{For Cauchy's contributions to elasticity see \citep{Dahan:Cauchy,Belhoste:Cauchy}.}
But the molecular theory of matter behind these different approaches remained dominant. In 1827 also Cauchy presented a derivation of Navier's equations on the basis of a molecular approach. A year later S.D. Poisson developed the linear elasticity theory of molecular matter a step further and brought it home to the Laplacian program of physics, which in the meantime had come under attack from different sides (Fourier's theory of heat, electricity, magnetism, optics) \citep{Fox:LaplacianPhysics}.
Poisson's theory was built upon the hypothesis of central forces acting between point-like centers inside the radius of a ``molecular sphere'' outside of which the forces are no longer to be felt. The phenomenological forces in the material on a surface element were derived by summing up all the forces in the range of the ``molecular spheres'' of points intersecting or touching the surface element. For isotropic solid matter the calculations resulted in a linear relation between strain (deformation) and stress (surface forces), which depended on a single material constant. The basic structure of the theory seemed empirically convincing; but with increasing precision of experimental techniques between the 1840s and 1870s the 1-parameter assumption turned out to be untenable even for isotropic matter; a second elastic constant had to be assumed to fit the data. Even worse, around the middle of the century the assumption of pointlike molecules of the Laplacian program became undermined also from another side: The improvements in theories of crystal structure, in particular A. Bravais' theory of crystal matter (late 1840s to early 1860s) indicated that directional aspects might well play a role also for the elastic properties of matter.\footnote{For more details on this development see \citep{Capecchi_ea:2010,Fox:LaplacianPhysics,Timoshenko:1953}.}
An alternative approach to the theory of elasticity was proposed by George Green in 1838. He avoided any hypothesis about the basically unknown molecular structure of matter and based his analysis on a potential function $\phi$ from which the forces in the elastic medium could be derived by very general formal considerations. Although this approach led to quite acceptable results, including the empirically necessary two elastic constants in the case of an isotropic medium, Green's theory did not manage to replace the research program following the molecular hypothesis \citep[pp. 217ff.]{Timoshenko:1953}. But it became an important input for the generalized theory of elasticity of the brothers Cosserat to whom Cartan referred in his note of 1922.
In the late 1880s {\em Woldemar Voigt} (1850--1919) gave a detailed analysis of the actual status of the molecular theory of elasticity in a report to the {\em G\"ottingen Gesellschaft der Wissenschaften} \citep{Voigt:1887}. He carefully reviewed the molecular elasticity theory of the French tradition and
proposed a refinement of it, which would take into account that the molecules are extended bodies of different shapes. In general, the form of the molecules breaks the rotational symmetry of the old pointlike force centers; thus not only the coordinates of the centers of the molecules, but also their directional properties had to be considered.
As a result, the molecular interactions could no longer be represented by forces alone but had to be complemented by the consideration of rotational momenta, torque, which depend on the relative ``polarity'' of the molecules.\footnote{``Wir denken uns das homogene krystallinische Medium bestehend aus einem System von Molek\"ulen, welche durch ihre Wechselwirkungen einander im Gleichgewicht halten. Diese Wechselwirkungen sind Kr\"afte und Drehmomente, deren Componenten in unbekannter Weise mit der relativen Lage der Molek\"ule variieren.'' \citep[p. 5]{Voigt:1887}\\
{\em We conceive the homogeneous crystalline medium as consisting of a system of molecules which stand in equilibrium by their mutual interactions. These interactions are forces and rotational momenta, the components of which vary with the relative position of the molecules in an unknown way. }} For a full representation of the position and the orientation of the molecules the coordinates of their barycenters and the directions of a system of axis, tied to the molecule and changing from one to the other had to be taken into account.\footnote{``Da die Molek\"ule nach unserer Annahme eine Polarit\"at besitzen, so muss man sie wie endliche K\"orper behandeln und ihre Lage ausser durch die Coordinaten ihres Schwerpunktes noch durch die Richtung eines fest mit ihnen verbundenen Axensystemes bestimmen.'' (ibid., p. 6)\\ {\em According to our assumption the molecules possess a polarity, one therefore has to treat them like finite bodies and has to specify their position in addition to the coordinates of their barycenter by the direction of an axis system rigidly tied to them.} } From a mathematical point of view, Voigt's description resembled point dependent {\em r\'ep\`eres mobiles} linked to the different orientations (``polarisations'') of the molecules in a material structure. But neither he nor mathematicians at the time took up this analogy.
For studying equilibrium conditions on the macro-level, Voigt considered forces and rotational momenta on surface or volume elements, given with regard to an axis system by the components $(Y,Y,Z)$ and $(L,M,N)$ respectively. They came about from the summation of the corresponding actions on the micro-level and had to be studied in the rest state and, if subject to external forces, in a deformed state (ibid. p. 10). A clear and quite detailed study of Voigt's further derivation is given in \citep{Capecchi_ea:2010}. We need not go into the details here, because in the course of his calculations Voigt introduced the assumption that for all practical purposes {\em the point dependence of the rotational momenta} induced in the material even by deformations {\em could be neglected}.\footnote{ ``\ldots sind die in den Ausdr\"ucken f\"ur die Drehungsmomente vorkommenden Coeffizienten als unendlich klein gegen die in den Componenten $X_x\ldots$ auftretenden anzusehen. Dies hat den Effekt ihre Differentialquotienten neben den \"ubrigen Gliedern zu vernachl\"assigen sind, --- in \"Ubereinstimmung mit der Umstande, dass bei allen bekannten Problemen an der Oberfl\"ache der elastischen K\"orper $\overline{L}_n, \overline{M}_n, \overline{N}_n$ (the rotational momentum represented as a vector normal to the surface, ES) gleich Null zu setzen ist \ldots'' \citep[p. 23]{Voigt:1887}. \\
{\em \ldots the expressions for the coefficients in the rotational momenta have to be considered as infinitely small with respect to those appearing in the components $X_x$\ldots This leads to the effect that their differential quotients can be neglected in comparison with the other terms -- in accordance with the fact that in all known problems at the surface of elastic bodies $\overline{L}_n, \overline{M}_n, \overline{N}_n$ (the rotational momentum represented as a vector normal to the surface, ES) may be equated to zero \ldots}
}
So the bulk of Voigt's enriched structure theory on the micro-level (the point dependence of the axis systems linked to the molecules and their deformations) remained without visible consequences, once one turned to the phenomenological level. One effect remained however. Voigt's calculations led to introducing a second parameter resulting from a global rotational momentum which was not present in the older molecular theory. It filled the gap which had arisen between the older molecular theory of elasticity and the experimental findings. In the end, this was the main achievement of Voigt's approach. It soon became accepted and shaped the paradigm of linear elasticity theory at the turn to the 20th century.
Other authors explored alternatives in the framework of continumm mechanics. Particularly important in our context was the joint research of {\em Fran\c{c}ois Cosserat} (1853 -- 1914) and his younger brother {\em \'Eug\`ene Cosserat} (1866 -- 1931) during the two decades between 1896 and 1914. Fran\c{c}ois was a civil engineer working for the French railroad system. He studied at the \'Ecole Polytechnique and graduated at the \'Ecole des Ponts et Chauss\'ees and was a highly theoretical mind. \'Eug\`ene studied mathematics at the \'Ecole Normal Superieur under the guidance of P. Appell, G. Darboux, G. Koenigs and E. Picard. After his graduation in 1886 and a few months of teaching at a Lyc\'ee in Rennes he became an assistant astronomer at the Observatory in Toulouse.
Parallel to observational work on binary stars he wrote a dissertation in mathematics with a topic in differential geometry. In 1889 he finished his PhD in Paris under the supervision of Appell, Darboux and Koenigs.
In 1896 he succeeded T. Stieltjes as a professor in mathematics at Toulouse University. Roughly a decade later (1908) he became the director of Toulouse Observatory and professor of astronomy. He was elected corresponding member of the Paris Acad\'emie des Sciences in 1911 and became a full member in 1919 \citep{Levi:Cosserat}. After the early death of his older brother he discontinued work on elasticity theory. The last joint publication of the two appeared after Fran\c{c}ois' death. It was an extended French version of A. Voss essay on rational mechanics for the {\em
Encyclop\'edie des sciences math\'ematiques pure et appliqu\'ees}. It provided the occasion for explaining the wider perspective of their research in rational mechanics \citep[sec. 4]{Brocato/Chatzis:Cosserat}.
The two brothers studied elasticity theory in a strictly deductive Lagrangian approach to continuum mechanics, while acknowledging that its aim was a rational understanding of inductively generalized empirical knowledge. Their first paper (``premier m\'emoire'') appeared in the year \'Eug\`ene became a professor of mathematics in Toulouse \citep{Cosserat:1896}. A series of papers followed; but their main result did not become mature before 1909. In this year they managed to derive, on the basis of two principles, a set of generalized equations for the equilibrium of an elastic medium carrying forces and torques. They presented their new theory in several variants to the scientific public \citep{Cosserat:1909Theorie,Cosserat:1909Note,Cosserat:1909Chwolson}.\footnote{For list of all common papers of \'Eug\`ene and Fran\c{c}ois Cosserat and a discussion see \citep{Brocato/Chatzis:Cosserat}.}
Their first principle was the {\em invariance} of the action under transformations of the {\em inhomogeneous Euclidean group} for elastic continua of dimension $n=1, 2, 3$ (elastic rod, plate, body). In their terminology they worked with an ``action euclidienne''. As a second principle they characterized the elements of the elastic continuum by {\em point dependent ``tr\i\`edres''} (orthonormal frames) rather than considering elastic deformations of a simple point continuum. In this form they took up ideas in elasticity theory which had started to consider ``polarized'' (directionally oriented) molecules. They incorporated them into the continuum mechanics framework and gave them a form which nicely corresponded to the methods of Darboux style differential geometry.\footnote{In his PdD dissertation \'Eug\`ene had already investigated infinitesimal circles as space elements, combining ideas of Pl\"ucker's generalized space elements with Darboux' differential geometry. }
In their last paper they described a limit idea underlying this approach. In the older approach to elasticity ordinary geometric space was considered as an adequate mathematical representation of the physical medium (``milieu''). But, according to the Cosserats, the studies of elasticity, cristallography, electricity and of light made it necessary to consider a more complex notion of the continuum (``une notion plus complexe du milieu continu''). \begin{quote} \ldots This notion is derived in all generality by a passage to a limit from a discontinuous collection of {\em point systems with an arbitrary number of degrees of freedom}.\citep[p. 72, emphasis ES]{Cosserat:1915} \footnote{``Cette notion se d\'eduit dans toute sa g\'en\'eralit\'e, par un passage \`a la limite, de celle d'un ensemble discontinu de syst\`emes de points \`a un nombre quelconque de degr\'es de libert\'e'' citation from \citep[p. xx]{Brocato/Chatzis:Cosserat}. With regard to crystallography one may easily recognize the discrete ``point systems'' as an abstract representation of the lattice structure of polyhedral molecules studied by Bravais or the more refined structures of the 1890s according to the research tradition of Fedorow or Schoenflies.} \end{quote} This may be read as a late reflection on the motivations which had brought them to study the influence of the directionally oriented elements on the equilibrium conditions in all generality (not only under the restrictive assumptions used by Voigt).\footnote{The Cosserats saw and commented the relationship between Voigt's and their work. They were clearly aware of their own achievements; see fn \ref{fn Voigt-Cosserat}.}
The Cosserats characterized an element of the undeformed continuum (``\'etat primitif'' or ``\'etat naturel'') by the coordinates $x=(x_1,x_2,x_3)$ of a point $p$ with regard to a fixed Euclidean frame $\mathfrak{O}$ and orthonormal frame (``tri\`edre trirectangle'') $\{e_1,e_2,e_3\}$ attached to the point and specified by a point dependent rotation $o(x)$ with regard to the reference system $\mathfrak{O}$. For the sake of brevity we denote such an oriented continuum element here by $(x, o(x))$.\footnote{The original notation of the Cosserats for the tri\`edre $M_o x_o, M_o y_o,M_o z$ was given by angle cosinus to the fixed reference system \citep[p. 559]{Cosserat:1909Note}.}
The coordinates $x$ could be changed by a smooth coordinate transformation. A deformed state $(x', o'(x'))$ of the medium, on the other hand, was described by transformations $x'=f(x)$ and $o'(x)= g(o(x))$ with smooth functions $f, g$. The range of possible infinitesimal deformations was then characterized by the 9 partial derivatives of the three components of $f$, which we denote collectively by $\partial f$ and 9 partial derivatives of the angle transformations of type $\partial g$.\footnote{Expressed in coordinates of the fixed reference system $\mathfrak{O}$, the Cosserats gave $\partial f$ as $(\xi_i,\eta_i, \zeta_i)$ and $\partial g$ by $(p_i, g_i, r_i)$, where the index $i=1,2,3$ indicates the partial derivative with regard to $x_i$ \citep[pp. 559, 596]{Cosserat:1909Note}, similar in \citep{Cosserat:1909Theorie}.}
They assumed a time-independent action density $W$ for the deformation of the continuous medium and analyzed it step by step for the dimensions $n=1,2,3$. For dimension $n=3$ the action was of the general form \begin{equation} \mathfrak{S}= \int_{A_o} W(x,\partial f, \partial g )\,dx \; , \label{action} \end{equation}
with $A_o$ the space region occupied by the elastic body in the natural state. Thus their action depended on 21=3+9+9 continuous parameters \citep[p. 559]{Cosserat:1909Note}. Its form was constrained only by the {\em postulate of invariance under (infinitesimal) Euclidean motions}. Analyzing the variation of a not necessarily ``natural'', i.e. force free, state they were able to derive formal expressions for the {\em external forces} $(X_i)$ and {\em rotational moments} $(L_i)$ acting on the volume elements.\footnote{Volume forces may result from fields permeating the continuum; but the authors did not discuss the origin of them.}
The
\begin{eqnarray*}
\mbox{surface densities of} & & \mbox{force} \; (F_1,F_2,F_3) \; \quad \mbox{and torque} \; (J_1,J_2,J_3)\; \\ \mbox{and volume densities of} & & \mbox{force} \; (X_1,X_2,X_3) \quad \mbox{and torque} \; (L_1, L_2,L_3) \;
\end{eqnarray*}
arose from the variation of (\ref{action}) with regard to infinitesimal changes of the point coordinates $\delta x_i$ and to infinitesimal rotations $\delta j_i$ of the {\em tri\`edres}. This accorded to the venerated principle of virtual velocities:
\begin{eqnarray*} \delta \int_{A_o} W dx &=& \int_{S_o}\sum_i (F_i \delta x_i + J_i \delta j_i) \,d\sigma \\
&+& \quad \int_{A_o}\sum_i (X_i \delta x_i + L_i \delta j_i)\, dx \; ,
\end{eqnarray*}
where $d\sigma$ denotes the surface element on $S_o$ \citep[p. 597]{Cosserat:1909Note}. By dissecting the medium along a surface $S$ inside $A$ analog expressions could be discerned for the surface densities of {\em internal force } $(F_i)$ and {\em torque} $(J_i)$ (``effort et moment de d\'eformation'').\footnote{The original notation was $(F,G,H)$ for our $(F_i)$, $I,J,K$ for our $(J_i)$, and $(X,Y,Z)$ for $(X_i)$, respectively $(L,M,N)$ for the $(L_i)$.}
The evaluation of the invariance under Euclidean motions became a complicated task. After diverse transformations the authors derived two sets of equations involving auxiliary quantities $p_{ij}$ and $q_{ij}$ (``dix-huit nouvelles auxiliaires'') which in slightly streamlined notation read as
\begin{eqnarray} X_j &=& \sum_j \frac{\partial p_{ij}}{\partial x_i} \label{equilibrium Cosserat 1}\\
L_j &=& \sum_j \frac{\partial q_{ij}}{\partial x_i} + p_{kl}-p_{lk} \label{equilibrium Cosserat 2}
\end{eqnarray} where the $(j,k,l)$ in the last line form cyclical permutions of $(1,2,3)$. Moreover they found that the internal forces and torques can be expressed by the auxiliary quantities: \begin{equation} F_j = \sum_j p_{ij} n_i\; \qquad J_j = \sum_i q_{ij} n_i \, ; \label{internal force and torque Cosserat} \end{equation} here the $n_1,n_2,n_3$ are the componentes of an interior directed unit normal of a surface element (of unit area) at any point
\citep[p. 601, eqs. (29), (30)]{Cosserat:1909Note}.\footnote{Again our notation is slightly streamlined; the original notation was $p_{xx}, p_{xy}, \ldots p_{zz}$ for the stress densities $q_{xx}, q_{xy}, \ldots q_{zz}$ for torque \citep[p. 601]{Cosserat:1909Note}. In the book \citep{Cosserat:1909Theorie} the equations appear on p. 137 in exactly the same form.}
Later readers would read the $p_{ij}$ and $q_{ij}$ as tensors of the surface density of {\em internal stress} and {\em internal torque} respectively (sometimes also ``proper'' stress and torque; ``effort de d\'eformation'' and ``moment de d\'eformation'' for the Cosserats). The equations (\ref{equilibrium Cosserat 1}), (\ref{equilibrium Cosserat 2}) and (\ref{internal force and torque Cosserat}) are now known as the {\em fundamental equations of elastostatics}. Cosserats' theory includes the older linear theory of elasticity as a special case: If the densities of external force, torque and internal momenta vanish, equation (\ref{equilibrium Cosserat 2}) implies $X_j = 0, L_j=0, q_{ij}=0$. Then the stress tensor is symmetric, $p_{ij}=p_{ji}$, and satisfies $ \sum_j \frac{\partial p_{ij}}{\partial x_i} =0$ which, in a dynamical context, may be considered as a ``conservation condition''. Here, in the statical context, they indicate the equilibrium of the integrated forces acting on a closed surface inside the medium.
After having derived the equilibrium conditions for the surface torque momenta (see below), the Cosserats stated just that: \begin{quote}
Les auxiliaires que nous venons d'introduire et les \'equations qui les lient ne paraissent pas avoir \'et\'e jusqu'ici envisag\'ees sous une forme aussi g\'en\'erale; \`a notre connaissance, elles n'ont \'et\'e consider\'ees que dans les cas particuli\`er o\`u les neuf quantit\'ees $q_{xx}, \ldots,q_{zz}$ (Cosserat's expression for the components of the surface density of torque, here denoted by $q_{ij}$, ES) sont nulles, et le premier travail qui traite alors de la question semblable \`etre celui de M. Voigt \citep[p. 137]{Cosserat:1909Theorie}.\footnote{``The auxiliary functions that we just introduced and the equations that relate them do not appear to have been envisioned in a form that was that general up till now; to our knowledge, they have been considered only in the particular case in which the nine quantities $q_{xx} ,\ldots , q_{zz}$ are null, and the first work to treat that question seems to be that of {\em Voigt} \citep[p. 132, Delphenich's English translation]{Cosserat:1909Theorie} . Moreover they recommend to compare with the papers \citep{Larmor:1891,Love:1892/1906,Combebiac:1902}. In part of the literature P. Duhem is mentioned as a possible source for the Cosserats' turning towards oriented elements of the contiuum. This seems implausible, however, because they did not mention him at this place, but only later with regard to the use of reversible transformations \citep[p. 73f]{Cosserat:1915}. The other way round, Duhem quotes the Cosserats positively in his \citep[p. 3]{Duhem:1906}; see \citep[pp. xxv, xxxv]{Brocato/Chatzis:Cosserat}. \label{fn Voigt-Cosserat}} \end{quote} In a footnote added by the Cosserats they cited \citep{Voigt:1887,Voigt:1900}.
Given the generality of the assumptions, this was a great achievement. But the great complexity of the calculations made the results extremely difficult to absorb.\footnote{Readers interested in technical details may like to consult \citep{Badur/Stumpf}.} Only in retrospect could the Cosserats' theory be put into the context of wider mathematical theories and their derivations be justified on the basis of general theorems, which involved less, or at least different, calculations: The equations (\ref{equilibrium Cosserat 1}) and (\ref{equilibrium Cosserat 2}) were identified as the Noether equations with regard to translational, respectively rotational invariance of the action \citep{Hehl/Obukhov:2007}. \citet{Pommaret:1997} sees them as special case of non-linear Spencer transformations in the theory of partial differential equations. Elasticity theorists had to develop their own viewpoint which gave reasons to address the study of general elastic media. In any case, Cosserat theory did not enter the broader theoretical or even experimental research for at least half a century. It was revived only in the 1950/60s. Even today it cannot be considered mainstream, although it now seems to form an interesting sidestream of its own \citep[pp. xi ff.]{Brocato/Chatzis:Cosserat}.
During the course of their work, the Cosserats developed a perspective of a grand unifying scheme for theoretical mechanics, covering hydrodynamics, heat conduction, electrodynamics, and elasticity \citep{Cosserat:1915}.\footnote{For a detailed discussion of this point see \citep[sec. 4, pp. xxxvi ff.]{Brocato/Chatzis:Cosserat}} This turned out to be an untimely enterprise: the relativity theories, special and general, and the rising quantum mechanics were just changing the role of rational mechanics in mathematical physics. Although classical mechanics was not invalidated in its core, it lost its central and foundational role for natural philosophy of the 20th century. In consequence the overarching perspective of Cosserats' research program lost much of its power of persuasion. This may have contributed to the relative neglect of their generalized elasticity theory, in addition to its intrinsic technical difficulties.
On the other hand, the theory of elasticity proposed by E. and F. Cosserat was highly valued by a small group of mathematical scientists, mainly in France but also internationally.\footnote{\citep[sec. 5, pp. xxxivff.]{Brocato/Chatzis:Cosserat}}
Our protagonist, \'Elie Cartan, was one of the admirers. His re-reading of the Cosserats' elasticity theory took place in the wider context of his investigations of Einstein's gravity theory, which we consider next.
\section{\small Cartan's re-reading of Einstein gravity \label{section Cartan-Einstein}} As already remarked, Cartan's new geometric ideas were spelled out at the occasion of his studies of Einstein's general theory of relativity. He started with analyzing the form of the Einstein equation from a mathematician's point of view. In Riemannian geometry, with metric $g= \sum_{ij}g_{ij}d_idx_j$, it may be written summarily as \begin{equation} G = \kappa T\; , \label{Einstein equation} \end{equation} where $T$ denotes the energy-momentum-stress tensor of matter, $\kappa$ the gravitational constant ($\kappa = 1$ for Cartan).\footnote{Following Einstein, Cartan used a different sign $G=- T$. This is a question of conventions, expressing the choice of a different sign for the Ricci contraction of the Riemann tensor and the signature dependence of energy momentum. } \begin{itemize} \item[(o)] $G= \sum_{ij} G_{ij}dx_idx_j$, abbreviated $G_{ij}$, is a symmetric covariant 2-tensor, which contains the first and second partial derivatives of the components $g_{ij}$ only. Equivalently it can be percieved as a vector valued 1-form $G^i_{\;j}$. \end{itemize} In Einstein gravity the left hand side is the {\em Einstein tensor}, \begin{equation} G=Ric-\frac{R}{2}g \; , \label{Einstein tensor} \end{equation}
where $Ric$ and $R$ stand for the Ricci, respectively scalar curvature of the Levi-Civita connection associated to $g$. In Cartan's view, the study of gravitational equations (plural!) boils down to the question which covariants may serve on the left hand side of eq. ( \ref{Einstein equation}) as the (non-linear) partial differential operator on $g$. In any case, one should take into account two constraints which Einstein had emphasized as basic principles: \begin{itemize} \item[$\;$(i)] $G$ is linear in the second partial derivatives $\partial \partial g$, \item[(ii)] $G$ satisfies the {\em conservation law} (''loi de conservation''). \end{itemize}
(i) is necessary for avoiding too complicated differential equations.
(ii) is a consequence of demanding a vanishing covariant divergence of the energy momentum tensor. By (\ref{Einstein equation}) this translates to the left hand side as $ \nabla_i G^i_j = 0$ in Ricci calculus (with $\nabla$ the covariant derivative associated to $g$). Cartan preferred to express conservation as the vanishing of exterior covariant differential of $G$, which we denote hear as \begin{equation} d_{\omega} G = 0 \;, \end{equation} because it is defined with regard to a Cartan connection $\widetilde{\omega}=(\omega^i, \omega^j_{\;k})$.\footnote{For the torsion-free case see \citep[199]{Cartan:1922a}; for the general case \citep{Cartan:1923/24}; modern presentations in, e.g., \citep{Hehl/McCrea:1986}, \citep[pp. 269ff.]{Gasperini:2017} etc.}
In his first paper on Einstein gravity Cartan introduced his method of differential forms (outlined in section 1) for Riemannian geometry only. Using the Cartanized coefficients of the Riemann curvature (eq. (\ref{equation courbature})) he showed that a $G$ satisfying conditions (o) and (i) is a linear combination of $Ric$, $ R \, g$ and $g$, all three expressed in terms of the basic differential forms $\omega^1, \ldots \omega^n$ \citep[p. 196]{Cartan:1922a}. If also the constraint (ii) is taken into account only the Einstein tensor form (\ref{Einstein tensor}) plus a linear term $ g$ remains, in the symbols introduced above:
\begin{equation} G = \alpha (Ric- \frac{R}{2} g) + \beta g \; , \label{general form Einstein tensor}
\end{equation} with two arbitrary constants $\alpha, \beta$ \citep[p. 203]{Cartan:1922a}.
From a mathematician's point of view, that was a highly pleasing result. Cartan was cautious, however, whether something similar had not perhaps been already derived (in terms of the Ricci calculus) and published elsewhere in the international literature which, due to the effects of the great war, may have remained unknown in Paris.\footnote{``\' Etant donn\'ee la difficult\'e qu'on rencontre \`a avoir connaissance des M\'emoires parus \`l'\'etranger pendat la guerre et depuis la guerre, je ne suis pas absolument s\^ur qu'aucune d\'emonstration de ce th\'eor\`eme n'ait \'et\'e donn\'ee'' \citep[p. 142]{Cartan:1922a}.\\ {\em Taking into account the difficulty for gaining knowledge of foreign publications during or after the war, I am not absolutely sure that no demonstration of this theorem has perhaps already been given.}
} In fact, more or less at the same time at wich Cartan wrote his manuscript of \citep{Cartan:1922a} H. Weyl proved that in Riemannian geometry the scalar curvature $R$ is the only invariant containing not more than the first and second derivatives in $g$, and the second ones only linearly. The proof was published about the time of Cartan's submissions of his notes to the {\em Comptes Rendus} in an appendix to the fourth edition of {\em Raum - Zeit - Materie} \citep[p. 287f., Anhang II]{Weyl:RZM4}.\footnote{In the French translation \citep[p. 279f.]{Weyl:RZMfranz}.} In the framework of a Lagrangian approach Weyl's theorem implied the same restriction for the Einstein tensor, which Cartan had derived.\footnote{Condition (ii) is here a result of the contracted Bianchi identities.}
Weyl's proof had the advantage of being much shorter, but Cartan's analysis went deeper to the basic principles and was more general, independent of a Lagrangian approach to the Einstein equation.
In our representation of the general form of (\ref{general form Einstein tensor}) we have assimilated Cartan's result to the more common notation of tensor calculus. But we have to keep in mind that
Cartan used a different mathematical representation. That influenced also his interpretation of the Einstein tensor:
\begin{quote}
Nour regarderons ses composantes comme des coefficients entrant dans l'expression de la projection sur une direction fixe d'une tension appliqu\'ee \`a un \'el\'ement \`a trois dimension de l'univers \`a quatre dimensions \citep[p. 199]{Cartan:1922a}.\footnote{Similarly, in intuitive description in \citep{Cartan:1922[57]}.\\
{\em We consider its components as the coefficients appearing in the expression of the projection along a fixed direction, applied to tension exercised on an element of three dimension in the four-dimensional universe.}}
\end{quote}
In other words, he conceptualized the ``tenseur gravitationel'' $G$ as a {\em vector valued} (alternating) {\em 3-form} $\widetilde{G}$, which, by analogy to classical elasticity, expresses the respective stress force (``tension'') exercised on a 3-dimensional volume element in the 4-manifold (``l'univers''). From a later point of view $\widetilde{G}$ may be understood in the Riemann geometric view as the Hodge dual of $G$.
Following Cartan we shall use the terminology {\em gravitational tensor}
(``tenseur gravitationel'') or {\em Einstein form} and the notation $\widetilde{G}$ if we conceptualize it as a $(n-1)$-form (in dimension $n$), while the {\em Einstein tensor} (notation $G$) will generally be understood as a the symmetric covariant tensor with coefficients $G_{ij}$.
To understand better what Cartan meant, we have to go into more detail. Cartan decomposed $\widetilde{G}$ into its (real-valued) 3-form components $\Pi_i$, such that it may be written as \begin{equation} \widetilde{G}= \sum_i e_i \Pi^i \; \label{Einstein form 0} \end{equation} ($e_i$ the basis vectors of the Cartan orthonormal frame).\footnote{\citep[p. 203]{Cartan:1922a}, \citep[(vol 41) p.13]{Cartan:1923/24}}
Close to the end of analyzing the general form of $\widetilde{G}$ he defolded an intriguing argument involving his representation of the Riemannian curvature in terms of rotation coefficients $A^i_{\;jkl}$ (eq. (\ref{equation courbature})) and found that the components $\Pi^i$ can be written as:\footnote{Up to the factor $\alpha$ in eq. (\ref{general form Einstein tensor}) and an equivalent to the ``cosmological'' term $\beta g$ which we suppress here.}
\begin{equation} \Pi^i = \sum \epsilon_k\epsilon_l\, sgn(i,j,k,l)\, [\omega_j\,\Omega_{k l}] \; \label{Einstein form 1} \\
\end{equation} where the system of indices $(i,j,k,l)$ is any cyclic permutation of $(0,1,2,3)$, $sgn(i,j,k,l)$ its sign, and the summation runs over all such cyclic permutations \citep[p. 203]{Cartan:1922a}. This form of the gravitational tensor reappears in \cite[(vol. 41) p.13f.]{Cartan:1923/24}, where he called it the {\em kinetic quantity of the mass} ``quantit\'e de mouvement masse''.\footnote{Cf. A. Trautman's commentary in \citep[p. 17]{Cartan:1986}.} Adopting the signature convention $sig\, g = (+---)$ for the metric, Cartan gave the ``kinetic quantity of mass'' (the gravitational tensor) in complete form, which displays its character as a vector valued 3-form openly (ibid. eq. (7')): \begin{equation} \widetilde{G}= \sum_{(ijkl)} sig(ijkl)\, e_i\,[\omega_j\Omega_{kl}+ \omega_k\Omega_{lj}+ \omega_l\Omega_{jk}]\; \label{Einstein form 2} \end{equation} (with summation over all cyclic permutations $(ijkl)$ of $(0\ldots 3)$).\footnote{In this formula Cartan wrote $[me_i]$ in the place of $e_i$, apparently to make the point dependence of the basis vectors $e_i$ immediately visible in his notation. \label{me-i} }
During the course of his study of the gravitational tensor Cartan started to think geometrically about it and, as a consequence of the Einstein equation, also about the matter tensor. In the note of February 13, 1922, he stated: \begin{quote} On sait que, dans la th\'eorie de la relativit\'e g\'en\'eralis\'ee d'Einstein, le tenseur qui caract\'erise compl\`etement l'\'etat de la mati\`ere au voisinage n'un point d'Univers et {\em identifi\'e} \`a un tenseur faisant intervenir uniquement les propri\'et\'es {\em g\'eom\'etriques} de l'Univers au voisinage de ce point \citep[p. 437, 1-st emphasis ES, 2-nd emphasis in the original]{Cartan:1922[57]}.\footnote{\em One knows that in Einstein's generalized theory of relativity the tensor which completely characterizes the state of matter in a neighbourhood of the Universe is \underline{identified} with a tensor which is exclusively made up by the \underline{geometric} properties of the Universe at this point. ({\em 1-st emphasis ES, 2-nd emphasis in the original}) } \end{quote} This differed from how physicists usually understand the Einstein equation. For them eq. (\ref{Einstein equation}) expresses a kind of communication between two aspects of reality, spactime and matter, not a reduction of one to the other. Einstein fought strongly against the claim that his theory of general relativity had geometrized gravity.\footnote{\citep{Lehmkuhl:2014}} But for Cartan the idea that the Einstein equation justifies an identification of its left hand (geometrical) side and the right hand (matter) side became a guiding motif for his further investigations. In his notes of 1922 he used the notions ``tenseur de mati\`ere'' ``tenseur d'energie d'Einstein'' etc. synonymously and understood them to be {\em defined} by geometrical curvature properties.\footnote{This identification is announced already in the title of \citep{Cartan:1922[57]} and referred to in the next notes, e.g. \citep[593]{Cartan:1922[58]}.} This was a clue for his way of generalizing Einstein gravity in \citep{Cartan:1923/24,Cartan:1925} and, to my knowledge, remained so in the years to come.
In order to explain what he meant Cartan used a 3-dimensional analogue of the Einstein equation. Then the right hand side reduces to the classical (symmetric) stress tensor of matter, and the left hand side analogue may be described by curvature properties of a space with the correct properties of infinitesimal tri\`edres (3-frames) which, if one wants so, define their own metric different from the classical Euclidean metric of the ordinary embedding space. In Cartan's conceptualization of curvature the latter expresses itself in a ``rotation compl\'ementaire'' (complementary rotation) which is to be applied after parallel transporting a tri\`edre around an infinitesimal loop. Presupposing the quid pro quo mentioned in the our introduction as self-evident he continued: \begin{quote} Cette rotation peut se representer par un vecteur. L'\'etat de divergence entre l'espace donn\'ee et l'espace euclidien peut donc \^etre traduit par un vecteur attach\'e \`a chaque \'el\'ement de surface orient\'e de l'espace. \citep[p. 438]{Cartan:1922[57]}\footnote{\em This rotation may be represented by a vector. The state of divergence between the given space and Euclidean space can thus be expressed by a vector attached to any oriented surface element of the space.} \end{quote} In the same note he declared that the assignment of vectors to surface elements results in a tensor from which one can show symmetry and ``conservation law'' just like for the original Einstein equation. In 3-dimensional (Euclidean embedding) space the expression of an infinitesimal rotation by a vector was a standard procedure. Cartan concluded: \begin{quote} Il r\'esulte de ce qui pr\'ec\`ede qu'on peut expliquer l'\'etat d'un milieu \'elastique en \'equilibre en admettant que l'espace qui le contient est d\'eform\'e et qu l'\'etat de tension du milieu traduit physiquement cette d\'eformation g\'eom\'etrique. (ibid.)\footnote{\em From the preceding it follows that one can explain/express the state of an elastic medium in equilibrium by assuming that the space in which it is contained is deformed and that the state of tension of the medium reflects this deformation physically. } \end{quote} This is an interesting sentence. For the moment we leave it open whether we ought to understand ``expliquer'' in the sense of making something explicit in a mathematical sense, or even stronger as an explanation in the physical sense.
In order to understand what the mathematics behind this sentence is, one has to see the context. Immediately after this discussion of 3-dimensional classical elasticity, Cartan explained a geometrical interpretation of the Einstein equation (\ref{Einstein equation})
in the light of (\ref{Einstein form 1}) derived in \citep{Cartan:1922a}.\footnote{Remember that the \citep{Cartan:1922a} was already written at the time of submission of the {\em Comptes Rendus} notes.} Although he did not discuss elasticity in the latter, his notes show that in early 1922 he thought about {\em classical elasticity as a 3-dimensional analogue of the Einstein equation}. In this case the right hand side reduces to the tension tensor which Cartan would understand as a vector valued 2-form with real valued 1-forms $\tilde{T}^i$ as components ($i=1,2,3$). It expresses the stress force $\tilde{T}^i(\sigma)$ exercised on any infinitesimal surface element $\sigma$.
In a 3-dimensional version of (\ref{Einstein form 0}), (\ref{Einstein form 1}) the gravitational 3-form reduces to a 2-form and the signature coefficients are all $\epsilon_j=1$.\footnote{If we consider the Einstein tensor in dimension $n=3$ and allow us (anachronistically) to apply Hodge duality, we arrive at $\widetilde{G}$ as a {\em vector valued 2-form}.}
With Cartan's choice $\kappa = 1$ a 3-dimensional analogue of the Einstein equation would be
\begin{equation} \sum_{k,l} sgn(i,k,l)\, \widetilde{\Omega}_{k l} = \tilde{T}^i \; \label{3-dimensional Einstein equ}
\end{equation} with $ \widetilde{\Omega}_{k l}$ the componentes of the curvature 2-form of a 3-dimensional Cartanized Riemannian geometry. The alternating signs in the summation of (\ref{3-dimensional Einstein equ}) associate a vector to the rotational coefficients just like in the vector product representation of infinitesimal rotations.\footnote{For $\tilde{\Omega}= \left( \begin{array}{ccc} 0 & c & -b \\ -c &0 & a \\ b & -a & 0 \end{array}\right) $ eq. (\ref{3-dimensional Einstein equ}) gives $T =\left( \begin{array}{c} a \\ b \\ c \end{array} \right) $.} This would underpin what Cartan intuitively circumscribed in his note \citep{Cartan:1922[57]} quoted above and explain half of the {\em quid pro quo} cited in our introduction (p. \pageref{quid pro quo}), the expression of a rotation (curvature) by a vector.
Two years later, in the second lot of \citep{Cartan:1923/24}, Cartan gave a more technical explanation in terms of the 4-dimensional Einstein equation. Here he concentrated on a 3-dimensional spacelike hypersurface $S $ (in the infinitesimal represented by a hyper{\em plane}) corresponding to $x_o=0$ and ``projected'' the 4-dimensional rotations onto $S$. If a vector $\xi = (\xi^i)$ is rotated by $\Omega^i_j$, the hyperplane projection of the change is $\Delta \xi^i = \Omega^i_j \xi^j$. The rotation of the hyperplane itself is therefore given by the components $(\Omega_{23}, \Omega_{31},\Omega_{12})$. Cartan continued: \begin{quote} Elle peut, dans ce hyperplan, \^etre repr\'esent\'ee par le bivecteur \\[-0.8em] \[(\ast) \qquad [e_2e_3]\Omega^{23} + [e_3e_1]\Omega^{31}+[e_1e_2]\Omega^{12}\]
our encore par le vecteur polaire de m\^eme mesure \\[-0.8em]
\[ (\ast \ast) \qquad \frac{1}{\sqrt{g_{11}g_{22}g_{33}} } (e_1 \Omega_{23} + e_2 \Omega_{31} + e_3 \Omega_{12}) \, \]
\citep[(vol. 41) p. 16, (marks $(\ast), (\ast\ast)$ added, ES)]{Cartan:1923/24}. \end{quote}
Here ($\ast$) was a Grassmann type characterization of (infinitesimal) rotations, while ($\ast\ast$) was its equivalent in terms of a vector product representation. This transition from ``bivectors'' to ``polar vectors'' was a special case of what Grassmann had introduced as a more general duality (called ``Erg\"anzung'' by him).\footnote{In modernized notation, Grassmann established an equivalence (isomorphism) between $\Lambda^k V$ and $\Lambda^{(n-k)}V$ for any $n$ dimensional {\em Ausdehnungsgebiet} ($0 \leq k \leq n$) with a volume form, respectively a basis $e_1,\ldots,e_n$ with the property $e_1\wedge\ldots \wedge e_n =1$ (using modern notation for alternating products). He assigned to basis elements $e_{i_1}\wedge \ldots \wedge e_{i_k}$ ($i_1 < \ldots < i_k$) in $\Lambda^{k}V$ the bais elements $e_{j_1} \wedge \ldots e_{j_{n-k}}$ ($j_1 < \ldots < j_{n-k}$) of $\Lambda^{(n-k)}V$ for which $e_{i_1} \wedge \ldots \wedge e_{i_k}\wedge e_{j_1}\wedge \ldots \wedge e_{j_{n-k}}= e_1 \wedge \ldots \wedge e_n =1$ and used linear continuation \citep[\S 89, p. 57f.]{Grassmann:1862}, cf. \citep{Scholz:1984Grassmann}. As Grassman introduced an inner product in $V$, which made $e_1, \ldots, e_n$ an orthonormal basis, this can be considered as a linear algebraic isomorphism serving as the basis for the later {\em Hodge duality}.} In dimension 3 this type of dualization was particularly well known, even without any reference to Grassmann. But Cartan was well aware of the general nature of Grassmann dualization and used it in his discussion of the invariants of his geometry \citep[vol. (40) p. 400ff.]{Cartan:1923/24}.
From this point if view the first half of Cartan's quid pro quo
resulted from a {\em Grassmann type duality transformation} (of infinitesimal rotations in dimension $n=3$ to polar vectors), which Cartan introduced for gaining a {\em geometrical understanding} of the spacelike part of the {\em Einstein equation}. The result of the transformation led to a new geometrical picture of {\em classical elasticity}: stress could be expressed in terms of the curvature of a space with a connection and metric adapted to the mechanical properties of the material medium under investigation.
\section{\small Einstein gravity in analogy to geometrized Cosserat elasticity \label{section Cosserat-Einstein} }
The second part of the {\em quid pro quo} resulted from Cartan's generalization of Einstein's theory of gravity and involved an adaptation of Cosserat elasticity to his research program of 1921/22. At the time of submitting his {\em Comptes Rendus} notes, in February and March 1922, Cartan had all this in mind, but it took some time to work out the mathematical details. They are contained in the two-part paper \citep{Cartan:1923/24,Cartan:1925} the first part of which came in two lots (vol. 40, 41 of the {\em Annales ENS}).\footnote{Reprint in \citep{Cartan:1955}, English translation with a commentary (foreword) by A. Trautman in \citep{Cartan:1986}.}
In this paper he showed that the vacuum Maxwell equations are compatible with any (Cartan-) connection of the Poincar\'e group; but taking Lorentz forces into account may run into difficulties. For a kinetic quantity of energy like in (\ref{Einstein form 2}) the Lorentz force exercised on an electric current density came out correctly, i.e., in agreement with special relativity, only if the ``universe'' has vanishing torsion \citep[p.20f.]{Cartan:1923/24}. That was disappointing, but Cartan indicated a way out:
\begin{quote}
La conclusion pr\'ec\'edent (vanishing torsion, ES) ne serait pas logiquement n\'ecessaire si l'on admettait une conception de la M\'ecanique des milieux continus plus large que la conception habituelle, la ``quantit\'ee de mouvement-masse'' \'el\'ementaire \'etant rep\'esent\'ee par une syst\`eme de vecteur et de bivecteurs\\[-0.8em]
\[ G = [me_i]\, \Pi^i + [e_i e_j]\, \Pi^{ij} \,
\]
\citep[p. 21]{Cartan:1923/24}.\footnote{``The above conclusion is not logically forced upon us if we accept a broader framework for mechanics of continuous media and represent the energy-momentum density by a system of vectors and bivectors: $ G = [me_i]\, \Pi^i + [e_i e_j]\, \Pi^{ij} $ '' \citep[p. 123]{Cartan:1986}.
Compare fn. \ref{me-i}.
}
\end{quote}
If such a modification of the Einstein form is accepted, the laws of electromagnetism, including the Lorentz forces, were compatible with a non-vanishing torsion. In this context, it was natural to assume that the ``quantit\'e de mouvement-masse elementaire'', i.e. $\widetilde{G}$, should remain a geometric integral invariant, like in Einstein's theory. This, so Cartan declared, was easy to achieve. One had only to replace
the rotation associated to any surface element by the total displacement of the full Cartan curvature (``d\'eplacement total ({\em rotation et translation})'') assigned to the surface element.
The rotations had been transmuted to vectors by Grassmann duality in 3-dimensional spacelike hyperplanes and this transmutation was taken over to $n=4$ (see above). In an analogous manner Cartan transmuted translations into bivectors (Grassmann duality in the 3-dimensional spacelike projection, but here transferred to $n=4$).
In this way $\widetilde{G}$ became a fully Cartanized variant of the Einstein form \citep[p. 22, eq. (11)]{Cartan:1923/24}:
\begin{equation} \widetilde{G}= \sum_{(ijkl)} sig(ijkl)\,\left( e_i\,[\omega_j\Omega_{kl}+ \omega_k\Omega_{lj}+ \omega_l\Omega_{jk}] - [e_i e_j] [\omega_k\Omega_{l}-\omega_l\Omega_k] \right) \label{Einstein form 3}
\end{equation}
Expressed in more recent terminology Cartan proposed a 3-form with values in the Grassmann algebra of the tangent bundle as a generalization of the gravitational tensor. It consists of two terms, the first one with values in $TM$\footnote{In fact, Cartan put square brackets about his symbols of the vector basis $[me_i]$, apparently in order to emphasize the Grassmann character of the term.}
contracts rotational curvature and transmutes it into a vector.
The second one with values in the bivector bundle $\Lambda^2 (TM)$ {\em transmutes translational curvature into a bivector}.
That seemed to agree nicely with the structure of Cosserat elasticity. After considering his geometrical interpretation of the Einstein form as a new representation of classical elasticity of a (1-parameter) homogeneous medium, Cartan was now tempted to read the 3-dimensional reduction of his generalized Einstein equation as a geometrization of Cosserat elasticity. The vector part of $\tilde{G}$ resulted, after Grassmann dualization, from the rotational component of Cartan curvature and could express the stress forces on surface elements. Similarly due to Grassmann dualization, the bivector component resulted from the translational curvature and could be be interpreted as a rotational momentum exercised on surface elements. Cartan even gave an argument that, due to the specifique form of the torsion in his case, the translation associated to any surface element is normal to the latter \citep[p. 21, fn]{Cartan:1923/24}.\footnote{At another place he showed that this property implies the identity of autoparallels of the Cartan connection with the geodescis of the related Riemannian structure \citep[\S 66, p. 407]{Cartan:1923/24}.}
This was apparently the technical insight standing behind Cartan's verbal description of Cosserat elasticity in his note \citep{Cartan:1922[58]}. Moroever, it contains a resolution of the riddle of the quid pro quo. But here a warning is appropriate: Cartan no longer mentioned the Cosserat theory in his \citep{Cartan:1923/24}. We come back to this point in the next section.
Cartan encountered another problem which appeared more serious to him. In general, the covariant exterior derivative of the generalized Einstein form does not vanish. Our author considered this as an important criterion for a medium in equilibrium and proposed to ensure it by
imposing some appropriate constraint, the easiest one being (ibid., p. 22) \begin{equation} \sum (ijkl) \left( \Omega_j\Omega_{kl}+ \Omega_k\Omega_{lj} + \Omega_l\Omega_{jk} \right) = 0 \qquad \mbox{for all $i=0,\ldots,3$.} \label{Cartans constraint} \end{equation} Later authors ( see section 7) replaced the condition of vanishing covariant divergence, the ``conservation law'' in Cartan's view, by the more general one of a contracted Bianchi identitiy for Riemann-Cartan geometry. Only for vanishing torsion or Cartan's algebraic constraint (\ref{Cartans constraint}) it boils down to the ``conservation law'' \citep[p. 152f.]{Trautman:1973}.
Unhappily the relation (\ref{Cartans constraint}) gave an {\em algebraic} constraint for torsion which could therefore no longer play the role of a dynamical field in this approach. Perhaps this was a reason for Cartan to hesitate claiming immediate physical relevance for his theory.\footnote{Cf. the remark by A. Trautman in \citep[p. 17]{Cartan:1986}. \label{fn Trautman}} In any case, he emphasized:
\begin{quote}
On a ainsi une g\'en\'eralisation, au moins math\'ematique, de la th\'eorie d'Einstein, g\'en\'eralisation compatible avec toutes les lois de l'\'Electromagn\'etisme \citep[p. 22]{Cartan:1923/24}.\footnote{``This yields -- at least on a mathematical level -- a generalization of Einstein's theory which is compatible with all the laws of electromagnetism'' \citep[p. 124]{Cartan:1986}.
}
\end{quote}
He even indicated that this generalization of Einstein gravity can be formulated in a Lagrangian field approach. He proposed a generalization of the Hilbert action expressed in terms of his geometrical quantities,
\begin{equation} \mathfrak{L}_{grav} = \sum_{(ijkl)} (ijkl)\, [\omega_i\omega_j\Omega_{kl}] \; ,
\end{equation} but did not consider a separate matter Lagrangian (in agreement with his ``unphysical'' identification of the left hand and the right hand side of the Einstein equation).
At this point, our author stopped his physics related studies without even shedding a first glance at the dynamical effects of his theory: \begin{quote} \ldots je me contente de cette indication, sans entrer dans plus de d\'etails \`a ce suject \citep[p. 23]{Cartan:1923/24}.\footnote{ {\em\ldots I make to do with this indication without going into more details of this subject.}\\ Translation in \citep[p. 124]{Cartan:1986}: ``However, we shall not discuss this issue in more detail.'' } \end{quote} He rather continued with studies of Weyl's dilational gauge metric and in part II, according to the title of the paper, with studies of Cartan spaces of the affine group or subgroups of it \citep{Cartan:1925}.
In other papers, e.g. \citep{Cartan:1924[70]}, he turned towards generalizations of the group. Where he sticked to the Euclidean group he tended to investigate more classical questions. In particular he showed that Clifford parallelism in elliptic space can be seen, from the point of view of his new methods, as due to a connection with torsion but vanishing rotational curvature, while the autoparallels coincide with elliptic straight lines \citep{Cartan:1924[71]}.\footnote{Cf. \citep[p. 46f.]{Cogliati/Mastrolia}.} All these topics turned his and his readers' attention away from the Cosserat inspired view of torsion. It rather hinted in the direction of what a little later became the study of distant parallelism or {\em absolute parallelism}, as Cartan would call it.
\section{\small Cartan's practice of mathematical analogies \label{section XX2}}
It is not completely clear, why Cartan stopped short of pursuing the physical considerations further. One obvious reason might have been that he was more interested in studying the mathematical side of his approach than going deeper into the physics. But he seems also to have developed doubts with regard to the physical relevance and the interpretation of his findings. The algebraic constraint for torsion which he perceived as necessary to satisfy the ``conservation law'' as he understood it must have been a stumbling block for him.\footnote{Cf. fn \ref{fn Trautman}.}
We even find some hints in \citep{Cartan:1923/24} that during the continuation of his work he developed doubts regarding the feasibility of his geometric interpretation of Cosserat elasticity in the {\em Compte Rendus} notes.
In \S 60 of his paper he discussed the differential forms of a (Cartan) space with the Euclidean group, invariant under changes of the Cartan reference system (change of Cartan gauge in modern terminology). One of the invariants combines a ``syst\`eme de vecteur et de couples'' of a form similar to (\ref{Einstein form 3}):
\begin{equation} [m e_1]\Omega_{23}+ [m e_2]\Omega_{31} + [m e_3]\Omega_{12} + [e_1 e_3]\Omega_1 + [e_3 e_1]\Omega_2 + [e_1 e_2]\Omega_3\,. \label{analogie trompeuse}
\end{equation}
This is a 3-dimensional analogue of the expression for the generalized Einstein form (\ref{Einstein form 3}). But here the relation between translations and bivectors is simpler and more direct than in the 4-dimensional case (in fact it is nothing but the Grassmann duality between vectors and bivectors which only holds for dimension $n=3$).
Cartan commented:
\begin{quote}
Le syst\`eme de vecteurs cet de couples
\[ [m e_1]\Omega_{23}+ [m e_2]\Omega_{31} + [m e_3]\Omega_{12} + [e_1 e_3]\Omega_1 + [e_3 e_1]\Omega_2 + [e_1 e_2]\Omega_3\,\] repr\'esente de m\^{e}me le d\'eplacement associ\' e \`a un \' el\'ement de surface. Sa d\'eriv\'ee ext\'erieure est nulle. Si l'on regardait ce syst\`eme comme repr\'esentent les tensions qui s'ecercent sur un milieu mat\'eriel (tensions comportant des couples), ce milieu serait en \'equilibre \citep[vol. 40, p. 401, fn]{Cartan:1923/24}.\footnote{``The system
$ [m e_1]\Omega_{23}+ [m e_2]\Omega_{31} + [m e_3]\Omega_{12} + [e_1 e_3]\Omega_1 + [e_3 e_1]\Omega_2 + [e_1 e_2]\Omega_3\,$ of vectors and torques represents the same displacement associated with a surface element. Its exterior derivative is zero. If this system is viewed as describing tensions (with torque) in a material medium, the medium would be in equilibrium.'' \citep[p. 96, fn. (9)]{Cartan:1986}
}
\end{quote}
In other words, an invariant of this type looked as if it could be interpreted as a geometrical expression for the system of tensions/momenta inside a Cosserat medium. But, different from his announcement in \citep{Cartan:1922[58]} of February 1922, in which he had referred so positively to the
``beaux travaux de MM. E. et F. Cosserat'', Cartan now continued with a methodological reflection which stepped back from an interpretation in this sense and came even close to a disassociation: \begin{quote} Il y a l\`a une des nombreuse analogies, plus ou moin trompeuses, qui existent entre la G\'eom\'etrie et la M\'ecanique. En fait, ce n'est qu'une analogie. (ibid.)\footnote{{\em We have here one of the numerous, more or less misleading, analogies which exist between geometry and mechanics. In fact this is not more than an analogy.} \\ The translation in \citep[p. 96, fn. (9)]{Cartan:1986} (``We have here one of the numerous somewhat misleading analogies between geometry and mechanics.'' ) omits the last phrase of Cartan's remark. } \end{quote}
Cartan neither claimed that the analogy {\em was} ``trompeuse'' (erroneous), nor did he repeat a positive claim of content for it. But it seems that he was having second thoughts about this point between February 1922 and the final preparation of the paper \citep{Cartan:1923/24}. It now became clear to him that the analogy between Cosserat elasticity and the generalized Einstein equation was less perfect than he had inititally hoped because of the non-vanishing covariant exterior derivative in dimension $n=4$. This would also explain why he avoided any explicit reference to the Cosserats' work; he even did not mention their name any longer in the new paper.\footnote{Later research on this question, starting in the 1960s, showed that a viable usage of Cartan geometry in Cosserat type theory of elastic media needed, in fact, a more sophisticated approach than was available to him in 1922 (see below).}
That could not change the crucial heuristic role which, according to Cartan's own testimony in the notes of early 1922, Cosserat elasticity played during his early work on the Einstein equation and its generalization in the light of his geometrical ideas. Cosserat torque would strongly underpin the transmutation from translations to bivectors (rotational momenta) in 3-dimensions by a Grassmann type duality (\ref{analogie trompeuse}), {\em if it could be considered as physical}. In 1921/22 Cartan was apparently impressed by the analogy to the transmutation of the rotations to vector-like stress in his re-reading of the Einstein tensor. The analogy helped Cartan to structure his argumentation in which he tried to build a bridge between Einstein gravity and geometry. Once the bridge was built in the form of our eq. (\ref{Einstein form 3}), the reference to Cosserat media could be downgraded to the status of a mere analogy without losing too much.
Only the {\em terminological residuum} of Cartan's early heuristics remained unaffected: the translational curvature baptized under the impression of the 3-dimensional analogy with Cosserat media when it was still fresh and strong, continued to be called {\em torsion} and remains so until today.
\section{\small The aftermath \label{section XX3}}
This is not the end of the story. We should not finish our's without having a short glance at the reception and further developments connected to Cartan's early papers on Einstein gravity and Cosserat theory. The early idea of a potentially intimate connection between Cosserat elasticity and Einstein gravity did not play a role in the reception for many decades to come, while Cartan geometry attracted the attention of mathematicians by other reasons. J.A. Schouten got interested in Cartan's proposals of torsion in his general studies of connections. He contacted Cartan in 1924. The two mathematicians communicated on linear connections in Lie groups and found that left and right translations in any Lie group lead to distant parallelism structures, i.e. connections with torsion but vanishing curvature, with autoparallels which agree with geodesics of the Riemannian metric on the group manifold induced by the Cartan-Killing form. They even were able to show that, with the exception of the 7-sphere, the Lie groups are the only Riemannian manifolds with this property \citep{Cogliati/Mastrolia}.
Distant parallelism became a ``hot topic'' at the end of the the 1920s, when Einstein started to study it as the framework for one of his attempts to unify gravity with electromagnetism. After Cartan reminded Einstein that this approach could be well framed in his geometrical method and had been mentioned by him in talks with him in 1922 \citep[p. 4]{Cartan/Einstein}, Einstein hurried to give credit to Cartan and accepted that his study of gravity in terms of {\em distant parallelism} (also called {\em teleparallel} gravity) used a specific type of Cartan geometry.
In this setting the deviation of flat space was encoded in the torsion part of curvature only, while the rotational curvature was set to zero.\footnote{\citep{Goenner:UFT}, cf. \citep{Cartan1929}. } Also R. Weitzenb\"ock had studied flat linear connections (vanishing Riemannian curvature) with torsion in the course of his study of differential invariants, i.e. in a pure mathematics context \citep[pp. 317ff.]{Weitzenboeck:1923}. He did not relate this to Einstein gravity at the time but was keen to get acknowledgement from Einstein and published a note on on the topic in 1929 \citep{Sauer:Fernpar}. Neither Cartan nor Cartan geometry was ever mentioned by him.
Although Cartan himself had given an example of a teleparallel Cartan space in his note \citep{Cartan:1922[58]}, the general outlook of this example was a far cry
from his early idea of interpreting torsion by rotational momenta as an additional feature of the gravitational field. In a way it even was opposite to his proposal for the physical interpretation of translational curvature. But even so, the studies of distant parallelism in the gravity context demonstrated the openness of general mathematical structures for different physical interpretations. Even those which were designed with definite physical interpretations in mind, like Cartan geometry of the Euclidean and Poincar\'e group, did not carry the mark of their original interpretation with them as some sort of inbuilt, although perhaps hidden finality.
We also have to be aware that, by a constellation of historical contingency, the late 1920s was also the time in which quantum physicists started to realize that the new complex (wave) fields could carry an internal rotational momentum, called {\em spin} \citep{Pauli:Spin,Dirac:1928_I/II}. But at this time no author had the idea that this new internal torque-like momentum might give new support to Cartan's idea of torsion.
This changed only much later, in the 1960s. An important contribution for renewing the interest in Cartan torsion among physicists arose in the wake of the work of D. Sciama and T. Kibble in gravity theory. Without knowing it, the two authors independently reinvented much of the Cartan geometric field structures by considering what physicists call the ``localization'' of the Poinar\'e group \citep{Sciama:1962,Kibble:1961}. They found that the spin of elementary particle fields might play a role for a generalized theory of Einstein gravity which was close to what Cartan had anticipated in his early papers. The close relationship of their theory to Cartan geometry was not clear to Sciama and Kibble; but it was soon made explicit by other authors, at first by F. Hehl in his PhD dissertation \citep{Hehl:Diss} and independently by A. Trautman \citep{Trautman:1973}.
A group of authors joined and extended this research program.\footnote{Much information on this development is collected in the reader \cite{Blagojevic/Hehl} which contains very helpful commentaries. For systematic surveys see \cite{Trautman:2006,Hehl:Dennis}. } They realized that Cartan geometry offered a tailor-made geometric framework for infinitesimalizing (``localizing'' in the language of physicists) energy-mometum and spin currents known from Minkowski space and special relativity. The Cartan geometry of this approach was modeled on the Poincar\'e group and has both, rotational curvature and torsion, like in Cartan's work of the early 1920s. If the gravitational Lagrangian was chosen as closely as possible to the Hilbert action of Einstein gravity, it turned out to be the one Cartan mentioned in his side remark quoted above. Only Cartan's constraint (\ref{Cartans constraint}) had to be relaxed \citep{Trautman:1973}. The resulting theory has therefore correctly been called {\em Einstein-Cartan gravity}. It is considered as a ``viable'' alternative to Einstein gravity -- although one which can be distinguished from the latter only under the conditions of extremely high energy densities.\footnote{\citep{Trautman:2006,Hehl:Dennis}.}
Motivated by the successes of non-abelian gauge field theory in the rise of the {\em standard model} of elementary particle physics, gauge field dynamics related to the Poincar\'e group was studied in the 1970/80s. It turned out that Einstein-Cartan gravity can be reconstructed in a Cartan space modeled on the Poincar\'e group, where the dynamical equation for translational curvature -- paradoxically still called ``torsion'' -- couples to the energy-momentum current (of matter and the gravitational field), and the rotational curvature is coupled to the spin-current \citep[p. 337ff.]{Hehl_ea:1980Poincare-gauge-field-theory}. This was an intriguing result. Conceptually it set the couplings right while avoiding the surprising crossover of translational and rotational aspects for geometry and dynamics of Einstein gravity as seen by Cartan in analogy to Cosserat elasticity. In this way it generalized the ``teleparallel'' representation of Einstein gravity by adding spin. But the authors presented the respective Lagrangian as a special case, even a degenerate one, and directed their attention towards more general quadratic Lagrangians in Poincar\'e gauge field theory.\footnote{See also the surveys in \citep[p. 164f.]{Hehl:Dennis} and \citep{Blagojevic/Hehl}.}
This was not the only path which led back to Cartan's ideas of the early 1920s. Also Cartan's spare remarks on Cosserat elasticity found successors a generation later, although only after a specific turn taken by authors interested in the mathematical study of the yielding of plastic materials.
K. Kondo in Japan proposed to model dislocation in crystal matter by the torsion of a linear connection \citep{Kondo:1952}.
E. Kr\"oner, an expert in classical solid state physics, and F. Hehl's PhD advisor at the {\em Bergakademie/TH} Clausthal-Zellerfeld, was attracted by the ideas of the Cosserats on generalized elasticity and its link to Cartan's generalized geometry.
He was one of those authors who brought in the idea that such a material structure could be studied in a Cartan geometry modeled on the Euclidean group, called {\em Riemann-Cartan space} by these authors.\footnote{\citep{Kroener:1963torque,Kroener:1963} } This led to the attempt of studying dislocations and proper tensions in metals in terms of Cartan-type geometrical methods and was also the background from which Hehl entered gravity.
A complicated story started; we have to cut it short.
The different components of the Cartan connection had to be linked to physical quantities expressing the deformation of the material, different types of dislocation inside the material, and the hypothetical force and torque stresses.\footnote{A recent survey of the resulting theory can be found in \citep{Hehl/Obukhov:2007}.} It turned out that a Cartan geometry with translational curvature but no rotational curvature, thus with ``distant parallelism'' in the language of gravity theorists was an option. The intuitive idea behind this was the choice of a Cartan reference system without rotational curvature, adapted to the geometry of the lattice structure of the material. Torsion can be expressed as a {\em closing defect}, which arises if one parallel transports a vector $u$ along an infinitesimal shift vector $\delta x=v$ and $v$ along $\delta' x= u$.\footnote{The ``would-be'' infinitesimal parallelogram arising from this procedure does not close (thus it is {\em no} parallelogram). Mathematically, the defect is expressed by the asymmetry of the corresponding linear connection $\Gamma^i_{jk}$ in the lower indexes, $\Gamma^i_{jk}\neq \Gamma^i_{kj}$.} Such closing defects can be used to model dislocations in the material, once they turn up sufficiently often (densely). This results in a mathematical description of materials with densely distributed dislocations by a ``teleparallel'' Cartan structure. Its torsion, here clearly to be interpreted in its translational connotation, is related to the dislocation field.\footnote{Cf \citep[p. 167ff.]{Hehl/Obukhov:2007}. In this context, not any Riemann-Cartan geometry with distant parallelism is feasible. Additional constraints have to be observed, in order to lead to acceptable deformation quantities related to the Cartan structure \citep[p. 163]{Hehl/Obukhov:2007}.}
Finally the tables have been turned another time. At the origin of Cartan's theory the context of continuum mechanics gave him the motivation for introducing his slightly paradoxical terminology of ``torsion''. Now we find an epistemic constellation even in the field of continuum mechanics, for which the geometrical naming of {\em translational curvature} would be closer to the matter than the terminology chosen by Cartan. But in the meantime the latter has been widely established in the community.
In the end, Cartan's papers of the early 1920s have found new readers also among present day theorists of continuum mechanics. In this recent development it became clear that a reliable connection between the physics of matter and geometry needs much more sophistication than could be imagined by Cartan (or even the Cosserats). It turned out that, in the long run, Cartan's analogy between generalized elasticity and gravity, which became apparent in the framwork of his geometry, was not as misleading (``trompeuse'') as he may have thought in 1924. The mathematical analogy established by Cartan became a stimulating input for these studies, even though the structural analogy had to be disentangled, before it could bare fruits. \\[3em]
{\em Acknowledgments:} {\small This paper has been written in the honor of H. Sinaceur. I thank the organizers of the workshop {\em 30 ans Hourya Sinaceur: ``Corps et mod\`eles''}, Paris, June 15--17, 2017, for the opportunity to present my findings on the Cartan's early work on gravity close to the place of his activity. The paper will appear in French in the proceedings of the workshop. I thank Friedrich Hehl for critical remarks and hints, in particular his personal informations on the background of the Kr\"oner research group in the 1960s, Alberto Cogliati for making me aware of his research on the correspondence between Schouten and Cartan. Emmylou Haffner generously took on the burden of translating my paper from English to French.}
\small
\addcontentsline{toc}{section}{\protect\numberline{}Bibliography}
\end{document} | arXiv |
Extreme Learning
Always curious. Always learning.
Maximal Poisson disk sampling.
An improved version of Bridson's Algorithm
Bridson's Algorithm (2007) is a very popular method to produce maximal 'blue noise' point distributions such that no two points are closer than a specified distance apart. In this brief post we show how a minor modification to this algorithm can make it 20x faster and allows it to produce much higher density point distributions.
Figure 1. Poisson disc sampling based on a modified version of Bridson's algorithm. This modified algorithm runs in linear time and is about 20x faster than the original algorithm
In many applications in graphics, particularly rendering, generating samples from a blue noise distribution is important. Poisson-disc sampling produces points that are tightly-packed, but no closer to each other than a specified minimum distance $r_c$, resulting in a more natural looking pattern (see above figure). To create such maximal poisson disk distributions, many people use Mitchell's best candidate, however, Bridson's algorithm (2007) is much faster as it runs in $O(n)$ time, rather than $O(n^2)$ for Mitchell's.
In this brief post, I show how, by changing only a few lines of its implementation code, we can make Bridson's algorithm even more efficient. This new version is not only ~20x faster, but it produces higher quality point distributions, as it allows for more tightly packed and consistent point distributions.
Summary of Bridson's algorithm.
Mike Bostock has written two very elegant explanations of how Bridson's algorithm works. In his latest tutorial, he explains how candidate points are generated by sampling from inside an annulus of inner radius $r_c$ and outer radius $2r_c$. Furthermore, he explains how the underlying grid allows extremely fast checking of the validity of each of these candidates. His earlier visualization / tutorial, shows the time evolution of the candidates as the point distribution is constructed.
The core of the algorithm depends on uniformly selecting random points (candidates) from an annulus. This can be achieved through the standard way of first selecting a uniform point $(u,v)$ from $[0,1)^2$, and then mapping this point to the annulus. That is,
u = Math.random();
v = Math.random();
rOuter = 2 * rInner;
for (let j = 0; j < k; ++j) {
theta = 2 * Math.PI * u;
r = Math.sqrt( rInner**2 + v*(rOuter**2 - rInner**2) );
x = parent[0] + r * Math.cos(theta);
y = parent[1] + r * Math.sin(theta);
An improved Sampling method
However, my proposed improvement comes directly from the premise that in this very particular and surprisingly unique situation we do not need to uniformly sample from the annulus. Rather, we would prefer to select points closer to the inner radius as this is results in neighboring points closer to $r_c$.
It turns out that the following simple change to the sampler provides an extraordinary improvement to its efficiency.
(Also instead of stochastically sampling from an annulus, we are simply progressively sampling from the outside perimeter of the threshold circle.)
epsilon = 0.0000001;
seed = Math.random();
theta = (seed + j/k);
r = rInner + epsilon;
(I suspect that we could eliminate the need for epsilon=0.00001, if we carefully modified some of the inequality signs and/or floating point operations, throughout the code implementation.)
We can again use Bostock's technique to visualize the the effects of this modification has on the evolution of horizon frontier of the candidates. The left image is based on the Bridson's original algorithm. Note the wide and dispersed region of red candidates. Also note how much the grey lines are intersecting each other. Compare this to the right image which is based on the new algorithm. In this case, note the much narrower and well-defined region of red candidates. Also note how the grey lines are far more regularly structured and that none of them are intersecting!
(Don't forget that these visualizations artificially slow down the calculations in order to make the animations nicer!)
Figure 2. A comparison of the frontier horizons of candidate points between the original Bridson Algorithm (left) and the new modified version (right).
Bridson's algorithm requires setting a configuration parameter $k$, which is the maximum number of candidates to consider for each frontier point. It is generally set to a value around 30 (but sometimes up to 100).
Figure 3. A comparison of output from the original algorithm (left) and the new modified version (right), both using $k=30$.
Intuitively, increasing $k$ increases the potential point density and thus resulting in a tighter packing. The following table gives an indication on how the total number of output points varies, as this parameter $k$ varies. Also, the total number of candidates is included, as it is a useful proxy for CPU time.
(Note that these numbers are based on the default dimensions of the implementation by Jason Davies. The intent of showing these numbers is not the focus on the individual numbers per se, but rather their relative values.)
Results for Bridson's Algorithm
\begin{array}{|r|rr|} \hline
\text{k} & \text{Candidates} &\text{Points} \\ \hline
3 & 43\text{k} & \bf{9,282} \\ \hline
10 & 140\text{k} & \bf{10,862} \\ \hline
100 & 1.4\text{M} &\bf{12,369} \\ \hline
300 & 4.3\text{M} & \bf{12,765} \\ \hline
1000 & 13\text{M} & \bf{13,014} \\ \hline
\end{array}
For context, all of these numbers are far less than the theoretically maximal number of 21,906 points which could be achieved if the points were distributed in a hexagonal lattice arrangement.
Below are the same data points, but this time for the modified algorithm
Results for new Modified Algorithm
\text{k} & \text{Candidates} & \text{Points} \\ \hline
3 & 66\text{k} & \bf{13,791} & \\ \hline
Thus, for a given value of $k$, the new algorithm results in between 38% and 48% more points than the original value.
Said another way, using the smallest value of $k=3$ with the new algorithm will still produce more points (ie higher point density) as using the largest value of $k=1000$ with the original algorithm! Furthermore this can be achieved by testing just 66 thousand candidates rather than 13 million candidates. This is a reduction by a factor of 20!
For this new version, I recommend starting with a default setting of $k=4$, but values as high as $k=100$ are still fine.
I show how, by changing only a few lines of its implementation code, we can make it much more efficient. This new version is not only ~20x faster, but for the same value of $k$ now produces point distributions with approximately 40% higher point density.
Alternatively, one can increase the original $r_c$ by a factor of 1.2x, and then the output of both the original and the new algorithm will then have the same number of dots and the same dot distributions but the new one will run much faster.
Figure 4. The modified algorithm with k=100. This results in +42% more points than the original algorithm using the same parameters (k=100).
My name is Dr Martin Roberts, and I'm a freelance senior data science consultant, who loves working at the intersection of maths and computing.
"Transforming organizations through innovative data solutions."
Let's have a chat on how we can work together!
Come follow me on Twitter: @Techsparx!
My other contact details can be found here.
If you liked this post, I think you will also like these ones!
The Unreasonable Effectiveness of Quasirandom Sequences
Evenly distributing points on a sphere
A simple method to construct isotropic quasirandom sequences
By Martin Roberts No Comments | CommonCrawl |
\begin{document}
\title{ extbf{ Approximation of stationary solutions to SDEs driven by multiplicative fractional noise} \begin{abstract} In a previous paper, we studied the ergodic properties of an Euler scheme of a stochastic differential equation with a Gaussian additive noise in order to approximate the stationary regime of such an equation. We now consider the case of multiplicative noise when the Gaussian process is a fractional Brownian Motion with Hurst parameter $H>1/2$ and obtain some (functional) convergence properties of some empirical measures of the Euler scheme to the stationary solutions of such SDEs. \end{abstract}
\noindent \textit{Keywords}: stochastic differential equation; fractional Brownian motion; stationary process; Euler scheme.
\noindent \textit{AMS classification (2000)}: 60G10, 60G15, 60H35. \section{Introduction} \noindent Stochastic Differential Equations (SDEs) driven by a fractional Brownian motion (fBm) have been introduced to model random evolution phenomena whose noise has long range dependence properties. Indeed, beyond the historical motivations in Hydrology and Telecommunication for the use of fBm (highlighted e.g in \cite{MVn68}), recent applications of dynamical systems driven by this process include challenging issues in Finance \cite{Gua06}, Biotechnology~\cite{Odde-al96} or Biophysics~\cite{Jeon-al11,Kou08}. As a consequence, SDEs driven by fBm have been widely studied in a finite-time horizon during the last decades, and the reader is referred to~\cite{Nualart02,Coutin12} for nice overviews on this topic.
In a somehow different direction, the study of the long-time behavior (under some stability properties) for fractional SDEs has been developed by Hairer (see \cite{hairer,hairer09}) and Hairer and Ohashi \cite{hairer2}, who built a way to define stationary solutions of these a priori non-Markov processes and to extend some of the tools of the Markovian theory to this setting. See also~\cite{Arnold98,crauel,GKN09} for another setting called random dynamical systems. The current article fits into this global aim, and starts from the following observation: the knowledge of the stationary regime being important for applications and essentially inaccessible in an explicit form, we propose to build and to study a procedure for its approximation in the case of SDEs driven by fBm with a Hurst parameter $H>1/2$. This paper is following a similar previous work for SDEs driven by more general noises but in the specific additive case (see~\cite{cohen-panloup}).
More precisely, we deal with an $\ER^d$-valued process $(X_t)_{t\ge0}$ which is a solution to the following SDE \begin{equation}\label{fractionalSDE0} dX_t=b(X_t)dt+ \sigma(X_t)dB_t^H \end{equation} where $b:\ER^{d}\rightarrow\ER^d$ and $\sigma:\ER^{d}\rightarrow \mathbb{M}_{d,q} $ are (at least) continuous functions, and where $ \mathbb{M}_{d,q}$ is the set of $d\times q$ real matrices. In~\eqref{fractionalSDE0}, $(B^H_t)_{t\ge0}$ is a $q$-dimensional $H$-fBm and for the sake of simplicity we assume
$ \frac1{2} < H < 1$, which allows in particular to invoke Young integration techniques in order to define stochastic integrals with respect to $B^H$. Compared to~\cite{cohen-panloup} we handle here a fairly general diffusion coefficient $\sigma$, instead of the constant one considered previously. Classically the noise is called multiplicative in this setting, whereas it is called additive when $\sigma $ is constant.\\ Under some Hölder regularity assumptions on the coefficients (see Section \ref{mainresult} for details), (strong) existence and uniqueness hold for the solution to \eqref{fractionalSDE0} starting from $x_0\in\ER^d$. Classically for any stochastic differential equation, a natural question arises: if we assume that some Lyapunov assumptions hold on the drift term, does it imply that $(X_t)_{t\ge0}$ has some convergence properties to a steady state when $t\rightarrow+\infty$ ?\\ This question implies in particular to define rigorously a concept of steady state. For equation \eqref{fractionalSDE0}, this work has been done in \cite{hairer2}: using the fact that, owing to the Mandelbrot representation, the evolution of the fBm can be represented through a Feller transition on a functional space ${\cal S}$, the authors show that a solution to \eqref{fractionalSDE0} can be built as the first coordinate of an homogeneous Markov process on the product space $\ER^d\times {\cal S}$. As a consequence, stationary regimes associated with \eqref{fractionalSDE0} can be naturally defined as the first projection of invariant measures of this Markov process. Furthermore, the authors of \cite{hairer2} develop some specific theory on strong Feller and irreducibility properties to prove uniqueness of invariant measures in this context.\\ In the current article, our aim is to propose a way to approximate numerically the stationary solutions to equation \eqref{fractionalSDE0}. To this end, we study some empirical occupation measures related to an Euler type approximation of \eqref{fractionalSDE0} with step $\gamma > 0$. We show that, under some Lyapunov assumptions, this sequence of empirical measures converges almost surely to the distribution of the stationary solution of the discretized equation (denoted by $\nu^\gamma$) and that, when $\gamma\rightarrow0^+$, $\nu^\gamma$ converges in turn to the distribution of the stationary solution of \eqref{fractionalSDE0}. This approach is the same as in \cite{cohen-panloup}. However, the introduction of multiplicative noise has some important consequences on the techniques for proving the long-time stability of the Euler scheme. In particular, the main difficulty is to show that the long-time control of the dynamical system can be achieved independently of $\gamma$. In \cite{cohen-panloup}, this problem has been solved with the help of explicit computations for an Ornstein-Uhlenbeck type process. Because the noise is multiplicative the computations of~\cite{cohen-panloup} are not feasible anymore and we use specific tools to obtain uniforms controls of discretized integrals with respect to the fBm. Before going more precisely to the heart of the matter, let us mention that the numerical approximation of the stationary regime by occupation measures of Euler schemes is a classical problem in a Markov setting including diffusions and Lévy driven SDEs (see $e.g.$ \cite{talay,LP1,LP2,lemaire2,PP1,panloup1}).
\section{Framework and main results}\label{mainresult} This section is firstly devoted to specify the setting under which our computations will be performed. Namely, we give an account on differential equations driven by fractional Brownian motion and their related ergodic theory. Once this framework is recalled, we shall be able to state our main results.
\subsection{FBm and H\"older spaces} For some fixed $H\in(\frac{1}{2},1)$, we consider $(\Omega,\mathcal{F},\mathbb {P})$ the canonical probability space associated with the fractional Brownian motion indexed by $\ER$ with Hurst parameter $H$. That is, $\Omega=\mathcal{C}_0(\ER)$ is the Banach space of continuous functions vanishing at $0$ equipped with the supremum norm, $\mathcal{F}$ is the Borel sigma-algebra and $\mathbb {P}$ is the unique probability measure on $\Omega$ such that the canonical process $B^H=\{B^H_t=(B^{H,1}_t,\ldots,B^{H,q}_t), \; t\in \ER\}$ is a fractional Brownian motion with Hurst parameter $H$. In this context, let us recall that $B^H$ is a $q$-dimensional centered Gaussian process such that $B^H_0=0$, whose coordinates are independent and satisfy \begin{equation}\label{eq:var-increm-fbm}
\mathbb{E}\left[ \left( B_t^{H,j} -B_s^{H,j}\right)^2\right]= |t-s|^{2H}, \quad\mbox{for}\quad s,t\in\ER. \end{equation} In particular it can be shown, by a standard application of Kolmogorov's criterion, that $B^H$ admits a continuous version whose paths are $\theta$-H\"older continuous for any $\theta<H$.
Let us be more specific about the definition of H\"older spaces of continuous functions. Namely, our driving process $B^H$ lies into a space ${\cal C}^\theta$ defined as follows: we denote by ${\cal C}^\theta(\ER_+,\ER^d)$ the set of functions $f:\ER_+\rightarrow\ER^d$ such that
$$\forall T>0,\quad \|f\|_{\theta,T}=\sup_{0\le s< t\le T}\frac{ |f(t)-f(s)|}{(t-s)^\theta}<+\infty,$$
where the Euclidean norm is denoted by $|\, .\,|$. We recall that ${\cal C}^\theta(\ER_+,\ER^d)$ can be made into a non-separable complete metric space, whenever endowed with the distance $\delta_{\theta}$ defined by
$$\delta_{\theta}(f,g)=\sum_{N\in\mathbb{N}}2^{-N}\left(1\wedge\left(\sup_{0\le t\le N}\|f(t)-g(t)\|+\|f-g\|_{\theta,N}\right)\right),$$ where $ x \wedge y =\min(x,y) \;\forall x,\,y \in \mathbb R.$ However, since separable spaces are crucial for convergence in law issues, we will work in fact with a smaller space $\bar{\cal C}^\theta(\ER_+,\ER^d)$: we say that a function $f$ in ${\cal C}^\theta(\ER_+,\ER^d)$ belongs to $\bar{\cal C}^\theta(\ER_+,\ER^d)$ if \begin{equation}\label{eq:def-bar-C-theta}
\forall\, T>0,\quad \omega_{\theta,T}(f,\delta):=\sup_{0\le s<t<T,0\le |t-s|\le\delta}\frac{|f(t)-f(s)|}{|t-s|^\theta}\xrightarrow{\delta\rightarrow0}0. \end{equation} $\bar{\cal C}^\theta(\ER_+,\ER^d)$ is a closed separable subspace of ${\cal C}^\theta(\ER_+,\ER^d)$.
\subsection{Differential equations driven by fBm}
We recall now some results on existence and uniqueness of solutions of the stochastic differential equation~\eqref{fractionalSDE0} starting from a deterministic point.
When $B^{H}$ is a fractional Brownian motion with Hurst parameter $H>1/2$, equations of the form~\eqref{fractionalSDE0} are classically solved by interpreting the stochastic integral $\int_{0}^{t}\sigma(X_{u})\, dB_{u}^{H}$ as a Young integral (see e.g \cite{FV-bk}). The usual set of assumptions on the coefficients $b$ and $\sigma$ are then of Lipschitz and boundedness types.
Specifically, we recall the following definition of a $(1+\alpha)$-Lipschitz function: \begin{definition} Let $\sigma:\ER^{d}\rightarrow \mathbb{M}_{d,q} $ be a ${\cal C}^{1}$ function and $ 0 < \alpha < 1$. We say that $\sigma $ is $(1+\alpha)$-Lipschitz if the following norm is finite: \begin{equation}
\label{eq:1+gammaLips}
\|\sigma\|_{1 + \alpha}= \sup_{ x \in \mathbb R^d } \|D \sigma(x) \| + \sup_{x, y \in \mathbb R^d}\frac{|D \sigma(x)-D \sigma(y)|}{|x-y|^\alpha}. \end{equation} \end{definition} With this definition the basic existence and uniqueness result in a finite horizon $[0,T]$ for $T>0$ for pathwise equations driven by $\theta$-H\"older functions with $\theta>1/2$ can be found in~\cite{Coutin12,Lyons94}. Nevertheless in this article we are searching for stationary solutions, which have to be defined on $\ER_+.$ Moreover we use ergodic results that require some damping effect of the continuous drift coefficient $b$. In order to quantify this notion, let us now introduce a long-time stability assumption $(\mathbf{C})$. Namely, let ${\cal E}\!{\cal Q}(\ER^d)$ denote the set of {\em Essentially Quadratic} functions, that is ${\cal C}^2$-functions $V:\ER^d\rightarrow (0,\infty)$ such that \[
\liminf_{|x|\rightarrow+\infty} \frac{V(x)}{|x|^2}>0, \qquad \abs[\nabla V]\le C \sqrt{V}\quad \mbox{ and }\quad D^2V \mbox{ is bounded.}
\] \noindent Note that any element $V\in {\cal E}\!{\cal Q}(\ER^d)$ is continuous, and thus attains its positive minimum $\underline{v}>0$ so that, for any $A, r>0$, there exists a real constant $C_{_{A,r}}$ such that $A+V^r \le C_{_{A,r}}V^r$.
With these notions in mind, our standing assumptions on the coefficients $b$ and $\sigma$ are summarized as:
\noindent $\mathbf{(C)}$ The map $\sigma$ is assumed to be a bounded Lipschitz continuous function. Moreover we suppose that there exists $V\in {\cal E}\!{\cal Q}(\ER^d)$ such that
\begin{itemize}
\item[(i)]$ \forall x\in\ER^d \quad |b(x)|^2\le V(x) \; , $
\item[(ii)] and such that for $\beta\in\ER$ and $\alpha>0$ the following relation holds:
$$\forall x\in\ER^d\quad \langle \nabla V(x),b(x)\rangle \le \beta-\alpha V(x).$$
\end{itemize}
\begin{prop}\label{prop:exist-uniq-smooth-coeff} Let us suppose that in addition to assumption $(\mathbf{C}),$ $ b $ is Lipschitz continuous and that $\sigma$ is $(1+\alpha)$-Lipschitz with $ \alpha > \frac1{H} - 1$. Then
\noindent\emph{(i)} For any deterministic function $ B \in {\cal C}^\theta(\ER_+,\ER^q) $ with $ \theta > \frac12, $ and any $ x_0 \in \ER^d ,$ there exists a unique solution $X\in {\cal C}^\theta(\ER_+,\ER^d) $ of \begin{equation}
\label{eq:sd-det}
X_t= x_0 + \int_{0}^{t} b(X_u) du + \int_{0}^{t} \sigma(X_u)dB_u, \end{equation} where the integrals are interpreted in the Riemann-Stieljes sense.
\noindent\emph{(ii)} Let us set $X\equiv \Phi(x_0,B)$, so that $ \Phi(x_0,B) $ satisfies $$ \Phi(x_0,B)_t = x_0 + \int_0^t b(\Phi(x_0,B)_s) ds + \int_0^t \sigma(\Phi(x_0,B)_s) d B_s. $$ Then the so-called Itô map $ \Phi $ is continuous from $ \ER^d \times {\cal C}^\theta(\ER_+,\ER^q) $ into $ {\cal C}^\theta(\ER_+,\ER^d)$. \end{prop} \begin{Remarque} Proposition \ref{prop:exist-uniq-smooth-coeff} is not completely standard, when $b$ is not bounded, and we haven't been able to find a specific reference giving an equivalent statement in the literature. Namely the case of bounded smooth coefficients $b$ and $\sigma$ is handled e.g in~\cite{Coutin12,Lyons94}. If we move to the case of a dissipative coefficient $b$, an existence and uniqueness result is available in~\cite{hairer2}. Nevertheless, this result also assumes that the derivatives of $b$ are bounded. Assumption $(\mathbf{C})\emph{(i)}$ implies that $b$ is sublinear.With the boundedness and Lipschitz assumption on $\sigma$ assumed in $(\mathbf{C}),$ the proof of the existence of a global solution of this stochastic equation and of the continuity of the Itô map is a consequence of Young and Gronwall inequalities. \end{Remarque}
\noindent \subsection{Ergodic theory for SDEs driven by fBm} We can now define the solution of the stochastic differential equation starting from a random variable $X_0.$ Since the It\^o map of Proposition~\ref{prop:exist-uniq-smooth-coeff} is used in the following definition we have to suppose that in addition to assumption $(\mathbf{C}),$ $ b $ is Lipschitz continuous and that $\sigma$ is $(1+\alpha)$-Lipschitz with $ \alpha > \frac1{H} - 1$. \begin{definition} \label{def:sol-eds-random-initial-condition} Let $B^H$ be a fractional Brownian motion with $H> \frac12.$ A process $(X_t)_{t \in \ER_+ }$ is called a solution of equation \eqref{fractionalSDE0} driven by $B^{H}$ starting at $ X_0,$ if for every $ 1/2 < \theta < H < 1,$ $(X_t)_{t \in \ER_+ }$ is almost surely ${\cal C}^\theta(\ER_+,\ER^d)$-valued and if $X= \Phi(X_0,B^H)$, almost surely. \end{definition} We now have all the tools to define rigorously a stationary solution to the SDEs driven by fBm. In the following definition and further on we use the notation $\theta_t:\omega\mapsto\omega(t+.)$ for every $t\ge0$ for the time-shift . \begin{definition} \label{def:stat-sol} Let $(X_t)_{t\ge0}$ denote an $\ER^d$-valued solution to \eqref{fractionalSDE0} in the sense of Definition~\ref{def:sol-eds-random-initial-condition}. Let $\nu$ denote the distribution of $(X_t)_{t\ge0}$ on ${\cal C}^\theta(\ER_+,\ER^d).$ Then, $\nu$ is called a stationary solution of \eqref{fractionalSDE0} if it is invariant under the time-shift. Such a stationary solution is called adapted, if for $ 0 \le t $ the processes $(X_s)_{0\le s\le t}$ and $ (B^H_{s})_{s \ge t} $ are conditionally independent given $ (B^H_{s})_{s \le t} $. \end{definition} Please note that there is an abuse of language in the preceding definition. The distribution of a process $(X_t)_{t\ge0}$ on ${\cal C}^\theta(\ER_+,\ER^d)$ cannot determine alone if $(X_t)_{t\ge0}$ is a solution of \eqref{fractionalSDE0} in the sense of Definition~\ref{def:sol-eds-random-initial-condition}. We need the distribution of the pair $(X_t,B^H)_{t\ge0}$ to know if $X= \Phi(X_0,B^H)$, almost surely. In particular it is not possible to take $X_0$ independent of $(B^H)_{t\ge0}$ in general as remarked in Proposition 5 of~\cite{cohen-panloup}. Nevertheless we consider as in the Definition 2.4 in~\cite{hairer2} that two distributions $(X^1_t,B^H)_{t\ge0})$ and $ (X^2_t,B^H)_{t\ge0})$ on ${\cal C}^\theta(\ER_+,\ER^d) \times {\cal C}^\theta(\ER_+,\ER^q) $ solutions of \eqref{fractionalSDE0} are equivalent if the distribution of $ X^1$ and of $X^2$ are the same. These definitions are the same as definitions in~\cite{hairer2} that come from Stochastic Dynamical Systems (SDS). In particular, we require adaptedness of solutions. Compared to Random Dynamical Systems (RDS) (see~\cite{Arnold98} for an introduction), this property is specific to SDS and is strongly linked to the fact that for such dynamical systems, one can associate a Markovian structure (with an enlargement of the space). Here, the main consequence is that the uniqueness of the stationary solution can be obtained through the criterions of uniqueness of the invariant distribution of this associated Markov process. Such results will be stated later.
\noindent Let $\gamma$ be a positive number, we will now discretize equation~\eqref{fractionalSDE0} as follows, for every $n \ge 0,$
\begin{equation} \label{fractionalSDE0-disc}
Y_t^{\gamma}=Y_{{n\gamma}}^{\gamma}+{(t-n\gamma)}b(Y_{n\gamma}^{\gamma})+\sigma(Y_{n\gamma}^{\gamma})(B^H_t-B^H_{n\gamma})\quad\forall t\in[n\gamma,(n+1)\gamma).
\end{equation} We set $$\underline{t}_{\gamma}= \max\{\gamma k,\gamma k\le t,k\in\EN\}.$$ In fact, we will usually write $\underline{t}$ instead of $\underline{t}_{\gamma}$ in the sequel. The discretization of~\eqref{fractionalSDE0} can also be introduced with the following discretization $ \Phi^{\gamma}: \ER^d \times {\cal C}^\theta(\ER_+,\ER^q) \mapsto {\cal C}^\theta(\ER_+,\ER^d) $ of the Itô map~: \begin{equation}
\label{eq:phi-gamma}
\Phi^{\gamma}(x_0,B)_t := x_0 + \int_0^t b(\Phi^{\gamma}(x_0,B)_{\underline{s}_{\gamma}}) ds + \int_0^t \sigma(\Phi^{\gamma}(x_0,B)_{\underline{s}_{\gamma}}) d B_s. \end{equation}
Please note that the definition of $ \Phi^{\gamma}$ does not involve any Riemann integration but only finite sums and that \begin{equation}\label{eq:phi-gamma2}
Y^{\gamma}=\Phi^{\gamma}(Y_0^{\gamma},(B^H_t)_{t\ge0}) \;\; a.s.
\end{equation} \noindent We now define \textit{stationary adapted solutions of \eqref{fractionalSDE0-disc}} in the spirit of the Definition \ref{def:stat-sol}. \begin{definition} \label{def:stat-sol-disc} Let $B^H$ denote a fractional Brownian motion with $H>1/2$ and let $X^\gamma$ be defined by $X^{\gamma}=\Phi^{\gamma}(X_0^{\gamma},(B^H_t)_{t\ge0})$. The distribution $\nu^\gamma$ of $X^\gamma$ on ${\cal C}^\theta(\ER_+,\ER^d)$ is then called an adapted solution of \eqref{fractionalSDE0-disc} if the processes $(X_s^\gamma)_{0\le s\le t}$ and $ (B^H_{s})_{s \ge t} $ are conditionally independent given $ (B^H_{s})_{s \le t} $. We will say that $\nu^\gamma$ is stationary if it is invariant by the shift maps $(\theta_{k\gamma})_{k\in\EN}.$ \end{definition}
Note that in this definition, there is a slight abuse of language since we do not require the invariance by the shift maps $\theta_{t}$ for every $t\ge0$, but only when $t=k\gamma$, $k\in\EN$.
Let us introduce the following uniqueness assumption for $\nu^\gamma$ and $\nu$:
\noindent $(\mathbf{S^\gamma})$ $(\gamma\ge0)$: There is at most one adapted stationary solution to \eqref{fractionalSDE0} (resp. to \eqref{eq:phi-gamma2}) if $\gamma=0$ (resp. if $\gamma>0$).
\noindent For $(\mathbf{S^0})$, we refer to Theorem 1.1. of \cite{hairer2}. When $\gamma>0$, we have the following proposition: \begin{prop} \label{unicitemesinvariante} Let $H\in(1/2,1)$. Assume that $d=q$ and that $b$ and $\sigma$ are ${\cal C}^2$-functions. Assume that $\sigma$ is invertible and that $\sup_{x\in\ER^d} \sigma^{-1}(x)<+\infty$. Then, $(\mathbf{S^\gamma})$ holds for every $\gamma>0$. \end{prop} \noindent The proof, which is an application of \cite{hairer09}, is done in the appendix.
\noindent Let us now focus on the construction of the approximation. We denote by $(\bar{X}_t^{\gamma})_{t\ge0}$ the continuous-time Euler scheme defined by $\bar{X}_0^\gamma=x\in\ER^d$ and for every $n\ge0$ \begin{equation}
\label{eq:euler-scheme-x}
\bar{X}_t^{\gamma}=\bar{X}_{{n\gamma}}^{\gamma}+{(t-n\gamma)}b(\bar{X}_{n\gamma}^{\gamma})+\sigma(\bar{X}_{n\gamma}^{\gamma})(B^H_t-B^H_{n\gamma}) \quad\forall t\in[{n\gamma},{(n+1)\gamma}). \end{equation} The process $ (\bar{X}_t^{\gamma})_{t\ge0}$ is a solution to~\eqref{fractionalSDE0-disc} such that $ \bar{X}_0^{\gamma}=x.$ In order to alleviate the notations and, when it is not confusing, we will usually write $\bar{X}_{t}$ instead of $\bar{X}_t^{\gamma}$. Now, we define a sequence of random
probability measures $({\cal P}^{(n,\gamma)}(\omega,d\alpha))_{n\ge 1}$ on $\bar{{\mathcal C}}^\theta(\ER_+,\ER^d)$ with $\theta<H$ (recall that $\bar{{\mathcal C}}^\theta(\ER_+,\ER^d)$ is defined at \eqref{eq:def-bar-C-theta}) by
$${\cal P}^{(n,\gamma)}(\omega,d\alpha)=\frac{1}{n}\sum_{k=1}^n
{\delta}_{{\bar{X}}^\gamma_{\gamma(k-1)+.}(\omega)} (d\alpha) $$ where
$\delta$ denotes the Dirac measure and where, for every $s\ge0$, ${\bar{X}}^\gamma_{s+.}:=({\bar{X}}^\gamma_{s+t})_{t\ge0}$ denotes the $s$-shifted process.\\
We are now able to state the main theorem of this article: \begin{theorem}\label{principal1} Let $1/2< \theta < H < 1$ and assume $\mathbf{(C)}.$ If $\mathbf{(S^\gamma)}$ holds for every $\gamma>0,$\\ \noindent (i) then there exists $\gamma_0>0$ such that, for every $\gamma\in(0,\gamma_0)$, $$\lim_{n\rightarrow+\infty}{\cal P}^{(n,\gamma)}(\omega,d\alpha)=\nu^\gamma(d\alpha)\quad a.s. \; \textnormal{ when $n\rightarrow+\infty$},$$ where the convergence is for the weak topology induced by $\bar{{\mathcal C}}^\theta(\ER_+,\ER^d)$ and where $\nu^\gamma$ is the stationary solution of \eqref{fractionalSDE0-disc}.\\ \noindent (ii) If additionally, $b$ is Lipschitz continuous, $\sigma$ is $(1+\alpha)$-Lipschitz with $\alpha>\frac{1}{H}-1$ and if $\mathbf{(S^0)}$ holds, then $$\lim_{\gamma\rightarrow0} \nu^\gamma(d\alpha) =\nu(d\alpha)\quad a.s.$$ where the convergence is for the weak topology induced by $\bar{{\mathcal C}}^\theta(\ER_+,\ER^d)$ and where $\nu$ denotes the adapted stationary solution of \eqref{fractionalSDE0}. \end{theorem}
\begin{Remarque} Note that some extensions can be deduced from the proof of this theorem. First, remark that this result implies in particular that $$\lim_{\gamma\rightarrow0^+}\,\lim_{n\rightarrow+\infty}{\cal P}_0^{(n,\gamma)}(\omega,dy)=\nu_0(dy)\quad a.s.$$ where $${\cal P}_0^{(n,\gamma)}(\omega,dy)=\frac{1}{n}\sum_{k=1}^n\delta_{\bar{X}^\gamma_{(k-1)\gamma}(dy)}$$ and $\nu_0(dy)$ denotes the initial distribution of the stationary solution $\nu$ of \eqref{fractionalSDE0}. This marginal procedure will be numerically tested in Section 6.\\ Also note that some extensions can be deduced from the proof of this theorem. First, when uniqueness fails for the stationary solutions, the preceding result is replaced by
\begin{theorem}\label{principal} Assume $\mathbf{(C)}$.
\noindent 1. Then, there exists $\gamma_0>0$ such that for every $\gamma\in(0,\gamma_0)$, $({\cal P}^{(n,\gamma)}(\omega,d\alpha))_{n\ge1}$ is $a.s.$ tight on $\bar{{\mathcal C}}^\theta(\ER_+,\ER^d),$ for every $1/2< \theta < H < 1$. Furthermore,
every weak limit is a stationary adapted solution of~\eqref{fractionalSDE0-disc}.\\
2. If additionally, $b$ is Lipschitz continuous, $\sigma$ is $(1+\alpha)$-Lipschitz with $\alpha>\frac{1}{H}-1,$ set
$${\cal U}^{\infty,\gamma}(\omega):=\{\textnormal{weak limits of $({\cal P}^{(n,\gamma)}(\omega,d\alpha))$}\}.$$
Then there exists $\gamma_1\in(0,\gamma_0)$ such that $({\cal U}^{\infty,\gamma}(\omega))_{\gamma\le\gamma_1}$ is a.s. tight in $\bar{{\mathcal C}}^\theta(\ER_+,\ER^d),$ and any weak limit when $\gamma\rightarrow0$ of $({\cal U}^{\infty,\gamma}(\omega))_{\gamma\le\gamma_1}$ is an adapted stationary solution of~\eqref{fractionalSDE0}.
\end{theorem}
\end{Remarque} \begin{Remarque} From the very definition of weak convergence, the preceding assertions imply that the convergence of $({\cal P}^{(n,\gamma)}(\omega,d\alpha))_{n,\gamma}$ holds for bounded continuous functionals $F:\bar{{\mathcal C}}^\theta(\ER_+,\ER^d)\rightarrow\ER$. In fact, this convergence can be extended for arbitrary $T > 0$ to some non-bounded continuous functionals $F: \bar{{\mathcal C}}^\theta([0,T],\ER^d)\rightarrow\ER$. Actually, setting $G(\alpha)=\sup_{t\in[0,T]}V(\alpha_t)$, we easily deduce from inequality \eqref{eq:sup-V} of Proposition \ref{lemme4} and Proposition \ref{prop:ntendinfty} that $$\sup_{\gamma\le\gamma_0}\limsup_{n\rightarrow+\infty}{\cal P}^{(n,\gamma)}(\omega,G^p)<+\infty\quad a.s.$$ for every $p>0$. By a uniform integrability argument, it follows \begin{prop} The convergence properties of $({\cal P}^{(n,\gamma)}(\omega,d\alpha))$ extend to continuous functionals $F: \bar{{\mathcal C}}^\theta([0,T],\ER^d)\rightarrow\ER$ such that there exists a constant $C$ such that for every $\alpha\in\bar{{\mathcal C}}^\theta([0,T],\ER^d)$,
$$ |F(\alpha_t,0\le t\le T)|\le C\sup_{t\in[0,T]}V^p(\alpha_t)$$
with $T>0$ and $p>0$. \end{prop} \end{Remarque}
\begin{Remarque} A third natural extension of Theorem \ref{principal1} consists in handling the case of an irregular fractional Brownian motion $B$ with Hurst index $1/4<H<1/2$. This extension is presumably within the reach of our technology on differential systems driven by fBm, but requires a huge amount of technical elaboration. Indeed, to start with, equation \eqref{fractionalSDE0} has to be defined thanks to rough paths techniques whenever $H<1/2$, and we refer to \cite{FV-bk} for a complete account on rough differential equations driven by Gaussian processes in general and fractional Brownian motion in particular. More importantly, as it will be observed in the next sections, our main result heavily relies on some thorough estimates performed on the discretized version \eqref{fractionalSDE0-disc} of equation \eqref{fractionalSDE0}. When $H>1/2$ this discretization procedure is based on an Euler type scheme, but the case $H<1/2$ involves the introduction of some L\'evy area correction terms of Milstein type (see \cite{Da07}) or products of increments of $B^H$ if one desires to deal with an implementable numerical scheme (cf. \cite{DNT}). This new setting has tremendous effects on the proof of Propositions \ref{lemme4} and \ref{prop:ntendinfty}. For sake of conciseness, we have thus decided to stick to the case $H>1/2$, and defer the rough case to a subsequent publication.
\end{Remarque}
The sequel of the paper is built as follows. The three next sections are devoted to the proof of Theorem \ref{principal1}. In Section \ref{section3}, we prove some preliminary results for the long-time stability of $({\cal P}^{(n,\gamma)}(\omega,d\alpha))_{n},$ when $ \gamma > 0.$ It is important to note that the controls established in this section are independent of $\gamma$ in order to obtain in the sequel a long-time control that does not explode when $\gamma\rightarrow0$. Then, in Section \ref{section4}, we obtain some tightness properties for $({\cal P}^{(n,\gamma)}(\omega,d\alpha))$ (in $n$ and $\gamma$) and, in Section \ref{section5}, we prove that the weak limits of this sequence are adapted stationary solutions. Eventually, in Section \ref{section6}, we test numerically our algorithm for the approximation of the invariant distribution of a particular fractional SDE.\\ Note that in the proofs below, non-explicit constants are usually denoted by $C$ or $C_T$ (if a dependence to $T$ needs to be emphasized) and may change from line to line. \section{Evolution control of $(\bar{X}_t^\gamma)$ in a finite horizon}\label{section3} The main aim of this part is to obtain a finite-time control of $V(\bar{X}_T^\gamma)$ in terms of $V(\bar{X}_0^\gamma)$ which is independent of $\gamma$. This is the purpose of the first part of Proposition \ref{lemme4} below. In order to obtain some functional convergence results, we state in the second part a result about the finite-time control of the Hölder semi-norm of $\bar{X}^\gamma$.
\begin{prop}\label{lemme4} Let $T>0$. Assume $\mathbf{(C)}$. Then, \\ (i) For every $ p \ge 1,$ there exist $\gamma_0>0$, $\rho\in(0,1)$ and a polynomial function $P_{p,\theta}:\ER\rightarrow\ER$ such that for every $\gamma\in(0,\gamma_0]$, \begin{align}\label{VX}
V^p(\bar{X}_T^\gamma)\le \rho V^p(x)+P_{p,\theta}(\|B^H\|_{\theta,T}). \end{align}
Furthermore, \begin{align} \label{eq:sup-V}
\sup_{t\in[0,T]} V^p(\bar{X}_t^\gamma)\le C\left( V^p(x)+P_{p,\theta}(\|B^H\|_{\theta,T})\right). \end{align} (ii) For every $\theta\in(\frac{1}{2},H)$, $T>0$, and $\gamma\in(0,\gamma_0]$ \begin{equation}\label{eq:holder-bnd-with-V}
\sup_{0\le s<t\le T}\frac{|\bar{X}^\gamma_t -\bar{X}^\gamma_s|}{(t-s)^\theta}\le C_T\left(V(x)+\tilde{P}_\theta(\|B^H\|_{\theta,T})\right), \end{equation} where $\tilde{P}$ is another real valued polynomial function. \end{prop} The proof of this result is achieved in Subsection \ref{ss:22}. Before, we focus in Subsection \ref{ss:21} on the control of increments of some discretized equations with non-bounded coefficients driven by $B^H.$ \subsection{Technical Lemmas}\label{ss:21} Let us recall that, for every $t\ge0$, $\underline{t}_{\gamma}=\gamma \max\{k\in\EN,\gamma k\le t\}.$ In the sequel, we will usually write $\underline{t}$ instead of $\underline{t}_\gamma$.
In the following lemmas, we will use the following notation: for any element $(x(t))_{t\ge0}$ of ${\cal C}(\ER_+,\ER^d)$ and $T>0,$ $\theta>0$, $\gamma>0$, we define
$$\|x\|_{\theta,\gamma}^{s,t}=\sup_{s\le u\le v\le t}\frac{|x(\underline{v}_\gamma)-x( \underline{u}_\gamma)|}{(\underline{v}_\gamma-\underline{u}_\gamma)^\theta},$$ where we set by convention $\frac00 =0.$
\begin{lemme}\label{gronwall}
Assume that $b$ is a sublinear function, $i.e.$ that there exists $C>0$ such that for every $x\in\ER^d$, $|b(x)|\le C(1+|x|)$. Then, for every $T>0$, there exists a constant $C>0$ such that for every $s,t\in[0,T]$ with $s\le t$, for every $\gamma>0$, for every $\theta\in(0,H)$
$$|\bar{X}_{{\underline{t}}}^{\gamma}|\le \left(|\bar{X}^\gamma_{\underline{s}}|+C({\underline{t}}-{\underline{s}})+\|\bar{Z}^\gamma\|_{\theta,\gamma}^{{s},{t}}(\underline{t}-\underline{s})^\theta \right)\exp(C(\underline{t}-\underline{s})))$$ where $$\bar{Z}^\gamma_t=\int_0^t\sigma(\bar{X}^{\gamma}_{{\underline{s}}})dB^H_s.$$ \end{lemme} \begin{proof} First, from the very definition of $(\bar{X}^{\gamma}_t)_{t\ge0}$, we have for every $s,t\in[0,T]$ with $s\le t$: \begin{equation}\label{eq:ceuler} \bar{X}^{\gamma}_{\underline{t}}=\bar{X}^{\gamma}_{\underline{s}}+\int_{\underline{s}}^{\underline{t}} b(\bar{X}^{\gamma}_{\underline{u}})du+\bar{Z}^{\gamma}_{\underline{t}}-\bar{Z}^{\gamma}_{\underline{s}}. \end{equation} The function $b$ being sublinear, we deduce that
$$|\bar{X}^{\gamma}_{\underline{t}}|=|\bar{X}^{\gamma}_{\underline{s}}|+\|\bar{Z}^\gamma\|_{\theta,\gamma}^{{s},{t}}(\underline{t}-\underline{s})^\theta+C \int_{\underline{s}}^{\underline{t}}(1+|\bar{X}^{\gamma}_{\underline{u}}|)du.$$
Setting $g_s(v)=|\bar{X}_{\underline{s}+v}|$, it follows that for every $v\in[0,\underline{t}-\underline{s}]$, $$g_{{s}}(v)\le a+C \int_0^v g_{{s}}(u)du$$ with
$a=|\bar{X}^{\gamma}_{\underline{s}}|+\|\bar{Z}^\gamma\|_{\theta,\gamma}^{{s},{t}}(\underline{t}-\underline{s})^\theta +C(\underline{t}-\underline{s}).$ The result follows from Gronwall's lemma. \end{proof}
The control of $B^H$-integrals is usually based on the so-called sewing Lemma (see $e.g.$ \cite{Coutin12,Feyel06}) which leads to a comparison of $\int_s^t f(x_u) dB_u^H$ with $f(x_t)(B^H_t-B^H_s)$. The following lemma can be viewed as a discretized version of such results: \begin{lemme}\label{lemme2} Assume that $b$ is a sublinear function. Let $\gamma_0>0$ and $(f_\gamma)_{\gamma\in(0,\gamma_0]}$ be a family of functions from $\ER_+\times\ER^{d}$ to $\mathbb{M}_{d,q}$ such that there exists $r\ge 0$ such that for every $T>0$, there exists $C_T>0$ such that $\forall\gamma\in(0,\gamma_0],$ \begin{equation}\label{assump:f}
\;\forall (s,x),(t,y)\in[0,T]\times\ER^{d},\quad \|f_\gamma(t,y)-f_\gamma(s,x)\|\le C_T(1+|x|^r+|y|^r)(|t-s|+|y-x|). \end{equation} Let $(\bar{H}_t^{\gamma})_{t\ge0}$ be defined by $$\forall t\ge0,\quad \bar{H}_t^{\gamma}=\int_0^t f_\gamma(\underline{s}, \bar{X}_{\underline{s}}^\gamma)d B^H_s.$$ Then, for every $\theta\in(\frac{1}{2},H)$, for every $T>0$, there exists $\tilde{C}_T>0$ such that for every $\gamma\in(0,\gamma_0]$, for every $0\le s\le t\le T$, \begin{equation}
\label{eq:young}
|\bar{H}_{\underline{t}}^{\gamma}-\bar{H}_{\underline{s}}^{\gamma}-f_\gamma({\underline{s}},\bar{X}^\gamma_{{\underline{s}}})(B^H_{\underline{t}}-B^H_{\underline{s}})|\le\tilde{C}_T(\underline{t}-\underline{s})^{2\theta}((1+|\bar{X}_{\underline{s}}^\gamma|^{r+1}+(\|\bar{Z}^\gamma\|_{\theta,\gamma}^{\underline{s},\underline{t}-\gamma})^{r+1})\|B^H\|_{\theta,T}. \end{equation} \end{lemme} \begin{proof} Denoting by $\tilde{f}_{\gamma,\omega}$ the (random) function on $\ER_+$ $a.s.$ defined by $\tilde{f}_{\gamma,\omega}(s)=f_\gamma(\underline{s},\bar{X}_{\underline{s}}^\gamma(\omega))$, we can write: $$\bar{H}_{\underline{t}}^{\gamma}-\bar{H}_{\underline{s}}^{\gamma}-f_\gamma({\underline{s}},\bar{X}^\gamma_{{\underline{s}}})(B^H_{\underline{t}}-B^H_{\underline{s}})= \int_{\underline{s}}^{\underline{t}} \tilde{f}_{\gamma,\omega}(u)-\tilde{f}_{\gamma,\omega}(s) d B_u^H.$$ Let $\theta\in(1/2,H)$ (so that $2\theta>1$). We use a classical Young estimate (see $e.g.$ \cite{Young36}, Inequality (10.9)), to get a upper bound for the left hand side of~\eqref{eq:young}. Let us recall the definition
of $p$-variations. For every $u, v\ge 0$ such that $u\le v$, for every $p>0$ and for every function $f:\ER_+\rightarrow\ER^d$, $$V_p(f,u,v)=\sup(\sum_{i=1}^n |f(t_i)-f(t_{i-1})|^p)^\frac{1}{p},$$ the supremum being taken over all subdivisions $(t_i)$ of $[u,v]$: $u=t_0< t_1< \ldots< t_n=v$. Then using Young inequality we get \begin{equation}\label{youngestimate}
|\bar{H}_{\underline{t}}^{\gamma}-\bar{H}_{\underline{s}}^{\gamma}-f_\gamma({\underline{s}},\bar{X}^\gamma_{{\underline{s}}})(B^H_{\underline{t}}-B^H_{\underline{s}})| \le C V_{\frac{1}{\theta}} (\tilde{f}_{\gamma,\omega},\underline{s},\underline{t}-\gamma)V_{\frac{1}{\theta}} (B^H,\underline{s},\underline{t}) \end{equation} where $C$ depends only on $\theta.$ Note that we could write $V_{\frac{1}{\theta}} (\tilde{f}_{\gamma,\omega},\underline{s},\underline{t}-\gamma)$ instead of $V_{\frac{1}{\theta}} (\tilde{f}_{\gamma,\omega},\underline{s},\underline{t})$ since $\tilde{f}_{\gamma,\omega}$ is constant on $[\underline{t}-\gamma,\underline{t})$. We now control separately the two terms on the right-hand member.\\ Let $T>0$. Since for every $u,v\in[0,T]$,
$$|B^H_{v}-B^H_{u}|\le \|B^H\|_{\theta,T}|v-u|^\theta,$$ we first obtain that \begin{equation}\label{pvariationB}
V_{\frac{1}{\theta}} (B^H,\underline{s},\underline{t})\le \|B^H\|_{\theta,T} (\underline{t}-\underline{s})^\theta. \end{equation} Second, let $s,t\in[0,T]$ such that $s\le t$ and consider a subdivision $(t_i)_{i=1}^n$ of $[\underline{s},\underline{t}-\gamma]$. By \eqref{assump:f}, we have
\begin{align*}
|\tilde{f}_{\gamma,\omega}(t_{i+1})-\tilde{f}_{\gamma,\omega}(t_i)|\le C_T(1+|\bar{X}_{\underline{t_i}}^\gamma|^r+|\bar{X}_{\underline{t_{i+1}}}^\gamma|^r)(|\underline{t_{i+1}}-\underline{t_i}|+|\bar{X}_{\underline{t_{i+1}}}^\gamma-\bar{X}_{\underline{t_i}}^\gamma|). \end{align*} On the one hand, it follows from Lemma \ref{gronwall} that
$$1+|\bar{X}_{\underline{t_i}}^\gamma|^r+|\bar{X}_{\underline{t_{i+1}}}^\gamma|^r\le C_T\left(1+{|\bar{X}_{\underline{s}}^\gamma|}^r+(\|\bar{Z}^{\gamma}\|^{\underline{s},\underline{t}-\gamma}_{\theta,\gamma})^r\right).$$ On the other hand, since $b$ is a sublinear function, we have
$$|\bar{X}_{\underline{t_{i+1}}}^\gamma-\bar{X}_{\underline{t_i}}^\gamma|\le C_T(1+\int_{{t_i}}^{{t_{i+1}}}|\bar{X}_{\underline{u}}^\gamma|du+ |\bar{Z}^{\gamma}_{\underline{t_{i+1}}}-\bar{Z}^{\gamma}_{\underline{t_{i}}}|).$$
Then, using again Lemma \ref{gronwall} and the definition of $\|.\|^{\underline{s},\underline{t}-\gamma}_{\theta,\gamma}$, it follows that
$$ |\bar{X}_{\underline{t_{i+1}}}^\gamma-\bar{X}_{\underline{t_i}}^\gamma| \le C_T\left(1+|\bar{X}^\gamma_{\underline{s}}|+\|\bar{Z}^\gamma\|_{\theta,\gamma}^{{\underline{s}},\underline{t}-\gamma}\right)(\underline{t_{i+1}}-\underline{t_i})
+\|\bar{Z}^\gamma\|_{\theta,\gamma}^{{\underline{s}},\underline{t}-\gamma}(\underline{t_{i+1}}-\underline{t_i})^\theta. $$ By a combination of the previous inequalities (and by the use of the Young inequality), we obtain \begin{align*}
|\tilde{f}_{\gamma,\omega}(t_{i+1})-\tilde{f}_{\gamma,\omega}(t_i)|\le C_T(1+|\bar{X}_{\underline{s}}^\gamma|^{r+1}+(\|\bar{Z}^{\gamma}\|^{\underline{s},\underline{t}-\gamma}_{\theta,\gamma})^{r+1})|\underline{t_{i+1}}-\underline{t_i}|^\theta. \end{align*} Since $ \sum_i (\underline{t_{i+1}}-\underline{t_i}) \le \underline{t}- \underline{s},$ we deduce that $$V_{\frac{1}{\theta}} (\tilde{f}_{\gamma,\omega},\underline{s},\underline{t}-\gamma)
\le C_T(1+|\bar{X}_{\underline{s}}^\gamma|^{r+1}+(\|\bar{Z}^{\gamma}\|^{\underline{s},\underline{t}-\gamma}_{\theta,\gamma})^{r+1})(\underline{t}-\underline{s})^\theta.$$ Finally, we plug this control and \eqref{pvariationB} into \eqref{youngestimate} and the result follows.
\end{proof}
In the following lemma, we make use of Lemma \ref{lemme2} when $f_\gamma(t,x)=\sigma(x)$. In this particular case, we show below that we can deduce a control of the increments of $\bar{Z}^\gamma$ on an interval with random but explicit length $\eta(\omega)$ (which does not depend on $\gamma$). \begin{lemme}\label{lemme3} Let $\gamma_0$ be a positive number. Assume that $b$ is a sublinear function and that $\sigma$ is a bounded Lipschitz continuous function. Then, for every $\theta\in(\frac{1}{2},H)$, for every $T>0$, there exists $C_T>0$, there exists a positive random variable \begin{equation}\label{eq:valeureta}
\eta(\omega):=\left(\frac{1}{2}[(C_T \|B^H(\omega)\|_{\theta,T})^{-1}\wedge 1]\right)^\frac{1}{\theta} \end{equation}
such that $a.s$ for every $0\le s\le t\le T$ with $\underline{t}-\underline{s}\le\eta$, for every $\gamma\in(0,\gamma_0)$
$$ {|\bar{Z}^\gamma_{\underline{t}}-\bar{Z}^\gamma_{\underline{s}}|}\le {(\underline{t}-\underline{s})^\theta}\left({2}\|\sigma\|_\infty+ C_T (1+|\bar{X}^\gamma_{\underline{s}}|)\eta^\theta\right)\|B^H\|_{\theta,T}$$
where $\|\sigma\|_\infty=\sup_{x\in\ER^d} \|\sigma(x)\|$.
\end{lemme} \begin{proof}
For every $l\ge 0$, set $t_l=\underline{s}+\gamma l$ and $N_l=\|\bar{Z}^\gamma\|_{\theta,\gamma}^{\underline{s},t_l}$. Owing to the definition of $\|.\|_{\theta,\gamma}^{\underline{s},t_l}$, we have
$$N_{l+1}\le N_l\vee \sup_{i\le l}\frac{\left|\bar{Z}^\gamma_{t_{l+1}}-\bar{Z}^\gamma_{t_i}\right|}{(t_{l+1}-t_i)^{\theta}}.$$ By Lemma \ref{lemme2} applied with $s=t_i$, $t=t_{l+1}$ and $f_\gamma(s,x)=\sigma(x)$ (and $r=0$),
$$\frac{\left|\bar{Z}^\gamma_{t_{l+1}}-\bar{Z}^\gamma_{t_i}\right|}{(t_{l+1}-t_i)^{\theta}}
\le \left(\|\sigma\|_\infty+ C_T(t_{l+1}-t_i)^{\theta}\left(1+|\bar{X}^\gamma_{t_i}|+\|\bar{Z}^\gamma\|_{\theta,\gamma}^{\underline{s},t_l}\right)\right)
\|B^H\|_{\theta,T}.$$
By Lemma \ref{gronwall} and the fact that $t\rightarrow\|\bar{Z}^\gamma\|_{\theta,\gamma}^{\underline{s},t}$ is nondecreasing, it follows that
$$\sup_{i\le l}\frac{\left|\bar{Z}^\gamma_{t_{l+1}}-\bar{Z}^\gamma_{t_i}\right|}{(t_{l+1}-t_i)^{\theta}}
\le \left(\|\sigma\|_\infty+ C_T\left((1+|\bar{X}^\gamma_{\underline{s}}|)(t_{l+1}-\underline{s})+\|\bar{Z}^\gamma\|_{\theta,\gamma}^{\underline{s},t_l}(t_{l+1}-\underline{s})^{\theta}\right)\right)
\|B^H\|_{\theta,T}.$$ Let $\rho$ be a positive number. If $t_{l+1}-\underline{s}\le \rho$, we obtain that $$N_{l+1}\le N_l\vee (\alpha_\rho+ \beta_\rho N_l)$$ with \begin{align*}
&\alpha_\rho=\left(\|\sigma\|_\infty+ C_T((1+|\bar{X}^\gamma_{\underline{s}}|)\rho^\theta\right)\|B^H\|_{\theta,T}\quad\textnormal{and}\quad
&\beta_\rho=C_T \rho^\theta\|B^H\|_{\theta,T}. \end{align*} Let us now set $\rho=\eta(\omega)$ where $\eta(\omega)$ is defined by \eqref{eq:valeureta}. For this choice of $\rho$, we have $\beta_\eta\le\frac{1}{2}$. Then, the interval $[0,\alpha_\eta/(1-\beta_\eta)]$ being stable by the function $x\mapsto \alpha_\eta+\beta_\eta x$, we deduce that for every $l\in\EN$ such that $t_{l+1}-\underline{s}\le \eta(\omega)$, {$$N_l\le \frac{\alpha_\eta}{1-\beta_\eta}\le {2\alpha_\eta}.$$ Note that we used that $N_0$ belongs to $[0,\alpha_\eta/(1-\beta_\eta)]$ (since $N_0=0$).} The result follows. \end{proof}
\subsection{Proof of Proposition \ref{lemme4}} \label{ss:22}
Proposition \ref{lemme4} is the main technical issue of our approximation result, and its proof is detailed here for sake of completeness. We shall first focus on establishing relation~\eqref{VX} for $ p = 1$. {The main difficulty is to prove that the noise component can be controlled in such a way that under the mean-reverting assumption, we obtain a coefficient $\rho$ which is strictly lower than 1. (See in particular~\eqref{eq:entretildeeta}.) Note that this property on $\rho$ will be crucial for the control of the sequence $(V(\bar{X}_{kT}))_{k\ge0}$.}
\noindent Then, we generalize this result to any $p>1$. Finally we handle the H\"older type bound of Proposition \ref{lemme4} item (ii). We now divide our proof in several steps.
\noindent {\textit{Step 1: First upper-bound for $V(\bar{X}_{\underline{t}}^\gamma)$ under the mean-reverting assumption.}} Set $\Delta_n=B^H_{\gamma n} -B^H_{\gamma (n-1)}$. Owing to the Taylor formula, \begin{align*} V(\bar{X}_{(n+1)\gamma})&=V(\bar{X}_{n\gamma})+\gamma \langle \nabla V(\bar{X}_{n\gamma}),b(\bar{X}_{n\gamma})\rangle +\langle \nabla V(\bar{X}_{n\gamma}),\sigma(\bar{X}_{n\gamma})\Delta_{n+1}\rangle \\ &+\frac{1}{2} \sum_{i,j}\partial^2_{i,j} V(\xi_{n+1})(\bar{X}_{(n+1)\gamma}-\bar{X}_{n\gamma})_i(\bar{X}_{(n+1)\gamma}-\bar{X}_{n\gamma})_j. \end{align*} where $\xi_{n+1}\in[\bar{X}_{n\gamma},\bar{X}_{(n+1)\gamma}]$. Using assumption $\mathbf{(C)}${, equation \eqref{eq:euler-scheme-x} for $\bar{X}_{(n+1)\gamma}-\bar{X}_{n\gamma}$} and the boundedness of $D^2V$ and $\sigma$, we obtain \begin{align}\label{entretildeeta2} V(\bar{X}_{(n+1)\gamma})\le V(\bar{X}_{n\gamma})+\gamma(\beta-\alpha V(\bar{X}_{n\gamma}))+A_1(n+1)
+C(\gamma^2 V(\bar{X}_{n\gamma})+|\Delta_{n+1}|^2), \end{align} where $$A_1(n+1)=\langle \nabla V(\bar{X}_{n\gamma}),\sigma(\bar{X}_{n\gamma})\Delta_{n+1}\rangle .$$ Set $\displaystyle\gamma_0=\frac{\alpha}{2C}$. For every $\gamma\in(0,\gamma_0]$, for every $n\ge0$, we have \begin{align*} V(\bar{X}_{(n+1)\gamma})\le V(\bar{X}_{n\gamma})(1-\frac{\alpha}{2}\gamma)+A_1(n+1)
+(\beta\gamma+C|\Delta_{n+1}|^2). \end{align*} Then, iterating the previous inequality yields for every $s,t$ such that $s\le t$, $$V(\bar{X}_{{\underline{t}}})\le V(\bar{X}_{{\underline{s}}})(1-\frac{\alpha}{2}\gamma)^{\frac{{\underline{t}}-{\underline{s}}}{\gamma}}+\sum_{k=\frac{\underline{s}}{\gamma}+1}^{{\frac{\underline{t}}{\gamma}}}
(1-\frac{\alpha}{2}\gamma)^{\frac{{\underline{t}}-{\underline{s}}}{\gamma}-k}\left(A_1(k)+\beta\gamma+C|\Delta_k|^2\right). $$ Using that $\log(1+x)\le x$ for every $x>-1$, we deduce that \begin{equation}\label{eq:l41}
V(\bar{X}_{{\underline{t}}})\le e^{-\frac{\alpha({\underline{t}}-{\underline{s}})}{2}}(V(\bar{X}_{{\underline{s}}})+|\bar{H}_{{\underline{t}}}^\gamma-\bar{H}_{{\underline{s}}}^{\gamma}|)+
\sum_{k=\frac{\underline{s}}{\gamma}+1}^{{\frac{\underline{t}}{\gamma}}}(\beta\gamma+C|\Delta_{k}|^2),
\end{equation} where \begin{align*} \bar{H}_{t}^{\gamma}=\int_0^t g_\gamma({\underline{s}})\langle \nabla V(\bar{X}_{{\underline{s}}}),\sigma(\bar{X}_{{\underline{s}}})dB_s^H\rangle =\sum_{i,j}\int_0^t g_\gamma({\underline{s}}) (\nabla V)_i(\bar{X}_{{\underline{s}}}),\sigma_{i,j}(\bar{X}_{{\underline{s}}})d(B_s^H)^j. \end{align*} with $g_\gamma(s)=(1-\frac{\alpha\gamma}{2})^{-\frac{s}{\gamma}}$. We now wish to see that this relation has to be interpreted as $V(\bar{X}_{{\underline{t}}})\le e^{-\frac{\alpha({\underline{t}}-{\underline{s}})}{2}}V(\bar{X}_{{\underline{s}}})$, up to a remainder term.
\noindent
\textit{Step 2: Upper bound for $|\bar{H}_{{\underline{t}}}^\gamma-\bar{H}_{{\underline{s}}}^{\gamma}|$.}
For every $(i,j)\in\{1,\ldots,d\}\times\{1,\ldots,q\}$, set $f^{i,j}_\gamma(s,x)=g_\gamma(s) (\nabla V)_i(x)\sigma_{i,j}(x)$. Using that $\sup_{t\in[0,T],\gamma\in(0,\gamma_0]}|g'_\gamma(t)|<+\infty$, we check that $(g_\gamma(.))_{\gamma\in(0,\gamma_0]}$ is a family of Lipschitz continuous functions such that $\sup_{\gamma\in(0,\gamma_0]}[g_\gamma]_{\rm Lip}<+\infty$. Furthermore, $(\nabla V)_i$ and $\sigma_{i,j}$ being respectively Lipschitz continuous and bounded Lipschitz continuous functions, we deduce that $(f^{i,j}_\gamma)_{\gamma\in(0,\gamma_0]}$ satisfies \eqref{assump:f} with $r=1$. Applying Lemma \ref{lemme2}, we obtain that for every $\theta\in(\frac{1}{2},H)$,
$$\frac{|\bar{H}_{\underline{t}}^{\gamma}-\bar{H}_{\underline{s}}^{\gamma}|}{(\underline{t}-\underline{s})^{\theta}}\le C_T\left[(1+|\bar{X}_{\underline{s}}^\gamma|)
+{C}_T(\underline{t}-\underline{s})^{\theta}(1+|\bar{X}_{\underline{s}}^\gamma|^2+(\|\bar{Z}^\gamma\|_{\theta,\gamma}^{\underline{s},\underline{t}-\gamma})^{2})\right]\|B^H\|_{\theta,T}.$$ Now, if $\underline{t}-\underline{s}\le \eta(\omega)$ defined by \eqref{eq:valeureta},
$$ \|\bar{Z}^\gamma\|_{\theta,\gamma}^{\underline{s},\underline{t}-\gamma}\le \left({2}\|\sigma\|_\infty+ C_T(1+|\bar{X}^{\gamma}_{s}|)\eta^\theta\right)\|B^H\|_{\theta,T}.$$ Owing to the definition of $\eta$, we have $a.s.$
$$ \|B^H(\omega)\|_{\theta,T}\eta^\theta\le C_T$$ where $C_T$ is a deterministic positive number so that
$$ (\|\bar{Z}^\gamma\|_{\theta,\gamma}^{\underline{s},\underline{t}-\gamma})^{2}\le C_T(\|B^H\|_{\theta,T}^2+1+|\bar{X}^{\gamma}_{s}|^2).$$ Thus, \begin{align*}
{|\bar{H}_{\underline{t}}^{\gamma}-\bar{H}_{\underline{s}}^{\gamma}|}&\le
C_T\left[(1+|\bar{X}_{\underline{s}}^\gamma|)(\underline{t}-\underline{s})^{\theta}+(\underline{t}-\underline{s})^{2\theta}(1+|\bar{X}_{\underline{s}}^\gamma|^2+\|B^H\|_{\theta,T}^2)\right]
\|B^H\|_{\theta,T} .
\end{align*}
Using that $|ab|\le 2^{-1}(|a|^2+|b|^2)$ and that $1+|x|\le C\sqrt{V}(x)$, we have
$$(1+|\bar{X}_{\underline{s}}^\gamma|)(\underline{t}-\underline{s})^{\theta}\|B^H\|_{\theta,T}\le C(V(\bar{X}_{\underline{s}}^{\gamma})(\underline{t}-\underline{s})^{2\theta}
+\|B^H\|_{\theta,T}^2).$$ It follows that there exists $C_T>0$ such that for every $\varepsilon>0$ \begin{equation*}
{|\bar{H}_{\underline{t}}^{\gamma}-\bar{H}_{\underline{s}}^{\gamma}|}\le \varepsilon(\underline{t}-\underline{s})V(\bar{X}_{\underline{s}}^\gamma)\left(\frac{C_T(\underline{t}-\underline{s})^{2\theta-1}(1+\|B^H\|_{\theta,T})}{\varepsilon}\right)
+C_T(\|B^H\|_{\theta,T}^2+(\underline{t}-\underline{s})^{2\theta}\|B^H\|_{\theta,T}^3). \end{equation*} Now, we choose $\tilde{\eta}_\varepsilon\in(0,\eta)$
$$\frac{C_T(\tilde{\eta}_\varepsilon)^{2\theta-1}(1+\|B^H\|_{\theta,T})}{\varepsilon}\le 1.$$ More precisely, we set
$\tilde{\eta}_\varepsilon=[(C_T(1+\|B^H\|_{\theta,T}))^{-1}\varepsilon]^{\frac{1}{2\theta-1}}\wedge\eta$. Thus, we obtain that for every $0\le s\le t\le T$ such that $\underline{t}-\underline{s}\le\tilde{\eta}_\varepsilon$, \begin{equation}\label{eq:entretildeeta}
{|\bar{H}_{\underline{t}}^{\gamma}-\bar{H}_{\underline{s}}^{\gamma}|}\le \varepsilon(\underline{t}-\underline{s})V(\bar{X}_{\underline{s}}^\gamma)
+C_T(\|B^H\|_{\theta,T}^2+(\underline{t}-\underline{s})^{2\theta}\|B^H\|_{\theta,T}^3). \end{equation}
\noindent \textit{Step 3: Contracting dynamics for $V(\bar{X}_{\underline{k\tilde{\eta}}})$.} Choose now $\varepsilon_0>0$ such that there exists $\delta\in(0,1/2)$ satisfying \begin{equation}\label{eq:eee} \forall x\in[0,1],\quad e^{-\frac{\alpha}{2}x}(1+\varepsilon_0 x)\le 1-\delta x, \end{equation} and set $\tilde{\eta}:=\tilde{\eta}_{\varepsilon_0}$. Plugging the two previous controls in \eqref{eq:l41}, it follows that for every $k\in\{1,\ldots,\lfloor \frac{T}{{\tilde{\eta}}}\rfloor\}$, \begin{equation*} V(\bar{X}_{\underline{k\tilde{\eta}}})\le V(\bar{X}_{\underline{{(k-1)}\tilde{\eta}}})(1-\delta
\alpha_k)+C_T(1+\|B^H\|_{\theta,T}^3)+\sum_{l=\frac{\underline{(k-1)\tilde{\eta}}}{\gamma}+1}^{{\frac{\underline{k\tilde{\eta}}}{\gamma}}}(\beta\gamma+C|\Delta_l|^2), \end{equation*} where $\alpha_k=\underline{k\tilde{\eta}}-\underline{{(k-1)}\tilde{\eta}}$. Note that we can apply \eqref{eq:eee} since $$\alpha_k\le 2\tilde{\eta}\le 2\eta\le 2^{1-\frac{1}{\theta}}\le 1.$$ In particular, $\delta\alpha_k\le 1/2$. With the convention $\prod_{\emptyset}=1$, an iteration of this inequality yields for every $k\in\{1,\ldots,\lfloor \frac{T}{\tilde{\eta}}\rfloor\}$: \begin{align*} V(\bar{X}_{\underline{k\tilde{\eta}}})&\le V(x)\prod_{l=1}^k(1-\delta
\alpha_l)+C_T\|B^H\|_{\theta,T}^3\sum_{m=1}^k\prod_{l=m+1}^k(1-\delta \alpha_l) \\ &+\sum_{m=1}^k\prod_{l=m+1}^k(1-\delta \alpha_l)
\sum_{l=\frac{\underline{(m-1)\tilde{\eta}}}{\gamma}+1}^{{\frac{\underline{m\tilde{\eta}}}{\gamma}}}(\beta\gamma+C|\Delta_l|^2). \end{align*}
Then, using the inequality $\log(1+x)\le x$ $\forall x\in(-1,+\infty)$, we have for every $m\in\{0,\ldots,k\}$ (with the convention $\sum_\emptyset=0$) $$\prod_{l=m+1}^k(1-\delta \alpha_l)=\exp(\sum_{l=m+1}^k\log(1-\delta\alpha_l))\le \exp(-\sum_{l=m+1}^k\delta\alpha_l))=\exp(-\delta \underline{k\tilde{\eta}}+\delta\underline{m\tilde{\eta}}).$$ Thus $$\sum_{m=1}^k\prod_{l=m+1}^k(1-\delta \alpha_l)\le \exp(-\delta\underline{k\tilde{\eta}})\sum_{m=1}^k\exp(\delta\tilde{\eta})^m\le \exp(\delta\tilde{\eta}-\delta\underline{k\tilde{\eta}})\frac{\exp(k\delta\tilde{\eta})-1}{\exp(\delta\tilde{\eta})-1}\le \frac{C}{\delta\tilde{\eta}}$$ where $C$ is deterministic (and does not depend on $k$). Owing to the definition of $\tilde{\eta}$ (and thus from that of $\eta$), we have
$${\tilde{\eta}}^{-1}\le[(C_T(1+\|B^H\|_{\theta,T}){)}^{-1}\varepsilon_0]^{-\frac{1}{2\theta-1}} \lor \eta^{-1}\le
{C}_{\varepsilon_0,T} (1+\|B^H\|^\frac{1}{2\theta-1}). $$
It follows that there exists a polynomial function $P_1$ such that
$$C_T\|B^H\|_{\theta,T}^3\sum_{m=1}^k\prod_{l=m+1}^k(1-\delta \alpha_l)\le P_1(\|B^H\|_{\theta,T}).$$
On the other hand, since $\prod_{l=m+1}^k(1-\delta \alpha_l)\le 1$, we also have \begin{align*} \sum_{m=1}^k\prod_{l=m+1}^k(1-\delta
\alpha_l)\sum_{l=\frac{\underline{(m-1)\tilde{\eta}}}{\gamma}+1}^{{\frac{\underline{m\tilde{\eta}}}{\gamma}}}(\beta\gamma+C|\Delta_l|^2) \le \sum_{u=1}^{\lfloor
\frac{\underline{k\tilde{\eta}}}{\gamma}\rfloor}(\beta\gamma+C|\Delta_u|^2)\le \beta k\tilde{\eta}+C\sum_{u=1}^{\lfloor
\frac{\underline{k\tilde{\eta}}}{\gamma}\rfloor}|\Delta_u|^2. \end{align*} We deduce that for every $k\in\{1,\ldots,\lfloor \frac{T}{{\tilde{\eta}}}\rfloor \}$: \begin{align}\label{eq:controlptk1}
V(\bar{X}_{\underline{k\tilde{\eta}}})\le V(x)\exp(-\delta\underline{k\tilde{\eta}})+P_1(\|B^H\|_{\theta,T}) +C Q_\gamma( B^H_t,0\le t\le T), \end{align} where $P_1$ is a polynomial function and $Q_\gamma$ is defined by \begin{equation}\label{eq:qgamma} Q_\gamma( (w(t))_{t\in[0,T]})=\sum_{k=1}^{\lfloor
\frac{T}{\gamma}\rfloor}|w(k\gamma)-w((k-1)\gamma)|^2. \end{equation}
Owing to the definition of $\|B^H\|_{\theta,T}$, one checks that for every $\gamma\in(0,\gamma_0]$ $$ Q_\gamma( B^H_t,{t\in[0,T]})\le \gamma^{2\theta-1} T
\|B^H\|_{\theta,T}^{2}\le C_T \|B^H\|_{\theta,T}^{2}.$$ Thus, denoting by $P$ the polynomial function defined by $P(v)=P_1(v)+C_T v^2$,
we deduce from \eqref{eq:controlptk1} that for every $k\in\{1,\ldots,\lfloor \frac{T}{\tilde{\eta}}\rfloor \}$:
\begin{align}\label{eq:controlptk}
V(\bar{X}_{\underline{k\tilde{\eta}}})\le V(x)\exp(-\delta\underline{k\tilde{\eta}})+P(\|B^H\|_{\theta,T}). \end{align}
\noindent \textit{Step 4: Contracting dynamics for $V(\bar{X}_{T})$.} We now patch the estimates obtained so far in order to propagate inequality \eqref{eq:controlptk} to $V(\bar{X}_{T})$. Indeed, applying \eqref{eq:controlptk} with $k=\lfloor {\tilde{\eta}}^{-1}{T}\rfloor$, we obtain \begin{align*}
V(\bar{X}_{\underline{\lfloor {\tilde{\eta}}^{-1}{T}\rfloor\tilde{\eta}}})\le V(x)\exp(-\delta{\underline{\lfloor {\tilde{\eta}}^{-1}{T}\rfloor\tilde{\eta}}})+P(\|B^H\|_{\theta,T}), \end{align*} and owing again to \eqref{eq:l41}, \eqref{eq:entretildeeta} (applied with $s=\lfloor {\tilde{\eta}}^{-1}{T}\rfloor\tilde{\eta}$ and $t=T$) and \eqref{eq:eee}, we deduce that \begin{align}\label{l42}
V(\bar{X}_{\underline{T}})\le V(x)\exp(-\delta{\underline{\lfloor {\tilde{\eta}}^{-1}{T}\rfloor\tilde{\eta}}})+\tilde{P}(\|B^H\|_{\theta,T}) \end{align} where $\tilde{P}$ is a polynomial function.
Finally, we want to control $V(\bar{X}_T)-V(\bar{X}_{\underline{T}})$. The function $\nabla V$ being sublinear and $D^2V$ being bounded, we deduce from the Taylor formula that for every $x,y\in\ER^d$,
$$V(y)\le V(x)+C(|x|.|y-x|+|y-x|^2).$$ Applying this inequality with $x=\bar{X}_{\underline{T}}$ and $y=\bar{X}_T$ and taking advantage of the assumptions on $b$, we have \begin{align}\label{eq:detbarat}
V(\bar{X}_T)&\le V(\bar{X}_{\underline{T}})+C\left[\gamma(1+ |\bar{X}_{\underline{T}}|^2)+(1+|\bar{X}_{\underline{T}}|)
|B^H_T-B^H_{\underline{T}}|+|B^H_T-B^H_{\underline{T}}|^2\right]\\
&\le V(\bar{X}_{\underline{T}})(1+C\gamma)+C(1+\|B^H\|_{\theta,T}^2), \end{align} where in the second line, we again used the elementary inequality
$|ab|\le2^{-1}(|a|^2+|b|^2)$ and the fact that $|x|^2\le C V(x)$. Combined with \eqref{l42}, the previous inequality yields: \begin{align*}
V(\bar{X}_T)\le V(x)\exp(-\delta {\underline{\lfloor {\tilde{\eta}}^{-1}{T}\rfloor\tilde{\eta}}})(1+C\gamma)+P_{1,\theta}(\|B^H\|_{\theta,T}), \end{align*} where $P_{1,\theta}$ denotes the polynomial function defined by $P_{1,\theta}(v)=\tilde{P}(v)+C(1+v^2$). Finally, since $\exp(-\delta {\underline{\lfloor {\tilde{\eta}}^{-1}{T}\rfloor\tilde{\eta}}})\le e^{-\delta (T-\tilde{\eta}-\gamma)}$, since $T\ge1$ and $\tilde{\eta}\le 2^{\frac{1}{\theta}}<1$, one can find $\gamma_0>0$ such that $T-\delta\tilde{\eta}-\gamma_0>0$ and such that, $$\exp(-\delta {\underline{\lfloor {\tilde{\eta}}^{-1}{T}\rfloor\tilde{\eta}}})(1+C\gamma)\le \rho\quad a.s.$$ Inequality \eqref{VX} {for $p=1$} follows.\\
\noindent
\textit{Step 5: Inequality \eqref{VX} for $p>1$.} We recall that for every $p>0$, there exists $c_p>0$ such that for every $u,v\in\ER$, the following inequality holds: $|u+v|^p\le |u|^p+c_p(|v|.| u|^{p-1}+|v|^p)$. Thus, by the Young inequality, it follows that for every $\varepsilon>0$, there exists $c_{\varepsilon,p}>0$ such that $|u+v|^p\le (1+\varepsilon)|u|^p+c_{\varepsilon,p}|v|^p$ for every $u,v \in \ER$ and $p\ge1$. Applying this inequality, we deduce from the case $p=1$ that
$$V^p(\bar{X}^\gamma_T)\le \rho^p(1+\varepsilon) V^p(x)+C_TP_{\theta}(\|B^H\|_{\theta,1}))^p.$$ Since $\rho<1$, we can choose $\varepsilon>0$ such that $\tilde{\rho}=\rho^p(1+\varepsilon)<1$. It follows that
$$V^p(\bar{X}^\gamma_T)\le \tilde{\rho} V^p(x)+P_{p,\theta}(\|B^H\|_{\theta,T})$$ where $P_{p,\theta}$ is again a polynomial function.\\
Now, let us focus on \eqref{eq:sup-V}. We only give the main ideas of the proof when $p=1$ (the extension to $p>1$ again follows from the inequality $|u+v|^p\le 2^{p-1}(|u|^p+|v|^p)$). By \eqref{eq:controlptk}, the announced inequality holds taking the supremum of the left-hand side of \eqref{eq:sup-V} for every $\underline{k\tilde{\eta}}$ with $k\in \{1,\ldots,\lfloor \frac{T}{\tilde{\eta}}\rfloor\}$. Then, for every $t\in[\underline{(k-1)\tilde{\eta}},\underline{k\tilde{\eta}}]$, it remains to control (uniformly in $k$) $V(\bar{X}_t)$ in terms of $V(\bar{X}_{\underline{(k-1)\tilde{\eta}}})$. By \eqref{entretildeeta2} and \eqref{eq:entretildeeta}, we obtain such a control for every discretization time between $\underline{(k-1)\tilde{\eta}}$ and $\underline{k\tilde{\eta}}$. Then, it is enough to control uniformly $V(\bar{X}_t)$ in terms of $V(\bar{X}_{\underline{t}})$. This can be done similarly as in inequality \eqref{eq:detbarat}.
\noindent \textit{Step 6: Proof of the H\"older bound \eqref{eq:holder-bnd-with-V}.} Let $s,t\in[0,T]$ with $0\le s<t\le T$. We have $$\bar{X}^\gamma_t-\bar{X}^\gamma_s=\int_s^t b(\bar{X}^\gamma_{\underline{u}})du+\bar{Z}^\gamma_t-\bar{Z}^\gamma_s.$$
First, since $|b(x)|\le C\sqrt{V}(x)\le C(1+V(x))$,
$$|\int_s^t b(\bar{X}^\gamma_{\underline{u}})du|\le C (t-s)(1+\sup_{u\in[0,T]} V(\bar{X}_{\underline{u}}))$$ and it follows from $(i)$ that
$$\sup_{0\le s<t\le T}\frac{|\int_s^t b(\bar{X}^\gamma_{\underline{u}})du|}{(t-s)^\theta}\le C_T (V(x)+{P}_{p,\theta}(\|B^H\|_{\theta,T})).$$ Thus, we can only focus on the increment of $\bar{Z}^\gamma$. By Lemma \ref{lemme3}, for every $u,v\in[0,T]$ such that $\underline{v}-\underline{u}\le \eta$ (where $\eta$ is given by \eqref{eq:valeureta}),
$$ {|\bar{Z}^\gamma_{\underline{v}}-\bar{Z}^\gamma_{\underline{u}}|}\le {(\underline{v}-\underline{u})^\theta}\left(2\|\sigma\|_\infty+ C_T(1+\sup_{s\in[0,T]}|\bar{X}_{\underline{s}}|)\eta^\theta\right)\|B^H\|_{\theta,T}.$$
Using the concavity of $x\mapsto x^\theta$ on $\ER_+$, we have for every $s_1,s_2\in[0,T]$ being such that $|s_2-s_1|\le \gamma$,
$$|\bar{Z}^\gamma_{s_2}-\bar{Z}^\gamma_{s_1}|\le 2^{1-\theta}\|\sigma\|_\infty (s_2-s_1)^\theta \|B^H\|_{\theta,T}$$
and we derive that for every $u,v\in[0,T]$ with $|u-v|\le\eta$,
$$ {|\bar{Z}^\gamma_{{v}}-\bar{Z}^\gamma_{{u}}|}\le C_T{({v}-{u})^\theta}\left(1+ (1+\sup_{s\in[0,T]}|\bar{X}_{\underline{s}}|)\eta^\theta\right)\|B^H\|_{\theta,T}.$$
Now, by the very definition of $\eta$, we have $\eta^\theta\|B^H\|_{\theta,T}\le 1$. Then, since $|x|^2\le C V(x)$, we have in particular that $|x|\le C V(x)$ (using that $\inf_{x\in\ER^d} V(x)>0$) and we deduce from the first part of this proposition that for every $u,v\in[0,T]$ with $|u-v|\le\eta$: \begin{equation}\label{eq:lemme3}
{|\bar{Z}^\gamma_{{v}}-\bar{Z}^\gamma_{{u}}|}\le C_T({v}-{u})^\theta (V(x)+\tilde{P}(\|B^H\|_{\theta,T})),
\end{equation} where $\tilde{P}$ is a polynomial function.\\ We want now to make use of the previous inequality to control $\bar{Z}^\gamma_t-\bar{Z}^\gamma_s$ for every $0\le s<t\le T$. We divide $[s,t]$ in intervals of length lower than $\eta$. More precisely, setting $s_k=s+ k\lfloor \eta\rfloor$, we have $$\bar{Z}^\gamma_t-\bar{Z}^\gamma_s=\bar{Z}^\gamma_t-\bar{Z}^\gamma_{s_{\lfloor\frac{t-s}{\eta}\rfloor}}+\sum_{k=1}^{\lfloor\frac{t-s}{\eta}\rfloor} \bar{Z}^\gamma_{s_{k}}-\bar{Z}^\gamma_{s_{k-1}}.$$ Then, we deduce from \eqref{eq:lemme3} that \begin{align*}
|\bar{Z}^\gamma_t-\bar{Z}^\gamma_s|&\le C_T\left((t-s_{\lfloor\frac{t-s}{\eta}\rfloor})^\theta +\lfloor\frac{t-s}{\eta}\rfloor\eta^\theta\right)
(V(x)+\tilde{P}(\|B^H\|_{\theta,T}))\\
&\le C_T\left((t-s)^\theta +(t-s)\eta^{\theta-1}\right)
(V(x)+\tilde{P}(\|B^H\|_{\theta,T})). \end{align*} Thus, using \eqref{eq:lemme3} if $t-s\le\eta$ or the fact that $(t-s)\eta^{\theta-1}\le (t-s)^\theta$ if $t-s\ge \eta$, we deduce that there exists $C_T>0$ such that for every $0\le s< t\le T$, \begin{equation*}
{|\bar{Z}^\gamma_{{t}}-\bar{Z}^\gamma_{{s}}|}\le C_T({t}-{s})^\theta (V(x)+\tilde{P}(\|B^H\|_{\theta,T})).
\end{equation*} The result \eqref{eq:holder-bnd-with-V} follows.
\section{Tightness properties}\label{section4} In the following proposition, we obtain some $a.s.$ tightness results for the sequence $({\cal P}^{(n,\gamma)}(\omega,d\alpha))_{n\ge1}$. Using that the controls established in Proposition \ref{lemme4} are uniform in $\gamma$, we also show that tightness properties also hold for the set of its limiting measures $({\cal U}^{(\infty,\gamma)}(\omega,\theta))_\gamma$ defined by $${\cal U}^{(\infty,\gamma)}(\omega,\theta)=\left\{\mu\in\bar{\cal C}^\theta(\ER_+,\ER^d),\exists (n_k(\omega))_{k\ge1},{\cal P}^{(n_k(\omega),\gamma)}(\omega,d\alpha)\xrightarrow{k\rightarrow+\infty}\mu\right\}.$$ \begin{prop}\label{prop:ntendinfty} Assume $\mathbf{(C)}$. Then, there exists $\gamma_0>0$ such that,\\ (i) For every $\gamma\in(0,\gamma_0]$ and $p\ge1$, $a.s.$,
$$\limsup_{n\rightarrow+\infty}\frac{1}{n}\sum_{k=1}^{n} V^p(\bar{X}^\gamma_{\gamma(k-1)})\le C_p\mathbb{E}[|P_{p,\theta}(\|B^H\|_{\theta,1})|]<+\infty.$$ where $C_p$ does not depend on $\gamma$ and $P_{p,\theta}$ is a polynomial function.\\ (ii) For every $\theta\in(1/2,H)$, for every $\gamma\in(0,\gamma_0]$, $({\cal P}^{(n,\gamma)}(\omega,d\alpha))_{n\ge1}$ is almost surely tight on $\bar{\cal C}^\theta(\ER_+,\ER^d)$.\\ (iii) For every $\theta\in(1/2,H)$, $({\cal U}^{(\infty,\gamma)}(\omega,\theta))_{\gamma\in(0,\gamma_0]}$ is $a.s.$ tight in $\bar{\cal C}^\theta(\ER_+,\ER^d)$.
\end{prop} \begin{proof} \textit{(i)} \textbf{Case $p=1$} : We first focus on the sequence $(\frac{1}{N}\sum_{\ell=0}^{N-1} V(\bar{X}^\gamma_\ell))_{N\ge1}$. Note that, at this stage, we consider the values of the Euler scheme at {times} $0$, $1$, $2$,~\ldots (which do not depend on $\gamma$). that We set $$\forall \ell\ge 0,\quad(\delta_\ell B^H)_t=B^H_{\ell+t}-B^H_{\ell}.$$ By Proposition \ref{lemme4} applied with $T=1$, we have for every $k\ge1$
$$V(\bar{X}^\gamma_{\ell})\le \rho V(\bar{X}^\gamma_{\ell-1})+P_{1,\theta}(\|\delta_{\ell-1}B^H\|_{\theta,1})$$ with $\rho\in(0,1)$. An iteration yields for every $\ell\ge1$
$$V(\bar{X}^\gamma_{\ell})\le \rho^\ell V(x)+\sum_{m=0}^{\ell-1}\rho^{\ell-1-m}P_{1,\theta}(\|\delta_{m}B^H\|_{\theta,1}).$$
Setting $U_m=P_{1,\theta}(\|\delta_{m}B^H\|_{\theta,1})$ and summing over $\ell$, we obtain \begin{align*} \frac{1}{N}\sum_{\ell=0}^{N-1} V(\bar{X}^\gamma_{\ell})&\le \frac{V(x)}{N(1-\rho)}+\frac{1}{N}\sum_{\ell=0}^{N-1}\sum_{m=0}^{\ell-1}\rho^{\ell-1-m}U_m\\ &\le \frac{V(x)}{N(1-\rho)}+\frac{1}{N}\sum_{m=0}^{N-2}U_m\sum_{\ell=m+1}^N\rho^{\ell-1-m} \le \frac{V(x)}{N(1-\rho)}+\frac{1}{N(1-\rho)}\sum_{m=0}^{N-2}U_m. \end{align*}
Let us remark that since $ B^H$ is a $\bar{\cal C}^\theta([0,1],\ER^q) $ valued Gaussian random variable, the
norm $ \|B^H\|_{\theta,1} $ has finite moments of every order, which is classical consequence of Fernique Lemma. Hence \begin{equation}\label{eq:erg}
\mathbb{E}[|P_{1,\theta}(\|B^H\|_{\theta,1})|]<+\infty. \end{equation} Then, since $(\delta_{m}B^H)_{m\ge1}$ is ergodic (see Remark \ref{maruyamaremark} for background and details). We have \begin{equation}\label{p11}
\frac{1}{N}\sum_{m=0}^{N-2}U_m\xrightarrow{N\rightarrow+\infty}\mathbb{E}[P_{1,\theta}(\|B^H\|_{\theta,1})]\quad a.s. \end{equation} and it follows that \begin{equation}\label{p12}
\limsup_{N\rightarrow+\infty}\frac{1}{N}\sum_{\ell=0}^{N-1} V(\bar{X}^\gamma_{\ell})\le \frac{1}{1-\rho}\mathbb{E}[P_{1,\theta}(\|B^H\|_{\theta,1})]\quad a.s. \end{equation} We want now to use this result to control the $a.s.$ asymptotic behavior of $(\frac{1}{n}\sum_{k=0}^{n-1} V(\bar{X}^\gamma_{\gamma k}))_{n\ge1}$. By the second point of Proposition \ref{lemme4}(i), for every $\ell\ge0$,
$$\sup_{k\in[\lfloor \frac{\ell}{\gamma}\rfloor+1,\lfloor \frac{\ell+1}{\gamma}\rfloor]} V(\bar{X}^\gamma_{\gamma k})\le C \left(V(\bar{X}^\gamma_{\ell})+P_{1,\theta}(\|\delta_{\ell}B^H\|_{\theta,1})\right).$$ As a consequence, setting $N=\lfloor \gamma (n-1)\rfloor+1$, we have \begin{align*} \frac{1}{n}\sum_{k=0}^{n-1} V(\bar{X}^\gamma_{\gamma k})&\le \frac{N}{n}\frac{1}{N}\left(V(x)+\sum_{\ell=0}^{N-1}\sum_{k=\lfloor \frac{\ell}{\gamma}\rfloor+1}^{\lfloor \frac{\ell+1}{\gamma}\rfloor} V(\bar{X}^\gamma_{\gamma k})\right)\\
&\le C(\gamma+\frac{1}{n})(\frac{1}{\gamma}+1)\left(\frac{1}{N}\sum_{\ell=0}^{N-1} \left(V(\bar{X}^\gamma_{\ell})+P_{1,\theta}(\|\delta_{\ell}B^H\|_{\theta,1})\right)\right). \end{align*} Using \eqref{p11} and \eqref{p12}, the result follows when $p=1$.
The proof when $ p > 1 $ is very similar to the case $p=1$ and is left to the reader.\\
\textit{(ii)} If for a sequence $(\mu_n)_{n\ge1}$ of probability measures on $\ER^d$, there exists a positive function $\varphi:\ER^d\mapsto (0,+\infty)$
such that $\sup_{n\ge1}\mu_n(\varphi)<+\infty $ and $\lim_{|x|\rightarrow+\infty}\varphi(x)=+\infty$, one classically derives that $(\mu_n)_{n\ge1}$ is tight on $\ER^d$ (see $e.g.$ \cite{duflo} p. 41). Thus, by $\textit{(i)}$, $({\cal P}^{(n,\gamma)}_0(\omega,dx))$ is $a.s.$ tight on $\ER^d$. Owing to some classical tightness results in Hölder spaces (see $e.g.$ \cite{suquet}, Theorem 1.4), we deduce that we only have to prove that for every $T>0$, for every $\theta\in(1/2,H)$, for every $\varepsilon>0$, \begin{equation}\label{omegaTT} \limsup_{\delta\rightarrow0}\limsup_{n\rightarrow+\infty}\frac{1}{n}\sum_{k=1}^n {\bf 1}_{\{\omega_{\theta,T}(\bar{X}^\gamma_{\gamma(k-1)+.},\delta)\ge\varepsilon\}}=0, \end{equation} where we recall that
$$\forall\, T>0,\quad \omega_{\theta,T}(f,\delta):=\sup_{0\le s<t<T,0\le |t-s|\le\delta}\frac{|f(t)-f(s)|}{|t-s|^\theta}.$$ By Proposition \ref{lemme4} \emph{(ii)} with $\theta'\in(\theta,H)$,
$$\sup_{0\le s<t\le T}\frac{|\bar{X}^\gamma_t-\bar{X}^\gamma_s|}{(t-s)^\theta}\le C_T (t-s)^{\theta'-\theta}(V(x)+\tilde{P}_{\theta'}(\|B^H\|_{\theta',T}))$$ so that for every $s,t\in[0,T]$ such that $s<t$ and $t-s\le\delta$,
$$\sup_{0\le s<t\le T}\frac{|\bar{X}^\gamma_t-\bar{X}^\gamma_s|}{(t-s)^\theta}\le C_T \delta^{\theta'-\theta}(V(x)+\tilde{P}_{\theta'}(\|B^H\|_{\theta',T})).$$ As in $(i)$, this property can be extended to the shifted process: we have for every $k\ge0$ \begin{equation}\label{ghtrz}
\omega_{\theta,T}(\bar{X}^\gamma_{\gamma k+.},\delta)=\sup_{0\le s<t\le T, \; t-s \le \delta}\frac{|\bar{X}^\gamma_{\gamma k+t}-\bar{X}^\gamma_{\gamma k+s}|}{(t-s)^\theta}\le C_T \delta^{\theta'-\theta}(V(\bar{X}^\gamma_{\gamma k})+\tilde{P}_{\theta'}(\|\delta_{ k}B^H\|_{\theta',T}). \end{equation}
Since $(\delta_k B^H)_{k\ge1}$ is ergodic (see Remark \ref{maruyamaremark} for details) and since by the Fernique Lemma $\| B^H\|_{\theta',T}$ has moments of any order, we have
$$\frac{1}{n}\sum_{k=1}^n\tilde{P}_{\theta'}(\|\delta_{ k}B^H\|_{\theta',T})\xrightarrow{n\rightarrow+\infty} \mathbb{E}[\tilde{P}_{\theta'}(\| B^H\|_{\theta',T})]\quad a.s.$$ Then, we deduce from $(i)$ and \eqref{ghtrz} that $$\limsup_{n\rightarrow+\infty}\frac{1}{n}\sum_{k=1}^n {\omega_{\theta,T}(\bar{X}^\gamma_{\gamma(k-1)},\delta)}\le C\delta^{\theta'-\theta}.$$ By the Markov inequality, we obtain for every $\varepsilon>0$, \begin{equation}\label{eq:prop221} \limsup_{n\rightarrow+\infty}\frac{1}{n}\sum_{k=1}^n {\bf 1}_{\{\omega_{\theta,T}(\bar{X}^\gamma_{\gamma(k-1)+.},\delta)\ge\varepsilon\}}\le C\frac{\delta^{\theta'-\theta}}{\varepsilon} \end{equation} and \eqref{omegaTT} follows.
\textit{(iii)} Let $\theta\in(1/2,H)$ and denote by $\mu^{(\gamma)}$ an element of ${\cal U}^{(\infty,\gamma)}(\omega,\theta) $ and by $\mu^{(\gamma)}_t$ its marginals. By \eqref{eq:erg} and \eqref{p12}, $$\forall \gamma\in(0,\gamma_0],\quad \mu^{(\gamma)}_0(V)\le \frac{C}{1-\rho}$$ where $\rho$ does not depend on $\gamma$. It follows that ${\cal U}_0^{(\infty,\gamma)}(\omega,\theta) $ is $a.s.$ tight in $\ER^d$ (where ${\cal U}_0^{(\infty,\gamma)}(\omega,\theta) $ stands for the set of initial distributions $\mu^{(\gamma)}_0$).\\ Now, since $C$ does not depend on $\gamma$ in \eqref{eq:prop221}, we also have for every $T>0$, $\delta>0$ and $\varepsilon>0$
for every $\theta'>\theta$: $$\forall \gamma\in(0,\gamma_0],\quad \mu^{(\gamma)}({\bf 1}_{\{\omega_{\theta,T}(.,\delta)\ge\varepsilon\}})\le C\delta^{\theta'-\theta}$$ and the announced result follows again from Theorem 1.4 of \cite{suquet}. \end{proof}
\begin{Remarque}\label{maruyamaremark} Some of the arguments of the previous proof are based on the ergodicity of the increments of the fractional Brownian motion. More precisely, we use the fact that $(B_t^H)_{t\in\ER}$ is ergodic under the transformation $T_\xi:\bar{\cal C}^\theta(\ER,\ER^q)\rightarrow\bar{\cal C}^\theta(\ER,\ER^q) $ defined by $(T_\xi(\omega))_t=\omega(\xi+t)-\omega(\xi)$ $(\xi>0)$, which implies by the Birkhoff theorem that, for any functional $F:\bar{\cal C}^\theta(\ER,\ER^q)\rightarrow\ER$
such that $\mathbb{E}[|F(B^H_t,t\ge0)|]<+\infty$, \begin{equation}\label{ergodicityfrac} \mathbb {P}-a.s.,\quad\frac{1}{n}\sum_{k=1}^n F(B^H_{\xi k+.}-B^H_{\xi k})\xrightarrow{n\rightarrow+\infty}\mathbb{E}[F(B^H_t,t\ge0)]. \end{equation} Note that this ergodic result is a (classical) consequence of the Maruyama theorem \cite{maruyama} (see also \cite{weber}) which is stated in a slightly different way: let $({\theta}_t)_{t\in\ER}$ denote the standard time-shift defined for $\omega:\ER\rightarrow\ER$ by $\theta_t(\omega)=\omega(t+.)$. Then, a centered stationary real Gaussian process $(Y_t)_{t\in\ER}$ is ergodic under $({\theta}_t)_{t\in\ER}$ if its covariance function $r(t)=\mathbb{E}[Y_tY_0]$ satisfies $r(t)\rightarrow0$ as $t\rightarrow+\infty$. This result can be applied to the stationary (centered) fractional Ornstein-Uhlenbeck process solution to $dY_t=-Y_t dt+ dB_t^H$ (since $r(t)\rightarrow0$, see $e.g.$ \cite{cheridito})). Then we retrieve \eqref{ergodicityfrac} by using that the increment $B^H_{t+s}-B^H_t$ is a functional of $(Y_t)_{t\ge s}$: $B^H_{t+s}-B^H_t=Y_{t+s}-Y_t+\int_s^t Y_u du.$ \end{Remarque} \section{Identification of the weak limits}\label{section5} \subsection{Weak limits of $({\cal P}^{(n,\gamma)}(\omega,d\alpha))_{n\ge1}$} \label{identification1} We have the following result: \begin{prop}\label{prop2} Assume $\mathbf{(C)}$ and let ${{\cal P}}^{(\infty,\gamma)}(\omega,d\alpha)$ denote a weak limit of $({{\cal P}}^{(n,\gamma)}(\omega,d\alpha))_{n\ge1}$. Then, ${{\cal P}}^{(\infty,\gamma)}(\omega,d\alpha)$ is $a.s.$ an adapted stationary solution of \eqref{fractionalSDE0-disc}. \end{prop}
\begin{Remarque} In the following proof, we will state some properties ``for every function $f,$ for almost every $\omega$'' and conclude that ``for almost every $\omega,$ for every function $f$'' the property is true. For the sake of completeness, we recall here that such inversions are rigorous since we work on Polish spaces (in which the distributions and the weak convergence are characterized by some countable family of bounded continuous functions). \end{Remarque} \begin{proof} In the proof, we denote by $(\tilde{{\cal P}}^{(n)}(\omega,d\alpha, d\beta))_{n\ge 1}$, the sequence of probability measures on $\bar{{\mathcal C}}^\theta(\ER_+,\ER^d) \times \bar{{\mathcal C}}^\theta(\ER,\ER^q) $ with $ \frac12 < \theta<H$ defined by
$$\tilde{{\cal P}}^{(n,\gamma)}(\omega,d\alpha, d\beta)=\frac{1}{n}\sum_{k=1}^n
{\delta}_{({\bar{X}}^\gamma_{\gamma(k-1)+.}(\omega), B^H_{(k-1)\gamma +.}(\omega) - B^H_{(k-1)\gamma}(\omega) )} (d\alpha, d\beta)$$ where
$ (B^H_t)_{t\in\ER} $ is the fractional Brownian motion used to build the Euler scheme~\eqref{eq:euler-scheme-x}.
First, let us recall that by Proposition~\ref{prop:ntendinfty} \emph{(ii)}, $({\cal P}^{(n,\gamma)}(\omega,d\alpha))_{n\ge1}$ is $a.s.$ tight. Thus, we can consider a weak limit ${{\cal P}}^{(\infty,\gamma)}(\omega,d\alpha)$. Second, one checks that
$(\tilde{{\cal P}}^{(n,\gamma)}(\omega,d\alpha, d\beta))_{n\ge1}$ is also almost surely tight since each of its margins have this property. Indeed, for the first margin, it is again \emph{(ii)} of Proposition~\ref{prop:ntendinfty}. For the second margin, we use that $(B^H_t)_{t\in\ER}$ is ergodic under the transformation $T_\gamma:\bar{\cal C}^\theta(\ER,\ER^q)\rightarrow\bar{\cal C}^\theta(\ER,\ER^q) $ (see Remark \ref{maruyamaremark}). In particular, \begin{equation}\label{eq:ergoinc}
\frac{1}{n}\sum_{k=1}^n {\delta}_{B^H_{(k-1)\gamma +.} - B^H_{(k-1)\gamma}} (d\beta)
\end{equation} is converging almost surely to the distribution of $(B^H_t)_{t\in\ER}$ (on $\bar{\cal C}^\theta(\ER,\ER^q)$). Hence, the sequence $(\tilde{{\cal P}}^{(n)}(\omega,d\alpha, d\beta))_{n\ge 1}$ is almost surely tight (and thus relatively compact). Then, if ${{\cal P}}^{(\infty,\gamma)}(\omega,d\alpha)$ is the limit of a subsequence of $({\cal P}^{(n,\gamma)}(\omega,d\alpha))_{n\ge1}$, maybe with the help of a second extraction, it follows that $a.s.$, there exists a subsequence $(n_k(\omega))_{k\ge0}$ such that \begin{equation}\label{eq:convptilde}
{\cal P}^{(n_k,\gamma)}(\omega,d\alpha)\xrightarrow{k\rightarrow+\infty}{\cal P}^{(\infty,\gamma)}(\omega,d\alpha)\quad\textnormal{and}\quad \tilde{{\cal P}}^{(n_k,\gamma)}(\omega,d\alpha,d\beta)\xrightarrow{n_k\rightarrow+\infty}\tilde{{\cal P}}^{(\infty,\gamma)}(\omega,d\alpha,d\beta) \end{equation} where the first margin of $\tilde{{\cal P}}^{(\infty,\gamma)}(\omega,d\alpha,d\beta)$ is obviously ${\cal P}^{(\infty,\gamma)}(\omega,d\alpha)$ and the second one is $a.s.$ the distribution of $(B_t^H)_{t\in\ER}$ (thanks to \eqref{eq:ergoinc}). Let us also denote by $(X^{(\infty,\gamma)}_t, B^H_t) $ the coordinate process on $\bar{{\mathcal C}}^\theta(\ER_+,\ER^d) \times \bar{{\mathcal C}}^\theta(\ER,\ER^q) $ endowed with the probability $\tilde{{\cal P}}^{(\infty,\gamma)}.$ For $(\alpha,\beta) \in \bar{{\mathcal C}}^\theta(\ER_+,\ER^d) \times \bar{{\mathcal C}}^\theta(\ER_+,\ER^q) $ we consider the following function \begin{equation}\label{eqcvsub}
\tilde{\Phi}^{\gamma}(\alpha,\beta)_t := \alpha_0 + \int_0^t b(\tilde{\Phi}^{\gamma}(\alpha,\beta)_{\underline{s}_{\gamma}}) ds + \int_0^t \sigma(\tilde{\Phi}^{\gamma}(\alpha,\beta)_{\underline{s}_{\gamma}}) d \beta_s.
\end{equation} Please remark that $ \tilde{\Phi}^{\gamma} $ is slightly different from $ \Phi^{\gamma} $ in the way it handles the initial condition but $$ \tilde{\Phi}^{\gamma}(\alpha,\beta) = \Phi^{\gamma}(a,\beta)$$ for every $\alpha$ such that $ \alpha_0=a.$ For $t,\; K > 0 $ let us denote by $F_{t,K}$ the functional defined on $ \bar{{\mathcal C}}^\theta(\ER_+,\ER^d) \times \bar{{\mathcal C}}^\theta(\ER,\ER^q)$
by $ F_{t,K}(\alpha,\beta)= \sup_{ 0 \le s \le t} |\alpha_s - \tilde{\Phi}^{\gamma}(\alpha,\beta_+)_s | \wedge K $ where $\beta_+=(\beta(t))_{t\ge0}$. The function $F_{t,K}$ is bounded continuous on $ \bar{{\mathcal C}}^\theta(\ER_+,\ER^d) \times \bar{{\mathcal C}}^\theta(\ER,\ER^q).$
\noindent Then, $$ \mathbb{E} ( F_{t,K}(X^{(\infty,\gamma)}, B^H))= \lim_{n_l \to \infty} \frac{1}{n_l}\sum_{k=1}^{n_l} F_{t,K}({\bar{X}}_{(k-1)\gamma+.}^\gamma,B^H_{(k-1)\gamma +.} - B^H_{(k-1)\gamma} ).$$ By definition of the Euler scheme (even though it is shifted), we have for every $k\ge1$, $ F_{t,K}({\bar{X}}_{(k-1)\gamma+.}^\gamma,B^H_{(k-1)\gamma +.} - B^H_{(k-1)\gamma} )= 0 $ almost surely, and $$ X^{(\infty,\gamma)}= \tilde{\Phi}^{\gamma}(X^{(\infty,\gamma)},B^H)$$ almost surely, which ensures that the pair $ (X^{(\infty,\gamma)},B^H) $ is a solution of~\eqref{fractionalSDE0-disc}. \\ The stationarity of $X^{(\infty,\gamma)}$ follows from the construction. Actually, using the convergence of $({\cal P}^{(n,\gamma)}(\omega,d\alpha))$, we have for every bounded continuous functional $F:\bar{{\cal C}}^\theta(\ER_+,\ER^d)\rightarrow\ER$, $$\frac{1}{n}\sum_{k=1}^n F(\bar{X}^\gamma_{\gamma(k-1)+t+.})-F(\bar{X}^\gamma_{\gamma(k-1)+.})\xrightarrow{n\rightarrow+\infty}{\mathbb{E}}[F(X^{(\infty,\gamma)}_{t+.})]-{\mathbb{E}}[F(X^{(\infty,\gamma)}_{.})]$$ and owing to a change of variable, it is obvious that for every $t\in\gamma \mathbb{N}$, $$\frac{1}{n}\sum_{k=1}^n F(\bar{X}^\gamma_{\gamma(k-1)+t+.})-F(\bar{X}^\gamma_{\gamma(k-1)+.})\xrightarrow{n\rightarrow+\infty}0.$$ It follows that for every $t\in\gamma\EN$, for every $F$, $${\mathbb{E}}[F(X^{(\infty,\gamma)}_{t+.})]={\mathbb{E}}[F(X^{(\infty,\gamma)}_{.})].$$ This property implies that $X^{(\infty,\gamma)}$ is stationary.\\ We now focus on the adaptation of $X^{(\infty,\gamma)}$. In this step, we need to introduce, for a subset $D$ of $\ER$ that contains $0$, the Polish space ${\cal W}_{\theta,\delta}(D)$ that denotes the completion of ${\cal C}_0^{\infty}(D,\ER^q)$ (the space of ${\cal C}^\infty$-functions $f:D\rightarrow\ER^q$ with compact support and $f(0)=0$) for the norm
$$\|f\|=\sup_{s,t\in D}\frac{|f(t)-f(s)|}{|t-s|^\theta(1+|t|^\delta+|s|^{\delta})}.$$ This space is convenient to obtain some Feller properties for the conditional distribution of the fractional Brownian motion given its past. More precisely, by Lemmas 4.1 to 4.3 of \cite{hairer2}, the paths of $B^H$ belong $a.s.$ to ${\cal W}_{\theta,\delta}(\ER)$ when $\theta\in(1/2,H)$ and $\theta+\delta\in(H,1)$. Furthermore, setting $B^{H,u}_t=B^H_{t+u}-B^H_u$, we also deduce from these lemmas that for every non-negative $t$ and $T$,
$${\cal P}_T(\omega,.):={\cal L}((B^{H,t+T}_s)_{s\le 0}| (B^{H,t}_s)_{s\le0}=(\omega_s)_{s\le0})$$
is a Feller transition on ${\cal W}_{\theta,\delta}(\ER_{-})$ (which does not depend on $t$). \\
Let us now prove that $X^{(\infty,\gamma)}$ is adapted, $i.e.$ that for every $t\ge 0$, $(X^{(\infty,\gamma)}_s)_{s\le t}$ and $(B^H_s)_{s\ge t}$ are independent conditionally to $(B^H_s)_{s\le t}$. One can check that it is enough to prove that for every $t\ge0$ and (arbitrary large) $T\ge0$, $(X^{(\infty,\gamma)}_s)_{s\le t}$ and $(B^{H,t+T}_s)_{s\ge0}$ are independent conditionally to $(B^{H,t}_s)_{s\le 0}$ (using on the one hand that $(B^H_s)_{s\le t}$ is trivially $\sigma(B^H_s,s\le t)$-measurable and that for every $u\ge0$, $\sigma (B^{H,u}_s,s\le0)=\sigma(B^{H}_s,s\le u)$). {To prove this conditional independence property, it is now enough to} show that for every $t\ge 0$, for every $T\ge0$, for every bounded continuous functionals $f:\bar{\cal C}^{\theta}([0,t],\ER^d)\rightarrow\ER$, $g:{\cal W}_{\theta,\delta}(\ER_{-})\rightarrow\ER$ and $h:{\cal W}_{\theta,\delta}(\ER_{-})\rightarrow\ER$ \begin{equation}\label{ekek} \begin{split} {\mathbb{E}}[f(X^{(\infty,\gamma)}_{s},s\in[0, t])&g(B^{H,t+T}_{s},s\le 0) h(B^{H,t}_{s},s\le0)]\\ &={\mathbb{E}}[f(X^{(\infty,\gamma)}_{s},s\in[0, t])\psi^{g}(B^{H,t}_{s},s\le 0)h(B^{H,t}_{s},s\le 0)] \end{split} \end{equation}
where $\psi^g((\omega_{s})_{s\le 0})=\mathbb{E}[g(B^{H,t+T}_s,s\le 0)| (B^{H,t}_s)_{s\le 0}= (\omega_s)_{s\le 0}]={\cal P}_Tg((\omega_{s})_{s\le0})$. Since ${\cal P}_T(\omega,.)$ is Feller, $\psi^g$ is continuous on ${\cal W}_{\theta,\delta}(\ER_{-})$.\\ Then, using the ergodicity of the increments of $B^H$, we can show as in the beginning of the proof that $(\tilde{{\cal P}}^{(n,\gamma)}(\omega))_{n\ge1}$ is tight on $\bar{\cal C}^\theta(\ER_+,\ER^d)\times{\cal W}_{\theta,\delta}(\ER)$. Thus, there exists $a.s.$ a sequence $(n_k)$ such that $${\mathbb{E}}[f(X^{(\infty,\gamma)}_{s},s\le t)g(B^{H,t+T}_{s},s\le 0) h(B^{H,t}_{s},s\le 0)]=\lim_{k\rightarrow+\infty}\frac{1}{n_k}\sum_{k=1}^{n_k} H_{k-1} J_k$$ and such that
$${\mathbb{E}}[f(X^{(\infty,\gamma)}_{s},s\le t)\psi^g(B^{H,t}_{s},s\le 0) h(B^{H,t}_s,s\le 0)]=\lim_{k\rightarrow+\infty}\frac{1}{n_k}\sum_{k=1}^{n_k}
H_{k-1} \mathbb{E}[J_k|{\cal F}_{\gamma (k-1)+t}]$$ with ${\cal F}_u=\sigma(B_s^H,s\le u)$, $H_k=f(\bar{X}^\gamma_{\gamma k+s},s\le t) h(B^{H}_{\gamma (k-1)+s+t}-B^H_{\gamma (k-1)+t},s\le 0)$, and $J_k={g}( B^H_{\gamma (k-1)+s+t+T}-B^H_{\gamma (k-1)+t+T}, s\le 0)$. This implies that it is now enough to prove that
$$\frac{1}{n}\sum_{k=1}^n H_{k-1}\left(J_k-\mathbb{E}[J_k|{\cal F}_{\gamma (k-1)+t}]\right)\xrightarrow{n\rightarrow+\infty}0\quad a.s.$$ This point follows from a decomposition of the above sum in martingale increments and from classical martingale arguments (see proof of Proposition 6 of \cite{cohen-panloup} for a similar argument).
\end{proof} \subsection{Identification of limits when $\gamma \to 0^+$} In this part we fix a $H$-fractional Brownian motion $B^H$ on $ \bar{{\mathcal C}}^\theta(\ER,\ER^q) $ and we consider a pair $(X^{\infty,\gamma},B^H) $ on $ \bar{{\mathcal C}}^\theta(\ER_+,\ER^d) \times \bar{{\mathcal C}}^\theta(\ER_+,\ER^q) $ such that for each $ \gamma > 0 $ the joint distribution is given by Proposition~\ref{prop2}. \begin{prop}\label{gamma-0} Let $(\gamma_k) $ be a sequence converging to $ 0$ such that the distributions of $(X^{\infty,\gamma_k},B^H) $ are converging weakly on $ \bar{{\mathcal C}}^\theta(\ER_+,\ER^d) \times \bar{{\mathcal C}}^\theta(\ER,\ER^q) $ to $(X^{\infty},B^H).$ Then $X^{\infty} $ is a stationary adapted solution to~\eqref{fractionalSDE0} in the sense of Definition~\ref{def:stat-sol}. \end{prop} \begin{proof} Let us first introduce $$ \tilde{\Phi}(\alpha,\beta)_t := \alpha_0 + \int_0^t b(\tilde{\Phi}(\alpha,\beta)_{s}) ds + \int_0^t \sigma(\tilde{\Phi}(\alpha,\beta)_{s}) d \beta_s,$$ and remark that $ \tilde{\Phi}(\alpha,\beta) = \Phi(a,\beta),$ if $\alpha_0=a.$ We want to show that \begin{equation}
\label{eq:tilde-eds}
X^{\infty}= \tilde{\Phi}(X^{\infty},B^H) \end{equation} almost surely so that $(X^{\infty},B^H) $ is a solution to~\eqref{fractionalSDE0}. Let us rewrite the equation with the help of two continuous operators on $\bar{{\mathcal C}}^\theta(\ER_+,\ER^d) \times \bar{{\mathcal C}}^\theta(\ER_+,\ER^q) $ ~: $$ \Psi(\alpha,\beta)_t= \int_0^t b(\alpha_{s}) ds +\int_0^t \sigma(\alpha_{s}) d \beta_s,$$ and $$ \Delta(\alpha)_t= \alpha_t - \alpha_0.$$ Then equation~\eqref{eq:tilde-eds} is equivalent to \begin{equation}
\label{eq:tilde-eds-split}
\Delta(X^{\infty})= \Psi(X^{\infty},B^H). \end{equation} Let us also consider the discretization of $ \Psi $ $$ \Psi^{\gamma}(\alpha,\beta)_t= \int_0^t b(\alpha_{\underline{s}_{\gamma}}) d s +\int_0^t \sigma(\alpha_{\underline{s}_{\gamma}}) d \beta_s.$$ Obviously~\eqref{fractionalSDE0-disc} can be rewritten \begin{equation}
\label{eq:tilde-eds-split-disc}
\Delta(X^{\infty,\gamma})= \Psi^{\gamma}(X^{\infty,\gamma},B^H). \end{equation} \begin{lemme} \label{lem:psi-conv} Let $(\gamma_k)_{k\ge1}$ be a sequence converging to $ 0$ such that $(X^{\infty,\gamma_k},B^H)_{k\ge1} $ converges weakly on $ \bar{{\mathcal C}}^\theta(\ER_+,\ER^d) \times \bar{{\mathcal C}}^\theta(\ER,\ER^q) $ to $(X^{\infty},B^H).$ Then $\Psi^{\gamma_k}(X^{\infty,\gamma_k},B^H)$ converges weakly on $ \bar{{\mathcal C}}^\theta(\ER_+,\ER^d) $ to $\Psi(X^{\infty},B^H).$ \end{lemme} \begin{proof} Let $(\alpha,\beta) \in \bar{{\mathcal C}}^\theta(\ER_+,\ER^d) \times \bar{{\mathcal C}}^\theta(\ER_+,\ER^q).$ A classical result concerning the discretization of Young integrals shows that
$$ | \Psi(\alpha,\beta)_t- \Psi^{\gamma}(\alpha,\beta)_t| \le \|\alpha\|_{\theta,t} \|\beta\|_{\theta,t} \gamma^{2 \theta - 1} t.$$ See for instance~\cite{Coutin12}, Proposition 31 or~\cite{Young36}. Hence for $T > 0,$ \begin{equation}
\label{eq:int-Young-ineq}
\| \Psi(\alpha,\beta)- \Psi^{\gamma}(\alpha,\beta)\|_{\theta,T} \le \|\alpha\|_{\theta,T} \|\beta\|_{\theta,T} \gamma^{2 \theta - 1} T^{1-\theta}. \end{equation} Let $ F $ be any bounded $K$-Lipschitz functional on $ \bar{{\mathcal C}}^\theta([0,T],\ER^d), $ \begin{equation}
\label{eq:F1-ineq}
|\mathbb{E} (F(\Psi(X^{\infty,\gamma_k},B^H) ) - \mathbb{E} (F(\Psi(X^{\infty},B^H) )| \to 0 \end{equation} as $ k \to \infty.$ Then \begin{equation}
\label{eq:F2-ineq}
|\mathbb{E} (F(\Psi^{\gamma_k}(X^{\infty,\gamma_k},B^H) ) - \mathbb{E} (F(\Psi(X^{\infty,\gamma_k},B^H) )| \le K \mathbb{E} ( \|X^{\infty,\gamma_k}\|_{\theta,T} \|B^H\|_{\theta,T}) T^{1-\theta} \gamma_k^{2 \theta - 1}, \end{equation} and using Proposition~\ref{lemme4}\emph{(ii)} the left hand side of~\eqref{eq:F2-ineq} is converging to $0$ as $ k \to \infty.$ Combining~\eqref{eq:F1-ineq} and this last fact, we get the desired convergence in distribution. \end{proof} \noindent Let us start with \begin{equation}
\label{eq:concl} \Delta(X^{\infty,\gamma_k})=\Psi^{\gamma_k}(X^{\infty,\gamma_k},B^H), \end{equation} and let $ k \to \infty.$ By Lemma~\ref{lem:psi-conv}, the right hand side of~\eqref{eq:concl} converges to $ \Psi(X^{\infty},B^H) $ and the left hand side to $ \Delta(X^{\infty}),$ which, in turn, has the same distribution as $\Psi(X^{\infty},B^H).$\\
Now, let us prove that $X^\infty$ is stationary. It is enough to show that $\mathbb{E}[F(X^\infty_.)]=\mathbb{E}[F(X^{\infty}_{t+.})]$ for every $t\ge0$ and for every functional $F$ defined by $F(\alpha)=\prod_{k=1}^m f_i(\alpha_{t_i})$ where $f_1,\ldots, f_m$ denote Lipschitz continuous functions on $\ER^d$ and $t_1,\ldots,t_m$ belong to $\ER_+$. By Proposition \ref{prop2}, the distribution of $X^{\infty,\gamma}$ is invariant by the time-shift $(\theta_{k\gamma})$ for every $k\in\EN$ so that $\mathbb{E}[F(X^\infty_.)]=\mathbb{E}[F(X^{\infty}_{\underline{t}+.})]$. The result follows easily by checking that for every $T>0$,
$$\mathbb{E}[\sup_{u,v\in[0,T],|u-v|\le\gamma}|X^{\infty,\gamma}_v-X^{\infty,\gamma}_u|]\xrightarrow{\gamma\rightarrow0}0.$$ Finally, it remains to show that $(X^\infty,B^H)$ is adapted. Since $(X^{\infty,\gamma_k})$ converges in distribution to $X^\infty$ on $\bar{\cal C}^\theta(\ER_+,\ER^d)$ and since $B^H$ belongs to ${\cal W}_{\theta,\delta}$ (with $\theta\in(1/2,H)$ and $\theta+\delta\in(H,1)$), $(X^{\infty,\gamma'_k}, B^H)$ converges to $(X^\infty,B^H)$ for $\gamma'_k$ a subsequence of $ \gamma_k.$ Then, we can let $\gamma$ go to $0$ in equality \eqref{ekek} and the result follows. \end{proof} \section{Simulations}\label{section6} In this section, we give an illustration of the application of our procedure for a one-dimensional toy equation: $$dX_t=-X_tdt+(4+\cos(X_t))dB_t^H.$$ We propose to compute an estimation of the density of the (marginal) invariant distribution in this case. We denote it by $\nu^H_0$. By Theorem \ref{principal1}, for every bounded continuous function $f:\ER^d\rightarrow\ER$, $$\lim_{\gamma\rightarrow0}\lim_{n\rightarrow+\infty}{\cal P}^{(n,\gamma)}_0(\omega,f)=\nu_0^H(f).$$ \noindent The first step is to simulate the sequence $(B^H_{\gamma k}-B^H_{\gamma (k-1)})_{k=1}^n$. We use the Wood-Chan method (see \cite{wood}) which is based on the embedding of the covariance matrix of the fractional increments in a symmetric circulant matrix (whose eigenvalues can be computed using the Fast Fourier Transform).\\ Then, we compute $K_h\ast{\cal P}^{(n,\gamma)}_0$ where $K_h$ is the Gaussian convolution kernel defined by $K_h(x)=\frac{1}{\sqrt{2\pi}h}\exp(-\frac{x^2}{2h})$. Note that $K_h\ast{\cal P}^{(n,\gamma)}_0(x_0)={\cal P}^{(n,\gamma)}_0(K_h(x_0-.))$, where, for a measure $\mu,$ and a $\mu$-measurable function f, we set $\mu(f)=\int fd\mu$. In Figure \ref{figure1} is depicted the approximate density with the following choices of parameters
$$ n=10^7,\quad \gamma=0.05\quad h=0.2,\quad H=\frac{3}{4}.$$ We choose to compare it with the density of the invariant distribution when $H=1/2$. Note that in this case, the invariant distribution is (semi)-explicit (as for every ergodic one-dimensional diffusion) and is given by $$\nu^{\frac{1}{2}}_0(dx)=\frac{M(dx)}{M(\ER)}\quad\textnormal{where}\quad M(dx)=\frac{1}{(4+\cos x)^2}\exp\left(-\int_0^x\frac{2u}{(4+\cos u)^2}du\right)dx.$$ We observe that the distribution when $H=3/4$ has heavier tails than in the diffusion case. \begin{figure}\label{figure1}
\end{figure} Finally, in order to have a rough idea of the rate of convergence, we depict in Figure \ref{figure2} the approximate densities for different values of $n$ keeping the other parameters unchanged. \begin{figure}
\caption{Approximate density of $\nu_0^H$ for $n=10^5$ (dotted line), $n=10^6$ (dash-dotted line), $n=10^7$ (continuous line)}
\label{figure2}
\end{figure} \begin{Remarque} As mentioned before, this section is only an illustration. In fact, there are (many) numerical open questions. For the estimation of the error, it would be necessary for a function $f$ to get some rate of convergence results for ${\cal P}^{(n,\gamma)}_0(f)-\nu_H(f)$ (long-time error) and for $\nu^{H,\gamma}_0(f)-\nu_0^H(f)$ (discretization error) where $\nu^{H,\gamma}_0$ denotes the initial distribution of the stationary Euler scheme with step $\gamma$. Note that in the diffusion case, it can be shown (under some appropriate assumptions that the long time error is about $(\gamma n)^{-\frac{1}{2}}$ (see \cite{bhatta82} for the corresponding result in the continuous case) whereas the discretization error is $O(\gamma)$ (see \cite{talay}, Theorem 3.3 for a similar result with the Milstein scheme). Finally, even if the Wood and Chan simulation method is fast and exact, it requires a lot of memory because of the Fast Fourier Transform. On Matlab, for instance, this implies that we can not take $n$ greater than $2.10^7$. Thus, it could be interesting to study some discretization schemes based on some approximations of the fBm-increments simulated, which consumes less memory.
\end{Remarque} \section{Appendix} \textbf{Proof of Proposition \ref{unicitemesinvariante}} Let us show that $(\bar{X}_{\gamma k})$ is a \textit{skew-product} in the sense of~\cite{hairer09} as follows. For a fractional Brownian $B^H$ motion on $\ER$, set for every $n\in\mathbb{Z}$ $\Delta_n^{\gamma}=B^{H}_{(n+1)\gamma}-B^H_{n\gamma}$. Setting ${\cal W}:=(\ER^d)^{\mathbb {Z}_{-}}$, we then introduce the regular conditional probability $\bar{\cal P}^{\gamma}:{\cal W}\rightarrow{\cal M}_1(\ER^d)$ defined by\footnote{
Note that since $(\Delta_n^{\gamma})_{n\in\mathbb {Z}}$ is a stationary sequence, ${\cal L}(\Delta_1^{\gamma}|(\Delta_k^{\gamma})_{k\le0}=\omega)={\cal L}(\Delta_{n+1}^{\gamma}|(\Delta_{n+k}^{\gamma})_{k\le0}=\omega)$ for every $n\in\mathbb{Z}$.}:
$$\bar{\cal P}^{\gamma}(\omega)={\cal L}(\Delta_1^{\gamma}|(\Delta_k^{\gamma})_{k\le0}=\omega)$$ and denote by ${\cal P}^{\gamma}$ the Feller transition on ${\cal W}$ defined for every measurable function $f:{\cal W}\rightarrow \ER$ by ${\cal P}^{\gamma}f(\omega)=\int_{\ER^d} f(\omega\sqcup \tilde{\omega})\bar{\cal P}^{\gamma}(\omega,d\tilde{\omega})$ where for $\omega\in(\ER^d)^{\mathbb{Z}_-}$ and $\tilde{\omega}\in\ER^d$, $\omega\sqcup\tilde{\omega}=(\ldots,\omega_{_2},\omega_{_1},\omega_{0},\tilde{\omega})$. Setting ${ \Phi}^{\gamma}(x,\tilde{\omega})=x+\gamma b(x)+\sigma(x) \tilde{\omega}$ and $\mathbb {P}_H^{\gamma}:={\cal L}((\Delta_n)_{n\le0})$, we have defined a skew-product $({\cal W},\mathbb {P}_H^{\gamma},{\cal P}^{\gamma},\ER^d,{ \Phi}^{\gamma})$ with the transition operator ${\cal Q}^{\gamma}$ on $\ER^d\times{\cal W}$ defined by $${\cal Q}^{\gamma} f(x,\omega)=\int f(\Phi^\gamma(x,\omega')){\cal P}^{\gamma}(\omega,d{\omega}'),$$ which describes the dynamics of the Euler scheme. \\ Then, thanks to Theorem 1.4.17 of \cite{hairer09}, uniqueness of the adapted and stationary discrete Euler scheme $(\bar{X}_{\gamma k})$ (in distribution) holds, if the skew-product $({\cal W},\mathbb {P}_H^{\gamma},{\cal P}^{\gamma},\ER^d,{ \Phi}^{\gamma})$ is strong Feller and topologically irreducible (in the sense of Definition 1.4.6 and 1.4.7 of \cite{hairer09}).\\ First, write $\tilde{\omega}=(\tilde{\omega}^1,\ldots,\tilde{\omega}^q)$ and $\Phi^\gamma=(\Phi^\gamma_1,\ldots,\Phi^\gamma_d)$. Denote by $M^{\Phi}(x,\tilde{\omega})$ the (discrete) Malliavin covariance matrix of $\Phi$ defined by $$\forall (x,\tilde{\omega})\in\ER^d\times\ER^d\quad\textnormal{and}(i,j)\in\{1,\ldots,d\}^2,\quad M^{\Phi}_{i,j}(x,\tilde{\omega}):=\sum_{k=1}^d \partial_{\tilde{\omega}^k}\Phi^{\gamma}_i(x,\tilde{\omega})\partial_{\tilde{\omega}^k}\Phi^{\gamma}_j(x,\tilde{\omega}).$$ Thus, $M^\Phi(x,\tilde{\omega})=(\sigma\sigma^*)(x)$ and since $\sigma^{-1}$ is bounded (and continuous), it follows that $x\rightarrow({\rm det}(M^\Phi)^{-1}(x,\omega)$ is bounded continuous. Second, the functions $D_\omega\Phi$, $D_\omega D_x \Phi$ and $D^2_\omega \Phi$ are clearly bounded continuous. Finally, the sequence $((\Delta_n^\gamma)^1)$ has a spectral density $f$ that satisfies $\int_{-\pi}^\pi (f(x))^{-1}dx<+\infty$ (see $e.g.$ \cite{beran} for an explicit expression of $f$). Thus, it follows from Theorem 1.5.9 of \cite{hairer09} that the skew-product is strong Feller.\\ For the topological irreducibility, it is enough to show that for every $(x,\omega)\in\ER^d\times{\cal W}$, for every $(y,\varepsilon)\in\ER^d\times\ER_+^*$, ${\cal Q}(x,\omega,B(y,\varepsilon)\times{\cal W})>0$ . Since $\sigma$ is invertible, the map $\Phi$ is controllable in the following sense: $\Phi(x,\tilde{\omega}_x)=y$ has a (unique) solution $\tilde{\omega}\in\ER^q$, for every $x,y\in\ER^d$. Furthermore, $b$ and $\sigma$ being continuous, for every $\varepsilon>0$, there exists $r_\varepsilon$ such that for every $\tilde{\omega}\in B(\tilde{\omega}_x,r_\varepsilon)$, $\Phi(x,{\tilde{\omega}})\in B(y,\varepsilon)$. Thus $${\cal Q}(x,\omega,B(y,\varepsilon)\times{\cal W})\ge \bar{\cal P}(\omega,B(\tilde{\omega}_x,r_\varepsilon))>0,$$ since $\bar{\cal P}(\omega,.)$ is Gaussian with positive variance. This concludes the proof.\\
{\bf Acknowledgment}
We would like to thank the anonymous referee for his/her careful reading and his/her suggestions that helped us to improve the first version of this article.
\end{document} | arXiv |
# Linear programming: formulation and solution methods
Linear programming is a fundamental optimization technique that seeks to find the best solution to a mathematical problem by finding the optimal values for the variables that satisfy the constraints. It is widely used in various fields such as operations research, economics, and engineering.
In this section, we will cover the following topics:
- The concept of linear programming and its applications
- The standard form of linear programming problems
- The simplex method for solving linear programming problems
- The interior-point method for solving linear programming problems
Consider the following linear programming problem:
$$\begin{aligned}
\text{Maximize:} \quad & 3x_1 + 2x_2 \\
\text{Subject to:} \quad & x_1 + 2x_2 \le 10 \\
& 2x_1 + x_2 \le 8 \\
& x_1, x_2 \ge 0
\end{aligned}$$
This problem can be solved using the simplex method or the interior-point method.
## Exercise
Solve the linear programming problem given in the example using the simplex method.
To implement the simplex method in Python, you can use the `scipy.optimize.linprog` function. Here's an example:
```python
import numpy as np
from scipy.optimize import linprog
# Objective function coefficients
c = [-3, -2]
# Constraint matrix
A = [[1, 2], [2, 1]]
# Constraint values
b = [10, 8]
# Bounds for variables
x_bounds = (0, None)
y_bounds = (0, None)
# Solve the linear programming problem
res = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds], method='simplex')
# Print the optimal solution
print(res.x)
```
This code will output the optimal values for the variables `x_1` and `x_2`.
# Integer programming: formulation and solution methods
Integer programming extends linear programming by adding the constraint that the variables must be integers. This constraint makes the problem more challenging to solve, but it can be useful in solving optimization problems where the variables must be discrete.
In this section, we will cover the following topics:
- The concept of integer programming and its applications
- The branch-and-bound method for solving integer programming problems
- The cutting-plane method for solving integer programming problems
Consider the following integer programming problem:
$$\begin{aligned}
\text{Maximize:} \quad & 3x_1 + 2x_2 \\
\text{Subject to:} \quad & x_1 + 2x_2 \le 10 \\
& 2x_1 + x_2 \le 8 \\
& x_1, x_2 \in \mathbb{Z}
\end{aligned}$$
This problem can be solved using the branch-and-bound method or the cutting-plane method.
## Exercise
Solve the integer programming problem given in the example using the branch-and-bound method.
# Gradient descent: theory and implementation in Python
Gradient descent is an optimization algorithm that seeks to find the minimum of a function by iteratively moving in the direction of the steepest decrease. It is widely used in machine learning and deep learning.
In this section, we will cover the following topics:
- The concept of gradient descent and its applications
- The gradient of a function and its properties
- The implementation of gradient descent in Python
Consider the following function:
$$f(x) = x^2$$
The gradient of this function is:
$$\nabla f(x) = 2x$$
## Exercise
Implement the gradient descent algorithm in Python to minimize the function $f(x) = x^2$.
Here's an example implementation of the gradient descent algorithm in Python:
```python
import numpy as np
# Gradient function
def gradient(x):
return 2 * x
# Initial values
x = 5
learning_rate = 0.1
# Gradient descent iterations
for i in range(100):
x = x - learning_rate * gradient(x)
print(f"Iteration {i + 1}: x = {x}")
```
This code will output the values of `x` after each iteration of the gradient descent algorithm.
# Genetic algorithms: theory and implementation in Python
Genetic algorithms are a class of optimization algorithms inspired by the process of natural selection. They are used to find approximate solutions to optimization and search problems.
In this section, we will cover the following topics:
- The concept of genetic algorithms and their applications
- The components of a genetic algorithm, including chromosomes, fitness functions, and selection, crossover, and mutation operations
- The implementation of genetic algorithms in Python
Consider the following function to be optimized:
$$f(x) = x^2$$
The fitness function for this problem is:
$$F(x) = -f(x) = -x^2$$
## Exercise
Implement a genetic algorithm in Python to minimize the function $f(x) = x^2$.
Here's an example implementation of a genetic algorithm in Python:
```python
import numpy as np
# Fitness function
def fitness(x):
return -x**2
# Genetic algorithm implementation
# ...
# Run the genetic algorithm
solution = genetic_algorithm()
print(f"Optimal solution: x = {solution}")
```
This code will output the optimal solution found by the genetic algorithm.
# Simulated annealing: theory and implementation in Python
Simulated annealing is an optimization algorithm that uses a random search to find the minimum of a function. It is inspired by the annealing process in metallurgy, where a material is heated and then allowed to cool slowly.
In this section, we will cover the following topics:
- The concept of simulated annealing and its applications
- The components of a simulated annealing algorithm, including the temperature parameter and cooling schedule
- The implementation of simulated annealing in Python
Consider the following function to be optimized:
$$f(x) = x^2$$
## Exercise
Implement a simulated annealing algorithm in Python to minimize the function $f(x) = x^2$.
Here's an example implementation of a simulated annealing algorithm in Python:
```python
import numpy as np
# Objective function
def objective(x):
return x**2
# Simulated annealing algorithm implementation
# ...
# Run the simulated annealing algorithm
solution = simulated_annealing()
print(f"Optimal solution: x = {solution}")
```
This code will output the optimal solution found by the simulated annealing algorithm.
# Case studies and practical applications of optimization in Python
- Linear programming in supply chain optimization
- Integer programming in scheduling problems
- Gradient descent in training deep learning models
- Genetic algorithms in protein fold prediction
- Simulated annealing in solving the traveling salesman problem
# Performance evaluation and comparison of optimization algorithms
- The concept of performance evaluation for optimization algorithms
- The evaluation criteria for optimization algorithms, including running time, solution quality, and convergence rate
- The comparison of different optimization algorithms using these evaluation criteria
Consider the following linear programming problem:
$$\begin{aligned}
\text{Maximize:} \quad & 3x_1 + 2x_2 \\
\text{Subject to:} \quad & x_1 + 2x_2 \le 10 \\
& 2x_1 + x_2 \le 8 \\
& x_1, x_2 \ge 0
\end{aligned}$$
We can compare the performance of the simplex method and the interior-point method by evaluating their running time and solution quality.
## Exercise
Compare the performance of the simplex method and the interior-point method for solving the linear programming problem given in the example.
# Challenges and future directions in optimization
- The challenges in developing efficient and accurate optimization algorithms
- The future directions in optimization research, including new algorithm development, hybrid algorithms, and the integration of machine learning techniques
# Selected exercises and problem sets for practice
- Exercise 1: Solve a linear programming problem using the simplex method.
- Exercise 2: Solve an integer programming problem using the branch-and-bound method.
- Exercise 3: Implement a genetic algorithm to minimize a given function.
- Exercise 4: Compare the performance of gradient descent and simulated annealing for a given optimization problem.
- Exercise 5: Solve a combinatorial optimization problem using a suitable optimization algorithm.
## Exercise
Solve the linear programming problem given in Exercise 1 using the simplex method.
This exercise will test your understanding of the simplex method and your ability to implement the method in Python.
Here's an example implementation of the simplex method in Python:
```python
import numpy as np
from scipy.optimize import linprog
# Objective function coefficients
c = [-3, -2]
# Constraint matrix
A = [[1, 2], [2, 1]]
# Constraint values
b = [10, 8]
# Bounds for variables
x_bounds = (0, None)
y_bounds = (0, None)
# Solve the linear programming problem
res = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds], method='simplex')
# Print the optimal solution
print(res.x)
```
This code will output the optimal values for the variables `x_1` and `x_2`. | Textbooks |
Chladni patterns
At Climate Audit, Steve McIntyre has been frequently writing about Chladni patterns in connected with principal components of autocorrelated data. He developed it for the Steig paper, and was disappointed when J Climate wouldn't allow its inclusion in the response. And he's recently claimed an appearance in another paper.
Chladni patterns are modes of oscillation, originally of a vibrating plate. Now people more often think of a drum membrane, which is a slightly different wave equation, but the idea, and patterns, are similar.
I must admit that I hadn't heard of Chladni before Hans Erren drew attention to them in Steve's first post. Some interesting history there. But I am familiar with the modes in question.
Steve thinks that if a Chladni pattern emerges, that somehow means that the result is showing that rather than the information about the climate pattern being sought, so the information content is reduced. I don't agree - there are reasons why the patterns arise, and they are just as informative in PCA as they are in wave studies. I'll try to show why.
Warning - mathematics (and \(\LaTeX\)) after the jump.
Resonance and wave equations
Resonance is familiar with acoustics. If you speak in the open air, your voice propagates away in all directions, attenuating without reflection or selective amplification. If you stand in a partly enclosed cavity, your voice is slightly louder in certain frequency bands. In a totally enclosed bare room, you hear a characteristic booming response - some frequencies are much louder.
These are the ones that excite a resonant mode, in which the air oscillates, but the normal velocity at the boundary is zero. A 3D version of the modes of a vibrating stretched string, which has zero velocity at its ends.
The essential requirement is that the wave energy must be confined but not dissipated. That is why the resonance improves in deeper cavities, for example.
The wave equation for pressure, say, is (with c=speed of sound) $$\nabla^2 p = \frac{1}{c^2}\frac{\partial^2 p}{\partial t^2}$$ If you substitute a resonant mode \(p=P \sin(\omega t)\), then the equation is $$\nabla^2 P = -(\frac{\omega}{c})^2 P$$ P is the resonant mode, and so is an eigenvector of the Laplacian \(\nabla^2\). The resonant frequencies correspond to the eigenvalues.
For Chladni's plate the wave equation is more complicated, but the principle is the same.
Spatial autocorrelation.
Steve McI also described the simple Toeplitz autocorrelation coefficient matrix that you get in one dimension for a spatial model. With N+1 equally spaced points, the coefficients can be assumed to be powers of r - the correlation of adjacent sites. The matrix is: $$R = \left(\begin{array}{ccccc} 1 & r & r^2 & \ldots & r^N\\ r & 1 & r & r^2 & \ldots \\ r^2 & r & 1 & r & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots\\ r^N & \ldots & r^2 & r & 1 \end{array}\right) $$ The Toeplitz property is that all terms on each diagonal are the same. This correlation matrix has a simple inverse: $$R^{-1} = \left(\begin{array}{ccccc} q & -qr & 0 & \ldots & 0\\ -qr & 2q-1 & -qr & 0 & \ldots \\ 0 & -qr & 2q-1 & -qr & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & \ldots & 0 & -qr & q \end{array}\right),\quad q=\frac{1}{1-r^2} $$ Still a Toeplitz matrix, almost, but also banded - tridiagonal. The deviation from Toeplitz is at the top left and bottom right corner terms.
Relation to Laplacian and the wave equation
From the last equation, $$\begin{align} R^{-1} &= qr\left(\begin{array}{ccccc} 1/r & -1 & 0 & \ldots & 0\\ -1 & r+1/r & -1 & 0 & \ldots \\ 0 & -1 & r+1/r & -1 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & \ldots & 0 & -1 & 1/r \end{array}\right) \\ &= qr\left(\begin{array}{ccccc} 1 & -1 & 0 & \ldots & 0\\ -1 & 2 & -1 & 0 & \ldots \\ 0 & -1 & 2 & -1 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & \ldots & 0 & -1 & 1 \end{array}\right) + q(1-r)^2 \left(\begin{array}{ccccc} 1/(1+r^2) & 0 & 0 & \ldots & 0\\ 0 & 1 & 0 & 0 & \ldots \\ 0 & 0 & 1 & 0 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & \ldots & 0 & 0 & 1/(1+r^2) \end{array}\right) \end{align} $$ If you remember finite differences, the first matrix is just the negative of the second difference operator, corresponding to the 1D Laplacian. And the second is almost a multiple of the identity, and is small if r is close to 1.
That's the key to the connection between Chladni patterns and the autocorrelation matrix R. The inverse of R and the Laplacian of the wave equation differ by close to a multiple of the identity, which means they have almost the same eigenvectors. And the eigenvectors of R and \(R^{-1}\) are the same. R is symmetric and positive definite.
Well, an unsubtle difference is that Chladni oatterns are not 1D. But the same reasoning works - it's too messy to set out here.
The subtler difference is that the diagonal correction is not quite Toeplitz. That relates to the notion of boundary conditions for the wave equation, which is indeed critical for resonance.
At this stage I'll have to just arm-wave on that - it does in fact give the zero normal boundary condition, which is sufficient for resonance.
The same mathematics that gives Chladni for the wave equations gives similar eigenvectors for the spatially autocorrelated matrix. It isn't a spurious consequence. Consequently their appearance in, say, Steig et al 2009 doesn't mean that the PC's are "just Chladni" any more than they are just autocorrelation. Spatial autocorrelation is an essential part of the results, and the Chladni patterns reflect that.
I think that's enough heavy maths for now - I could later go more into the properties of Chladni patterns, and in particular into the implications for eigenvalue pairing (for PCs) which SM criticised in Steig et al. That would generate pretty pictures.
Meanwhile, I'll just relay these pictures of resonant modes from Wikipedia where more Chladni patterns on the disc and sphere can be found.
Posted by Nick Stokes at 9:35 PM 35 comments
Latex now works here
I have implemented Latex (the MathJax version). It seems to work in comments too. You can just write normal Latex between these symbols(remove the _s)
\_(...\_) for inline
\_[...\_] or $_$...$_$ for display equations - note the doubled $'s
The next post should give this a workout.
The Woody Guthrie Award - Bart Verheggen
Last November, I was very pleased and flattered when ScienceofDoom chose Moyhu as the recipient of the Woody Guthrie award.
This award has an interesting history - it was initiated as a commendation of a "thinking blogger". The award was passed to SoD from Skeptical Science. I think both have been highly respected for their dedication to dispassionate analysis, and Moyhu has sought to live up to that.
I think Bart impressed all sides of the climate debate with his hosting of the very popular discussion on comparing temperature indices, which evolved into a debate about time series, unit roots and random walks. Strong views were expressed by well-informed commenters led by VS. Bart kept the discussion on track and constructive, with 2186 comments made. I think it is the best technical climate discussion I have seen.
But his blog isn't just technical - there is thoughtful, well-informed and inclusive discussion of policy as well. Bart always takes a broad view.
And there are brilliant cartoons as well.
I commend Bart's blog to readers, and I am very pleased to pass on the Award to him. | CommonCrawl |
communications earth & environment
Methane emissions from agricultural ponds are underestimated in national greenhouse gas inventories
Half of global methane emissions come from highly variable aquatic ecosystem sources
Judith A. Rosentreter, Alberto V. Borges, … Bradley D. Eyre
Separating natural from human enhanced methane emissions in headwater streams
Yizhu Zhu, J. Iwan Jones, … Mark Trimmer
Global regulation of methane emission from natural lakes
Lúcia Fernandes Sanches, Bertrand Guenet, … Francisco de Assis Esteves
Disproportionate increase in freshwater methane emissions induced by experimental warming
Yizhu Zhu, Kevin J. Purdy, … Mark Trimmer
Significant methane ebullition from alpine permafrost rivers on the East Qinghai–Tibet Plateau
Liwei Zhang, Xinghui Xia, … Peter A. Raymond
Contribution of oxic methane production to surface methane emission in lakes and its global importance
Marco Günthel, Daphne Donis, … Kam W. Tang
Florida's urban stormwater ponds are net sources of carbon to the atmosphere despite increased carbon burial over time
Audrey H. Goeckner, Mary G. Lusk, … Joseph M. Smoak
Warming reshapes methane fluxes
Kuang-Yu Chang
Modeling suggests fossil fuel emissions have been driving increased land carbon uptake since the turn of the 20th Century
Christopher R. Schwalm, Deborah N. Huntzinger, … Yaxing Wei
Martino E. Malerba ORCID: orcid.org/0000-0002-7480-47791,
Tertius de Kluyver2,
Nicholas Wright3,
Lukas Schuster1 &
Peter I. Macreadie ORCID: orcid.org/0000-0001-7362-08821
Communications Earth & Environment volume 3, Article number: 306 (2022) Cite this article
Agricultural ponds have some of the highest methane emissions per area among freshwater systems, and these anthropogenic emissions should be included in national greenhouse gas inventories. Here we deliver a continental-scale assessment of methane emissions from agricultural ponds in the United States and Australia. We source maps of agricultural ponds, compile a meta-analysis for their emissions and use published data to correct for temperature and the relative contributions of two methane fluxes (diffusion and ebullition). In the United States, 2.56 million agricultural ponds cover 420.9 kha and emit about 95.8 kt year−1 of methane. In Australia, 1.76 million agricultural ponds cover 291.2 kha and emit about 75.1 kt year−1 of methane. Despite large uncertainties, our findings suggest that small water bodies emit twice as much methane than is currently accounted for in national inventories. Managing these systems can reduce these emissions while benefiting productivity, ecosystem services, and biodiversity.
Globally, aquatic systems contribute to half of total natural and anthropogenic emissions of methane (CH4)1, a powerful greenhouse gas (GHG) with much higher warming potential than carbon dioxide (CO2)2,3. Small aquatic habitats (<0.1 ha in area) emit disproportionally more methane per unit area than larger lakes, contributing to ca. 37% of total lentic methane emissions, despite occupying <10% of the global freshwater surface area of lakes and ponds4. Many of these small systems are human-constructed to secure water for crops and livestock, and to support the ever-increasing demand for agricultural production5,6,7. This proliferation of agricultural water bodies is likely to affect global biogeochemical cycles significantly, but the evidence is lacking.
Agricultural ponds (also known as farm dams, impoundments, or dugouts) are small, constructed waterbodies (typically between 0.01 and 1 ha in surface area) with some of the highest methane emissions per area among freshwater ecosystems8,9,10. These recently discovered emissions are boosted by unusually high concentrations of fertiliser and manure runoffs, which increases organic matter and creates the ideal conditions for methane production8,9. Also, these systems are typically shallow and can warm up rapidly, boosting metabolic rates, bacterial build-up, and methanogenesis. For example, Ollivier et al.9 estimated that farm ponds produce 3.43 times more CO2-eq (methane + carbon dioxide) emissions per area than reservoirs.
Importantly, emissions from agricultural ponds are of anthropogenic nature and should therefore be included in national carbon inventories submitted to the United Nations Framework Convention on Climate Change (UNFCCC) under the Paris agreement11. The Intergovernmental Panel on Climate Change (IPCC) recently rectified their guidelines to encourage the inclusion of agricultural ponds as "Other Constructed Waterbodies" in National Greenhouse Gas Inventories12. Yet, there is little data on the abundance and distribution of agricultural ponds in most of the world6, and this knowledge gap complicates their inclusion in national GHG inventories.
Here we deliver a first-order assessment of methane emissions from agricultural ponds in the United States and Australia (see Fig. S1 for the model diagram). We leveraged two mapping programs6,13 to identify 4.17 million agricultural ponds in 1.75 million kha (17.5 million km2) of land across both countries (Fig. 1a, b). We merged this dataset with annual temperatures (Fig. 1c, d), and we conducted a meta-analysis to quantify average methane emissions from agricultural ponds (N = 286, Fig. 2). Then, we used a published dataset1 to calibrate the effects of temperature on methane emissions (N = 257, Fig. S2), and the relative contributions of ebullitive and diffusive methane fluxes (N = 164, Fig. S3). We calibrated a model to map temperature-adjusted methane emissions associated with agricultural ponds in the United States and Australia. Finally, we compared our results with the figures reported in the latest national GHG inventories reported to UNFCCC for 2020.
Fig. 1: Covariates (at 5 arcmin resolution) used to parametrise our model on methane emissions from agricultural ponds.
a Densities of agricultural ponds in the United States (2.56 millions) from Moore et al.13. b Densities of agricultural ponds in Australia (1.76 millions) from AusDams in Malerba et al.6. c Annual median temperatures in the United States. d Annual median temperatures in Australia. All temperatures are calculated using 10 years of weekly data from MODIS Terra Land Surface in Wan et al.32.
Fig. 2: Meta-analysis on total methane emissions (diffusion + ebullition) from agricultural ponds.
We compiled values from the scientific literature, supplemented with new data from 11 sites in temperate Australia (see labels for sample size; N). We standardised all rates to 15 °C (using Eq. 1 and Fig. S2) and accounted for ebullition and diffusion (using Fig. S3). The colours of the density functions indicate the climate. Black points indicate either the IPCC emission factor recommended for constructed waterbodies across all climates (Table 7.12 in Lovelock et al.14), or the geometric mean calculated from all data compiled in this study—with error bars representing the 95% confidence intervals. Box-and-whiskers show the distribution of the compiled data divided by climate.
Our meta-analysis showed that total temperature-adjusted methane emissions (diffusion + ebullition) from agricultural ponds are variable, with fluxes spanning from <1 to >103 kg CH4 ha−1 year−1 (Fig. 2). On average, we predict that agricultural ponds at 15 °C should emit 204 kg CH4 ha−1 year−1 (95% C.I.: 83–521; median: 157.7). Our estimate is within 12% of the relevant IPCC emission factor for "freshwater and brackish ponds" of 183 (95% C.I.: 118–228) kg CH4 ha−1 year−1 12. Yet, the proposed IPCC emission factor is temperature-independent and will underpredict emissions in warmer climates. For example, our model predicts that a farm dam at 30 °C should, on average, emit 405 (95% C.I.: 164–1037; median: 314.2) kg CH4 ha−1 year−1, which is twice as much as the IPCC emission factor.
In the United States, 2.56 million agricultural ponds cover 420.9 kha (Fig. 1a) and emit an estimated 95.8 kt CH4 year−1 (95% C.I.: 61–157; Fig. 3a). In Australia, 1.76 million agricultural ponds cover 291.2 kha (Fig. 1b) and emit 75.1 kt CH4 year−1 (95% C.I.: 47–123; Fig. 3b). Assuming a global warming potential of 28 times that of CO2 over a 100-year time scale (following IPCC Fifth Assessment Report3), these methane emissions are equivalent to 4.79 Mt CO2-eq year-1 (95% C.I.: 3.01–7.86; see Fig. S4 for the hotspots of methane emissions from agricultural ponds in the United States and Australia).
Fig. 3: Methane emissions from agricultural ponds in the United States and Australia.
a Spatial model at five arcmin resolution for annual methane emissions for agricultural ponds in the United States (total of 96 kt CH4 year−1). b Spatial model at five arcmin resolution for annual methane emissions for agricultural ponds in Australia (total of 75 kt CH4 year−1). c Emissions from agricultural ponds calculated in this study (with error bars representing the 95% confidence intervals) compared against those from "Other Constructed Waterbodies" under "Land Use, Land Use Change, and Forestry" reported to UNFCCC15 in 2020 for the United States and Australia. Percentages indicated above bars quantify the relative decrease from estimates in this study to those in UNFCCC reports. See Fig. S4 for the hotspots of methane emissions in the United States and Australia.
After the 2019 Refinement of IPCC guidelines, states are recommended to include methane emissions from all constructed ponds smaller than 8 ha for agriculture, recreation, and aquaculture in UNFCCC GHG inventories (as "Other Constructed Waterbodies" under "Land Use, Land Use Change, and Forestry")14. For 2020, the United States reported 173.1 kha of pond area emitting 43.75 kt CH4 (average emission factor of 252.78 kg CH4 ha−1 year−1), and Australia reported 316.4 kha of pond area emitting 40.73 kt CH4 [average emission factor of 128.75 kg CH4 ha−1 year−1; see Table 4(II) of the Common Reporting Format by UNFCCC15]. Our analysis suggests that these emissions are underestimated by around half. Specifically, emissions reported to UNFCCC for all constructed waterbodies smaller than 8 ha in the United States (43.75 kt CH4 year−1) and Australia (40.73 kt CH4 year−1) are 46% and 54% lower than our estimates for methane emissions from agricultural ponds between 0.01 and 1 ha in the United States (95.8 kt CH4 year−1) and in Australia (75.1 kt CH4 year−1), respectively (Fig. 3c). Part of this discrepancy may be that guidelines for national GHG inventories allow separating methane emissions of agricultural ponds (under "Other Constructed Waterbodies") from those of animal manure contamination in agricultural ponds (under "Manure Management"). Unfortunately, national inventories lack details on the methods for accounting for manure in agricultural ponds.
Agriculture contributes to 36% and 47% of all methane emissions in the United States (10.04 Mt CH4 year−1) and Australia (2.08 Mt CH4 year−1), respectively—mainly through enteric fermentation and manure breakdown16. However, these calculations omit emissions from agricultural ponds associated with rearing livestock, which could be non-trivial. Thus, it will be important for future studies to quantify the relative contributions of agricultural ponds to the total carbon footprint of animal agriculture.
It is important to note that there are several sources of error in our calculations (Fig. S5). Of the parameters in our model, estimates for the average methane flux, temperature sensitivity, and the contribution of methane ebullition have the most significant uncertainties (CV between 20% and 28%), mainly because these estimates are based on relatively small sample sizes (Fig. S5). Therefore, future work should prioritise improving current estimates using on-the-ground measurements from agricultural ponds across different climates. Conversely, predictions for pond distribution (CV of 10%) were more accurate because of large-scale assessments using satellite data (Fig. S5).
There are other sources of relevant emissions from agricultural ponds that our model fails to capture. In particular, while methane is often the most prominent GHG associated with these ponds (e.g., 83-94% of CO2-eq flux in Ollivier et al.9), our analysis ignored the contributions of other types of GHG—such as carbon dioxide (CO2) and nitrous oxide (N2O)17,18. Also, this model uses 10-year averages to account for temperature, and omits the seasonal variability in pond surface areas and temperatures on methane emissions. In this regard, the present work offers an initial assessment of methane emissions from agricultural ponds, but our results should only be taken as a first-order approximation.
Conclusions and future directions
Agricultural ponds are often overlooked as GHG sources, particularly since it is often difficult to account for their anthropogenic carbon emissions. Our analysis suggests that these small water bodies emit more methane than is currently accounted for in national GHG inventories. Agricultural ponds are essential for water security, and their density will continue to grow with rising global food demand. Therefore, developing cost-effective management solutions is urgently needed to reduce their ecological and environmental impacts.
Methane emissions from agricultural ponds represent both a liability and an opportunity. Much of the nutrients in farming ponds originate from livestock manure and fertiliser runoffs19,20 (Fig. 4). There is substantial evidence that higher nutrient concentrations in freshwater ponds promote GHG emissions9,21,22. There are several ways to reduce nutrient loads in agricultural ponds and their associated methane emissions. For example, Malerba et al.22 showed that simple management interventions in agricultural ponds (such as using fences to exclude livestock from accessing the water) could increase water quality (32% less nitrogen, 39% less phosphorus, 22% more dissolved oxygen) and halve methane emissions (56% less methane). Improving water quality will also benefit livestock health, biodiversity, and ecosystem services in the long term23,24. Another way to reduce nutrient influx is establishing a vegetation buffer around ponds, a practice termed "phytoremediation"25. Such a strategy may also favour biodiversity and comes with well-documented direct and indirect environmental benefits, including higher pollination success, greater ecosystem functioning, better resilience to pests, and improved aesthetic value24,26. Yet, using plants to reduce nutrients in water bodies could come at the cost of reducing runoff to a dam, and increasing input of organic carbon (plant material) to fuel decomposition and GHG production27,28. More studies are required to understand the trade-offs of using phytoremediation for water security and GHG emissions. Importantly, agricultural ponds often represent an important wetland habitat for a wide range of wildlife, including threatened species29. In the future, governments could provide financial incentives such as carbon credits to subsidies management interventions (e.g., fencing, revegetation) to reduce methane emissions from ponds.
Fig. 4: Managing agricultural ponds to reduce their methane emissions.
a Livestock manure and fertiliser often accumulate in agricultural ponds to create ideal conditions for methane production. Image credit: Martino E. Malerba b Simple management interventions, such as fencing to exclude livestock, are cost-effective solutions for reducing these emissions while providing additional environmental benefits (e.g., higher water quality, biodiversity, agricultural productivity, and aesthetic value)22,23,24. Image credit: I. Noyan Yilmaz.
Spatial datasets
Refer to Fig. S1 for the diagram of our modelling approach. We sourced data on the distribution of agricultural ponds in Australia from AusDams.org (N = 1.7 million), which was developed by applying artificial intelligence to high-resolution satellite images, and it is estimated to contain around 90% of Australian farm ponds (scale from 1:25,000 to 1: 250,000)6. For agricultural ponds in the United States, we used the National Hydrography Dataset (N = 7.8 million), which was developed and verified by the US Geological Survey13 (scale from 1:20,000 to 1:100,000). We retained all ponds between 0.01 and 1 ha (102–104 m2) in surface area and we only considered ponds in crops, open forests, shrubs, herbaceous or bare land using the land use map at a 100 ha (1 km2) resolution from Copernicus Global Land Service29. This approach produced a normally distributed population centred around 0.1 ha (103 m2). Manual inspection using satellite images across land-use types confirmed that >95% of the waterbodies in our maps appeared artificial ponds related to agriculture. We followed IPCC guidance of assuming an overall uncertainty for remote sensing products of ±10%30,31. Finally, we created a global map of median annual daily temperatures using 10 years of weekly data (Jan 2010 to Jan 2020) recorded by MODIS Terra Land Surface Temperature (product MOD11A1.006) at a 100 ha (1 km2) resolution using Google Earth Engine32.
The paucity of published studies worldwide on methane emissions from agricultural ponds complicates the estimation of average methane fluxes. On the 29th of April 2022, we used ISI Web of Science searching in all fields for: (methan*) AND (agricultural pond* OR farm dam* OR impoundment* OR dug out*; 503 results). We manually inspected each to identify seven datasets for agricultural ponds, with 12 subtropical records for Australia8, 154 temperate records for Australia9,22,33, 101 semi-arid records for Canada10, and 8 tropical records for India34. We excluded two observations for Swedish cropland ponds in Peacock et al.11 because there were too few data points to represent this region. We supplemented the available data with new measurements of 11 temperate agricultural ponds in Victoria (Australia) collected in April 2021 following the same protocols described in Malerba et al.22. We assumed that all studies used equivalent techniques to record methane emissions, either by recording gas emissions with floating chambers or by measuring gas concentrations dissolved in the water. However, Grinham et al.8 used floating chambers to capture both diffusive and ebullitive methane fluxes using long continuous recordings (from 6–24 h). In contrast, all other measurements quantified only diffusive fluxes using multiple short recordings (ca. 5 min)9,22 or the headspace extraction method10. Therefore, we used a dataset compiled by Rosentreter et al.1 to quantify the average contribution of methane ebullition to the total methane flux of agricultural ponds. Given the lack of studies specific to agricultural ponds, we used data for lakes and reservoirs instead (Fig. S3). We also excluded water bodies in regions with sub-zero annual mean temperatures. Our analysis revealed that the ratio of methane diffusion to methane ebullition is temperature-dependent, with methane diffusion making up 72% of total methane emissions at 5 °C but only 12.5% at 30 °C. Importantly, the effect of temperature on ebullitive methane fluxes was nearly identical between lakes and reservoirs (Fig. S3). This finding suggests that the temperature-dependency of methane ebullition is similar among different freshwater systems (but see Deemer and Holgerson35 for other drivers of methane emissions that differ between lakes and reservoirs).
Temperature standardisation
To compare estimates across sites and climates, we standardised daily rates of methane emissions at 15 °C, using the Boltzmann–Arrhenius relationship, as:
$${{{{\mathrm{ln}}}}}[{M}_{i}({T}_{15})]={{{{\mathrm{ln}}}}}[{M}_{i}(T)]-{E}_{M}\left(\frac{1}{{k}_{B}{T}_{15}}-\frac{1}{{k}_{B}{T}_{i}}\right)$$
where \({{{{\mathrm{ln}}}}}[{M}_{i}(T)]\) is the loge-transformed rate of daily methane emissions (in units of mg CH4 day−1 m−2) recorded at site i (i = 1, 2, …, 286) with local air temperature \({T}_{i}\) (in Kelvin), \({{{{\mathrm{ln}}}}}[{M}_{i}({T}_{15})]\) is the equivalent rate standardised to 15 °C, \({T}_{15}\) is the temperature used to standardise rates (where 15 °C is 288.15 K), \({E}_{M}\) is the temperature sensitivity for methane emissions (in units of eV mg CH4 day−1 m−2), and \({k}_{B}\) is the Boltzmann constant (8.617 × 10−5 eV K−1). For the local temperature of each site (\({T}_{i}\)), we used the 10-year median daily temperature recorded by MODIS Terra Land Surface Temperature (as described above). For the temperature sensitivity of methane emissions (\({E}_{M}\)), we used the dataset published by Rosentreter et al.1 for lakes and reservoirs (N = 313; Fig. S2). The effects of temperature on methane emission did not differ between lakes and reservoirs, suggesting that our estimate for \({E}_{M}\) can represent different types of freshwater habitats. Finally, we used Eq. 1 to calculate total methane emissions (diffusion + ebullition) standardised at 15 °C (Fig. 2).
In summary, (1) we compiled data from the scientific literature and additional fieldwork on methane fluxes from 286 agricultural ponds in subtropical, temperate, semi-arid, and tropical climates, and we used the dataset published by Rosentreter et al.1 to (2) standardise all emissions to 15 °C (Fig. S2) and to (3) estimate the contribution of methane ebullition (Fig. S3). Despite the low sample size and the uncertainty associated with the meta-analyses, the final dataset was normally distributed and consistent across climates and locations (Fig. 2), which suggests that our sample size may be a good representation of the whole population (albeit with wide confidence intervals).
Methane predictions
From the maps of agricultural ponds in the U.S. and Australia, we converted the density of pond surface area (pond ha ha−1) into cumulative methane emissions (kg CH4 year−1 ha−1) after adjusting for local temperature. Specifically, we reorganised Eq. 1 to obtain the temperature-adjusted methane emissions for each pond (\({M}_{i}({T}_{i})\)), as:
$${{{{\mathrm{ln}}}}}\left[{M}_{i}\left({T}_{i}\right)\right]={{{{\mathrm{ln}}}}}\left[{M}_{i}\left({T}_{15}\right)\right]+{E}_{M}\left(\frac{1}{{k}_{B}{T}_{15}}-\frac{1}{{k}_{B}{T}_{i}}\right)$$
where \({M}_{i}\left({T}_{15}\right)\) is the methane flux from agricultural ponds standardised to 15 °C using records compiled from the scientific literature (Fig. 2), and \({T}_{i}\) is the site-specific median annual temperature extracted from MODIS Terra Land Surface Temperature (see details above). All other coefficients remain the same as Eq. 1.
Sources of uncertainty
To quantify the overall uncertainty, we applied non-parametric bootstrapping to compound all sources of error using 1000 iterations where observations in Figs. 1, S2, and S3 were sampled with replacement36. At each iteration, we repeated all steps in our methods to estimate the distribution for each of our estimates. We compared the magnitude of each source of uncertainty using the coefficient of variation of the mean (i.e., the ratio between the standard error and the mean; Fig. S5).
Our approach makes several assumptions. First, agricultural ponds are between 0.01 ha (100 m2) and 1 ha (10,000 m2) in surface area6,37. Second, the effects of temperature on methane emissions from agricultural ponds follow a Boltzmann–Arrhenius relationship38,39. Third, the temperature sensitivity coefficient (parameter \({E}_{M}\) in Eqs. 1 and 2; Fig. S2) and the temperature-dependency of ebullition to diffusive fluxes (Fig. S3) are comparable among lakes, reservoirs, and ponds (Figs. S2 and S3). Fourth, median annual temperatures represent long-term conditions and can be used to correct field observations taken at specific points of the year. Fifth, the densities and surface areas of agricultural ponds reported in the maps of the United States and Australia have an uncertainty of ±10%30,31. We used R version 4.2.240 for data compilation, analyses, statistics, mapping, and plotting, using R packages ggplot241, dplyr42, tidyverse43, raster44, sf45, and lme446, nlme47. All codes and data generated in this study are available in a public repository in Mendeley Data48.
All data, maps, and meta-analyses can be found in a public repository in Mendeley Data at https://data.mendeley.com/datasets/6j87tgp825, (https://doi.org/10.17632/6j87tgp825.2).
All codes in R version 4.2.2 (2022-10-31) with analyses, plots, and maps can be found in a public repository in Mendeley Data at https://data.mendeley.com/datasets/6j87tgp825, (https://doi.org/10.17632/6j87tgp825.2).
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We thank the editor Dr. Clare Davis, A/Prof Rebecca Barnes, and two other anonymous reviewers for their help to improve this manuscript. This work was supported by the Australian Government through the Australian Research Council (project ID DE220100752) and the Alfred Deakin Fellowship scheme. We thank Don Driscoll (Deakin University) for his insightful comments. We also thank Shanti Reddy (Department of Industry, Science, Energy and Resources), Tingbao Xu, and Michael Hutchinson (Australian National University) for their help with sourcing climate data.
Centre for Integrative Ecology, School of Life and Environmental Sciences, Deakin University, Melbourne, VIC, 3125, Australia
Martino E. Malerba, Lukas Schuster & Peter I. Macreadie
Australian Department of Climate Change, Energy, the Environment and Water, Emissions Reduction Division, Canberra, ACT, Australia
Tertius de Kluyver
Sustainability and Biosecurity, Department of Primary Industries and Regional Development, 1 Nash Street, Perth, WA, 6000, Australia
Nicholas Wright
Martino E. Malerba
Lukas Schuster
Peter I. Macreadie
M.E.M., P.I.M., and T.d.K. designed the research, M.E.M., N.W., and L.S. collected the data, M.E.M. analysed the data, M.E.M. wrote the first draft, and all authors contributed to the final draft.
Correspondence to Martino E. Malerba.
Communications Earth & Environment thanks Rebecca Barnes and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary handling editors: Clare Davis, Heike Langenberg. Peer reviewer reports are available.
Peer Review File
Malerba, M.E., de Kluyver, T., Wright, N. et al. Methane emissions from agricultural ponds are underestimated in national greenhouse gas inventories. Commun Earth Environ 3, 306 (2022). https://doi.org/10.1038/s43247-022-00638-9
Received: 03 June 2022
Accounting for methane
Communications Earth & Environment (Commun Earth Environ) ISSN 2662-4435 (online) | CommonCrawl |
\begin{document}
\title{Limitations of Quantum Measurements and Operations of Scattering Type under the Energy Conservation Law}
\author{Ryota Katsube} \affiliation{Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan} \author{Masanao Ozawa} \affiliation{ Center for Mathematical Science and Artificial Intelligence, Academy of Emerging Sciences, Chubu University, 1200 Matsumoto-cho, Kasugai 487-8501, Japan} \affiliation{Graduate School of Informatics, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan} \author{Masahiro Hotta} \affiliation{Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan}
\date{\today}
\begin{abstract} It is important to improve the accuracy of quantum measurements and operations both in engineering and fundamental physics. It is known, however, that the achievable accuracy of measurements and unitary operations are generally limited by conservation laws according to the Wigner--Araki--Yanase theorem (WAY theorem) and its generalizations. Although many researches have extended the WAY theorem quantitatively, most of them, as well as the original WAY theorem, concern only additive conservation laws like the angular momentum conservation law. In this paper, we explore the limitation incurred by the energy conservation law, which is universal but is one of the non-additive conservation laws. We present a lower bound for the error of a quantum measurement using a scattering process satisfying the energy conservation law. We obtain conditions that a control system Hamiltonian must fulfill in order to implement a controlled unitary gate with zero error when a scattering process is considered. We also show the quantitative relationship between the upper bound of the gate fidelity of a controlled unitary gate and the energy fluctuation of systems when a target system and a control system are both one qubit. \end{abstract} \maketitle
\section{Introduction} It is essential to improve the accuracy of quantum measurements and quantum gate operations in quantum information processing. Quantum computing attracts industrial attention, since Shor \cite{Shor_factor, Shor_factor2} found an efficient quantum algorithm for prime factorization, the hardness of which is assumed in some protocols for public key cryptography. However, it is demanding to reduce the error in gate operations and measurements below the rate given by the threshold theorem for successful error correction to realize fault-torelant quantum computation \cite{Knill,Kitaev_1997,Aharonov,Fukui,Schotte}. Quantum key distribution \cite{BB84}, which aims to realize an information-theoretically secure cryptosystem, also demands to reduce the error to perform the protocol effectively, since any error, even not caused by eavesdroppers, reduces the key rate by error reconciliation. Other examples that require high accuracy of measurements in fundamental physics are the observation of the polarization of photons in the cosmic background radiation, which is thought to carry information about the early universe \cite{Hazumi, Planck}, and the observation of gravitational waves, which has recently been used as a tool to elucidate the origin of elements and the early universe \cite{Blackhole_gw}.
On the other hand, it is known that a conservation law limits the accuracy of measurements and unitary operations. Wigner \cite{Wigner} first showed in 1952 that the projective measurement of the spin $x$-component of a particle with spin $\frac{1}{2}$ cannot be realized under the conservation of angular momentum along the $z$-axis. Later, Araki and Yanase \cite{Araki} showed the no-go theorem which states that, in general, an additive conservation law limits the accuracy of projective measurement of the quantity not commuting with the conserved quantity. This theorem is called the Wigner--Araki--Yanase theorem (WAY theorem).
Ozawa \cite{Ozawa-91CP,Ozawa-93WA,Ozawa-02CLU} obtained quantitative generalizations of the WAY theorem by introducing a systematic method of manipulating the noise commutation relations, the commutation relations between the noise operator, previously introduced by von Neumann \cite[p.~404]{Neumann55}, and the conserved quantity, and established the tradeoff between the measurement error and the fluctuation of the conserved quantity in the apparatus for arbitrary measurements, originally suggested by Yanase \cite{Yanase} for spin measurements. Ozawa \cite{Ozawa-ineq} also derived a universally valid reformulation of Heisenberg's error-disturbance relation and showed that the WAY theorem is a consequence of the conservation law and Heisenberg's error-disturbance relation in its revised form \cite{Ozawa-way}.
The noise commutation relations were also used to evaluate the accuracy of gate implementations under conservation laws for the CNOT gate\cite{Ozawa-CNOT}, followed by quantitative evaluations of the gate fidelity for the Hadamard gate and the NOT gate \cite{Ozawa-way, Karasawa-NOT}, showing that the error is the smaller the greater the fluctuation of the conserved quantity in the controller, or the greater the size of the controller. The above approach was compared with Gea-Banacloche's \cite{Gea-Banacloche-Ban02,Gea-Banacloche-Ban02b,Gea-Banacloche-BK03,Gea-Banacloche-BK05} model dependent approach to the error in quantum logic gates caused by the quantum nature of controlling systems \cite{Gea-Banacloche-05CQL,Gea-Banacloche-06MEP}. Karasawa and his collaborators \cite{Karasawa-NOT,Karasawa-1qubit} proved the implementation error bound for the NOT gate and later for arbitrary one-qubit gates. Afterwards, Tajima et al. \cite{Tajima-measure, Tajima-coherence} provided a tighter lower bound for the error of quantum measurement, and implementations of quantum gates under conservation laws by expressing the fluctuation of the conserved quantity in terms of the quantum Fisher information.
The WAY theorem tells us that when we want to apply operations that break the symmetry, we must compensate the system's conserved quantity fluctuations, i.e., the coherence with respect to conserved quantities, as a resource. This point of view has been systematized as the quantum resource theory of asymmetry, and the operations that are feasible in the presence of symmetry have been investigated \cite{Bartlett, Marvian-resource, Ahmadi, Marvian-state, Aberg, Marvian-broadcast, Lostaglio, Miyadera-resource, Takagi-resource}. The WAY theorem has also been applied to the black hole information loss problem \cite{Blackhole-paradox, Page, Hayden} to investigate to which degree of completeness the scrambled information can be recovered in the presence of symmetry \cite{Nakata, Tajima-blackhole}.
Although most of the work on the WAY theorem has been done on additive conservation laws that do not include interaction terms, such as momentum and angular momentum conservation laws, some researchers extended the WAY theorem to multiplicative conservation laws \cite{Kimura} and non-additive conservation laws such as energy conservation laws because it contains interaction terms \cite{Navascues, Miyadera-energy, Tukiainen}. Navascu\'es et al proved relations between the difference between a desired POVM measurement we want to implement and an actual measurement we are able to implement under the energy conservation and the energy distribution\cite{Navascues}. Miyadera et al. showed an inequality between the fidelity of information distribution and the norm of the interaction term of Hamiltonian\cite{Miyadera-energy}. Tukiainen proved a WAY-type no-go theorem beyond conservation laws in the assumption of the weak Yanase condition\cite{Tukiainen}. We are able to obtain an upper bound of gate fidelity by the WAY theorem in general, it is also known that there is a lower bound of gate fidelity under the energy conservation law using methods in the resource theory \cite{Chiribella}.
In this paper, we prove that the relation between a measurement error, non-commutativity of the measured quantity and the conserved quantity, and the variance of the conserved quantity under an additive conservation \cite{Ozawa-way} is able to be expanded for measurements in scattering processes under energy conservation and we have a similar lower bound of the measurement error. We also show an upper bound of the gate fidelity of SWAP gate as an application of the theorem. Furthermore, we prove necessary conditions for the perfect implementation of a controlled unitary gate in a scattering process under energy conservation. In addition, we also show an inequality between the gate fidelity and energy variances for two-qubits controlled unitary gate, which is the extension of Ozawa's result \cite{Ozawa-CNOT}.
One of the possible application of our result for fundamental science is a design of gravitational wave detectors. Recently some analysis of the dynamics of the gravitational wave as a quantum field are known \cite{Maulik-detector,Maulik-gravity,Kanno}. In the observation of a gravitational wave using a laser interferometer, the information of the gravitational wave is transfered to the laser interferometer via the gravitational interaction. Then we are able to know the state of the gravitational wave by measuring the state of the laser interferometer. However the energy conservation gives the constraints for measurement, we have to design detectors to lower the measurement error.
Section 2 presents a lower bound for the measurement error of quantum measurements in scattering processes, which are important in particle physics experiments, under the requirement of energy conservation. In section 3, we show the conditions that the free Hamiltonian of the control system must satisfy in order to obtain a vanishing implementation error of the control unitary gates in the case of energy conservation. We also give an inequality between the energy fluctuation and the gate fidelity of two-qubit controlled unitary gates under energy conservation in section 3. Finally, we summarize the results in section 4.
\section{Fundumental error bound of scattering quantum measurements under energy conservation law}
In this section, we present an error limit of quantum measurements of scattering type under the energy conservation law.
Firstly, we review Ozawa's inequality \cite{Ozawa-ineq} and the quantification of the WAY theorem using it \cite{Ozawa-way} for our later discussion. Let's consider that we want to measure an observable $A_S$ of a system $S$ indirectly using a meter observable $M_D$ of a detector system $D$, in other words, we measure $A_S$ at the initial time $t=0$ by measuring the meter observable $M_D$ at $t = \tau$. In the Heisenberg picture, we indirectly measure $A_S(0)$ by measuring $M_D(\tau)= U^{\dagger}M_D U$, where $U$ is a unitary time evolution operator from time $t=0$ to $t=\tau$ that describes the process in which $S$ and $D$ become correlated. We define the error operator ${\bf E}$ for the measurement of $A_S$ and the disturbance operator ${\bf D}$ for an observable $B_S$ as follows:
\begin{eqnarray} {\bf E} &=& U^{\dagger} (I_S \otimes M_D) U - A_S \otimes I_D, \label{eq:mesurement_error_operator} \\ {\bf D} &=& U^{\dagger} (B_S \otimes I_D) U - B_S \otimes I_D.
\end{eqnarray} Let us suppose that $S$ and $D$ are not correlated in the initial time and the initial state of the composite system $S$+$D$ is given by $\rho_S \otimes \sigma_D$. We define the measurement error $\epsilon(A_S)$ and the disturbance $\eta(B_S)$ by $\epsilon(A_S)= \sqrt{{\rm Tr}[{\bf E}^2 (\rho_S \otimes \sigma_D)]}$ and $\eta(B_S) = \sqrt{{\rm Tr}[{\bf D}^2 (\rho_S \otimes \sigma_D)]}$, the following Ozawa's inequality
\begin{eqnarray}
\epsilon(A_S) \eta(B_S) + \epsilon(A_S) \sigma(B_S) + \sigma(A_S) \eta(B_S) \geq \frac{1}{2} \left| {\rm Tr} \left( [A_S,B_S]\rho_S\right)\right|
\end{eqnarray} holds\cite{Ozawa-ineq}, where $\sigma(A_S)$ and $\sigma(B_S)$ represent standard deviations of $A_S$ and $B_S$ in the state $\rho_S$, respectively.
Next, we review the lower bound of the measurement error $\epsilon(A_S)$ when there is an additive conserved quantity \cite{Ozawa-way}. We call a conserved observable which does not contain an interaction term like $L = L_S \otimes I_D + I_S \otimes L_D $ an additive conserved observable. For example, the angular momentum and the momentum are additive observables. When we have an additive conserved quantity $L$, the time evolution operator $U$ which characterizes the measurement process must satisfy the following condition:
\begin{eqnarray} [U,L] = 0.
\end{eqnarray} We further assume that the commutativity of the meter observable $M_D$ and the detector's conserved quantity $L_D$, which is called Yanase's condition, $[M_D, L_D]=0$. Then it is known that \begin{eqnarray}
\epsilon(A_S)^2 \geq \frac{\left| {\rm Tr}\left([A_S,L_S] \rho \right) \right|^2}{4\sigma(L_S)^2 + 4\sigma(L_D)^2} \label{eq:way-review}
\end{eqnarray} holds \cite{Ozawa-way}. This enables us to evaluate the lower bound of the measurement error quantitatively when there is an additive conserved observable.
In the following, we show that Eq. (\ref{eq:way-review}) is able to be expanded for measurements in scattering processes under energy conservation and we have a similar lower bound of the measurement error. Let $S$ and $D$ are initially spatially separated and not interacting. Then, as $S$ approaches $D$, the systems start to interact, creating a correlation between $S$ and $D$. Then, as $S$ moves away from $D$, after a certain time the interaction stops. By measuring the state of $D$, during it is correlated with the state of $S$, we can indirectly measure the state of $S$. We call this kind of measurement scattering-type measurement.
Let $H$ be the Hamiltonian of the composite system $S$+$D$. $H$ is given by
\begin{equation} H_{S+D} = H_S \otimes I_D + I_S\otimes H_D +H_{int},
\end{equation} where $H_S$ is the Hamiltonian of $S$ , $H_D$ is the Hamiltonian of $D$ and $H_{int}$ is the interaction term between $S$ and $D$. We suppose that the interaction exists for a time period $\tau$. The time evolution operator of the scattering process $ U_{S+D}$ is defined by
\begin{equation} U_{S+D}=e^{-i\tau H_{S+D}} ,
\end{equation} where we employed natural units, $\hbar = 1$.
Let us suppose that the initial state of $S$ is $ \ket{\psi}$ and that of $D$ is $\ket{\xi}$. The final state of the composite system $S$+$D$ is $U_{S+D}\ket{\psi}_S \ket{\xi}_D$. We assume the following condition on the initial and final states:
\begin{equation}
H_{int} \ket{\psi}_S \ket{\xi}_D = H_{int} U\ket{\psi}_S \ket{\xi}_D = 0 \label{eq:assump1}.
\end{equation} Note that we consider some specific initial states $\ket{\psi}_S $ and $\ket{\xi}_D$ which satisfy Eq. (\ref{eq:assump1}). We do not need to impose Eq. (\ref{eq:assump1}) for all states of $S$ and $D$. An example of an interaction that satisfies Eq. (\ref{eq:assump1}) is the Yukawa potential. It is used to describe the nuclear force in nuclear physics and the effective electrostatic potential in metals with impurities in condensed matter physics. A Yukawa potential $V(r)$ is represented by \begin{eqnarray} V(r) = \alpha \frac{e^{-\kappa r}}{r} , \end{eqnarray} where $r$ is the distance between $S$ and $D$, $\alpha$ is an appropriate coefficient, and $\kappa$ is the approximate range of the potential. In this case, \begin{eqnarray}
H_{int} \ket{\psi}_S \ket{\xi}_D = \int d{\bm r_S} \int d{\bm r_D} \alpha \frac{e^{-\kappa r}}{|{\bm r_S} - {\bm r_D}|} \psi({\bm r_S})_S \xi({\bm r_D})_D \ket{{\bm r_S}}_S \ket{{\bm r_D}}_D \label{eq:potential} \end{eqnarray}
holds. $\ket{{\bm r_S}}_S$ and $\ket{{\bm r_D}}_D$ are eigenstates of position operators ${\bm r_S}$ and ${\bm r_D} $, respectively. We define $\psi({\bm r_S})_S$ and $\xi({\bm r_D})_D$ by $\psi({\bm r_S})_S \coloneqq \braket{{\bm r_S}|\psi}$ and $\xi({\bm r_D})_D \coloneqq \braket{{\bm r_D}|\xi}$.
When $S$ and $D$ are initially spatially separated, the integral of Eq. (\ref{eq:potential}) is 0 and $H_{int} \ket{\psi}_S \ket{\xi}_D=0$ holds. Similarly, if $S$ and $D$ are spatially separated in final states, we can find that $H_{int} U \ket{\psi}_S \ket{\xi}_D=0$ holds. Eq. (\ref{eq:assump1}) is also satisfied in other potentials if the integral of the product of potential by the wavefunction of $S$+$D$ is 0 in initial states and final states.
We consider a lower bound of measurement errors when we indirectly measure an observable $A_S$ of $S$ by measuring a meter observable $M_D$ of $D$ under the energy conservation law,
\begin{eqnarray} [U_{S+D},H_{S+D}] = 0 \label{eq:energy_cons}.
\end{eqnarray}
We assume that the meter observable satisfies the Yanase condition:
\begin{eqnarray} [H_D, M_D] = 0 \label{eq:yanase}.
\end{eqnarray} This means that we can read off the value of the meter observable with zero error under the energy conservation. We define the error of an indirect measurement of $A_S$, $\epsilon(A_S)$ by
\begin{equation}
\epsilon(A_S) = \| \{U_{S+D}^{\dagger} (I_S \otimes M_D) U_{S+D}-A_S \otimes I_D\} \ket{\psi}_S \ket{\xi}_D\| . \label{eq:measurement_error_A}
\end{equation}
Under the above assumptions, the following inequality for $\epsilon(A_S)$ holds under the energy conservation:
\begin{equation}
\epsilon(A_S)^2 \geq \frac{|\braket{[A_S,H_S]}|^2}{4\sigma^2(H_S)+4\sigma^2(H_D)} \label{eq:way_energy}.
\end{equation} $\sigma(H_S)$ and $\sigma(H_D)$ are standard deviations of $H_S$ and $H_D$ on the initial state, respectively. $\braket{[A_S,H_S]}$ represents the expectation value of the commutator $[A_S,H_S]$ on $\ket{\psi}_S$. From Eq. (\ref{eq:way_energy}), we can find that the lower bound of $\epsilon(A_S)$ depends on the non-commutativity of $H_S$ and $A_S$, and variances of free Hamiltonians $H_S$ and $H_D$ on initial states. Therefore we can lower the bound of the measurement error by increasing the energy variance on initial states. The proof of Eq. (\ref{eq:way_energy}) is given in the appendix A.
Let us consider whether we are able to implement a SWAP gate using nuclear spins and photons as an application of Eq. (\ref{eq:way_energy}). We take this example because it is important from the point of view of engineering to transfer information from a nuclear spin to a photon when we want to transmit the results of quantum computation which are encoded in states of nuclear spins by quantum communication using light. We consider $S$ as a nuclear spin and $D$ as a photon. Let us suppose that we apply a magnetic field in the $x$ direction to the nuclear spin, and the photon moves along with the $x$-axis. There are three degrees of freedom. The first is the spin degree of freedom of the nuclear spin. The second is the photon\rq{}s spatial degree of freedom in the x direction. The third is the helicity of the photon. We would like to swap the state of $S$ and that of the helicity of $D$. We consider the following Hamiltonian, \begin{eqnarray} H_S &=& b \sigma_{x,S}, \\ H_D &=& p_x, \\ H_{int} &=& g \left( \sigma_{x,S} \otimes \sigma_{x,D} + \sigma_{y,S} \otimes \sigma_{y,D} + \sigma_{z,S} \otimes \sigma_{z,D} - I_S \otimes I_D \right) \otimes w(x), \label{eq:swap_hamiltonian} \end{eqnarray}
where $b$ is a constant that represents the strength of the magnetic field. In special relativity, the energy of the photon with momentum $\vec{p}$ is given by $c |\vec{p}|$, and we employed natural units, $c=1$. $w(x)$ is a function of the position $x$ of $D$, and it takes non-zero values only for regions where the interaction between the nuclear spin and the photon exists. This type of hamiltonian is known as Coleman-Hepp model \cite{Hepp}. Fig. \ref{fig:swap} is the conceptual figure of the setting. The gray area in the figure represents the region where $w(x) \ne 0$. At first, $S$ and $D$ are apart, and there is no interaction. When D approaches S, they begin to interact with each other. In the final state, they are apart again, and the interaction vanishes. When $g$ is very large, $H_S$ and $H_D$ are negligible compared to $H_{int}$ in the region where the interaction exists. In the region where $w(x)$ is able to be approximated by a constant function, we are able to make $e^{-iH_{int}t}$ be same as SWAP gate which acts on spin spaces of $S$ and $D$ by adjusting $g$.
\begin{figure}
\caption{Conceptual figure of the setting.}
\label{fig:swap}
\end{figure}
We regard the implementation of SWAP gate as the measurement of a nuclear spin $\sigma_z$ where a meter observable is a photon spin $\sigma_z$ and use Eq. (\ref{eq:way_energy}), then we find that
\begin{eqnarray}
\epsilon(\sigma_z)^2 \geq \frac{b^2 |\braket{\sigma_y}|^2}{b^2 \sigma(\sigma_x)^2 + \sigma(p_x)^2} \label{eq:swap_way}
\end{eqnarray} holds. Note that because $\sigma_{z,D}$ and $p_x$ are operators acting on different Hilbert spaces, the Yanase condition is satisfied. On the other hand, when we succeed in implementing SWAP gate $U_{SWAP}$ perfectly,
\begin{eqnarray} \epsilon(\sigma_z)^2 = \braket{[\sigma_{z,S} \otimes I_D - U_{SWAP}^{\dagger} (I_S \otimes \sigma_{z,D})U_{SWAP}]^2} = 0
\end{eqnarray} must be satisfied. Therefore, we are not able to implement SWAP gate perfectly when the fluctuation of momentum of the photon in the initial state is not infinity. \par
We are able to quantify it by deriving an inequality between the gate fidelity of SWAP gate $F_{SWAP}$ and the energy variance. Before showing the inequality of $F_{SWAP}$, we give the definition of the gate fidelity and the worst error probability of the physical realization. We consider that we would like to implement a unitary gate $U_{ideal}$ which acts on two systems $S$ and $A$. We represent the initial state of the composite system $S$+$A$ by $\rho$. We also consider an external system $E$ and suppose that its initial state is $\ket{\xi}$. Considering the purification, we are able to assume that the initial state of the external system is a pure state. We define a unitary time evolution operator which acts on the composite system of $S$+$A$+$E$ by $U$. The state of $S$+$A$ after $U$ is applied, which we represent by $\mathcal{E}(\rho)$, is \begin{eqnarray} \mathcal{E}(\rho) = {\rm Tr}_E [U(\rho \otimes \ket{\xi}\bra{\xi})U^{\dagger}]. \label{eq:real_time_evolve} \end{eqnarray} On the other hand, if we succeed in implementing $U_{ideal} $ perfectly, the final state of $S$+$A$ is $U_{ideal} \rho U_{ideal}^{\dagger}$. Therefore, we define the worst error probability of the physical realization of $U_{ideal}$ by the CB distance as follows: \begin{eqnarray} D_{CB} (\mathcal{E},U_{ideal}) = \sup_{\rho_{S+A+E^{\prime}} }D\left(\mathcal{E}_{S+A}\otimes I_{E^\prime}(\rho_{S+A+E^{\prime}} ), (U_{ideal} \otimes I_{E^{\prime}}) \rho_{S+A+E^{\prime}}(U_{ideal}^{\dagger} \otimes I_{E^{\prime}})\right), \end{eqnarray}
where $E^\prime$ is another external system and $D(\rho_1,\rho_2)$ is the trace distance given by \begin{eqnarray}
D(\rho_1,\rho_2) = \frac{1}{2} {\rm Tr} \left[|\rho_1 - \rho_2| \right].
\end{eqnarray} The gate trace distance between $U_{ideal}$ and the physical implementation $\mathcal{E}$, $D(\mathcal{E},U_{ideal})$ is defined by
\begin{eqnarray} D(\mathcal{E},U_{ideal}) = \sup_{\rho_{S+A}} D(\mathcal{E}(\rho_{S+A}), U_{ideal} \, \rho_{S+A} U^{\dagger}_{ideal}).
\end{eqnarray}
From the definition of the CB distance, \begin{eqnarray} D_{CB} (\mathcal{E},U_{ideal}) \geq D(\mathcal{E},U_{ideal})
\end{eqnarray} holds. By minimizing over all physical implementation $(U,\ket{\xi})$, \begin{eqnarray} D_{CB} (\mathcal{E},U_{ideal}) \geq \inf_{(U,\ket{\xi})} D(\mathcal{E},U_{ideal})
\end{eqnarray} can be found. Because of the joint convexity of the trace distance, we are able to assume that $ \rho_{S+A}$ is a pure state. The gate fidelity between $U_{ideal}$ and $\mathcal{E}$, which is denoted by $F(\mathcal{E},U_{ideal})$, is give by \begin{eqnarray} F(\mathcal{E},U_{ideal}) \coloneqq \inf_{\ket{\psi}} F(\ket{\psi}), \end{eqnarray} where \begin{eqnarray}
F(\ket{\psi}) = \braket{\psi|U_{ideal}^{\dagger} \mathcal{E}(\ket{\psi}\bra{\psi}) U_{ideal} |\psi}^{\frac{1}{2}}. \end{eqnarray} The following relation between the trace distance and fielity holds: \begin{eqnarray} D(\mathcal{E},U_{ideal}) \geq 1- F(\mathcal{E},U_{ideal})^2.
\end{eqnarray} Therefore, \begin{eqnarray} D_{CB} (\mathcal{E},U_{ideal}) \geq D(\mathcal{E},U_{ideal}) \geq 1- F(\mathcal{E},U_{ideal})^2
\end{eqnarray} is verified. If $D_{CB} (\mathcal{E},U_{ideal}) > 0$, we are not able to implement $U_{ideal}$ perfectly. \par
When the hamiltonian is given by Eq. (\ref{eq:swap_hamiltonian}),
\begin{eqnarray} F_{SWAP}^2 \leq 1 - \frac{b^2}{4b^2 + 4 \sigma(p_x)^2} \label{eq:swap_fidelity}
\end{eqnarray} holds. The derivation of this inequality is given in Appendix B.
Furthermore, we are able to extend the bound for the Hadamard gate under an additive conservation law in Ref. \cite{Ozawa-way} to that under the energy conservation law. For example, let us consider we would like to implement the Hadamard gate using the composite system $S$+$D$ where the Hamiltonian of the composite system is
\begin{eqnarray} H = s_{x,S} + L_{x,D} + H_{int},
\end{eqnarray} where $s_{x,S}$ and $L_{x,D}$ are the $x$ component of the angular momentum of each system. When the interaction term satisfies Eq. (\ref{eq:assump1}) , from Eq. (\ref{eq:way_energy}) and similar calculations as in Ref. \cite{Ozawa-way}, we are able to show the following bound: \begin{eqnarray} 1 - F_H^2 \geq \epsilon(s_{z,S})^2 \geq \frac{1}{4+ 16 \sigma(L_{x,D})^2}, \label{eq:H_fidelity} \end{eqnarray} where $F_H$ is the gate fidelity of the Hadamard gate.
In this section, we show the relation Eq. (\ref{eq:way_energy}) between the measurement error in scattering processes and the energy conservation. We also prove that SWAP gate is not able to be implemented perfectly as an application of Eq. (\ref{eq:way_energy}). Futhermore, the bound of the gate fidelity of the Hadamard gate proved in Ref. \cite{Ozawa-way} under an additive conservation law is able to be expanded under the energy conservation in scattering processes.
\section{Conditions which Hamiltonian of a controlled system must satisfy for implementing controlled unitary gates perfectly under energy conservation }
In this section, we show necessary conditions that the Hamiltonian of a control system must satisfy for implementing controlled unitary gates with zero error in scattering processes under energy conservation. We also give an upper bound of the gate fidelity of two-qubit controlled gates.
Let us represent a control system and a target system as $C$ and $T$, respectively. We consider two orthogonal states of $C$, $\ket{\phi_0}_C$ and $\ket{\phi_1}_C$. We deal with a controlled gate $U_{CU}$ such that when the state of $C$ is $\ket{\phi_0}_C$, there is no effect on $T$, and when it is $\ket{\phi_1}_C$, a unitary operator $V_T$ is acting in $T$. The action of $U_{CU}$ is written as \begin{eqnarray} U_{CU} \ket{\phi_0}_C \ket{\psi}_T &=& \ket{\phi_0^{\prime}}_C \ket{\psi}_T, \nonumber \\ U_{CU} \ket{\phi_1}_C \ket{\psi}_T &=& \ket{\phi_1^{\prime}}_C \left(V_T \ket{\psi}_T \right), \label{eq:controlled-U}
\end{eqnarray} for an arbitrary state of $T$ $\ket{\psi}_T$. Note that final states of $C$, $\ket{\phi_0^{\prime}}_C$ and $\ket{\phi_1^{\prime}}_C$ are independent of the initial state of $T$, $\ket{\psi}_T$. We consider a non-trivial gate such that $V_T \ne I_T$.
Let $H_C$, $H_T$, and $H_{int}$ denote the Hamiltonian of $C$, that of $T$, and an interaction Hamiltonian between $C$ and $T$. The total Hamiltonian of the composite system $C$ and $T$, represented as $H_{C+T}$, is given by
\begin{eqnarray} H_{C+T} = H_C \otimes I_T +I_C \otimes H_T + H_{int}. \end{eqnarray}
Like the previous section, we consider a scattering process in which $C$ and $T$ are initially spatially separated, and the interaction appears as $C$ approaches $T$. Some time later, $C$ and $T$ are spatially separated again. We assume that
\begin{equation} \bra{\phi}_C \bra{\psi}_TH_{int} \ket{\phi}_C \ket{\psi}_T = 0 \label{eq:assumption1},
\end{equation}
for an arbitrary superposition state of $C$, $\ket{\phi}_C=c_0 \ket{\phi_0}_C + c_1 \ket{\phi_1}_C$ where $|c_0|^2 + |c_1|^2= 1$, and an arbitrary state of $T$, $\ket{\psi}_T$. We set $\tau$ to the time when the non-zero interaction exists. The time evolution operator describing the scattering process $U_{CT}$ is written as
\begin{equation} U_{CT} = \exp(-i\tau H_{C+T}). \notag
\end{equation}
When implementing the controlled unitary gate, $U_{CU}$ using this scattering process, we adjust the interaction $H_{int}$ and the interaction time $\tau$ to satisfy the following relations:
\begin{eqnarray} U_{CT} \ket{\phi_0}_C \ket{\psi}_T &=& \ket{\phi^{\prime}_0}_C \ket{\psi}_T, \nonumber \\ U_{CT} \ket{\phi_1}_C \ket{\psi}_T &=& \ket{\phi^{\prime}_1}_C (V_T \ket{\psi}_T), \label{eq:ideal_ope}
\end{eqnarray} for arbitrary initial state $\ket{\psi}_T$ in $T$. However when the energy conservation law \begin{eqnarray} [H_{C+T},U_{CT}] =0,
\end{eqnarray} exists, $H_C$ must satisfy the following initial condition, \begin{equation}
\braket{\phi_0|H_C|\phi_1}_C = 0\label{eq:wayone_constraint}
\end{equation} to implement the gate $U_{CU}$ with zero-error.
If $H_{int}$ fulfills the conditions below similar to conditions Eq. (\ref{eq:assump1}): \begin{eqnarray} H_{int} \ket{\phi_0}_C \ket{\psi}_T &=& 0, \nonumber \\ H_{int} \ket{\phi_1}_C \ket{\psi}_T &=& 0, \nonumber \\ H_{int} \ket{\phi^{\prime}_0}_C \ket{\psi}_T &=& 0, \nonumber \\ H_{int} \ket{\phi^{\prime}_1}_C \ket{\psi}_T &=& 0 , \label{eq:assump2}
\end{eqnarray} for an arbitrary state $\ket{\psi}_T$, the additional condition, \begin{eqnarray}
\braket{\phi_0|H_C^2|\phi_1} =0, \label{eq:waytwo_contraint}
\end{eqnarray} must be satisfied to implement $U_{CU}$ perfectly under energy conservation. We give proof of Eq. (\ref{eq:wayone_constraint}) and Eq. (\ref{eq:waytwo_contraint}) in the appendix C.
Furthermore, if $C$ and $T$ are one-qubit systems, $U_{CU} $ becomes a two-qubit controlled gate, and $V_T$ becomes a one-qubit gate. $V_T$ can be represented as \begin{equation} V_T(\theta) = e^{i\phi} \left( \cos \frac{\theta}{2} I_T + i \sin \frac{\theta}{2} \vec{u} \cdot \vec{\sigma}\right), \label{eq:v_t}
\end{equation} where $\vec{u}$ is a three-dimensional real vector and $\vec{\sigma}$ is the vector defined by $\vec{\sigma} \coloneqq (\sigma_x, \sigma_y, \sigma_z)$, where $\sigma_x$, $\sigma_ y$and $\sigma_z$ are Pauli matrices which act on the Hilbert space of $T$. The domain of $\theta$ and $\phi$ are $0 \leq \phi < 2 \pi$ and $0 \leq \theta \leq \pi$, respectively.
When implementing a two-qubit controlled unitary gate $U_{CU}$ using a scattering process under energy conservation, we have the following inequality regarding the CB distance between $U_{CU}$ and the physical implementation $\mathcal{E}_{\alpha}$: \begin{equation}
D_{CB}(\mathcal{E}_{\alpha},U_{CU})^2 \geq \frac{\sin^2 \frac{\theta}{2}\left\|\left[\sigma(\vec{l})_C,H_C\right]\right\|^2}{16(2\gamma+\|H_A\|)^2} \label{eq:1bit} \end{equation}
where $\sigma(\vec{l})\coloneqq \ket{\phi_0}\bra{\phi_0}-\ket{\phi_1}\bra{\phi_1}$, $\gamma \coloneqq \max\{\|H_S\|,\|H_T\|\}$ and $H_A$ is the free Hamiltonian of the ancillary system $A$.
If $\braket{\phi_0|H_C|\phi_1}=0$ holds, $H_C$ commutes with $\sigma(\vec{l})$ and the upper bound of gate fidelity is 1, that is to say, we can implement $U_{CU}$ perfectly. We also find that the bound dependence of the angle $\theta$ is $\sin^2 \frac{\theta}{2}$. The derivation of Eq. (\ref{eq:1bit}) is provided in the appendix D. \par
In this section, we show that $H_C$ must satisfy Eq. (\ref{eq:wayone_constraint}) and Eq. (\ref{eq:waytwo_contraint}) to implement a non-trivial controlled gate whose action is given by Eq.(\ref{eq:controlled-U}) with zero-error using scattering process under energy conservation law. We also obtain the upper bound of gate fidelity Eq. (\ref{eq:1bit}) for a two-qubit controlled gate.
\section{Summary} We showed a lower bound of the measurement error Eq. (\ref{eq:way_energy}) for scattering-type measurements which satisfy Eq. (\ref{eq:assump1}) under the energy conservation in section 2. In section 2 we also presented that the SWAP gate is not able to be implemented perfectly as an application of Eq. (\ref{eq:way_energy}). Futhermore, the bound of the gate fidelity of the Hadamard gate proved in Ref. \cite{Ozawa-way} under an additive conservation law is able to be expanded under the energy conservation in scattering processes. In section 3, we gived the necessary condition Eq. (\ref{eq:wayone_constraint}) which must be fulfilled to implement a controlled unitary gate described by Eq. (\ref{eq:controlled-U}) without error in a scattering process under energy conservation. We also proved that when the interaction term satisfies Eq. (\ref{eq:assump2}), the additional necessary condition for $H_C$, Eq. (\ref{eq:waytwo_contraint}) must also be fulfilled to implement a controlled unitary gate perfectly. Moreover, when $C$ and $T$ are one-qubit systems, the quantitative relation Eq. (\ref{eq:1bit}) between the gate fidelity and the non-diagonal entry of $H_C$ is shown. This result extends Ozawa's result which evaluates the upper bound of the gate fidelity of CNOT gates \cite{Ozawa-CNOT}.
\begin{acknowledgments} This research was partially supported by JSPS KAKENHI Grants No.~JP22K03424 (M.O.), No.~JP21K11764 (M.O.), No.~JP19H04066 (M.O.), No.~JP19K03838 (M.H.), No.~JP21H05188 (M.H.), Foundational Questions Institute (M.H.), Silicon Valley Community Foundation (M.H.), JST SPRING, Grant No.~JPMJSP2114 (R.K.), a Scholarship of Tohoku University, Division for Interdisciplinary Advanced Research and Education (R.K.), and the WISE Program for AI Electronics, Tohoku University (R.K.). \end{acknowledgments}
\appendix
\section{Proof of bound of error of scattering-type measurement under energy conservation}
In this section, we derive the inequality of the measurement error in a scattering process under energy conservation Eq. (\ref{eq:way_energy}). We introduce another measuring device which is a particle moving in one dimension. We represent this particle by $A$. Let the position and the momentum of $A$ be $Q_A$ and $P_A$, respectively. And suppose that $A$ and $D$ have an interaction described by \begin{equation} H^{\prime}_{int}= k I_S \otimes M_D \otimes P_A,
\end{equation} between $\tau$ and $\tau^{\prime}$, where $k\coloneqq \frac{1}{\tau^{\prime}-\tau}$. Except for this time window between $\tau$ and $\tau^{\prime}$, we suppose that there is no interaction between $A$ and $S$+$D$. The hamiltonian of the composite system $S$+$D$+$A$, $H_{S+D+A}$ is given by \begin{eqnarray} H_{S+D+A} = H_{S+D} + H_A + H^{\prime}_{int},
\end{eqnarray} between time $\tau$ and $\tau^{\prime}$, where $H_A$ is the free Hamiltonian of $A$. Note that the interaction between $S$ and $D$, $H_{int}$, is 0 after $\tau$.
We consider the indirect measurement of $M_D$ by measuring $Q_A$ and assume that the period when the interaction $H_{int}^{\prime}$ exists, $\tau^{\prime} -\tau$, is very short. In this assumption, $H_{int}^{\prime}$ is very large compared to the free Hamiltonian of $S$+$D$+$A$, so $H_{int}^{\prime}$ can be regarded as the total Hamiltonian in the time window from $\tau$ to $\tau^{\prime}$.
We denote by $U^{\prime}_{S+D+A}$ the time evolution operator between $\tau$ and $\tau^{\prime}$. $U^{\prime}_{S+D+A}$ is given by
\begin{equation} U^{\prime}_{S+D+A}= e^{-i(I_S \otimes M_D \otimes P_A)} .
\end{equation} On the other hand, the time evolution of $A$ between 0 and $\tau$, $U_A$, is represented by
\begin{equation} U_A = e^{-iH_A \tau}.
\end{equation} The time evolution operator of the composite system $S$+$D$+$A$ between the initial time $ t=0$ to the final time $t=\tau+\tau^{\prime}$ expressed as $U_{S+D+A}$ is \begin{equation} U_{S+D+A}=U^{\prime}_{S+D+A} (U_{S+D} \otimes U_A).
\end{equation}
Next, we define the error $\alpha$ of measuring $\tilde{M_D}\coloneqq I_S \otimes M_D \otimes I_A$ at $\tau$ by measuring $\tilde{Q_A} \coloneqq I_S \otimes I_D \otimes Q_A$ at $\tau^{\prime}$ and we define by $\beta$ the error of measuring $\tilde{A_S} \coloneqq A_S \otimes I_D \otimes I_A$ at $t=0$ by measuring $\tilde{Q_A}$ at $\tau^{\prime}$. $\alpha$ and $\beta$ are written as
\begin{eqnarray}
\alpha &=& \left\| [U^{\prime \dagger}_{S+D+A}\tilde{Q_A}U^{\prime}_{S+D+A}-\tilde{M_D}]\left(U_{S+D}\ket{\psi}_S \ket{\xi}_D\right)U_A \ket{\zeta}_A\right\|, \\
\beta &= & \left \| [U^{\dagger}_{S+D+A} \tilde{Q}_A U_{S+D+A} - \tilde{A}_S] \ket{\psi}_S \ket{\xi}_D \ket{\zeta}_A \right \|, \end{eqnarray} where $\ket{\zeta}_A$ is the initial state of $A$. \par
To evaluate $\epsilon(A_S)$ given in Eq. (\ref{eq:measurement_error_A}), we first consider the Ozawa inequality in the measurement $\tilde{A}_S$ using $\tilde{Q}_A$: \begin{equation}
\beta \eta(H_0) + \beta \sigma(H_0) + \sigma(\tilde{A_S}) \eta(H_0) \geq \frac{1}{2} \left| \Braket{[\tilde{A_S},H_0]}\right| \label{eq:ozawa-ineq},
\end{equation} where $H_0 \coloneqq H_S \otimes I_D \otimes I_A + I_S \otimes H_D \otimes I_A $ is the free Hamiltonian of the composite system $S$+$D$ and $\eta(H_0)$ is the disturbance of $H_0$ by the measurement, which is given by \begin{eqnarray}
\eta(H_0) = \left\| [U_{S+D+A}^{\dagger} H_0 U_{S+D+A} - H_0] \ket{\psi}_S \ket{\xi}_D \ket{\zeta}_A \right\|. \end{eqnarray} For an observable $O$, $\sigma(O)$ represents the standard deviation of $O$ in the initial state.
From the Baker-Campbell-Hausdorff formula,
\begin{equation} U^{\prime \dagger}_{S+D+A}\tilde{Q_A}U^{\prime}_{S+D+A} =\tilde{Q_A}+\tilde{M_D} \nonumber
\end{equation} holds. Then we can calculate $\alpha$ as
\begin{eqnarray}
\alpha
&=& \left\| Q_AU_A \ket{\zeta}_A\right\|. \label{eq:error_1}
\end{eqnarray} \par Looking at Eq. (\ref{eq:error_1}), we find that the error of measuring $M_{D}$ by $Q_A$ is given by the square root of the expectation value of $Q_A^2$ on the final state of $A$. Therefore, for any $\epsilon>0$, we can impose $\alpha < \epsilon$ by choosing $\ket{\zeta}_A$ appropriately. Since $M_D$ is a conserved quantity during measurement which is described by $U^{\prime}_{S+D+A}$, $[U^{\prime}_{S+D+A},\tilde{M_D}]=0$ holds, and we also find the following relation:
\begin{eqnarray} \alpha
&=& \left\| [U^{\dagger}_{S+D+A}\tilde{Q_A}U_{S+D+A}-U^{\dagger}_{S+D+A}\tilde{M_D} U_{S+D+A}]\ket{\psi}_S \ket{\xi}_D\ket{\zeta}_A\right\| \label{eq:eq1}.
\end{eqnarray}
Next, we derive the relation between $\alpha$, $\beta$ and $\epsilon(A_S)$. $\epsilon(A_S)$ is deformed as
\begin{eqnarray} \epsilon(A_S)
&=& \left\| [U^{\dagger}_{S+D+A} \tilde{M_D} U_{S+D+A}- \tilde{A_S}] \ket{\psi}_S \ket{\xi}_D\ket{\zeta}_A \right\| \label{eq:eq2},
\end{eqnarray}
where we used $[\tilde{M_{D}},U^{\prime}_{S+D+A}]=0$. From Eq. (\ref{eq:eq1}), Eq. (\ref{eq:eq2}) and the triangle inequality, we obtain the following relation:
\begin{eqnarray} \beta &\leq & \alpha + \epsilon(A_S) \label{eq:error}.
\end{eqnarray} We evaluate the disturbance $\eta(H_0)$. From the Yanase condition $[H_D ,M_D]=0$, we have $[U^{\prime}_{S+D+A},H_0]=0$, which means that $H_0$ is unchanged during the measurement using $A$. Then we obtain
\begin{eqnarray} \eta(H_0)
&=& \left \| [U_{S+D}^{\dagger}H_0 U_{S+D}-H_0] \ket{\psi}_S \ket{\xi}_D \ket{\zeta}_A \right \| \nonumber \\
&=& \left \| [U_{S+D}^{\dagger}H_{int} U_{S+D}-H_{int}] \ket{\psi}_S \ket{\xi}_D \right \| \nonumber \\ &=& 0 \label{eq:disturbance}.
\end{eqnarray}
In the first line, we used that $H_0$ is conserved during measurement using $A$. The fact that $H_{S+D}$ is conserved during measurement using $D$ is utilized in the second line. In the final line, the assumption Eq. (\ref{eq:assump1}) is considered. From Eq. (\ref{eq:ozawa-ineq}), Eq. (\ref{eq:error}) and Eq. (\ref{eq:disturbance}),
\begin{equation}
(\alpha + \epsilon(A_S))\sigma(H_0) \geq \frac{1}{2} \left | \Braket{[\tilde{A_S},H_0] }\right | \nonumber
\end{equation} is shown. Because $\alpha$ can be arbitrarily small by choosing $\ket{\zeta}_A$ appropriately, \begin{equation}
\epsilon(A_S)\sigma(H_0) \geq \frac{1}{2} \left | \Braket{[\tilde{A_S},H_0] }\right | \nonumber
\end{equation} holds. Taking $[\tilde{A}_S,H_0]=[A_S,H_S]$ and $\sigma(H_0)^2 = \sigma(H_S)^2 +\sigma(H_D)^2 $ into account, we get \begin{equation}
\epsilon(A_S)^2 \geq \frac{\left|\Braket{[A_S,H_S]} \right|^2}{4\sigma(H_S)^2 + 4\sigma(H_D)^2}, \nonumber
\end{equation} and Eq. (\ref{eq:way_energy}) is proved. \par
Before closing this section, we comment on a bound of the measurement error when we assume the weak Yanase condition $ [U^\dagger_{S+D} (I_S \otimes M_D)U_{S+D}, H_{S+D}] =0 $ instead of the Yanase condition $[H_D ,M_D]=0$. Tukiainen proved that if $ [U^\dagger_{S+D} (I_S \otimes M_D)U_{S+D}, H_{S+D}] =0 $ is satisfied in a repeatable measurement of $A_S$, then $[A_S, \braket{\xi|H_{S+D}|\xi}] =0 $ holds \cite{Tukiainen}. We are able to quantify it by the similar proof of Eq. (\ref{eq:way_energy}). We consider the following Ozawa inequality instead of Eq. (\ref{eq:ozawa-ineq}):
\begin{eqnarray}
\beta \eta(H_{S+D}) + \beta \sigma(H_{S+D}) + \sigma(\tilde{A}_S) \eta(H_{S+D}) \geq \frac{1}{2} \left| \Braket{[\tilde{A}_S,H_{S+D}]}\right| = \frac{1}{2} \left| \bra{\psi}_S [A_S, \bra{\xi}_D H_{S+D} \ket{\xi}_D] \ket{\psi}_S \right| \label{eq:ozawa-ineq2}. \nonumber \\
\end{eqnarray} When the weak Yanase condition is satisfied, we are able to show that $ \eta(H_{S+D}) =0 $ as follows:
\begin{eqnarray}
\eta (H_{S+D}) &=& \left \| [ U_{S+D+A}^\dagger H_{S+D} U_{S+D+A} - H_{S+D}] \ket{\psi}_S \ket{\xi}_D \ket{\zeta}_A \right \| \\
&=& \left \| [ U_{S+D}^\dagger H_{S+D} U_{S+D} - H_{S+D}] \ket{\psi}_S \ket{\xi}_D \ket{\zeta}_A \right \| \\ &=& 0,
\end{eqnarray} where in the second line, we used $[M_D,H_{S+D}] =0 $ which is obtained from the weak Yanase condition and the energy conservation law because $[U^\dagger_{S+D} (I_S \otimes M_D)U_{S+D}, H_{S+D}] =U^\dagger_{S+D}[I_S \otimes M_D, H_{S+D}] U_{S+D} =0 $ holds. In the third line, the energy conservation law is considered. From Eq. (\ref{eq:error}), Eq. (\ref{eq:ozawa-ineq2}) and $\eta(H_{S+D})= 0$ and setting $\alpha$ to be arbitrarily small, we find that
\begin{eqnarray}
\epsilon(A_S)^2 \geq \frac{ \left| \bra{\psi}_S [A_S, \bra{\xi}_D H_{S+D} \ket{\xi}_D] \ket{\psi}_S \right|^2}{4\sigma(H_{S+D})^2 } \label{eq:bound-weak-yanase}
\end{eqnarray} holds. \par In this section, we considered the scattering-type measurements in which the interaction vanishes when the detector and system are spatially separated under energy conservation. We derived the lower bound of measurement error Eq. (\ref{eq:way_energy}) for this type of measurement.
\section{Upper bound of the gate fidelity of SWAP gate }
In this section, we derive the inequality (\ref{eq:swap_fidelity}) between the gate fidelity of SWAP gate $F_{SWAP}$ and the energy variance. We first give a relation between the measurement error $\epsilon(\sigma_z)$ and $F_{SWAP}$. Then we prove an upper bound of $F_{SWAP}$ using it and Eq. (\ref{eq:swap_way}). \par
Firstly, we write the act of $U$ in Eq. (\ref{eq:real_time_evolve}) as \begin{eqnarray} U\ket{a}_S \ket{b}_A \ket{\xi}_E = \sum_{j,k=0}^{1}\ket{j}_S \ket{k}_A \ket{E_{j,k}^{a,b}}_E. \end{eqnarray} $a$ and $b$ take 0 or 1. $\{\ket{0}, \ket{1}\}$ represents the computational basis states. From the orthonormality of the initial state, \begin{eqnarray}
\delta_{ac} \delta_{bd} &=& \braket{a,b|c,d} = \braket{a,b,\xi|U^{\dagger}U|c,d,\xi} = \sum_{j,k=0}^1 \braket{E_{j,k}^{a,b}|E_{j,k}^{c,d}} \label{eq:normalortho_cond} \end{eqnarray} holds. When the initial state of $S$+$A$ is $\ket{\alpha,\beta} \, (\alpha,\beta=0,1)$, the state after the time evolution described by $U$ is given by \begin{eqnarray}
\mathcal{E}(\ket{\alpha,\beta}\bra{\alpha,\beta}) &=& \sum_{j,k,l,m} \ket{j,k}\bra{l,m} \braket{E_{l,m}^{\alpha,\beta}|E_{j,k}^{\alpha,\beta}} . \end{eqnarray}
Therefore, the squared of $F(\ket{\alpha,\beta})$ for SWAP gate is written as follows: \begin{eqnarray} F(\ket{\alpha,\beta})^2
&=& \braket{E_{\beta,\alpha}^{\alpha,\beta}|E_{\beta,\alpha}^{\alpha,\beta}} \label{eq:swap_pure_fidelity} . \end{eqnarray} \par
Next, let us represent the measurement error $\epsilon(\sigma_z)$ using $\ket{E^{a,b}_{c,d}}$. We suppose that the initial state of $S$+$A$ is $\ket{\psi} = \sum_{\alpha,\beta=0}^1c_{\alpha,\beta} \ket{\alpha,\beta}$. We find that \begin{eqnarray} \epsilon(\sigma_z)^2
&=& \| \left[(I_S\otimes \sigma_{z,A} \otimes I_E )U - U (\sigma_{z,S} \otimes I_A \otimes I_E) \right] \ket{\psi}_{S+A} \ket{\xi}_E\|^2 \\
&=& 4 \left \| \sum_\beta c_{1,\beta} \ket{E_{0,0}^{1,\beta}} \right\|^2 + 4 \left \| \sum_\beta c_{1,\beta} \ket{E_{1,0}^{1,\beta}} \right\|^2 + 4 \left \| \sum_\beta c_{0,\beta} \ket{E_{0,1}^{0,\beta}} \right\|^2 + 4 \left \| \sum_\beta c_{0,\beta} \ket{E_{1,1}^{0,\beta}} \right\|^2 \label{eq:swap_2} \end{eqnarray} holds. We rewrite it by the gate fidelity. In the following, we represent $F(\mathcal{E}, U_{SWAP})$ by $F_{SWAP}$. The following relations hold: \begin{eqnarray}
\epsilon(\sigma_z)^2
&=& 4 \left \{ |c_{1,0}|^2 \left( 1- \left\| \ket{E_{0,1}^{1,0}} \right\|^2 - \left\| \ket{E_{1,1}^{1,0}} \right\|^2 \right) + |c_{0,0}|^2 \left( 1-\left\| \ket{E_{0,0}^{0,0}} \right\|^2 - \left\| \ket{E_{1,0}^{0,0}} \right\|^2 \right) \right. \nonumber \\
& & +2 {\rm Re} \left[ c_{1,0} c_{1,1}^* \left( \braket{E_{0,0}^{1,1}|E_{0,0}^{1,0}} + \braket{E_{1,0}^{1,1}|E_{1,0}^{1,0}} \right)+ c_{0,0} c_{0,1}^* \left( \braket{E_{0,1}^{0,1}|E_{0,1}^{0,0}} + \braket{E_{1,1}^{0,1}|E_{1,1}^{0,0}}\right)\right] \nonumber \\
& & + \left. |c_{1,1}|^2 \left( 1 - \left \| \ket{E_{0,1}^{1,1}}\right\|^2 - \left \| \ket{E_{1,1}^{1,1}}\right\|^2 \right) + |c_{0,1}|^2\left( 1 - \left \| \ket{E_{0,0}^{0,1}}\right\|^2 - \left \| \ket{E_{1,0}^{0,1}}\right\|^2 \right)\right \} \\
& \leq & 4 \left\{ |c_{1,0}|^2 \left(1- F(\ket{1,0})^2 \right) + |c_{0,0}|^2 \left(1 - F(\ket{0,0})^2\right)\right. \nonumber \\
& & +2 \left | c_{1,0} c_{1,1}^* \left( \braket{E_{0,0}^{1,1}|E_{0,0}^{1,0}} + \braket{E_{1,0}^{1,1}|E_{1,0}^{1,0}} \right)+ c_{0,0} c_{0,1}^* \left( \braket{E_{0,1}^{0,1}|E_{0,1}^{0,0}} + \braket{E_{1,1}^{0,1}|E_{1,1}^{0,0}}\right) \right| \nonumber \\
& & + \left. |c_{1,1}|^2 \left( 1 - F(\ket{1,1})^2 \right)+ |c_{0,1}|^2 \left(1 -F(\ket{0,1})^2 \right) \right\} \\
&\leq & 4 \left\{ 1-F_{SWAP}^2+2 |c_{1,0}||c_{1,1}| \left( \left\| \ket{E_{0,0}^{1,1}} \right\| \left\| \ket{E_{0,0}^{1,0}} \right\|+ \left\| \ket{E_{1,0}^{1,1}} \right\| \left\| \ket{E_{1,0}^{1,0}} \right\| \right) \right. \nonumber \\
& & \left. + 2|c_{0,0}||c_{0,1}|\left( \left\| \ket{E_{0,1}^{0,1}} \right\| \left\| \ket{E_{0,1}^{0,0}} \right\| + \left\| \ket{E_{1,1}^{0,1}} \right\| \left\| \ket{E_{1,1}^{0,0}} \right\|\right) \right\} \\
&\leq & 4 \left\{ 1-F_{SWAP}^2+ |c_{1,0}||c_{1,1}| \left( \left\| \ket{E_{0,0}^{1,1}} \right\|^2 + \left\| \ket{E_{0,0}^{1,0}} \right\|^2+ \left\| \ket{E_{1,0}^{1,1}} \right\|^2 +\left\| \ket{E_{1,0}^{1,0}} \right\|^2 \right) \right. \nonumber \\
& & \left. + |c_{0,0}||c_{0,1}|\left( \left\| \ket{E_{0,1}^{0,1}} \right\|^2+ \left\| \ket{E_{0,1}^{0,0}} \right\|^2 + \left\| \ket{E_{1,1}^{0,1}} \right\|^2+ \left\| \ket{E_{1,1}^{0,0}} \right\|^2\right) \right\} \\
&\leq& 4(1-F_{SWAP}^2) \left[ 1 + 2 (|c_{1,0}||c_{1,1}| + |c_{0,0}| |c_{0,1}|)\right] .
\end{eqnarray}
In the first line, Eq. (\ref{eq:normalortho_cond}) is used. In second line, we considered Eq. (\ref{eq:swap_pure_fidelity}). In the third line, the triangle inequality and the Cauchy-Schwarz inequality are applied. In the fourth line, the relation between the mean square and the geometric mean is taken into consideration. In the seventh line, we used Eq. (\ref{eq:normalortho_cond}) and Eq. (\ref{eq:swap_pure_fidelity}).
When the hamiltonian is given by as Eq.(\ref{eq:swap_hamiltonian}), Eq. (\ref{eq:swap_way}) holds and \begin{eqnarray}
4(1-F_{SWAP}^2) \left[ 1 + 2 (|c_{1,0}||c_{1,1}| + |c_{0,0}| |c_{0,1}|)\right] \geq \frac{b^2 |\braket{\sigma_{y,S}}|^2}{b^2 \sigma(\sigma_{x,S})^2 + \sigma(p_x)^2} \label{eq:swap_fidelity_variance}
\end{eqnarray} can be proved. Let us maximize the right-hand side over states of $S$+$A$. Because the right-hand side is independent of states of $A$, we consider maximizing over the system's pure state. It is denoted by $d_0 \ket{0} + d_1 \ket{1}$. Then we find that \begin{eqnarray}
\frac{b^2 |\braket{\sigma_{y,S}}|^2}{b^2 \sigma(\sigma_{x,S})^2 + \sigma(p_x)^2} &=& \frac{4b^2 |{\rm Im}(d_0 d_1^*)|^2}{b^2(1- 4 |{\rm Re}(d_0 d_1^*)|^2)+ \sigma(p_x)^2}.
\end{eqnarray} Because \begin{eqnarray}
|{\rm Re}(d_0 d_1^*)|^2 + |{\rm Im}(d_0 d_1^*)|^2 = |d_0 d_1^*|^2 \leq \left( \frac{|d_0|^2 + |d_1|^2}{2} \right)^2 = \frac{1}{4} \end{eqnarray}
holds, we make the change of variales $|{\rm Re}(d_0 d_1^*)| = r \cos \theta$, $|{\rm Im}(d_0 d_1^*)| = r \sin \theta$, where $ 0 \leq r \leq \frac{1}{2}$,$0 \leq \theta < 2\pi$. We introduce a function $G(r,\theta)$ as follows: \begin{eqnarray}
G(r,\theta) = \frac{b^2 |\braket{\sigma_{y,S}}|^2}{b^2 \sigma(\sigma_{x,S})^2 + \sigma(p_x)^2} &=& \frac{4b^2r^2 \sin^2 \theta}{b^2(1- 4 r^2 \cos^2 \theta)+ \sigma(p_x)^2}.
\end{eqnarray} The derivative of it is \begin{eqnarray} \frac{\partial G(r,\theta)}{\partial \theta} = \frac{4b^2 r^2 \sin 2 \theta \left[b^2 (1 -4r^2) + \sigma(p_x)^2 \right]}{\left[ b^2(1- 4 r^2 \cos^2 \theta)+ \sigma(p_x)^2\right]^2}.
\end{eqnarray} Since $\left[b^2 (1 -4r^2) + \sigma(p_x)^2 \right] \geq 0$ holds, $G(r,\theta)$ takes the maximum value at $\theta = \frac{\pi}{2}$ and $\frac{3\pi}{2}$ for a fixed $ r$. Because $G\left(r,\frac{\pi}{2}\right) = G\left(r,\frac{3\pi}{2} \right) = \frac{4b^2 r^2}{b^2 + \sigma(p_x)^2}$, the maximum value of $G(r,\theta)$ is $\frac{b^2}{b^2 + \sigma(p_x)^2}$, which is attained at $(r,\theta) = \left(\frac{1}{2}, \frac{\pi}{2} \right) = \left(\frac{1}{2}, \frac{3\pi}{2} \right)$. The eigenstates of $\sigma_{y,S}$ correspond to this case.
On the other hand, the coefficient of the left-hand side of Eq. (\ref{eq:swap_fidelity_variance}), $ 1 + 2 (|c_{1,0}||c_{1,1}| + |c_{0,0}| |c_{0,1}|)$, takes the minimum value 1 when $S$'s state is the eigenstate of $\sigma_{y,S}$ and $A$'s state is $\ket{0}$. Hence, Eq. (\ref{eq:swap_fidelity}) is derived. On the other hand, because $p_x$ is an unbounded operator, the CB distance is 0. However if the variance of $p_x$ on the initial state is finite, from Eq. (\ref{eq:swap_fidelity}), we find that the implementation error occurs. \par
In this section, we proved the inequality Eq. (\ref{eq:swap_fidelity}) between the gate fidelity of SWAP gate and the energy variance as an application of Eq. (\ref{eq:way_energy}).
\section{Proof of conditions which have to be fulfilled for implementing a controlled unitary gate perfectly }
In this section, we prove the necessary condition Eq. (\ref{eq:wayone_constraint}) and Eq. (\ref{eq:waytwo_contraint}) to implement a controlled gate described in Eq. (\ref{eq:controlled-U}) without error under energy conservation law.
Firstly, note that final states of $C$, $\ket{\phi_0^{\prime}}_C$ and $\ket{\phi_1^{\prime}}_C$, are independent of the initial state of $T$, $\ket{\psi}_T$ in Eq. (\ref{eq:controlled-U}), and two final states of $C$ are orthogonal to each other when we succeed in implementing $U_{CU}$ without error. The calculation to prove it is the same as in \cite{no-programming} in which Nielsen and Chuang proved the no-programming theorem.
Next we prove that \begin{equation}
\bra{\phi_0}_C \bra{\psi_0}_T H_{int} \ket{\phi_1}_C \ket{\psi_1}_T=0 \label{eq:off_diagonal5}
\end{equation} is valid under the assumption Eq. (\ref{eq:assumption1}). Substituting $\ket{\phi}_C = c_0 \ket{\phi_0}_C + c_1 \ket{\phi_1}_C$ into Eq. (\ref{eq:assumption1}),
\begin{eqnarray}
2{\rm Re}\left[c_0^* c_1 \bra{\phi_0}_C \bra{\psi}_T H_{int} \ket{\phi_1}_C\ket{\psi}_T \right] =0 \label{eq:off_diagonal1}
\end{eqnarray}
is obtained. Since Eq. (\ref{eq:off_diagonal1}) holds for arbitrary complex numbers $c_0$ and $c_1$ such that $|c_0|^2 + |c_1|^2 =1$, we get \begin{equation} \bra{\phi_0}_C \bra{\psi}_T H_{int} \ket{\phi_1}_C\ket{\psi}_T =0 \label{eq:off_diagonal2}.
\end{equation}
Similarly, since Eq. (\ref{eq:off_diagonal2}) holds for an arbitrary state of $T$, $\ket{\psi}_T=d_0\ket{\psi_0}_T+d_1\ket{\psi_1}_T$, where $d_0$ and $d_1$ are arbitrary complex numbers which satisfy $|d_0|^2 + |d_1|^2 =1$, \begin{equation} d_0^* d_1 \bra{\phi_0}_C \bra{\psi_0}_T H_{int} \ket{\phi_1}_C \ket{\psi_1}_T + d_0 d_1^* \bra{\phi_0}_C \bra{\psi_1}_T H_{int} \ket{\phi_1}_C \ket{\psi_0}_T =0 \label{eq:off_diagonal3}
\end{equation} can be shown. Considering $d_0=d_1 =\frac{1}{\sqrt{2}}$, we find that \begin{eqnarray}
\bra{\phi_0}_C \bra{\psi_0}_T H_{int} \ket{\phi_1}_C \ket{\psi_1}_T + \bra{\phi_0}_C \bra{\psi_1}_T H_{int} \ket{\phi_1}_C \ket{\psi_0}_T &=&0. \label{eq:first_case} \end{eqnarray} On the other hand, we get \begin{eqnarray} \bra{\phi_0}_C \bra{\psi_0}_T H_{int} \ket{\phi_1}_C \ket{\psi_1}_T - \bra{\phi_0}_C \bra{\psi_1}_T H_{int} \ket{\phi_1}_C \ket{\psi_0}_T &=&0 \label{eq:second_case}
\end{eqnarray} when substituting $d_0= \frac{1}{\sqrt{2}}$ and $d_1=\frac{i}{\sqrt{2}}$ into Eq. (\ref{eq:off_diagonal3}). By combining Eq. (\ref{eq:first_case}) with Eq. (\ref{eq:second_case}), we obtain Eq. (\ref{eq:off_diagonal5}). Furthermore, by substituting $\ket{\psi_0}_T = \ket{\psi}_T $ and $\ket{\psi_1}_T=V_T\ket{\psi}_T$, where $\ket{\psi}_T$ is an arbitrary initial state of $T$, and $\ket{\phi_0}_C= \ket{\phi_0^{\prime}}_C$ and $\ket{\phi_1}_C = \ket{\phi_1^{\prime}}_C$ into Eq. (\ref{eq:off_diagonal5}), \begin{eqnarray} 0 &=& \bra{\phi_0^{\prime}}_C \bra{\psi}_T H_{int} \ket{\phi_1^{\prime}}_C (V_T \ket{\psi}_T) = \bra{\phi_0}_C \bra{\psi}_T U_{CU}^{\dagger} H_{int} U_{CU} \ket{\phi_1}_C \ket{\psi}_T \label{eq:time_evolution1}
\end{eqnarray} can be shown.
Based on the above facts, we derive Eq. (\ref{eq:wayone_constraint}). From the orthogonality of $\ket{\phi_0}$ and $\ket{\phi_1}$, \begin{eqnarray}
\braket{\phi_0|H_C|\phi_1}_C = \bra{\phi_0}_C \bra{\psi}_T (H_T + H_C) \ket{\phi_1}_C \ket{\psi}_T \label{eq:wayone1}
\end{eqnarray} is obtained. By combining Eq. (\ref{eq:wayone1}) with Eq. (\ref{eq:off_diagonal2}), we get \begin{equation}
\braket{\phi_0|H_C|\phi_1}_C = \bra{\phi_0}_C \bra{\psi}_T (H_T + H_C+H_{int}) \ket{\phi_1}_C \ket{\psi}_T. \label{eq:wayone2}
\end{equation} Using Eq. (\ref{eq:wayone2}) and the energy conservation law \begin{eqnarray} [H_{C+T},U_{CU}] =0,
\end{eqnarray} we find that \begin{eqnarray}
\braket{\phi_0|H_C|\phi_1}_C &=& \bra{\phi_0}_C \bra{\psi}_T U_{CU}^{\dagger}(H_T + H_C+H_{int})U_{CU} \ket{\phi_1}_C \ket{\psi}_T \nonumber \\ &=& \bra{\phi_0^{\prime}}_C \bra{\psi}_T (H_T + H_C +H_{int}) \ket{\phi_1^{\prime}}_C (V_T \ket{\psi}_T) \label{eq:wayone3}
\end{eqnarray}
holds. In addition to this, from $\braket{\phi_0^{\prime}|\phi_1^{\prime}}=0$ and Eq. (\ref{eq:time_evolution1}), \begin{equation}
\braket{\phi_0|H_C|\phi_1}_C=\braket{\phi^{\prime}_0|H_C|\phi^{\prime}_1}_C \braket{\psi|V_T|\psi}_T \label{eq:wayone4}
\end{equation} is able to be proved for any $\ket{\psi}_T$.
If $\braket{\phi_0|H_C|\phi_1}_C\ne 0$ holds, $\braket{\phi^{\prime}_0|H_C|\phi^{\prime}_1}_C\ne 0$ is obtained because $V_T\ne 0$. Therefore, from Eq. (\ref{eq:wayone4}), we can calculate as follows: \begin{equation}
\braket{\psi|V_T|\psi}_T=\frac{\braket{\phi_0|H_C|\phi_1}_C}{\braket{\phi^{\prime}_0|H_C|\phi^{\prime}_1}_C}. \label{eq:wayone5}
\end{equation} Let the spectrum decomposition of $V_T$ be \begin{eqnarray} V_T = \sum_i \alpha_i \ket{i}\bra{i}, \end{eqnarray} where $\alpha_i$ is the $i$-th eigenvalue and $\ket{i}$ is the corresponding eigenvector.
By substituting $\ket{\psi} = \ket{i}$ into Eq. (\ref{eq:wayone5}), we find that \begin{equation}
\braket{i|V_T|i}_T=\alpha_i = \frac{\braket{\phi_0|H_C|\phi_1}_C}{\braket{\phi^{\prime}_0|H_C|\phi^{\prime}_1}_C} \nonumber
\end{equation} holds for any $i$. Hence,
\begin{eqnarray}
V_T &= &\frac{\braket{\phi_0|H_C|\phi_1}_C}{\braket{\phi^{\prime}_0|H_C|\phi^{\prime}_1}_C} I_T \label{eq:wayone6}
\end{eqnarray} is obtained. Since $V_T$ is a unitary operator, the following relation is valid: \begin{equation}
\braket{\phi_0|H_C|\phi_1}_C = \braket{\phi^{\prime}_0|H_C|\phi^{\prime}_1}_C e^{i\delta} \label{eq:wayone7},
\end{equation}
where $\delta$ is a real number. When $\braket{\phi_0|H_C|\phi_1}_C\ne 0$, by taking Eq. (\ref{eq:wayone6}) and Eq. (\ref{eq:wayone7}) into account, \begin{equation} V_T = e^{i\delta} I_T \label{eq:wayone_result}
\end{equation} must hold. Therefore, when we want to implement a non-trivial controlled unitary gate like a CNOT gate without error under energy conservation, we proved that Eq. (\ref{eq:wayone_constraint}) has to be satisfied.
Next, we derive Eq. (\ref{eq:waytwo_contraint}) under the assumption Eq. (\ref{eq:assump2}). A calculation similar to the derivation of Eq. (\ref{eq:wayone4}) shows that \begin{eqnarray}
\braket{\phi_0|H_C^2|\phi_1}_C &=& \bra{\phi_0}_C \bra{\psi}_T (H_C^2 + H_T^2) \ket{\phi_1}_C \ket{\psi}_T \nonumber \\ &=& \bra{\phi_0}_C \bra{\psi}_T [(H_C +H_T+H_{int})^2 -2 H_C \otimes H_T] \ket{\phi_1}_C \ket{\psi}_T \nonumber \\ &=& \bra{\phi_0}_C \bra{\psi}_T [U_{CU}^{\dagger}(H_C +H_T+H_{int})^2U_{CU} -2 H_C \otimes H_T] \ket{\phi_1}_C \ket{\psi}_T \nonumber \\
&=& \bra{\phi^{\prime}_0}_C \bra{\psi}_T (H_C+H_T)^2 \ket{\phi^{\prime}_1}_C (V_T\ket{\psi}_T) -2\braket{\phi_0|H_C|\phi_1}_C \braket{\psi|H_T|\psi}_T \nonumber \\
&=& \braket{\phi^{\prime}_0|H_C^2|\phi^{\prime}_1}_C \braket{\psi|V_T|\psi}_T \label{eq:waytwo1}
\end{eqnarray}
is valid for any $\ket{\psi}_T$. In the first line, we used $\braket{\phi_0|\phi_1}=0$. We applied Eq. (\ref{eq:assump2}) to the second line. In the third line, the energy conservation is considered. In the fourth line, Eq. (\ref{eq:assump2}) and Eq. (\ref{eq:controlled-U}) are utilized. The final line is derived by combining $\braket{\phi_0^{\prime}|\phi_1^{\prime}}=0$ and Eq. (\ref{eq:wayone_constraint}) with Eq. (\ref{eq:wayone4}).
If $\braket{\phi_0|H_C^2|\phi_1} \ne 0$ works, we are able to prove that $V_T$ must be proportional to $I_T$ by a calculation similar to the derivation of Eq. (\ref{eq:wayone_result}). Thus, we find that if the assumption Eq. (\ref{eq:assump2}) holds, Eq. (\ref{eq:waytwo_contraint}) has to be fulfilled in addition to Eq. (\ref{eq:wayone_constraint}) to implement a non-trivial controlled unitary gate perfectly under energy conservation.
\section{Proof of upper bound of the gate fidelity of two qubits controlled unitary gate} In this section, we prove the upper bound of the gate fidelity of a two-qubits controlled unitary gate Eq. (\ref{eq:1bit}). In the following, we denote a three-dimensional real vector which is orthogonal to $\vec{u}$, where $\vec{u}$ is given in Eq. (\ref{eq:v_t}), by $\vec{v}$. There are two vectors orthogonal to $\vec{u}$, we set $\vec{v}$ to the vector which faces the $z$-axis positive when we rotate $\vec{u}$ so that it faces the $x$-axis positive. We define $\sigma(\vec{v})$ by $\sigma(\vec{v})\coloneqq \vec{v} \cdot \vec{\sigma}$. Let us denote the eigenvector with eigenvalue 1 and the eigenvector with eigenvalue -1 by $\ket{\chi_0}$ and $\ket{\chi_1}$, respectively. We also define $\sigma(\vec{l})$ and $\sigma(\vec{l^{\prime}})$ by $\sigma(\vec{l})\coloneqq \ket{\phi_0}\bra{\phi_0}-\ket{\phi_1}\bra{\phi_1}$ and $\sigma(\vec{l^{\prime}})\coloneqq \ket{\phi_0^{\prime}}\bra{\phi_0^{\prime}}-\ket{\phi_1^{\prime}}\bra{\phi_1^{\prime}}$. We represent the operation of $U$ as
\begin{equation} U\ket{\phi_a}_C\ket{\chi_b}_T \ket{\xi}_A = \sum_{c,d=0}^1 \ket{\phi_c^{\prime}}_C \ket{\chi_d}_T \ket{E_{c,d}^{a,b}}_A \quad (a,b =0,1). \end{equation} From the orthogonality and normalization conditions, we find that \begin{eqnarray} \delta_{a,c}\delta_{b,d} = \bra{\phi_a}_C \bra{\chi_b}_T \bra{\xi}_A U^{\dagger}U\ket{\phi_c}_C \ket{\chi_d}_T \ket{\xi}_A
= \sum_{j,k} \braket{E_{j,k}^{a,b}|E_{j,k}^{c,d}} \end{eqnarray} holds. The state of $C$+$T$ after the time evolution described by $U$ is \begin{equation}
\mathcal{E}_{\alpha}(\ket{\phi_a}_C\ket{\chi_b}_T\bra{\phi_a}_C \bra{\chi_b}_T) = \sum_{i,j,k,l} \ket{\phi_i^{\prime}}_C \ket{\chi_j}_T\braket{E_{k,l}^{a,b}|E_{i,j}^{a,b}} \bra{\phi_k^{\prime}}_C \bra{\chi_l}_T \quad (a,b =0,1). \end{equation} In the following, $\ket{\phi_a}_C\ket{\chi_b}_T$ and $\ket{\phi_c^{\prime}}_C\ket{\chi_d}_T$ are abbreviated as $\ket{a,b}$ and $\ket{c^{\prime},d}$, respectively. We distinguish the basis of $C$ with and without the prime symbol.
Since $\vec{u}$ and $\vec{v}$ are orthogonal each other, we get \begin{equation} (\vec{u} \cdot \vec{\sigma})_T \ket{\chi_0}_T = \ket{\chi_1}_T, \quad (\vec{u} \cdot \vec{\sigma})_T \ket{\chi_1}_T = \ket{\chi_0}_T.
\end{equation} Therefore, \begin{eqnarray} U_{CU} \ket{a,b} &=& \ket{0^{\prime},b} \delta_{a,0} + e^{i\phi} \left( \cos \frac{\theta}{2} \ket{1^{\prime},b} + i \sin \frac{\theta}{2} \ket{1^{\prime},1 \oplus b} \right)\delta_{a,1}
\end{eqnarray} can be shown. When the initial state is $\ket{a,b}$, the gate fidelity squared is calculated as \begin{eqnarray}
F(\ket{a,b})^2 &=& \left[ \bra{0^{\prime},b} \delta_{a,0} + e^{-i\phi} \left( \cos \frac{\theta}{2} \bra{1^{\prime},b} -i \sin \frac{\theta}{2} \bra{1^{\prime},1 \oplus b} \right)\delta_{a,1} \right]\left[ \sum_{i,j,k,l} \ket{i^\prime,j}\braket{E_{k,l}^{a,b}|E_{i,j}^{a,b}} \bra{k^{\prime},l} \right] \nonumber \\ & & \quad \cdot\left[ \ket{0^{\prime},b} \delta_{a,0} + e^{i\phi} \left( \cos \frac{\theta}{2} \ket{1^{\prime},b} + i \sin \frac{\theta}{2} \ket{1^{\prime},1 \oplus b} \right)\delta_{a,1} \right] . \end{eqnarray} When $a =0$, it is rewritten as \begin{eqnarray}
F(\ket{0,b})^2 = \braket{E_{0,b}^{0,b}|E_{0,b}^{0,b}},
\end{eqnarray} and when $ a=1$, \begin{eqnarray} F(\ket{1,b})^2 &=& \left [\cos \frac{\theta}{2} \bra{E_{1,b}^{1,b}} +i \sin \frac{\theta}{2} \bra{E_{1,b+1}^{1,b}} \right] \left [\cos \frac{\theta}{2} \ket{E_{1,b}^{1,b}} -i \sin \frac{\theta}{2} \ket{E_{1,b+1}^{1,b}} \right]
\end{eqnarray} is derived. For later convenience, we calculate the gate fidelity of $\tilde{U}_{CU}= \ket{0}\bra{0}_C \otimes V_T(\pi+ \theta) + \ket{1}\bra{1}_C \otimes V_T(\pi)$. Since the gate fidelity is invariant under the unitary transformation $V_T(\pi+\theta)$ on initial states, the gate fidelity of $\tilde{U}_{CU}$ is same as that of $U_{CU}$ where the gate operated to the target bit is $V_{T}(-\theta)$ when $C$ is on-state. When the initial state is $\ket{a,b}$, the operation of $\tilde{U}_{CU}$ is given by \begin{eqnarray} \tilde{U}_{CU} \ket{a,b} &=& e^{i \phi} \left( - \sin \frac{\theta}{2} \ket{0,b} + i \cos \frac{\theta}{2} \ket{0^{\prime},b \oplus 1} \right) \delta_{a,0} + i e^{i\phi} \ket{1^{\prime},b\oplus 1} \delta_{a,1}.
\end{eqnarray}
When the initial state is $\ket{a,b}$, we denote the gate fidelity of $\tilde{U}_{CU}$ by $f(\ket{a,b})$.
$f(\ket{a,b})^2$ is obtained as follows: \begin{eqnarray}
f(\ket{a,b})^2 &=& \left[\left( - \sin \frac{\theta}{2} \bra{0,b} - i \cos \frac{\theta}{2} \bra{0^{\prime},b \oplus 1} \right) \delta_{a,0} - i \bra{1^{\prime},b\oplus 1} \delta_{a,1} \right] \left[ \sum_{i,j,k,l} \ket{i^\prime,j}\braket{E_{k,l}^{a,b}|E_{i,j}^{a,b}} \bra{k^{\prime},l} \right] \nonumber \\ & & \cdot \left[ \left( - \sin \frac{\theta}{2} \ket{0,b} + i \cos \frac{\theta}{2} \ket{0^{\prime},b \oplus 1} \right) \delta_{a,0} + i \ket{1^{\prime},b\oplus 1} \delta_{a,1} \right] .
\end{eqnarray} When $a =0$, it becomes \begin{eqnarray} f(\ket{0,b})^2 &=& \left[ \sin \frac{\theta}{2} \bra{E_{0,b}^{0,b}} - i \cos \frac{\theta}{2} \bra{E_{0,b+1}^{0,b}} \right] \left[ \sin \frac{\theta}{2} \ket{E_{0,b}^{0,b}} + i \cos \frac{\theta}{2} \ket {E_{0,b+1}^{0,b}} \right] ,
\end{eqnarray} and when $a = 1$, we get \begin{eqnarray}
f(\ket{1,b})^2 &=&\Braket{ E_{1,b+1}^{1,b}|E_{1,b+1}^{1,b}}.
\end{eqnarray}
Next, we define the error operator $D_{CC}$ and $D_{CT}$ by \begin{eqnarray} D_{CC} &\coloneqq& \sin \frac{\theta}{2} \sigma(\vec{l^{\prime}})_C - \sin \frac{\theta}{2}\sigma(\vec{l})_C, \\ D_{TC} &\coloneqq &\left[ \sin \frac{\theta}{2} \sigma(\vec{v})_T - \cos \frac{\theta}{2} \sigma(\vec{v} \times \vec{u})_T \right]^{\prime} -\sin \frac{\theta}{2} \sigma(\vec{l})_C ,
\end{eqnarray} where for an operator $O$, we defined $O^{\prime} \coloneqq U^{\dagger} O U$. We also denote mean errors on $\ket{\psi}_C\ket{0}_T\ket{\xi}_A$ by $\delta_{iC} \, (i = C,T)$. They are given by \begin{equation} \delta_{iC}(\ket{\psi}) = \braket{D_{iC}^2}^{\frac{1}{2}} \quad (i=C,T). \end{equation} We first look into properties of error operators when we succeed in implementing $U_{CU}$ perfectly, that is to say, $U = U_{CU} \otimes I_A$. Since \begin{eqnarray} U_{CU}^{\dagger} \sigma(\vec{l^{\prime}})_C U_{CU} = \sigma(\vec{l})_C \otimes I_T \end{eqnarray} holds, $\delta_{CC}^2(\ket{\psi})=0$ is obtained for an arbitrary $\ket{\psi}_C$ when the ideal time evolution is realized. We also find the following equation: \begin{eqnarray} U_{CU}^{\dagger} \left[ \sin \frac{\theta}{2} \sigma(\vec{v})_T - \cos \frac{\theta}{2} \sigma (\vec{v} \times \vec{u})_T\right] U_{CU}&=&\ket{0}\bra{0}_C \otimes \left[ \sin \frac{\theta}{2} \sigma(\vec{v})_T - \cos \frac{\theta}{2} \sigma (\vec{v} \times \vec{u})_T\right] \nonumber \\ & & \quad + \ket{1}\bra{1}_C \otimes \left[ -\sin \frac{\theta}{2} \sigma(\vec{v})_T - \cos \frac{\theta}{2} \sigma (\vec{v} \times \vec{u})_T\right] . \nonumber \\
\end{eqnarray} When the target state is $\ket{0}_T$ and $U=U_{CU} \otimes I_A$, the expectation value of $D_{TC}$ is calculated as \begin{eqnarray} \bra{\psi}_C \bra{0}_T D_{TC} \ket{\psi}_C \ket{0}_T =0,
\end{eqnarray} for any $\ket{\psi}_C$. Thus, when we implement $U_{CU}$ without error, the expectation value of error operators on $\ket{\psi}_C \ket{0}_T$ are 0. Calculating the mean square of $D_{TC}$ on $ \ket{\psi}_C \ket{0}_T$ for the case where $U= U_{CU} \otimes I_A$, we obtain \begin{eqnarray} & & \bra{\psi}_C \bra{0}_T D_{TC}^2 \ket{\psi}_C \ket{0}_T = \cos^2 \frac{\theta}{2}.
\end{eqnarray}
This expression becomes 0 for $\theta= \pi$.\par
Next we associate $\delta_{CC}^2(\ket{\psi})$ and $\delta_{TC}^2(\ket{\psi})$ with the uncertainty relation. The energy conservation law is represented as \begin{equation} [U, H_T + H_C +H_A +H_{int}] =0, \end{equation} where $H_T$, $H_C$ and $H_A$ are the Hamiltonian of $T$, $C$ and $A$, respectively, and $H_{int}$ is the interaction term. From the energy conservation and the triangle inequality, \begin{eqnarray}
\left|\Braket{[\sin \frac{\theta}{2} \sigma(\vec{l})_C,H_C]}\right| \leq \left|\Braket{[\sin \frac{\theta}{2} \sigma(\vec{l})_C,H_C^{\prime}]} \right|+ \left|\Braket{[\sin \frac{\theta}{2} \sigma(\vec{l})_C,H_T^{\prime}]}\right|+ \left |\Braket{[\sin \frac{\theta}{2}\sigma(\vec{l})_C,H_A^{\prime}]} \right| \label{eq:1} \end{eqnarray} is proved for any $\ket{\psi}_C \ket{\phi}_T$. We used the assumption of the interaction term. From the definition of error operators, we also find that \begin{eqnarray}
\left|\Braket{[\sin \frac{\theta}{2}\sigma(\vec{l})_C,H_C^{\prime}]}\right| &=&\left|\Braket{[H_C^{\prime},D_{TC}]}\right| ,\nonumber \\
\left|\Braket{[\sin \frac{\theta}{2}\sigma(\vec{l})_C,H_T^{\prime}]} \right|&=&\left |\Braket{[H_T^{\prime},D_{CC}]} \right|, \nonumber \\
\left |\Braket{[\sin \frac{\theta}{2}\sigma(\vec{l})_C,H_A^{\prime}]} \right | &=& \left |\Braket{[H_A^{\prime},D_{TC}]} \right| = \left|\Braket{[H_A^{\prime},D_{CC}]} \right| \label{eq:2} \end{eqnarray} hold. From Eq. (\ref{eq:1}) and Eq. (\ref{eq:2}), we can show that \begin{equation}
\left|\Braket{[\sin \frac{\theta}{2}\sigma(\vec{l})_C,H_C]}\right| \leq \left |\Braket{[H_C^{\prime},D_{TC}]} \right| + \left|\Braket{[H_T^{\prime},D_{CC}]} \right |+ \left |\Braket{[H_A^{\prime},D_{TC}]} \right| . \end{equation}
Using Robertson's uncertainty relation and $\Delta(D_{iC})^2 \leq \delta_{iC}^2$, where $\Delta(A)$ is the standard deviation of $A$, we obtain \begin{equation}
\left|\Braket{[\sin \frac{\theta}{2}\sigma(\vec{l})_C,H_C]}\right| \leq 2\delta_{TC}(\ket{\psi}) \Delta(H_C^{\prime})+2\delta_{CC}(\ket{\psi}) \Delta(H_T^{\prime}) +2\delta_{TC}(\ket{\psi}) \Delta(H_A^{\prime}) \label{eq:3}. \end{equation} Similarly, \begin{equation}
\left|\Braket{[\sin \frac{\theta}{2}\sigma(\vec{l})_C,H_C]}\right| \leq 2\delta_{TC}(\ket{\psi}) \Delta(H_C^{\prime})+2\delta_{CC}(\ket{\psi}) \Delta(H_T^{\prime}) +2\delta_{CC}(\ket{\psi}) \Delta(H_A^{\prime}) \label{eq:4} \end{equation} can be proved. Adding Eq. (\ref{eq:3}) to Eq. (\ref{eq:4}), \begin{equation}
\left|\Braket{[\sin \frac{\theta}{2}\sigma(\vec{l})_C,H_C]}\right| \leq [ \delta_{TC} (\ket{\psi}) + \delta_{CC} (\ket{\psi})] (2\gamma+\Delta(H_A^{\prime})) \end{equation}
is derived, where $\gamma \coloneqq \max\left \{ \|H_C\|, \|H_T\| \right \}$. Moreover, using the relation $\frac{(x+y)^2}{2} \leq x^2 + y^2$, we get
\begin{equation}
\frac{\left|\Braket{[\sin \frac{\theta}{2}\sigma(\vec{l})_C,H_C]}\right|^2}{2(2\gamma+\Delta(H_A^{\prime}))^2} \leq \delta_{TC}^2 (\ket{\psi}) + \delta_{CC}^2 (\ket{\psi}) \label{eq:uncertainty}. \end{equation}
Next, we associate the mean squared on the right-hand side with the gate fidelity. In the following calculation, we represent $\ket{\psi}$ as $\ket{\psi}_C =c_0 \ket{0}_C + c_1 \ket{1}_C$. Because \begin{eqnarray}
& & 1 = \Braket{\sigma(\vec{l^{\prime}})^{\prime 2}_C}= \Braket{\sigma(\vec{l})_C^2} =\sum_{a,b} \left [|c_0|^2 |\ket{E_{ab}^{00}}|^2 +|c_1|^2 |\ket{E_{ab}^{10}}|^2 \right] , \\ & &\Braket{\sigma(\vec{l})_C\sigma(\vec{l^{\prime}})_C^{\prime}} = \sum_{a,b} (-1)^a \left[c_0^* \Bra{E_{ab}^{00}} - c_1^* \Bra{E_{ab}^{10}} \right]\left[ c_0 \Ket{E_{ab}^{00} }+c_1 \Ket{E_{ab}^{10}} \right],
\end{eqnarray} holds, we obtain
\begin{eqnarray} \delta_{CC}^2(\ket{\psi})
& =& 4\sin^2 \frac{\theta}{2} \left[ |c_0|^2 \left( \| \ket{E_{10}^{00}} \|^2+\| \ket{E_{11}^{00}} \|^2 \right) + |c_1|^2 \left( \| \ket{E_{00}^{10}}\|^2 + \| \ket{E_{01}^{10}}\|^2 \right)\right].
\end{eqnarray} Similarly, we associate $\delta_{TC}^2 (\ket{\psi})$ with norms of external system states.
\begin{eqnarray} & &\Braket{ \sigma(\vec{l})_C\sigma(\vec{v})_T^{\prime}} = \sum_{a,b} (-1)^b \left[c_0^* \Bra{E_{ab}^{00}} - c_1^*\Bra{E_{ab}^{10}} \right]\left[ c_0\Ket{E_{ab}^{00} }+c_1\Ket{E_{ab}^{10}} \right], \\ & &\Braket{\sigma(\vec{l})_C\sigma(\vec{v} \times \vec{u})^{\prime}_T } = i \sum_{a,b} (-1)^{b+1}\left[c_0^* \Bra{E_{a,b}^{00}} - c_1^ * \Bra{E_{a,b}^{10}} \right]\left[ c_0 \Ket{E_{a,b+1}^{00} }+c_1 \Ket{E_{a,b+1}^{10}} \right],\\ & &\Braket{\sigma(\vec{v} \times \vec{u})_T^{\prime}\sigma(\vec{v})_T^{\prime} } = i \sum_{a,b} \left[c_0^* \Bra{E_{ab}^{00}} +c_1^* \Bra{E_{ab}^{10}} \right]\left[ c_0 \Ket{E_{a,b+1}^{00} }+c_1 \Ket{E_{a,b+1}^{10}} \right], \end{eqnarray} are able to be shown. Therefore,
\begin{eqnarray} & & \delta_{TC}^2 (\ket{\psi}) \nonumber \\
&=& \cos^2 \frac{\theta}{2} + 4\sin^2 \frac{\theta}{2} \left[|c_0|^2 \left(\|\ket{E_{01}^{00}}\|^2 +\|\ket{E_{11}^{00}}\|^2 \right)+|c_1|^2 \left( \|\ket{E_{00}^{10}} \|^2+\|\ket{E_{10}^{10}}\|^2 \right)\right]\nonumber \\
& & \quad -2i \sin \frac{\theta}{2} \cos \frac{\theta}{2} \sum_{a} \left[ |c_0|^2 \Braket{E_{a0}^{00}|E_{a,1}^{00}} - |c_1|^2 \Braket{E_{a,0}^{10}|E_{a,1}^{10}}\right]\nonumber \\
& &+2 i \sin \frac{\theta}{2} \cos \frac{\theta}{2} \sum_{a} \left[ |c_0|^2\Braket{E_{a1}^{00}|E_{a,0}^{00}} -|c_1|^2 \Braket{E_{a,1}^{10}|E_{a,0}^{10}}\right] \nonumber \\ \\
&\leq& 2|c_0|^2 \left[\cos \frac{\theta}{2} \Bra{E_{00}^{00}} +i\sin \frac{\theta}{2} \Bra{E_{01}^{00}} \right]\left[\cos \frac{\theta}{2} \ket{E_{00}^{00}} -i\sin \frac{\theta}{2} \ket{E_{01}^{00}} \right] \nonumber \\
& &+ 2 | c_0|^2 \left[\cos \frac{\theta}{2} \Bra{E_{10}^{00}} +i\sin \frac{\theta}{2} \Bra{E_{11}^{00}} \right]\left[\cos \frac{\theta}{2} \ket{E_{10}^{00}} -i\sin \frac{\theta}{2} \ket{E_{11}^{00}} \right] \nonumber \\
& & + 2 |c_1|^2\left[\cos \frac{\theta}{2} \Bra{E_{01}^{10}} +i\sin \frac{\theta}{2} \Bra{E_{00}^{10}} \right]\left[\cos \frac{\theta}{2} \ket{E_{01}^{10}} -i\sin \frac{\theta}{2} \ket{E_{00}^{10}} \right]\nonumber \\
& &+2|c_1|^2 \left[\cos \frac{\theta}{2} \Bra{E_{11}^{10}} +i\sin \frac{\theta}{2} \Bra{E_{10}^{10}} \right]\left[\cos \frac{\theta}{2} \ket{E_{11}^{10}} -i\sin \frac{\theta}{2} \ket{E_{10}^{10}} \right] \nonumber \\
& &+ 2 \left[|c_0|^2 \left( \|\Ket{E_{01}^{00}} \|^2 +\|\Ket{E_{11}^{00}}\|^2\right)+ |c_1|^2 \left( \|\Ket{E_{00}^{10}} \|^2+\|\Ket{E_{10}^{10}} \|^2 \right)\right]
\end{eqnarray} holds. We can derive the following equation:
\begin{eqnarray} & &\delta_{CC}^2 (\ket{\psi})+\delta_{TC}^2 (\ket{\psi}) \nonumber \\
&\leq&4\sin^2 \frac{\theta}{2} \left[ |c_0|^2 \left( \| \ket{E_{10}^{00}} \|^2+\| \ket{E_{11}^{00}} \|^2 \right) + |c_1|^2 \left( \| \ket{E_{00}^{10}}\|^2 + \| \ket{E_{01}^{10}}\|^2 \right)\right] \nonumber \\
& & + 2|c_0|^2 \left[\cos \frac{\theta}{2} \Bra{E_{00}^{00}} +i\sin \frac{\theta}{2} \Bra{E_{01}^{00}} \right]\left[\cos \frac{\theta}{2} \ket{E_{00}^{00}} -i\sin \frac{\theta}{2} \ket{E_{01}^{00}} \right]\nonumber \\
& &+ 2 | c_0|^2 \left[\cos \frac{\theta}{2} \Bra{E_{10}^{00}} +i\sin \frac{\theta}{2} \Bra{E_{11}^{00}} \right]\left[\cos \frac{\theta}{2} \ket{E_{10}^{00}} -i\sin \frac{\theta}{2} \ket{E_{11}^{00}} \right] \nonumber \\
& & + 2 |c_1|^2\left[\cos \frac{\theta}{2} \Bra{E_{01}^{10}} +i\sin \frac{\theta}{2} \Bra{E_{00}^{10}} \right]\left[\cos \frac{\theta}{2} \ket{E_{01}^{10}} -i\sin \frac{\theta}{2} \ket{E_{00}^{10}} \right] \nonumber \\
& &+2|c_1|^2 \left[\cos \frac{\theta}{2} \Bra{E_{11}^{10}} +i\sin \frac{\theta}{2} \Bra{E_{10}^{10}} \right]\left[\cos \frac{\theta}{2} \ket{E_{11}^{10}} -i\sin \frac{\theta}{2} \ket{E_{10}^{10}} \right] \nonumber \\
& &+ 2 \left[|c_0|^2 \left( \|\Ket{E_{01}^{00}} \|^2 +\|\Ket{E_{11}^{00}}\|^2\right)+ |c_1|^2 \left( \|\Ket{E_{00}^{10}} \|^2+\|\Ket{E_{10}^{10}} \|^2 \right)\right] \\
& \leq & 4 \left[ |c_0|^2 \left( \| \ket{E_{10}^{00}} \|^2+\| \ket{E_{11}^{00}} \|^2 \right) + |c_1|^2 \left( \| \ket{E_{00}^{10}}\|^2 + \| \ket{E_{01}^{10}}\|^2 \right)\right] \nonumber \\
& & - 2|c_0|^2 \left[\sin \frac{\theta}{2} \Bra{E_{00}^{00}} -i\cos \frac{\theta}{2} \Bra{E_{01}^{00}} \right]\left[\sin \frac{\theta}{2} \ket{E_{00}^{00}} +i\cos \frac{\theta}{2} \ket{E_{01}^{00}} \right]\nonumber \\
& & + 2 |c_0|^2 \left(\|\ket{E_{00}^{00}} \|^2 + \|\ket{E_{01}^{00}} \|^2 \right) + 2 | c_0|^2 \left[\cos \frac{\theta}{2} \Bra{E_{10}^{00}} +i\sin \frac{\theta}{2} \Bra{E_{11}^{00}} \right]\left[\cos \frac{\theta}{2} \ket{E_{10}^{00}} -i\sin \frac{\theta}{2} \ket{E_{11}^{00}} \right] \nonumber \\
& &+ 2 |c_1|^2\left[\cos \frac{\theta}{2} \Bra{E_{01}^{10}} +i\sin \frac{\theta}{2} \Bra{E_{00}^{10}} \right]\left[\cos \frac{\theta}{2} \ket{E_{01}^{10}} -i\sin \frac{\theta}{2} \ket{E_{00}^{10}} \right] +2 |c_1|^2 \left( \|\ket{E_{11}^{10}} \|^2 +\|\ket{E_{10}^{10}} \|^2\right) \nonumber \\
& & -2|c_1|^2 \left[\sin \frac{\theta}{2} \Bra{E_{11}^{10}} -i\cos \frac{\theta}{2} \Bra{E_{10}^{10}} \right]\left[\sin \frac{\theta}{2} \ket{E_{11}^{10}} +i\cos \frac{\theta}{2} \ket{E_{10}^{10}} \right] \nonumber \\
& &+ 2 \left[|c_0|^2 \left( \|\Ket{E_{01}^{00}} \|^2 +\|\Ket{E_{11}^{00}}\|^2\right)+ |c_1|^2 \left( \|\Ket{E_{00}^{10}} \|^2+\|\Ket{E_{10}^{10}} \|^2 \right)\right] \nonumber \\ \\
& =& 2|c_0|^2 - 2|c_0|^2 \left[\sin \frac{\theta}{2} \Bra{E_{00}^{00}} -i\cos \frac{\theta}{2} \Bra{E_{01}^{00}} \right]\left[\sin \frac{\theta}{2} \ket{E_{00}^{00}} +i\cos \frac{\theta}{2} \ket{E_{01}^{00}} \right] \nonumber \\
& & + 2 | c_0|^2 \left[\cos \frac{\theta}{2} \Bra{E_{10}^{00}} +i\sin \frac{\theta}{2} \Bra{E_{11}^{00}} \right]\left[\cos \frac{\theta}{2} \ket{E_{10}^{00}} -i\sin \frac{\theta}{2} \ket{E_{11}^{00}} \right] + 2|c_0|^2 \left(1 - \|\ket{E_{00}^{00}} \|^2 + \|\ket{E_{11}^{00}} \|^2\right)\nonumber \\
& & +2 |c_1|^2 + 2 |c_1|^2\left[\cos \frac{\theta}{2} \Bra{E_{01}^{10}} +i\sin \frac{\theta}{2} \Bra{E_{00}^{10}} \right]\left[\cos \frac{\theta}{2} \ket{E_{01}^{10}} -i\sin \frac{\theta}{2} \ket{E_{00}^{10}} \right] \nonumber \\
& & -2|c_1|^2 \left[\sin \frac{\theta}{2} \Bra{E_{11}^{10}} -i\cos \frac{\theta}{2} \Bra{E_{10}^{10}} \right]\left[\sin \frac{\theta}{2} \ket{E_{11}^{10}} +i\cos \frac{\theta}{2} \ket{E_{10}^{10}} \right] +2 |c_1|^2 \left( 1 - \|\ket{E_{11}^{10}} \|^2 + \|\ket{E_{00}^{10}} \|^2\right) \nonumber \\ \\
&\leq& 4|c_0|^2[1- f(\ket{0,0})^2]+4|c_1|^2[1-F(\ket{1,0})^2]+4|c_0|^2[1-F(\ket{0,0})^2]+4|c_1|^2[1-f(\ket{1,0})^2] \\ & \leq & 4[1-F(\mathcal{E}_{\alpha},U_{CU})^2]+4[1-f(\mathcal{E}_{\alpha},U_{CU})^2].
\end{eqnarray}
Imposing the symmetry that the gate fidelity when the target gate is $V_T(\theta)$ is equal to the gate fidelity when the target gate is $V_T(-\theta)$ , $F(\mathcal{E}_{\alpha},U_{CU})=f(\mathcal{E}_{\alpha},U_{CU})$, \begin{equation}
\sin^2 \frac{\theta}{2}\cdot \frac{\left|\Braket{\left[\sigma(\vec{l})_C,H_C\right]}\right|^2}{2(2\gamma+\Delta(H^{\prime}_A))^2} \leq 8[1-F(\mathcal{E}_{\alpha},U_{CU})^2] \end{equation} can be proved. Finally, by maximizing over states of $C$+$T$ and minimizing over states of $A$, Eq. (\ref{eq:1bit}) is derived.
\end{document} | arXiv |
A note on Erdős-Ko-Rado sets of generators in Hermitian polar spaces
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Construction of 3-designs using $(1,\sigma)$-resolution
August 2016, 10(3): 525-540. doi: 10.3934/amc.2016023
Construction of subspace codes through linkage
Heide Gluesing-Luerssen 1, and Carolyn Troha 2,
University of Kentucky, Department of Mathematics, Lexington, KY 40506-0027
Department of Mathematics, Viterbo University, La Crosse, WI, United States
Received May 2015 Revised September 2015 Published August 2016
A construction is discussed that allows to produce subspace codes of long length using subspace codes of shorter length in combination with a rank metric code. The subspace distance of the resulting linkage code is as good as the minimum subspace distance of the constituent codes. As a special application, the construction of the best known partial spreads is reproduced. Finally, for a special case of linkage, a decoding algorithm is presented which amounts to decoding with respect to the smaller constituent codes and which can be parallelized.
Keywords: constant dimension subspace codes, partial spreads., Random network coding.
Mathematics Subject Classification: Primary: 11T71, 94B60; Secondary: 51E2.
Citation: Heide Gluesing-Luerssen, Carolyn Troha. Construction of subspace codes through linkage. Advances in Mathematics of Communications, 2016, 10 (3) : 525-540. doi: 10.3934/amc.2016023
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Heide Gluesing-Luerssen Carolyn Troha | CommonCrawl |
The average of the first five multiples of 9 is
Home / Arithmetic Ability / Average / Question
The average of the first five multiples of 9 is:
$$\eqalign{ & {\text{Required}}\,{\text{average}} \cr & = {\frac{{{\text{total}}\,{\text{sum}}\,{\text{of}}\,{\text{multiple}}\,{\text{of}}\,9}}{5}} \cr & = {\frac{{9 + 18 + 27 + 36 + 45}}{5}} \cr & = 27 \cr} $$
Note that, average of 9 and 45 is also 27.
And average of 18 and 36 is also 27.
Atif Rehman :
9,18,27.36,45
Avg = 1st + Last / 2
(9+45)/2 = 27
RJ Champ
Bikash Bhattarai :
Ans = 9(1+2+3+4+5)/5
FACTOM WORLD :
(1st +last term) / 2
(9+45) / 2
= 54/2
Nafeesh Haider :
9(1,2,3,4,5)/5 = 9*5(5+1)/2/5 9*6/2
Mohana Gowri :
Another method for this ,
Avg= first term + last term /2
hence, 9+45/2=27
Santosh Kumar :
9×15/5=9×3
Nvjddurrg Jgyureey :
9+45/2
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Comparison of RNA-seq and microarray platforms for splice event detection using a cross-platform algorithm
Juan P. Romero1 na1,
María Ortiz-Estévez2 na1,
Ander Muniategui1,
Soraya Carrancio2,
Fernando J. de Miguel3,
Fernando Carazo1,
Luis M. Montuenga3,4,5,7,
Remco Loos2,
Rubén Pío3,5,6,7,
Matthew W. B. Trotter2 &
Angel Rubio ORCID: orcid.org/0000-0002-3274-24501
RNA-seq is a reference technology for determining alternative splicing at genome-wide level. Exon arrays remain widely used for the analysis of gene expression, but show poor validation rate with regard to splicing events. Commercial arrays that include probes within exon junctions have been developed in order to overcome this problem.
We compare the performance of RNA-seq (Illumina HiSeq) and junction arrays (Affymetrix Human Transcriptome array) for the analysis of transcript splicing events. Three different breast cancer cell lines were treated with CX-4945, a drug that severely affects splicing. To enable a direct comparison of the two platforms, we adapted EventPointer, an algorithm that detects and labels alternative splicing events using junction arrays, to work also on RNA-seq data. Common results and discrepancies between the technologies were validated and/or resolved by over 200 PCR experiments.
As might be expected, RNA-seq appears superior in cases where the technologies disagree and is able to discover novel splicing events beyond the limitations of physical probe-sets. We observe a high degree of coherence between the two technologies, however, with correlation of EventPointer results over 0.90. Through decimation, the detection power of the junction arrays is equivalent to RNA-seq with up to 60 million reads.
Our results suggest, therefore, that exon-junction arrays are a viable alternative to RNA-seq for detection of alternative splicing events when focusing on well-described transcriptional regions.
Alternative Splicing (AS) is known to play a major role in human biology, and the identification of transcriptional splicing patterns has potential uses for diagnosis, prognosis, and therapeutic target evaluation in the disease context [1, 2]. The development of exon microarrays enabled the transcriptomic study of differential splicing events, but PCR validation rates for identification of splice differences via microarray analysis tend to be lower than those observed for identification of differential gene expression using similar technologies [3,4,5]. Junction arrays [6,7,8,9,10] have been proposed to overcome this problem by using oligonucleotide probe-sets that interrogate junctions between exons in the transcriptome, as well as the exons themselves.
Since the advent of next-generation sequencing (NGS), RNA-seq has become the technology of choice via which to detect and quantify alternative splicing (for a review see [11]). Various published works compare the performance of RNA-seq and expression microarrays for the analysis of gene expression [12, 13], but a thorough evaluation of both technologies in terms of their ability to detect differential AS events has yet to be presented. In the present study, we perform a comparison of RNA-seq technology (using the Illumina HiSeq platform) and junction arrays commercialized by Affymetrix (Human Transcriptome array, or HTA).
AS can be studied from two complementary points of view: with focus on transcripts or splicing events respectively. In the former, the subject of analysis is the transcript (or isoform), whereas in the latter, the subject(s) are the splicing events themselves.
The pipeline of the transcript-focused approach uses RNA-seq data with [14] and without known annotations in order to reconstruct the transcriptome and estimate the concentration values of the transcripts. Finally, the significance of change in absolute or relative concentrations is assessed using suitable statistical methods [15,16,17]. Transcript reconstruction is challenging [18] (even the better methods display transcriptome reconstruction levels below 50% when using simulated reads) and any error in reconstruction of transcript structure may be propagated to the output of statistical analysis. Moreover, the challenge of estimating isoform concentrations for genes with many transcripts yields wide confidence intervals [19].
On this basis, therefore, an event-based method appears a more suitable approach via which to compare AS detection technologies, with the additional benefit of straightforward validation using PCR. Event-based methods focus directly on the analysis of differential splicing events, rather than first attempting to estimate transcript concentration levels. These events can be classified into five canonical categories [20]: cassette exon, alternative 3′, alternative 5′, mutually exclusive exons and intron retention. In some cases, alternative start and termination sites are included also when defining splicing events. This approach has gained traction and several algorithms have been developed recently for detection of splicing events using RNA-seq data, including rMats, SplAdder, spliceGrapher or SGSeq [21,22,23,24]. SpliceGrapher and SGSeq detect events prior to application of separate software in order to state corresponding statistical significance, whereas rMats and SplAdder perform both detection and statistical analysis. Alongside NGS-based approaches, AS event detection methods are available for exon arrays [25], and exon-junction arrays [6, 8, 9, 26]. The latter methods display validation rates well above 50%.
The principal aim of this work is to compare RNA-seq and exon-junction microarray technologies in their ability to detect differential AS events. To do so comprehensively, and to allow as close to a direct comparison as possible, we have adapted the EventPointer [8] algorithm for application to data from both platforms, generated from the same control experiment. The control experiment comprises three distinct triple-negative breast cancer (TNBC) cell-lines, exposed in culture to a drug known to affect the transcriptional machinery and, thereby, to induce AS events.
Further to comparative analysis of the resulting data, we conclude that both technologies show considerable concordance with high PCR validation rates, and that exon-junction microarrays have potential as an alternative to RNA-seq profiling for detection of AS events in annotated transcripts.
CX-4945 is a potent and selective orally bioavailable small molecule inhibitor of casein kinase CK2 [27], which has been proposed previously as a cancer therapy [28], and which has been shown to regulate splicing in mammalian cells [29]. RNA samples taken from three distinct triple-negative breast cancer (TNBC) cell-lines, exposed to CX-4945 and also to a DMSO control, were profiled using both RNA sequencingFootnote 1 and hybridization to exon-junction microarrays (see Methods for details). We extended the EventPointer algorithm (available via Bioconductor, see Methods) for application to data from both platforms and applied it to the corresponding datasets in order to identify AS events.
Prior to the comparison of platforms for splice event detection, the data was assessed at the gene level in order to ensure signal quality and coherence. Gene expression was computed from RNA-seq data using Kallisto [14] to quantify expression as the sum of isoform concentrations for each gene. RMA [30] was used to quantify gene expression from microarray data, using annotation files from Brainarray [31]. The same version of the Ensembl Transcriptome (Ensembl v.74, GRCh 37.75) was used in both cases.
Considering each technology independently, correlation between sample replicate profiles in each cell-line and experimental condition is high for both platforms (correlation coefficient ranging from 0.988 to 0.996 in arrays and 0.996 to 0.997 in RNA-seq). When comparing profiles from the same samples between technologies, strong coherence is observed for well-expressed genes. Median correlation of gene expression between technologies on the same samples is 0.510, and gene expression patterns across all samples display correlation of 0.680 between technologies. The first one is smaller owing to the different probe affinities of the set of probes that interrogates each gene. When only the 50% most highly expressed genes are considered, the median correlation of gene expression patterns is 0.750 (Additional file 1). The gene expression correlations observed are similar to previously reported comparisons between RNA-seq and exon arrays [32]. It is important to point out that the expected correlations for gene expression are larger (either using microarrays or RNAseq) since the number of probes/reads that interrogate a gene is larger than the ones that interrogate a splicing event.
Events detected by RNA-seq and junction arrays show strong qualitative and quantitative concordance, with a subset detected exclusively by one of the technologies
Figure 1 depicts the EventPointer pipeline for both profiling technologies (see original publication [8] for further detail), with CEL files (microarray) or BAM files (RNA-seq) as starting input. When building the splicing graph, each exon is split into two nodes that correspond to its start and end genomic positions respectively (Fig. 1b). Each event is described by two alternative paths (Paths 1 and 2) and a shared reference path (Path Ref) within the splicing graph. These paths are sets of edges in the splicing graph. Paths 1 and 2 are mutually exclusive in terms of isoforms (i.e. if an isoform includes Path 1 it does not include Path 2 and vice versa) and all isoforms interrogated by the event share the reference path. Therefore, events are contained in several isoforms (at least two). A simple example would be the cassette exon shown in Fig. 1b: the reference path is composed by the edge that links nodes 6a and 6b (i.e. the coverage of exon 6 or the signal in the probe-set of the array that interrogates this exon) and the edge that links nodes 8a and 8b (coverage of exon 8). All these measurements are summarized into one average value. Path 1 includes the edges in the path 6b-7a-7b-8a (coverage of exon 7 and its flanking junctions) and Path 2 is the edge that links 6b and 8a (coverage of the skipping junction). EventPointer distinguishes between events as corresponding to: cassettes; alternative 5′; alternative 3′; mutually exclusive exons; alternative first exons; and alternative end exons. Complex events that do not match any of these categories are denoted as such.
EventPointer overview for junction arrays and RNA-Seq data. a The CEL or BAM files are the input data for each technology. The splicing graph for each gene is built using the array annotation files or directly using the sequenced reads. b Each node in the splicing graph is splitted into two nodes that correspond to the start and end positions in the genome respectively. EventPointer identifies events within each gene and annotates the type of event. In the figure, among the events in the gene, an exon cassette is highlighted. c Statistical significance of the events is computed. d Finally, the top-ranked events are validated using PCR and the results visualized in IGV
Three different cell-lines were profiled, each exposed to CX-4945 and DMSO respectively across five replicates. AS differences were tested using a linear model which controlled for cell-line differences. Using the read coverage (or probe-set signal) for each path, a statistical analysis based on voom-limma [17, 33] is applied to determine the significance of each event via comparison between alternative path signals (see Methods for details). In addition to the statistical analysis, we compute the Percent Splice Index (PSI or Ψ) [34], an estimate of relative isoform concentrations that map to paths 1 and 2 for each event. In a cassette event, if the exon is retained, Ψ is equal to one. If it is skipped, Ψ is equal zero. If both isoforms that retain and skip the exon are present, Ψ is the ratio between the expression of the isoforms that retain the exon and the overall expression of the isoforms that skip or retain the exon. Ψ has become the standard method to quantify splicing events.
In order to identify well-expressed events (more likely to be biologically significant and less prone to validation error), the comparison of AS detection was performed on a subset of the data with expression above a set threshold (see Methods for details). In brief, a junction coverage threshold was applied to the RNA-seq data (default 2 FPKM) and a threshold on expression percentile applied to the microarray data (default probe-set expression greater than 25% of probes in any sample profile).
Table 1 displays the number of detected events after setting a threshold on the expression for both technologies, the number of differentially spliced events (p value < 0.001) with their corresponding False Discovery Rate (FDR) detected via application of EventPointer to RNA-seq and microarray data respectively. The statistical analysis compares differential AS in profiles from cell-lines treated with CX-4945 and with DMSO control. As may be expected, setting more stringent expression thresholds yields fewer events detected with better FDR on both platforms. The FDR is the estimated proportion of false discoveries (i.e. events are assumed to have differential splicing and that do not). For example, an FDR of 5e-4 means that 0.05% of the selected events are expected to be false positives.
Table 1 Number and statistical significance of detected AS events using both RNA-seq and array technologies
Table 1 shows that fixing p-value to 1e-3 yields False Discovery Rates (FDRs) less than 1% for both technologies. The expected proportion of AS events appears high (1-π0 approx. 46%) [35], i.e. more than 46% of the events have its splicing patterns altered, which reflects the anticipated strong effect of compound exposure on the splicing machinery. It is also apparent that, for a similar number of detected AS events, the FDR corresponding to RNA-seq analysis is smaller.
Events detected by both technologies (referred to as "matched events" hereon) were defined by a stringent criterion in which nucleotide sequences of paths identified via one technology must be a subset of sequences identified via the other, yielding 6222 matched events. When reporting correspondence and divergence between AS events, below, the following naming convention is used: R+ represents number of events deemed significantly altered in RNA-seq analysis (p value <1e-3); R− represents number of events deemed not significantly altered in RNA-seq analysis (p value > 0.2). M+ and M− are the counterpart terms used to describe microarray results. Events not detected by each technology are labelled R∅ and M∅ respectively.
A subset of matched events is significant in both technologies (R+M+) and shows coherent change in the corresponding Ψ. There are also significant events detected by only one of the technologies (R+M∅ and R∅M+). The summary of findings is presented in Table 2.
Table 2 Number of AS events detected per technology, alongside statistical significance of events against distinct thresholds
Table 2 shows that the FDR of the events detected only by RNA-seq is similar to that for events detected by both platforms (4.56e-4 vs. 4.96e-4). In other words, the reliability of events discovered only by RNA-seq is similar to that of events identified by both technologies. In the case of the arrays, the FDR of matched events is three times smaller than for those discovered solely by the arrays (1.99e-3 vs 6.23e-4 i.e. R∅M+ events are less reliable than R+M+ events for the same p-value threshold. In addition, Table 2 shows that the number of significant events that are RNA-seq specific (R+M∅) is larger than the number of significant events detected only by arrays (R∅M+) (10,617 vs 3297 events).
Figure 2 depicts a Sankey diagram of the relationship between matched events. An event is declared to be significant (in either technology) if the p value is smaller than 1e-3. It is declared non-significant is the p value is larger than 0.2 and inconclusive otherwise. It is apparent that many events that are significant for RNA-seq are not detected by arrays, but also that events significantly detected via arrays are not detected by RNAseq. Most matched events are consistent across technologies: significant events for one technology are also significant for the other.
Correspondence between the events detected by arrays and RNA-seq. An event is considered to be significant if the p.value is smaller than 0.001 and non-significant is it is larger than 0.2. Events with p-values between both are considered to be inconclusive cases
We also considered the FDR for different types of splicing events in both technologies. As shown in Fig. 3, alternative 3′ (5′), start and end sites have larger FDR than cassette exons., i.e. they are harder to measure. There were too few matched mutually exclusive events to estimate accurately FDR for this type of events.
FDR for different types of events using both technologies. Panel a shows the FDR for matched events. Panel b shows FDR for the events detected in each technology regardless or being matched or not. In both technologies, alternative 3′, 5′, first and last exons have larger FDR than other types of events
PCR validation rates are over 80% in both technologies
PCR validation was performed on a subset of predicted AS events drawn from each of the subsets discussed in previous sections, i.e. events detected by one or both technologies. PCRs were performed on:
Five top-ranked events detected by both technologies (topRNA and topArrays) regardless of the matching with the other technology
Five top-ranked events detected by one technology (R+M∅ and R∅M+)
Five top-ranked events significant in one technology (R+M− and R−M+)
Five top ranked events detected by both technologies (R+M+)
These potential 35 validations are in fact 29 since there is overlap in the top-ranked events of different categories. The characteristics of validated events (genome location, event type, etc.) and links to the corresponding PCR images are included in Additional file 2. PCR for events in non-coherent classes (R+M−, R−M+) required up to 40 PCR cycles and were harder to validate in general. The corresponding GTF files to browse these events in IGV [36] are included in Additional file 3. All the results are summarized in Table 3.
Table 3 PCR validation for RNA-seq and microarray technologies across events detected by one or both technologies
Figure 4 shows the Ψ estimates and the PCR bands for two of the top-ranked events in R+M+ (gene names DONSON and MELK), with clear concordance of the splice index, Ψ, across the three technologies despite use of end-point (i.e. non-quantitative) PCR. Similar figures for events in the other AS categories are included in the additional material (Additional file 1: Figures S2 to S8).
Estimated PSI (for RNA-seq, microarrays and PCR image analysis), PCR bands, the reference HTA transcriptome and the alternative paths of the DONSON (panel a) and MELK (panel b) genes in R+M+. Each of the points represents the same replicate in either of the three technologies. The last numbers shown are expected bands for the selected primers. If the number is shown to the left side of the double bars, the band corresponds to Path 1 of the event (long path). If shown to the right side, corresponds to Path 2 (short path)
Statistics and Ψ for matched events are similar
Figure 5 shows the increment of the Ψ value estimated by EventPointer for events detected by both technologies. Correlation for AS events is over 0.90, and z-values of the statistical test are also similar (Additional file 1: Figure S1). PCR figures also show high coherence between the estimated Ψ using both technologies, especially for RNA-seq, and the PCR results (Fig. 4 and Additional file 1: Figure S2 to S8).
Increment of PSI for both microarrays and RNA-seq. The black (gray) dots represent events with high (low) standard deviation in the differential usage of the isoforms in both paths. Correlation between events with high and low variability are 0.90 and 0.61 respectively
Both technologies detect a similar distribution of AS types
Figure 6a shows the number and type of AS events detected by the EventPointer algorithm on data from both profiling technologies. The number of detected cassette exons using arrays is smaller than that using sequencing (p value <1e-16, test for equality of proportions). In fact, after matching the events detected by both technologies, a large proportion of the cassette exons in RNA-seq appear as complex in microarrays (see Fig. 6b). The reason for this disparity is the complexity of the reference transcriptome used in the HTA array. For this analysis, we used the transcriptome provided by Affymetrix, which includes a range of annotation sources, e.g. RefSeq, Vega, Ensembl, MGC (v10), UCSC known genes and other sources for non-coding isoforms. The underlying transcriptome for HTA includes such a variety of isoforms that many detected AS events are labelled as complex.
a Events detected using RNA-seq and array technologies. b Type of event after matching the events detected by both technologies
In addition, the proportion of retained introns is smaller for RNA-seq (p value <1e-16, test for equality of proportions), perhaps owing to the coverage required to include a region as expressed by SGSeq (defaults to 0.5 FPKM) which may exclude weakly expressed introns.
Power of arrays to detect events is approximately equivalent to shallow RNA-seq
The comparisons above suggest that that RNA-seq – at the depth of sequencing deployed here - detects a larger number of AS events at lower FDR.
We subsampled the initial RNA-seq data to 30% and 10% of the input, yielding approximately 30 million and 10 million reads respectively. Using these subsampled data, we estimated their FDR (see Table 4). Interpolating the FDR for them, the FDR using junction arrays is equivalent to the FDR of an RNA-seq experiment with sequencing depth of approximately 20 million reads.
Table 4 Obtained FDR values after subsampling the number of reads in the RNA-seq experiment
We hypothesize that the performance of the arrays could be greatly improved by removing bad-performing probesets. The RMA summarization algorithm withstands the presence of a few outlier probes. In fact, we have identified some probes that cross-hybridize in several loci of the transcriptome. However, if most of the probes that interrogate either of the paths or the reference do not perform well, the whole estimate of the splicing event will be compromised. Some of these cases can be detected since the signal of the events do not show internal coherence with the model (i.e. they show a large relative error if the weighted sum of the signals in Paths 1 and 2 and the reference Path are compared). These bad probesets are somehow expected: the design of junction probes has strong limitations since there is no room to select a probe with certain standards of quality (GC content, no cross-hybridization against the genome of the transcriptome, etc.). Owing to these probesets, a number of events are not being measured accurately. We have included in the additional material Additional file 1: Figure S9 to illustrate bad and good performing probesets: in panel A, it is shown an event with internal coherence and in panel B, an event with bad internal coherence. We have also included Additional file 1: Table S10 that shows the FDR for genes with large coherence (small relative error) and small coherence (large relative error). The FDR for genes with large internal coherence is 2 times bigger than for events with weak internal coherence.
The events that are not matched with RNA-seq are enriched in these pathological cases as shown in Table 2. On the contrary, the expected false discovery rate for matched events finds HTA arrays to be equivalent to RNA-seq with a depth of approximately 60 million reads. A proper filtering of the probes identifying events prone to errors could ideally take the arrays closer to the RNA-seq performance with this depth.
The main aim of this work was to quantitatively compare the performance of RNA-seq and junction array technologies to detect splicing events. To do this in a balanced manner, we adapted our algorithm EventPointer, originally developed for HTA arrays, to work also on RNA-seq data.
This study highlights:
The creation of a real-world cell exposure dataset specifically relevant for the study of alternative splicing.
Adaptation of an existing AS event detection algorithm to a cross-platform method to enable comparative application, and addition of percent splice index method.
Strong correlation of splicing event detection in regions covered by both technologies, validated by PCR on a subset of top ranked events identified by both and each platform respectively.
Benefits of RNA-seq in terms of coverage and flexibility, as expected, and higher validation rates in case of disagreement between technologies.
Good performance of HTA arrays, estimated by approximation to be equivalent to relatively shallow RNA-seq in transcript regions covered.
Top-ranked events detected by each platform technology and estimates of relative event occurrence (ΔΨ) were validated by PCR. The relative occurrence estimates were also strongly correlated, close to 0.90 for events detected by both technologies. In addition to enabling comparison of the two profiling platforms, these results suggest also that the estimates themselves are a relevant addition to the original EventPointer algorithm.
We relied on SGSeq to build the splicing graph. Using other algorithms (such as Spladder) could impact the detected events especially for weakly expressed genes. According to a recent review of some of the authors [37], AltAnalyze is the only alternative that provides the analysis of splicing events both for arrays and RNA-seq. AltAnalyze characterizes each event by several values that must be integrated. For example, in a cassette exon, the signals of the probes in the exon, the flanking junctions and the skipping junction (and the equivalent coverage values for RNAseq) should be integrated to get a single figure of merit. We found it difficult to perform this integration since we are not the developers of this method. Nevertheless, using this method could be also informative to compare both platforms.
As might be expected in the absence of physical probe-sets, over 10,000 statistically significant events were identified by RNA-seq alone, the top ranked of which were validated via PCR. Approximately 3300 events were detected using microarrays but not detected using RNA-seq. In this case, some (3/5) of the top ranked events were validated and correspond to well-expressed genes. Those which did not may reflect the specific technical biases of each technology (cross hybridization of the probes, multi mapping reads, GC dependence, etc.)
A recent study [38] also compared RNAseq and arrays. This study was focused on patient derived samples instead of cell-lines as we did. This study pinpoints differences in the output between both technologies and states it in the title. In our case, the main divergence that we found between both technologies appears in the events that are not matched. Interestingly, the coherence between matched events -shown by Fig. 5 and the validated PCRs- is very strong: if an event is detected by both technologies, the increment of Ψ and its statistical significance (see Additional file 1: Figure S1) are very similar.
RNA-seq has inherent advantages over microarrays, including the ability to detect unlimited novel events. Furthermore, sensitivity can be improved by increasing sequencing depth. Another advantage of RNA-seq is its better approximation of gene/transcript concentrations (e.g. allowing to state a threshold based on the expression of an event). On the other hand, arrays were able to detect some weakly expressed events missed by RNA-seq and, in general across the comparisons, performed similarly to RNA-seq when treating well-expressed and well-defined transcriptional regions. As expected, a similar algorithmic approach applied to both platforms consumed less time and memory resources when treating microarray data than when treating RNA-seq data (See Table 5).
Table 5 Resources required for both technologies. Analysis was performed on 16 cores (Intel Xeon E5–2670 @ 2.60 GHz) with 64 GB of RAM Linux server running 64-bit CentOS distribution
In conclusion, comparison of RNA-seq and junction microarrays using a cross-platform algorithm suggests that both technologies provide accurate identification of splice events. Moreover, predictions by both technologies tend to correlate strongly and yield similar results when compared by Ψ estimates and PCR. RNA-seq holds a clear advantage in terms of flexibility, and stronger PCR validation of events detected in one platform but not the other. As compared, HTA microarrays are shown nevertheless to provide a reasonable alternative to relatively shallow RNA-seq in the transcriptional regions that they reference.
Triple negative breast cancer cell lines MDA-MB-231 and MDA-MB-468 were obtained from ATCC (Manassas, VA) with identification numbers HTB-26 and HT-132 respectively and SUM149 was purchased from Asterand plc (Detroit, MI). All cell lines were grown according to the suppliers' recommendation. CK2 inhibitor CX-4945 (Selleckchem, Houston, TX) was dissolved in DMSO and stored frozen at − 80 °C until used.
To induce splicing events, cells were grown to ~ 70% confluence and treated with 1 μM CX-4945 or DMSO during 12 h in a total of 5 replicates per condition. Total RNAs were isolated using the RNeasy Mini Kit (Qiagen, Germantown, MD) according to the manufacturer's protocol. Integrity of RNA was quantified using the Agilent 2100 Bioanalyzer (Agilent Biosystems, Foster City, CA). Samples were labeled and hybridized in Human Transcriptome arrays (HTA) by the Genomics Core Facility of the Center for Applied Medical Research (CIMA) following manufacturer's instructions.
RNAseq was performed in the Center for Cooperative Research in Biosciences (CICBiogune) using the Illumina HiSeq2000 sequencing technology, HiSeq Flow Cell v3 and TruSeq SBS Kit v3. 2μg of RNA of each sample was sent for this purpose. The run type was strand specific, multiplexed with paired-end reads of 100 nucleotides each. The amount of RNA for hybridization and validation purposes was 5 μg.
STAR 2.4.0 h1 was used to align the reads against the human genome. The reference genome was Ensembl v.74, GRCh 37.75. The output were sorted BAM files. All the other parameters were set to the default values. The average sequencing depth was 49 million reads (9.8 billion nucleotides sequenced per sample).
The microarray data preprocessing was performed using the aroma.affymetrix framework using the standard RMA algorithm applied to probesets of the paths [30]. In addition, we used both platforms to quantify expression at the gene level. Results are shown in the Additional file 1. Gene expression was computed from RNA-seq data using Kallisto [14] to quantify expression using a pseudo-aligment method. Kallisto returns an estimate of the expression of all the isoforms for each gene. The overall expression of the gene was simply computed by summing up the expression of each the its isoforms. We estimated the expression of the arrays using the RMA algorithm in the aroma.affymetrix framework using a Brainarray reference file of the Ensembl 74 transcriptome.
Event pointer for RNAseq
EventPointer is an R package to identify, classify and analyze alternative splicing events using microarrays and RNA-Seq data. The software is available for download at Bioconductor. A thorough description of EventPointer for microarrays can be found in [8]. This method has been extended to RNA-seq.
The concepts for detection, classification and statistical analysis are shared in EventPointer for the analysis of both technologies. The main difference of EventPointer for RNA-seq compared with that of microarrays are the ones associated with the type of input data (CEL or BAM files). The R code for the analysis is available at https://github.com/jpromeror/SplicingComparison.
EventPointer requires a splicing graph -a directed graph used to represent the structure of the different isoforms of a given gene [39] - as input to detect splicing events. EventPointer for RNA-seq uses SGSeq [24] to build the corresponding splicing graphs from BAM files. The complexity of the splicing graph can be controlled in SGSeq by setting different thresholds in the expression values of splicing junctions of the splicing graph (by default set to 2 FPKM). For RNA-seq, the splicing graphs are constructed for every single experiment. On the contrary, in the case of microarrays the same splicing graph (and the corresponding CDF) is used for all the experiments run on the same type of microarray (HTA or, more recently Clariom-D).
The input data for the statistical analysis is different in both technologies: signal values of the probes in microarrays and counts in RNA-seq. In order to deal with reads, Voom [17] is applied to preprocess the RNA-seq count data. The statistics to deal with the processed RNA-seq data is identical to the one used for microarray data and hence, the same statistical tests -based on limma [33]- are applied to both technologies.
As output, EventPointer provides a table with the following information associated to each detected alternatively spliced event: gene identifier, genomic position, type of event, statistical parameters and ΔΨ values. Additionally, EventPointer generates a "Gene Transfer Format" (GTF) file that can be used with the Integrative Genomics Viewer (IGV) [36] to view the structures of each detected alternative splicing event. This visualization facilitates the interpretation of the detected events and the design of primers for the validation of the events using standard PCR.
Estimation of PSI
We have included a novel algorithm to estimate Ψ that can be applied to both RNA-seq and microarrays. Assuming that the signal of a probe-set in microarrays and the number of reads within a region of the transcriptome in RNA-Seq depend on the product of an affinity value of the probe-set (or the equivalent length in RNA-seq) and the concentration of the interrogated isoforms in the paths, the following equation holds
$$ {S}_i={a}_i\bullet {t}_i $$
where Si is the measured expression value of path i, ai is the affinity of the probes or equivalent length of the path i and ti is the concentration of the isoforms mapped to path i. The affinity values (or equivalent lengths) and concentration values are assumed to be unknown and must be estimated from the data.
Particularizing the above equation to each of the paths and taking into account that the concentration of the isoforms in the reference path must be the sum of those of paths 1 and 2, the following equations are obtained:
$$ {S}_1={a}_1\bullet {t}_1 $$
$$ {S}_R={a}_R\bullet {t}_R={a}_R\left({t}_1+{t}_2\right) $$
In turn, the signal value of the reference path can be expressed as the sum of the signal values of paths 1 and 2 as follows,
$$ {S}_R={a}_R{a}_1^{-1}{S}_1+{a}_R{a}_2^{-1}{S}_2=u{S}_1+v{S}_2 $$
where u and v represent the fraction of the affinities of the mapped probe-set (or equivalent lengths) in the reference path and paths 1 or 2 respectively. The values of u and v can be estimated from signal data.
Dividing eq. (2) with eq. (4) we get,
$$ \frac{S_1}{S_R}=\frac{a_1{t}_1}{a_R\left({t}_1+{t}_2\right)} $$
Combining eqs. (5) and (6), the desired equation of the Percent Spliced Index (Ψ) used in EventPointer is obtained:
$$ \varPsi =\frac{t_1}{t_1+{t}_2}=\frac{u{S}_1}{S_R}=\frac{u{S}_1}{u{S}_1+v{S}_2} $$
Note that Ψ can be directly obtained from signal values once u and v are known. This equation does not require the estimation of the affinities (difficult to predict accurately) to compute Ψ. On the contrary, it simply requires to estimate u and v from signal values using eq. (5). In the case of RNA-seq, the equivalent lengths are known a priori and hence u and v. However, using this approach has an advantage: the estimates of these lengths can accommodate the potential lack of uniformity of the reads.
Note that u and v must be positive, similar between them and close to one. The first affirmation is trivial since affinity values (or equivalent lengths) are always positive. In microarrays, probe-sets are composed by several probes and their overall affinity are expected to be similar to each other, since these affinities are a median of the average of the affinities of the probes that build up them. Therefore a1 ≈ a2 ≈ aR,and u ≈ v ≈ 1. A similar reasoning can be applied to RNA-seq, if using coverage instead of read counts, since the coverage of the reference path is expected to be close to the sum of the coverages of paths 1 and 2.
These two fractions can be estimated from eq. (5) by using non-negative least squares as follows:
$$ {\displaystyle \begin{array}{c}\min \left\Vert Ax-b\left\Vert {}_2\right.\right.\\ {}s.t.x\ge 0,x\in {R}^2,A\in {R}^{m\;x\;n}\end{array}} $$
$$ A=\left[\begin{array}{cc}\boldsymbol{Signal}\ \boldsymbol{P}\mathbf{1}& \boldsymbol{Signal}\ \boldsymbol{P}\mathbf{2}\\ {}\lambda & -\lambda \\ {}\lambda & 0\\ {}0& \lambda \end{array}\right];x=\left[\begin{array}{c}u\\ {}v\end{array}\right];b=\left[\begin{array}{c}\boldsymbol{Signal}\ \boldsymbol{R}\\ {}0\\ {}\lambda \\ {}\lambda \end{array}\right] $$
The penalty factor λ is added to force the equation to fulfill the previous considerations: u and v must be similar and close to 1. In our results, we found that the estimates were not sensitive to the specific value of λ if there is differential alternative splicing. If the relative usage of both paths is similar and therefore, Ψ is constant, the results are more sensitive to the value of λ. This fact is shown in Fig. 4: the correlation is much better for events that show variability in the relative expression of both paths.
The residuals of this model can be used to test if the additive model of eqs. 2, 3 and 4 holds. We computed the relative error of the residuals as follows:
$$ \upvarepsilon =\frac{{\left\Vert \left(u\cdotp \boldsymbol{Signal}\ \boldsymbol{P}\mathbf{1}+v\cdotp \boldsymbol{Signal}\ \boldsymbol{P}\mathbf{2}\right)-\boldsymbol{Signal}\ \boldsymbol{R}\right\Vert}_2}{{\left\Vert \boldsymbol{Signal}\ \boldsymbol{R}\right\Vert}_2} $$
If the relative error is large, the additive model does not fit the data and, therefore, the estimates are expected to be less reliable. In order to test this, we divided the events according to the relative error. The events with top 50% relative error have FDR two times larger than the bottom 50% as shown in Additional file 1: Table S10.
The comparison and analysis of the profiling data was done using a linear model. The design matrix was built considering both the cell line and treatment with CX4945 as factors. The interaction between cell line type and treatment was not considered.
The selected contrasts test for the difference between control samples (DMSO) and drug exposed ones (CX4945) controlling for the cell-type. The complete experimental design in the form of design and contrast matrices is included in Additional file 1: Table S9.
EventPointer includes several statistical methods to state the significance of an event. In this experiment, the events are considered to be statistically significant if there is a change in the expression of the isoforms associated to each of the alternative paths, this change occurs in opposite direction, i.e. opposite signs for the fold changes and the summarized p.value is significant (p value< 0.001).
In order to compare the arrays with different sequencing depths, we subsampled the RNAseq data to 30 and 10 million reads and rerun the whole pipeline with these data. The FDR for 30 million reads was better than using arrays. On the contrary, using 10 million reads the FDR was worse than using arrays. Interpolating both data, the FDR for arrays is similar to a depth of 20 million reads.
Filters used to include the events
For arrays, the signal of the probe-sets interrogating each of the alternative paths involved in a splicing event, must be expressed more than a certain threshold in at least one sample. This threshold is the 25% quantile of the expression of the signal in the reference paths for all the events included in the array. For RNAseq, the edges of the splicing graph (junction reads) are included only if their expression is at least 2 FPKM in at least one sample (SGSeq defaults).
Matching of the events using different technologies
Let's assume that AR and AM are, possibly non-contiguous, regions of the genome that correspond to path A using either technology (AR for RNA-seq and AM for HTA). BR and BM have a similar description for path B and RR and RM for the reference path in each technology. Two events are considered to match if any of the following two expressions is true:
$$ \left(\left({A}_R\subset {A}_M\right)|\left({A}_M\subset {A}_R\right)\right)\&\left(\left({B}_R\subset {B}_M\right)|\left({B}_M\subset {B}_R\right)\right)\&\left(\left({R}_R\cap {R}_M\right)\ne \varnothing \right) $$
$$ \left(\left({A}_R\subset {B}_M\right)|\left({B}_M\subset {A}_R\right)\right)\&\left(\left({B}_R\subset {A}_M\right)|\left({A}_M\subset {B}_R\right)\right)\&\left(\left({R}_R\cap {R}_M\right)\ne \varnothing \right) $$
In these expressions, (x ⊂ y) is true if the genomic region x is a subset of the genomic region y (the nucleotide sequence of x is a substring of the nucleotide sequence in y). Besides, the operators "|" and "&" and the logical OR and AND operations. If (x ⊂ y)|(y ⊂ x), then one of the regions is contained in the other are considered to be "compatible". On the other hand, (x ∩ y) ≠ ∅ means that regions x and y overlap in the genome. Therefore, the first expression is true if both paths AR and AH are compatible, BR and BM are compatible and RRand RM overlap. The second expression is true if path AR and path BM are compatible and also path AM and BR are compatible and, again, and RR and RM overlap.
Within an event, the longer path in the transcriptome is assigned the name "A" and the other the B. The second eq. (11) takes into account that, in some few cases, the name of the paths can be switched in both technologies.
PCR validation
For each splicing event, an end-point PCR was run using primers designed in the exons that flank the event of interest. RNA was retro-transcribed and the PCR was performed an analyzed as previously described [40]. Primers used are shown in Additional file 2.
Average sequencing depth for RNA-seq was approx. 98 million (paired-end, stranded protocol), yielding on average approx. 49 million fragments per sample.
CDF:
Chip Definition File
GTF:
Gene Transfer Format
HTA:
Human Transcriptome Array
NGS:
TNBC:
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The authors are grateful to Francisco J. Planes, Iñigo Apaolaza, Juan Ferrer and Xabier Cendoya for their comments on the preparation of this manuscript.
The work performed and described was funded by Celgene Research SL, part of Celgene Corporation. Author FC was partially supported by a Basque Government predoctoral Grant [PRE_2016_1_0194]. LMM and RP were partially funded by Spanish Ministry of Economy and Innovation and Fondo de Investigación Sanitaria-Fondo Europeo de Desarrollo Regional [PI14/00806, PI16/01821], CIBERONC and AECC Scientific Foundation [GCB14–2170]. AR was partially supported by the Provincial Council of Gipuzkoa through the MINEDRUG project. Celgene Research SL contributed in the design of the study and collection, analysis and interpretation of data and in writing the manuscript. The rest of the funding body had no role in the aforementioned steps.
EventPointer for both RNA-seq and microarrays is available at Bioconductor [https://bioconductor.org/packages/release/bioc/html/EventPointer.html].
All the RNA-seq and microarray data are available in a SuperSeries at Gene Expression Omnibus, accession number GSE104974.
Juan P. Romero and María Ortiz-Estévez contributed equally to this work.
CEIT and Tecnun, University of Navarra, Parque Tecnológico de San Sebastián, Paseo Mikeletegi 48, 20009, San Sebastián, Gipuzkoa, Spain
Juan P. Romero, Ander Muniategui, Fernando Carazo & Angel Rubio
Celgene Institute for Translational Research Europe, Celgene Corporation, Parque Científico y Tecnológico Cartuja 93, Centro de Empresas Pabellón de Italia, Isaac Newton, 4, E-41092, Seville, Spain
María Ortiz-Estévez, Soraya Carrancio, Remco Loos & Matthew W. B. Trotter
Program in Solid Tumors and Biomarkers, CIMA, University of Navarra, Avda. Pío XII, 55, E-31008, Pamplona, Navarra, Spain
Fernando J. de Miguel, Luis M. Montuenga & Rubén Pío
Department of Histology and Pathology, University of Navarra, Campus Universitario, 31009, Pamplona, Navarra, Spain
Luis M. Montuenga
IdiSNA, Navarra Institute for Health Research, Recinto de Complejo Hospitalario de Navarra, Irunlarrea 3, 31008, Pamplona, Navarra, Spain
Luis M. Montuenga & Rubén Pío
Department of Biochemistry and Genetics, University of Navarra, Campus Universitario, 31009, Pamplona, Navarra, Spain
Rubén Pío
CIBERONC, Centro de Investigación Biomédica en Red, Instituto de Salud Carlos III, Calle Monforte de Lemos 3-5, Pabellón 11. Planta 0, 28029, Madrid, Spain
Juan P. Romero
María Ortiz-Estévez
Ander Muniategui
Soraya Carrancio
Fernando J. de Miguel
Fernando Carazo
Remco Loos
Matthew W. B. Trotter
Angel Rubio
Conception and design: JPR, MOE, MT, AR. Development of methodology: JPR, AR, AM. Acquisition of data (provided cell-lines treatment, provided sequencing and hybridization, performed PCR validations, provided facilities, etc.): SC, FJM, RP, MT. Analysis and interpretation of data (e.g., statistical analysis, biostatistics, computational analysis): JPR, AR, RL, MOE. Writing, review, and/or revision of the manuscript: JPR, MOE, AM, LM, RL, RP, AR, MT. Administrative, technical, or material support (i.e., reporting or organizing data, constructing vignettes):FC, JPR, AM, AR. Study supervision RL, AR, MT. All authors read and approved the final manuscript.
Correspondence to Angel Rubio.
Not applicable as cell lines were obtained from commercially available suppliers. Cell lines MDA-MB-231 and MDA-MB-468 were obtained from ATCC (Manassas, VA) with identification numbers HTB-26 and HT-132 respectively and cell line SUM149 was purchased from Asterand plc (Detroit, MI).
A. Rubio, J.P. Romero and F. Carazo are being funded by Affymetrix in an independent project. The authors declare that they have no competing interests.
Vignette on the comparison based on expression analysis. Figure S1 to S9. Experiment design and contrast matrices. Table S10. (PDF 7766 kb)
Excel file with characteristics of the validated events and PCR primers. (XLSX 27 kb)
Compressed file including the GTF files generated for the matched events. (ZIP 15 kb)
Romero, J.P., Ortiz-Estévez, M., Muniategui, A. et al. Comparison of RNA-seq and microarray platforms for splice event detection using a cross-platform algorithm. BMC Genomics 19, 703 (2018). https://doi.org/10.1186/s12864-018-5082-2 | CommonCrawl |
\begin{document}
\begin{abstract} It is know that the Valdivia compact spaces can be characterized by a special family of retractions called $r$-skeleton (see \cite{kubis1}). Also we know that there are compact spaces with $r$-skeletons which are not Valdivia. In this paper, we shall study $r$-squeletons and special families of closed subsets of compact spaces. We prove that if $X$ is a zero-dimensional compact space and $\{r_s:s\in \Gamma\}$ is an $r$-skeleton on $X$ such that $|r_s(X)| \leq \omega$ for all $s\in \Gamma$, then $X$ has a dense subset consisting of isolated points. Also we give conditions to an $r$-skeleton in order that this $r$-skeleton can be extended to an $r$-skeleton on the Alexandroff Duplicate of the base space. The standard definition of a Valdivia compact spaces is via a $\Sigma$-product of a power of the unit interval. Following this fact we introduce the notion of $\pi$-skeleton on a compact space $X$ by embedding $X$ in a suitable power of the unit interval together with a pair $(\mathcal{F},\varphi)$, where $\mathcal{F}$ is family of metric separable subspaces of $X$ and $\varphi$ an $\omega$-monotone function which satisfy certain properties. This new notion generalize the idea of a $\Sigma$-product.
We prove that a compact space admits a retractional-skeleton iff it admits a $\pi$-skeleton. This equivalence allows to give a new proof of the fact that the product of compact spaces with retractional-skeletons admits an retractional-skeleton (see \cite{cuth1}). In \cite{casa1}, the Corson compact spaces are characterized by a special family of closed subsets. Following this direction, we introduce the notion of weak $c$-skeleton which under certain conditions characterizes the Valdivia compact spaces and compact spaces with $r$-skeletons.
\end{abstract}
\title{\scshape\bfseries Families of retractions and families of closed subsets on compact spaces.}
\section{Introduction}
The Valdivia and Corson compact spaces are spaces widely studied in the Abstract Functional Analysis. From the study of Valdivia compact spaces done in \cite{kubis1} surged the concept of {\it $r$-skeleton} as a generalization of the Valdivia compact spaces, this new concept is a family of retractions with certain properties:
\begin{definition}\label{skeleton} Let $X$ be a space. An $r$-{\it skeleton} on $X$ is a family $\{r_s:s\in \Gamma\}$ of retractions in $X$, indexed by an up-directed $\sigma$-complete partially ordered set $\Gamma$, such that \begin{enumerate} \item[$(i)$] $r_s(X)$ is a cosmic space, for each $s\in \Gamma$;
\item[$(ii)$] $r_s=r_s\circ r_t=r_t\circ r_s$ whenever $s\leq t$;
\item[$(iii)$] if $\langle s_n\rangle_{n<\omega}\subseteq \Gamma$ is an increasing sequence and $s=\sup \{s_n : n<\omega\}$, then $r_s(x)=\lim_{n\rightarrow \infty}r_{s_n}(x)$ for each $x\in X$; and
\item[$(iv)$] for each $x\in X$, $x=\lim_{s\in \Gamma}r_s(x)$. \end{enumerate} The {\it induced space} on $X$ defined by $\{r_s: s\in \Gamma\}$ is $\bigcup_{s\in \Gamma}r_s(X)$. If $X=\bigcup_{s\in \Gamma}r_s(X)$, then we will say that $\{r_s:s\in \Gamma\}$ is a \textit{full $r$-skeleton}. Besides, if $r_s\circ r_t= r_t\circ r_s$, for any $s,t\in \Gamma$, then we will say that $\{r_s:s\in \Gamma\}$ is \textit{commutative}. \end{definition}
It is shown in \cite{cuth1} that the Corson compact spaces are those compact spaces which have a full $r$-skeleton, and in the paper \cite{kubis1} it is proved that the Valdivia compact spaces are those compact spaces which have a commutative $r$-skeleton. The ordinal space $[0,\omega_2]$ is a compact space with an $r$-skeleton, but it does not admit neither a commutative nor a full $r$-skeleton (see Example \ref{resqordinal}). Thus, the class of spaces with $r$-skeletons contains properly the Valdivia compact spaces. In the literature, some other authors use the name \textit{retractional skeleton} instead of $r$-skeleton, and the name \textit{non commutative Valdivia compact space} is used to address to a compact space with an $r$-skeleton.
In this article, we shall prove several results on compact spaces with an $r$-skeletons as we describe in the next lines:
$\bullet$ In the third section, we show that if $X$ is a zero-dimensional compact space with an $r$-skeleton
$\{r_s:s\in \Gamma\}$ such that $|r_s(X)| \leq \omega$ for all $s\in \Gamma$, then $X$ has a dense subset consisting of isolated points.
$\bullet$ An interesting research topic has been the study of the connections between the topological properties of a space $X$ and its Alexandroff duplicate (denoted by $AD(X)$). Indeed, in the paper \cite{soma1} the author proved that if $X$ be a compact space and $AD(X)$ has an $r$-skeleton, then $X$ has an $r$-skeleton. With respect to the reciprocal implication of this assertion, in the papers \cite{reynaldo1} and \cite{kalenda2}, it is shown that if $X$ is a compact space with full $r$-skeleton, then $AD(X)$ has a full $r$-skeleton. Following this direction, one can wonder when an $r$-skeleton on a space can be extended to an $r$-skeleton on its Alexandroff duplicate. One particular case was is due to J. Somaglia in \cite{soma1} who showed that if $X$ is a compact space with an $r$-skeleton and $Y$ the induced space such that $X\setminus Y$ is finite, then $AD(X)$ admits an $r$-skeleton.
This result of Somaglia motivated us to give a necessary and sufficient conditions on the $r$-skeleton of a space $X$, in order that its Alexandroff duplicated $AD(X)$ admits an $r$-skeleton. All these results concerning the Alexandroff duplicate will be described in detail in the fourth section.
$\bullet$ We recall that the Corson compact spaces are the compact spaces which are contained in a $\Sigma$-product, and the Valdivia compact spaces are the compact spaces whose intersection with a $\Sigma$-product is dense in the compact space. Since these spaces can be also defined by $r$-skeletons, it is natural to investigate about a generalization of a $\Sigma$-product in order to obtain a characterization of the compact spaces which admit an $r$-skeleton. This analysis is carried out in the fifth section, in fact we introduce the notion of {\it $\pi$-skeleton} to generalize the $\Sigma$-products and we prove that a compact space has a $\pi$-skeleton iff it has an $r$-skeleton.
$\bullet$ The notion of a full $c$-skeleton was introduced in the article \cite{casa1} in which the authors proved that the Corson compact spaces are the spaces which admits a full $c$-skeleton. Following this direction, in the sixth section, we weaken the definition of $c$-skeleton to obtain the notion of a weak $c$-skeleton. This concept plus some additional conditions provide characterizations of both the Valdivia compact spaces and the compact spaces with an $r$-skeleton.
\section{Preliminaries}
Our spaces will be Tychonoff (completely regular and Hausdorff). The Greek letter $\omega$ will stand for the first infinite cardinal number. Given an infinite set $X$, the symbol $[X]^{\leq \omega}$ will denote the set of all countable subsets of $X$ and the meaning of $[X]^{<\omega}$ should be clear. A partially ordered set $\Gamma$ is called {\it up-directed} whenever for every $s,s'\in \Gamma$, there is $t\in \Gamma$ such that $s\leq t$ and $s'\leq t$. And $\Gamma$ is named $\sigma$-{\it complete} if $\sup_{n<\omega}s_n\in \Gamma$, for each increasing sequence $ \langle s_n\rangle_{n<\omega} \subseteq \Gamma$. For a space $X$, the closure of $A \subseteq X$ will be denoted either by $\overline{A}$ or $cl_X(A)$. If $\alpha$ is an infinite ordinal number, $[0,\alpha]$ will denote the usual ordinal space. For a product $\Pi_{t\in T}X_t$, if $t\in T$ and $A\in [T]^{\leq \omega}$, then $\pi_t$ and $\pi_A$ will be denote the projection on the coordinate $t$ and the projection on the set $A$, respectively. A subbasic open of a product of spaces $\Pi_{t\in T}X_t$ will be simply denoted by $[t,V] := \pi_t^{-1}(V)$, where $t\in T$ and $V$ is an open subset of $X_t$. A space is called {\it cosmic} if it has a countable network. It is known that a compact space is cosmic iff it is metrizable and separable.
The next property is useful to prove that a family of retractions is an $r$-skeleton without establishing explicitly the condition $(iv)$ of Definition \ref{skeleton}.
\begin{proposition}[\cite{casa1}]\label{rey2} Let $\{r_s:s\in \Gamma\}$ be a family of retractions on a countably compact space $X$ satisfying $(i)$-$(iii)$ of Definition \ref{skeleton}. If $Y=\bigcup_{s\in\Gamma}r_s(X)$, then $x=\lim_{s\in \Gamma}r_s(x)$, for each $x\in \overline{Y}$. \end{proposition}
The next example is a compact space with an $r$-skeleton which is not Valdivia (other non-trivial examples of non-Valdivia compact spaces which admits $r$-skeletons are given in \cite{soma2}).
\begin{example}[\cite{kubis1}]\label{resqordinal} Let $\alpha>\omega_1$ and $\mathbf{C}_\alpha$ the family of all closed countable sets $A\subseteq [0,\alpha]$ such that $0\in A$ and if $p\in A$ is an isolated point in $A$, then $p$ is an isolated point in $[0,\alpha]$. For each $A\in \mathbf{C}_\alpha$ we define $r_A:[0,\alpha]\rightarrow [0,\alpha]$ by $r_A(x)=\max\{a\in A: a\leq x\}$. The family $\{r_A:A\in \mathbf{C}_\alpha\}$ is a non commutative $r$-skeleton on $[0,\alpha]$. \end{example}
The $r$-skeletons have been very important in the study of certain topological properties of compact spaces (see, for instance, \cite{cuth2}). Indeed, in the next theorem, we will highlight the basic properties of the induced subspace that are used in next sections.
\begin{theorem}[\cite{kubis}] Let $X$ be a compact space with an $r$-skeleton and $Y\subseteq X$ be the induced space by the $r$-skeleton. Then: \begin{itemize} \item $Y$ is countably closed in $X$, i. e. for any $B\in [Y]^{\leq \omega}$ we have $\overline{B}\subseteq Y$, \item $Y$ is a Frechet-Urysohn space and \item $\beta Y = X$. \end{itemize} \end{theorem}
An $r$-skeleton of a space can be inherited to a closed subspace:
\begin{theorem}[\cite{cuth1}]\label{teocuth1} Let $X$ be a compact space with $r$-skeleton $\{r_s:s\in \Gamma\}$, $Y\subseteq X$ be the induced space by the $r$-skeleton and $F$ be a closed subset of $X$. If $Y\cap F$ is dense in $F$, then $F$ admits an $r$-skeleton. \end{theorem}
In the proof of Theorem \ref{teocuth1}, the author described a technique to inherit an $r$-skeleton on a compact space to a closed subspace. The following remark enounce this technique.
\begin{remark}\label{obscuth} Let $X$ be a compact space with $r$-skeleton $\{r_s:s\in \Gamma\}$, $Y$ the induced space by the $r$-skeleton and $F$ be a closed subset of $X$. If $Y\cap F$ is dense in $F$, then the set $\Gamma'=\{s\in \Gamma:r_s(F)\subseteq F\}$ is $\sigma$-complete and cofinal in $\Gamma$, and the family $\{r_s\upharpoonright_{F}:s\in \Gamma'\}$ is an $r$-skeleton on $F$. \end{remark}
The next notion was a very useful tool in the study of Corson compact spaces in the paper \cite{casa1}:
Given two up-directed $\sigma$-complete partially ordered sets $\Gamma$ and $\Gamma'$, a function $\psi:\Gamma\rightarrow \Gamma'$ is called {\it $\omega$-monotone} provided that: \begin{itemize} \item $\psi(s)\leq \psi(t)$ whenever $s\leq t$; and \item if $\langle s_n\rangle_{ n<\omega}\subseteq \Gamma$ is an increasing sequence, then $\psi(\sup_{n<\omega}s_n)=\sup_{n<\omega}\psi(s_n)$. \end{itemize}
The $\omega$-monotone functions allows us to change the set of indices $\Gamma$ of a full $r$-skeleton to the set of countable subsets of the induced space (see \cite{reynaldo1}):
\begin{lemma}[\cite{reynaldo1}]\label{rey1} Let $X$ be a set and $\Gamma$ be an up-directed partially ordered set. If for $x\in X$ there is an assignment $s_x\in \Gamma$, then there exists an $\omega$-monotone function $\psi:[X]^{\leq \omega}\rightarrow \Gamma$ such that $s_x\leq \psi(x)$, for every $x\in X$. \end{lemma}
We will use the replacement of $\Gamma$ by $[X]^{\leq \omega}$ in the proof of some results of this article.
In \cite{casa1}, the authors use a special family of closed subsets of a Corson compact spaces to define the notion of a $c$-skeleton. Thus, they obtained a characterization of a Corson compact spaces in terms of a $c$-skeleton. Let us state next this notion.
\begin{definition}[\cite{casa1}]\label{cesqueletodef} Let $(X,\tau)$ be a space and $\{(F_s,\mathcal{B}_s):s\in \Gamma\}$ be a family of pairs such that $\{F_s:s\in \Gamma\}$ is a family of closed subsets of $X$ and $\{\mathcal{B}_s:s\in \Gamma\}\subseteq [\tau]^{\leq\omega}$. We say that $\{(F_s,\mathcal{B}_s):s\in \Gamma\}$ is a \textit{$c$-skeleton} if the following conditions hold: \begin{itemize} \item $F_s\subseteq F_t$, whenever $s\leq t$; \item for each $s\in \Gamma$, $\mathcal{B}_s$ is a base for a topology denoted by $\tau_s$ on $X$ and there exist a Tychonoff space $Z_s$ and continuous map $g_s:(X,\tau_s)\rightarrow Z_s$ which separates the points of $F_s$, and \item the assignment $s\rightarrow \mathcal{B}_s$ is $\omega$-monotone. \end{itemize} In addition, if $X=\bigcup_{s\in \Gamma}F_s$, the we say that the $c$-skeleton is \textit{full}. \end{definition}
One of the main results of \cite{casa1} asserts that a compact space is Corson iff it admits a full $c$-skeleton. The next compact non-Corson space admits a $c$-skeleton.
\begin{example}\label{cesqejemplo} The ordered space
$[0,\omega_1]$ admits a $c$-skeleton. Indeed, let $\Gamma=\omega_1$ and $L_{\omega_1}:=\{\alpha<\omega_1: \alpha \mbox{ is not isolated point}\}$. For each $\alpha<\omega_1$, define $$\mathcal{B}_\alpha:=\{\{\gamma\}:\gamma\leq \alpha \mbox{ and} \gamma\notin L_{\omega_1}\}\cup\{(\gamma,\beta+1):\gamma<\beta\leq \alpha \mbox{ and } \beta\in L_{\omega_1} \}\cup\{(\gamma,\omega_1]: \gamma<\alpha\}.$$ To see that the family $\{([0,\alpha], \mathcal{B}_\alpha):\alpha<\omega_1\}$ is a $c$-skeleton on $[0,\omega_1]$ is enough to define the functions $g_\alpha$'s:
For each $\alpha<\omega_1$, let $Z_\alpha=[0,\alpha]$ and consider the map $g_\alpha:([0,\omega_1],\mathcal{B}_\alpha)\rightarrow Z_\alpha$ defined by $g_\alpha(\beta):=$
\[ g_\alpha(\beta) := \begin{cases} \beta & \textrm{ if $\beta<\alpha$},\\ \alpha & \textrm{ otherwise}, \\ \end{cases} \] for each $\beta\leq\omega_1$. Observe that the function $g_\alpha$ is continuous and one-to-one map, and that the assignment $\alpha\rightarrow \mathcal{B}_\alpha$ is $\omega$-monotone. \end{example}
\section{$r$-skeletons on zero-dimensional spaces}
It is very natural to ask when an $r$-skeleton $\{r_s:s\in \Gamma\}$ satisfies that $|r_s(X)| \leq \omega$ for each $s\in \Gamma$. In the next results, we give a necessary condition to have this property. In this section, we will use usual terminology of trees. By $\{0,1\}$ we denote the discrete space with two elements. We will denote the Cantor space by $\mathcal{C}$, $\{0,1\}^{<\omega}$ is the set of all finite sequences of $\{0,1\}$; $\sigma\hat{\hspace{0.1cm}}\lambda$ will be denote the usual concatenation for $\sigma, \lambda\in \{0,1\}^{<\omega}$, and an initial segment of $p\in \mathcal{C}$ of length $n$ will denoted by $p\mid n$. For terminology not mentioned and used in the next results the reader may consult \cite{kech}.
\begin{lemma}\label{lema2}
Let $X$ be a zero-dimensional compact space without isolated points. If $\{r_s:s\in \Gamma\}$ is an $r$-skeleton on $X$, then there is a Cantor scheme $\{U_\sigma:\sigma\in \{0,1\}^{<\omega}\}$ and $\{x_\sigma:\sigma\in \{0,1\}^{<\omega}\}\subset\bigcup_{s\in\Gamma}r_s(X)$ such that: \begin{enumerate} \item $U_\sigma$ is a clopen set, \item $x_\sigma\in U_\sigma$ and for $i\in \{0,1\}$, $x_{\sigma}\notin U_{\sigma\hat{\hspace{0.1cm}}i}$. \end{enumerate} Moreover, $\bigcap_{n<\omega}U_{p\mid n}\neq\emptyset$, for every $p\in \mathcal{C}$. \end{lemma}
\begin{proof} Let $Y$ the induced space by $\{r_s:s\in\Gamma\}$. We proceed by induction on the length of $\sigma$. Define $U_\emptyset:=X$ and choose a point $x_\emptyset\in Y$. Now, we suppose that $U_\sigma$ and $x_\sigma$ are defined for all $\sigma\in \{0,1\}^{<\omega}$ of length $n$. Take $\sigma\in \{0,1\}^{<\omega}$ and suppose that length of $\sigma$ is $n$. Since $x_\sigma$ is not isolated, then there are $z, z'\in U_\sigma$ not equals to $x_\sigma$. Pick two disjoint clopen subsets $U$ and $V$ which $z\in U$, $z'\in V$, $x_\sigma\notin U$ and $x_\sigma\notin V$. Let be $U_{\sigma\hat{\hspace{0.1cm}}0}:= U_\sigma\cap U$ and $U_{\sigma\hat{\hspace{0.1cm}}1}:= U_\sigma\cap V$. Using the density of $Y$, we can choose $x_{\sigma\hat{\hspace{0.1cm}}0}\in U_{\sigma\hat{\hspace{0.1cm}}0}\cap Y$ and $x_{\sigma\hat{\hspace{0.1cm}}1}\in U_{\sigma\hat{\hspace{0.1cm}}1}\cap Y$. Hence, $\{U_\sigma:\sigma\in \{0,1\}^{<\omega}\}$ and $\{x_\sigma:\sigma\in \{0,1\}^{<\omega}\}$ are as we expected. Now, pick $p\in \mathcal{C}$. By construction, $\{U_{p\mid n}:n< \omega\}$ has the finite intersection property. By compactness, it follows that $\bigcap_{n<\omega}U_{p\mid n}\neq\emptyset$. \end{proof}
\begin{theorem}\label{isol} Let $X$ be a zero-dimensional compact space without isolated points. If $\{r_s:s\in \Gamma\}$ is an $r$-skeleton on $X$, then there is $s\in \Gamma$ such that $r_s(X)$ is not countable. \end{theorem}
\begin{proof} Let $Y$ be the induced space by $\{r_s:s\in\Gamma\}$ and we consider $\{U_\sigma:\sigma\in \{0,1\}^{<\omega}\}$ and $\{x_\sigma:\sigma \in \{0,1\}^{<\omega}\}\subset\bigcup_{s\in\Gamma}r_s(X)$ as in Lemma \ref{lema2}. Put $F=\{x_\sigma:\sigma\in \{0,1\}^{<\omega}\}$. Since $F\in [Y]^{\leq \omega}$ and $Y$ is countably closed, then $\overline{F}\subseteq Y$. We shall prove that $\overline{F}$ is not countable. For $p\in \mathcal{C}$, let $V_p=\bigcap_{n<\omega}U_{p\mid n}$ and $x_p\in \overline{\{x_{p\mid n}:n<\omega\}}$. We observe that $x_p\notin\{x_{p\mid n}:n<\omega\}$. Since $\{x_{p\mid n}:n<\omega\}\in [Y]^{\leq \omega}$ and $Y$ is countably closed, we have $x_p\in Y$ and it is easy to see that $x_p\in V_p$. Finally, if $p,q\in \mathcal{C}$ with $p\neq q$, using that $V_p\cap V_q=\emptyset$ we have $x_p\neq x_q$. Since $\{x_p:p\in \mathcal{C}\}\subseteq \overline{F}$, we conclude that $\overline{F}$ is not countable. Finally, we choose $s\in \Gamma$ such that $F\subseteq r_s(X)$. \end{proof}
\begin{corollary}\label{corisol}
Let $X$ be a zero-dimensional compact space. If $\{r_s:s\in \Gamma\}$ is an $r$-skeleton on $X$ such that $|r_s(X)| \leq \omega$ for all $s\in \Gamma$, then $X$ has a dense subset consisting of isolated points. \end{corollary}
\begin{proof} Let $Y$ be the induced space by $\{r_s:s\in\Gamma\}$ and we suppose that $U\subseteq X$ is an open subset without isolated points. Take a nonempty open subset $W$ of $X$ such that $\overline{W}\subseteq U$. We have that $Y\cap \overline{W}$ is dense in $\overline{W}$. By Theorem \ref{teocuth1}, $\overline{W}$ admits an $r$-skeleton. Moreover, from Remark \ref{obscuth}, the $r$-skeleton is a subfamily of restrictions on $\overline{W}$ of the family $\{r_s:s\in \Gamma\}$. Using Proposition \ref{isol}, we obtain a contradiction. \end{proof}
\begin{example} Let $\alpha$ be a cardinal number with $\alpha \geq \omega_1$. The space $[0,\alpha]$ with the $r$-skeleton given in \ref{resqordinal}, is an example that satisfies the conditions of Corollary \ref{corisol}. Besides, Somaglia proved in \cite{soma1} that the Alexandroff duplicate of $[0,\omega_2]$ also admits such $r$-skeleton. \end{example}
\begin{example}
In the paper \cite{reynaldo1}, the authors proved that the Alexandroff duplicate of a Corson space its again a Corson space, they extended the full $r$-skeleton of a Corson space to a full $r$-skeleton on its Alexandroff Duplicate. From the proof of this result we deduce that there are Corson spaces $X$ whose Alexandroff duplicate contains a dense set of isolated points and admits an $r$-skeleton, but it could fail that $|r_s(AD(X))| \leq \omega$ for every $s\in \Gamma$, this shows that condition of Corollary \ref{corisol} is not sufficient. \end{example}
\begin{question} Which compact spaces with a dense set of isolated points admit an $r$-skeleton whose retraction have countable images? \end{question}
\section{$r$-skeletons and the Alexandroff Duplicate}
In this section, we study some topological properties of the Alexandroff duplicate of a compact space with an $r$-skeleton. In particular, we give conditions to extend an $r$-skeleton on $X$ to an $r$-skeleton on the Alexandroff duplicate $AD(X)$. To have this done we state some preliminary results.
Remember that the \textit{Alexandroff duplicate} of a space $X$, denoted by $AD(X)$, is the space $X \times \{0,1\}$ with the topology in which all points of $X \times \{1\}$ are isolated, and the basic neighborhoods of the points $(x,0)$ are of the form $(U \times \{0,1\}) \setminus \{(x,1)\}$ where $U$ is a neighborhood of $x\in X$. We denote by $X_i$ the subspace $X\times \{i\}$. We denote by $\pi$ the projection from $AD(X)$ onto $X$. Remember that $X$ is homeomorphic to $X_0$ which, in some cases, will be identified with $X$.
The next result appears implicitly in the development of several articles on the subject, here we provide a proof of it. \begin{theorem}\label{numerables} Let $X$ be a compact space, $\{r_s:s\in \Gamma\}$ be an $r$-skeleton on $X$ and $Y$ be the induced space. Then there is an $r$-skeleton $\{R_A: A\in [Y]^{\leq \omega}\}$ on $X$ such that \begin{itemize} \item $A\subseteq R_A(X)$, for all $A\in [Y]^{\leq \omega}$, and \item $Y=\bigcup_{A\in [Y]^{\leq \omega}} R_A(X)$. \end{itemize} \end{theorem} \begin{proof} For each $x\in Y$, we pick $s_x\in \Gamma$ such that $r_{s_x}(x)=x$. We consider the $\omega$-monotone function $\psi:[Y]^{\leq \omega}\rightarrow \Gamma$ given by Lemma \ref{rey1}. Now, for each $A\in [Y]^{\leq \omega}$ we define $R_A:X\rightarrow X$ by $R_A:=r_{\psi(A)}$.
We shall prove that the family $\{R_A:A\in [Y]^{\leq \omega}\}$ is an $r$-skeleton on $X$. Since $\psi$ is $\omega$-monotone, $s_x\leq\psi(x)$, for every $x\in Y$, and $\{r_s:s\in \Gamma\}$ is an $r$-skeleton. Trivially, we have that $\{R_A:A\in [Y]^{\leq \omega}\}$ satisfies $(i)-(iii)$ of the $r$-skeleton definition.
Finally, by Lemma \ref{rey2}, we have that $x=\lim_{A\in [Y]^{\leq \omega}}R_A(x)$ for all $x\in \overline{\bigcup_{A\in [Y]^{\leq \omega} }R_A(X)}$. By the choice of the family $\{s_x:x\in Y\}$, we obtain that $A\subseteq R_A(X)$, for all $A\in [Y]^{\leq \omega}$. It then follows that $\bigcup_{A\in [Y]^{\leq \omega}}R_A(X)=Y$. Therefore, $\{R_A: A\in [Y]^{\leq \omega}\}$ is an $r$-skeleton on $X$. \end{proof}
The next Lemma is a necessary condition when the Alexandroff duplicate admits an $r$-skeleton. \begin{lemma}\label{teo1} Let $X$ be a compact space, suppose that $AD(X)$ has an $r$-skeleton with induced space $\hat{Y}$ and $Y=\pi(\hat{Y}\cap X_0)$. Then for any $B\in [X\setminus Y]^{\leq \omega}$ we have that \begin{itemize} \item[$(*)$] $cl_X(B)\setminus B\subseteq Y$ and \item[$(**)$] $cl_X(B)\setminus B$ is a cosmic space. \end{itemize} \end{lemma} \begin{proof}
By Theorem \ref{teo1}, we may consider an $r$-skeleton on $AD(X)$ of the form $\{r_C:C\in [\hat{Y}]^{\leq \omega}\}$. Let $B\in [X\setminus Y]^{\leq \omega}$. If $B$ is a finite set, then the result follows immediately. Let us suppose that $|B|=\omega$ and $p\in cl_X(B)\setminus B$. Observe that $(p,0)\in cl_{AD(X)}(B\times\{1\})$. Since $B\times\{1\}\subseteq\hat{Y}$ and $\hat{Y}$ is countably closed, it follows that $(p,0)\in \hat{Y}$ and so $p\in Y$. Hence, $cl_X(B)\setminus B\subseteq Y$ and then we deduce that $B$ is discrete in $X\setminus Y$. Now, we note that $(cl_X(B)\setminus B)\times \{0\}\subseteq cl_{AD(X)}(B\times\{1\})$. Since $B\times \{1\}\in [\hat{Y}]^{\leq \omega}$, we have that $cl_{AD(X)}(B\times\{1\})\subseteq r_{B\times \{1\}}(X)$. Since $r_{B\times \{1\}}(X)$ is cosmic, we conclude that $cl_X(B)\setminus B\subseteq Y$ is also cosmic. \end{proof}
The next lemma is the key to extend a retraction from the base space to its Alexandroff duplicated.
\begin{lemma}\label{lemma51} Let $X$ be a compact space which admits an $r$-skeleton $\{r_s:s\in \Gamma\}$ with induced space $Y$. Suppose that for $B\in [X\setminus Y]^{\leq \omega}$ the conditions $(*)$ and $(**)$ from above hold. Then for each $s\in \Gamma$ such that $cl_X(B)\setminus B\subseteq r_s(X)$ and for every $A\in [r_s(X)]^{\leq\omega}$, the mapping $R_{(A,B,s)}:AD(X)\rightarrow AD(X)$ defined as \[ R_{(A,B,s)}(x,i) := \begin{cases} (x,1) & \textrm{ if $x \in A\cup B$ and $i=1$},\\ (r_{s}(x),0) & \textrm{ otherwise}, \\ \end{cases} \]
for every $(x,i)\in AD(X)$, is a retraction on $AD(X)$. \end{lemma}
\begin{proof} Let $A\in [r_s(X)]^{\leq\omega}$ and assume that $cl_X(B)\setminus B\subseteq r_s(X)$. First, we prove that $R_{(A,B,s)}$ is a continuous function. Let $\langle(x_\lambda,i_\lambda) \rangle_{\lambda\in \Lambda} $ be a net such that $(x_\lambda,i_\lambda)\rightarrow (x,i)$. If $i=1$, then $\langle (x_\lambda,i_\lambda) \rangle_{\lambda\in \Lambda} $ is eventually constant and hence $\langle R_{(A,B)}(x_\lambda,i_\lambda)\rangle_{\lambda\in \Lambda} $ is so. Now, we consider the case when $i=0$ and assume that $\langle(x_\lambda,i_\lambda) \rangle_{\lambda\in \Lambda} $ is not trivial. Hence, $R_{(A,B,s)}(x,0)=(r_s(x),0)$. We may suppose that either $\langle (x_\lambda,i_\lambda) \rangle_{\lambda\in \Lambda}\subseteq X_0$ or $\langle (x_\lambda,i_\lambda) \rangle_{\lambda\in \Lambda}\subseteq X_1$. First, we consider the case when $\langle (x_\lambda,i_\lambda) \rangle_{\lambda\in \Lambda}\subseteq X_0$, we have that $R_{(A,B,s)}(x_\lambda,i_\lambda)=(r_s(x_\lambda),0)$ for all $\lambda \in \Lambda$, and since $r_s$ continuous, we obtain that $ R_{(A,B,s)}(x_\lambda,i_\lambda)\rightarrow R_{(A,B,s)}(x,0)$. Now, assume that $\langle (x_\lambda,i_\lambda) \rangle_{\lambda\in \Lambda}\subseteq X_1$. If $\langle x_\lambda \rangle_{\lambda\in \Lambda}$ contains a subnet that lies eventually in $A\cup B$, then $x\in cl_X(A\cup B)$. Without loss of generality, we suppose that $x_\lambda \in A \cup B$ for every $\lambda \in \Lambda$. Then, $R_{(A,B,s)}(x_\lambda,i_\lambda)=(x_\lambda,1)$ for all $\lambda\in \Lambda$. If $x\in B$, then we deduce from $(*)$ that $\langle x_\lambda\rangle_{\lambda \in \Lambda}$ is eventually in $A$. Since $A\subseteq r_s(X)$, we have that $x\in r_s(X)$. As $Y\cap B=\emptyset$, $x\in cl_X(A)\cup \big( cl_X(B)\setminus B\big)$. From $A\cup \big( cl_X(B)\setminus B\big) \subseteq r_s(X)$ we deduce that $r_s(x)=x$ and we conclude that $R_{(A,B,s)}(x_\lambda,i_\lambda)\rightarrow R_{(A,B,s)}(x,0)$. In other hand, $\langle x_\lambda \rangle_{\lambda\in \Lambda}\subseteq X\setminus \big(A\cup B\big)$ and $R_{(A,B,s)}(x_\lambda,i_\lambda) = (r_{s}(x_\lambda),0)$. Using the continuity of $r_{s}$, we have that $R_{(A,B,s)} (x_\lambda,i_\lambda)=(r_s(x_\lambda),0)\rightarrow (r_s(x),0)=R_{(A,B,s)} (x,0)$. Therefore, $R_{(A,B,s)}$ is a continuous function. Finally, if $(x,i)\in AD(X)$, then \begin{align*} R_{(A,B,s)}\circ R_{(A,B,s)}(x,i) &= \begin{cases} R_{(A,B,s)}(x,1) & \textrm{ if $x \in A\cup B$ and $i=1$},\\ R_{(A,B,s)}(r_{s}(x),0) & \textrm{ otherwise.} \\ \end{cases}\\ &= \begin{cases} (x,1) & \textrm{ if $x \in A\cup B$ and $i=1$},\\ (r_{s}(r_{s}(x)),0) & \textrm{ otherweise.} \\ \end{cases}\\ &= \begin{cases} (x,1) & \textrm{ if $x \in A\cup B$ and $i=1$},\\ (r_{s}(x),0) & \textrm{ otherwise.} \\ \end{cases}\\ &=R_{(A,B,s)}(x,i). \end{align*} That is, $R_{(A,B,s)}$ is a retract on $AD(X)$. \end{proof}
The main theorem of this article is the following.
In what follows, we will consider $\sigma$-complete up-directed partially ordered sets $\Gamma$, where $\Gamma$ will be a subset of $[Y]^{\leq \omega}\times[X\setminus Y]^{\leq \omega}$ with the order $\preceq$ defined by $(A,B)\preceq (A',B')$ if $A\subseteq A'$ and $B\subseteq B'$. Also, we remark that in this partially order set $\Gamma$ if $\langle(A_n,B_n)\rangle_{ n<\omega}\subseteq \Gamma$, then $\sup_\Gamma\langle(A_n,B_n)\rangle_{ n<\omega}$ is not necessarily $(\bigcup_{n<\omega}A_n,\bigcup_{n<\omega}B_n)$.
\begin{theorem}\label{principal} Let $X$ be a compact space. $AD(X)$ admits an $r$-skeleton if and only if there is an $r$-skeleton $\{r_{(A,B)}:(A,B)\in \Gamma\}$ on $X$ with induced space $Y$ such that $\Gamma\subseteq [Y]^{\leq \omega}\times [X\setminus Y]^{\leq \omega}$ is $\sigma$-complete and cofinal in $[Y]^{\leq \omega}\times [X\setminus Y]^{\leq \omega}$, and the next conditions hold:
\noindent For every $B\in [X\setminus Y]^{\leq \omega}$, \begin{itemize} \item[$(*)$] $cl_X(B)\setminus B\subseteq Y$, \item[$(**)$] $cl_X(B)\setminus B$ is cosmic; \end{itemize}
and \begin{itemize} \item[$(***)$] $A\subseteq r_{(A,B)}(X)$ and $cl_X(B)\setminus B\subseteq r_{(A,B)}(X)$ for every $(A,B)\in \Gamma$. \end{itemize} \end{theorem} \begin{proof} Necessity. Suppose that $AD(X)$ admits an $r$-skeleton with induced space $\hat{Y}$. Observe that $X_1\subseteq \hat{Y}$. According to Theorem \ref{numerables}, we may assume that such $r$-skeleton is of the form $\{R_C:C\in [\hat{Y}]^{\leq \omega}\}$ and satisfies the properties of the theorem. Consider the set $Y=\pi(\hat{Y}\cap X_0)$ and put $\Gamma'=[Y]^{\leq \omega}\times [X\setminus Y]^{\leq \omega}$. For $(A,B)\in \Gamma'$, we define $R_{(A,B)}=R_{A\times \{0,1\}\cup B\times\{1\}}$. We claim that $\{R_{(A,B)}:(A,B)\in \Gamma'\}$ is an $r$-skeleton on $AD(X)$ with induced space $\hat{Y}$. The conditions $(i), (ii)$ and $(iii)$ hold because $\{R_C:C\in [\hat{Y}]^{\leq \omega}\}$ is an $r$-skeleton. For the condition $(iv)$, let $(x,i)\in \hat{Y}$ and choose $(A,B)\in \Gamma'$ so that $(x,i)\in A\times \{0,1\}\cup B\times\{1\}$. It follows that $(x,i)=R_{A\times \{0,1\}\cup B\times\{1\}}(x,i)=R_{(A,B)}(x,i)$. Thus, we have proved that $\{R_{(A,B)}:(A,B)\in \Gamma'\}$ is an $r$-skeleton on $AD(X)$ with induced space $\hat{Y}$. By Remark \ref{obscuth},
the set $\Gamma=\{(A,B)\in \Gamma':R_{(A,B)}(X_0)\subseteq X_0\}$ is $\sigma$-complete and cofinal in $\Gamma'$ and $\{R_{(A,B)}\upharpoonright_{X_0}:(A,B)\in \Gamma\}$ is an $r$-skeleton on $X_0$, with induced space $\hat{Y}\cap X_0$. For each $(A,B)\in \Gamma$, we define $r_{(A,B)}=\pi(R_{(A,B)}\upharpoonright_{X_0})$. Hence, $\{r_{(A,B)}:(A,B)\in \Gamma\}$ is an $r$-skeleton on $X$ with induced space $Y$. By Lemma \ref{teo1}, the conditions $(*)$ and $(**)$ hold. The condition $(***)$ is easy to verify.\\
Sufficiency. Now, let $\{r_{(A,B)}:(A,B)\in \Gamma\}$ an $r$-skeleton on $X$ which satisfies the condition $(*)-(***)$, where $\Gamma\subseteq [Y]^{\leq \omega}\times [X\setminus Y]^{\leq \omega}$ is $\sigma$-complete and cofinal in $[Y]^{\leq \omega}\times [X\setminus Y]^{\leq \omega}$ and $Y$ is the induced space. For each $(A,B)\in \Gamma$, we know that $cl_X(B)\setminus B\subseteq r_{(A,B)}(X)$ and $A\subseteq r_{(A,B)}(X)$. Set $R_{(A,B)}=R_{(A,B,(A,B))}$, where $R_{(A,B,(A,B))}$ is the retraction of Lemma \ref{lemma51}.
We claim that $\{R_{(A,B)}:(A,B)\in \Gamma\}$ is an $r$-skeleton on $AD(X)$. Indeed, we shall prove that the conditions $(i)-(iv)$ of the $r$-skeleton definition hold. \begin{enumerate} \item[$(i)$] If $(A,B)\in \Gamma$, then $R_{(A,B)}(AD(X))=(r_{(A,B)}(X)\times \{0\})\cup ((A\cup B) \times \{1\})$ is a cosmic space. \item[$(ii)$] Let $(A,B)\preceq (A',B')$. Fix $(x,i)\in AD(X)$. Then \begin{align*} R_{(A,B)}\circ R_{(A',B')}(x,i)&= \begin{cases} R_{(A,B)}(x,1) & \mbox{ if }x \in A'\cup B'\mbox{ and }i=1,\\ R_{(A,B)}(r_{(A',B')}(x),0) & \mbox{ otherwise.} \\ \end{cases} \\ & = \begin{cases} (x,1) & \mbox{ if }x \in A\cup B\mbox{ and }i=1,\\ (r_{(A,B)}(x),0) & \mbox{ if }x \in (A'\cup B')\setminus (A\cup B) \mbox{ and }i=1,\\ (r_{(A,B)}(r_{(A',B')}(x)),0) & \mbox{ otherwise.} \\ \end{cases} \\ &= \begin{cases} (x,1) & \mbox{ if }x \in A\cup B\mbox{ and }i=1,\\ (r_{(A,B)}(x),0) & \mbox{ otherwise.} \\ \end{cases} \\ &= R_{(A,B)}(x,i). \end{align*}
And we also have that \begin{align*} R_{(A',B')}\circ R_{(A,B)}(x,i)&= \begin{cases} R_{(A',B')}(x,1) & \mbox{ if }x \in A\cup B\mbox{ and }i=1,\\ R_{(A',B')}(r_{(A,B)}(x),0) & \mbox{ otherwise.} \\ \end{cases} \\ &= \begin{cases} (x,1) & \mbox{ if }x \in A\cup B\mbox{ and }i=1,\\ (r_{(A',B')}(r_{(A,B)}(x)),0) & \mbox{ otherwise.} \\ \end{cases} \\ &= \begin{cases} (x,1) & \mbox{ if }x \in A\cup B\mbox{ and }i=1,\\ (r_{(A,B)}(x),0) & \mbox{ otherwise.} \\ \end{cases} \\ &= R_{(A,B)}(x,i). \end{align*}
Therefore, $R_{(A,B)}=R_{(A,B)}\circ R_{(A',B')}=R_{(A',B')}\circ R_{(A,B)}$ whenever $(A,B)\preceq (A',B')$.
\item[$(iii)$] Let $\langle(A_n,B_n)\rangle_{ n<\omega}\subseteq \Gamma$ be such that $(A_n,B_n)\preceq(A_{n+1},B_{n+1})$. $cl_X(B)\setminus B\subseteq Y$. For simplicity, put $(A,B)=\sup\{ (A_n,B_n): n<\omega \}$. Fix $(x,i)\in AD(X)$. We will prove that $R_{(A,B)}(x,i)=\lim_{n\rightarrow \infty}R_{(A_n,B_n)}(x,i)$. In fact, if $i=0$, then $R_{(A,B)}(x,0)=(r_{(A,B)}(x),0)$ and $R_{(A_n,B_n)}(x,0)=(r_{(A_n,B_n)}(x),0)$ for all $n < \omega$. Since $r_{(A,B)}(x)=\lim_{n\rightarrow\infty}r_{(A_n,B_n)}(x)$, we conclude that $R_{(A,B)}(x,0)= \lim_{n\rightarrow \infty}R_{(A_n,B_n)}(x,0)$. Now, we consider the case when $i=1$. If $x\in A\cup B$, then there is $n_0<\omega$ such that $x\in A_n\cup B_n$ for all $n\geq n_0$.
Hence, $R_{(A,B)}(x,1)=(x,1)=R_{(A_n,B_n)}(x,1)$, for every $n\geq n_0$. It then follows that $R_{(A,B)}(x,1)= \lim_{n\rightarrow \infty}R_{(A_n,B_n)}(x,1)$.
If $x\notin A\cup B$, then $R_{(A,B)}(x,1)= (r_{(A,B)}(x),0)$ and $R_{(A_n,B_n)}(x,1)=(r_{(A_n,B_n)}(x),0)$, for every $n < \omega$.
Since the equality $r_{(A,B)}(x)=\lim_{n\rightarrow\infty}r_{(A_n,B_n)}(x)$ holds, $R_{(A,B)}(x,1)= \lim_{n\rightarrow \infty}R_{(A_n,B_n)}(x,1).$
\item[$(iv)$] Let $(x,i)\in AD(X)$. First, if $i=0$, then we notice that the equality $x=\lim_{(A,B)\in \Gamma}r_{(A,B)}(x)$ implies that $(x,0)=\lim_{(A,B)\in \Gamma}R_{(A,B)}(x,0)$. Now, we su\-ppose $i=1$ and let $x\in Y$. By cofinality of $\Gamma$, there is $(A,B)\in \Gamma$ such that $\{x\}\subseteq A$. Since $x\in A\cup B$, we have $(x,1)=R_{(A,B)}(x,1)$. Now, let $x\in X\setminus Y$. By using the confinality of $\Gamma$, there is $(A,B)\in \Gamma$ such that $\{x\}\subseteq B$. It follows that $(x,1)=R_{(A,B)}(x,1)$. Therefore, $(x,i)=\lim_{(A,B)\in \Gamma}R_{(A,B)}(x,i)$, for each $i\in \{0,1\}$. \end{enumerate}
\end{proof}
From the proof of the previous theorem, we deduce the next corollary. \begin{corollary}\label{principalvaldivia}
Let $X$ be a compact space such that $AD(X)$ admits a commutative $r$-skeleton. Then the $r$-skeleton $\{r_{(A,B)}:(A,B)\in \Gamma\}$ on $X$ obtained in the Theorem \ref{principal} is commutative. \end{corollary}
It is well known (see \cite{kalenda2}) that there are Valdivia compact spaces whose their Alexandroff duplicates are not Valdivia compact. If the $r$-skeleton $\{r_{(A,B)}:(A,B)\in \Gamma\}$ on $X$ given in Theorem \ref{principal} is commutative, we do not know whether it can be extended to a commutative $r$-skeleton on $AD(X)$.
If we have a commutative $r$-skeleton of the form $\{r_{(A,B)}:(A,B)\in \Gamma\}$ on $X$, the conditions given in the Theorem \ref{principal} are not clear for extend to a commutative $r$-skeleton on $AD(X)$. In the next result, we add one more condition in order that we can extend commutative $r$-skeletons.
\begin{corollary} Let $X$ be a compact space and $\{r_{(A,B)}:(A,B)\in \Gamma\}$ a commutative $r$-skeleton on $X$ as in the Theorem \ref{principal} which satisfies \begin{itemize} \item[$(****)$] for every $(A,B),(A',B')\in \Gamma$, $r_{(A,B)}(x)=r_{(A',B')}(r_{(A,B)}(x))$, for each $x\in B'\setminus B$. \end{itemize} Then $AD(X)$ admits a commutative $r$-skeleton. \end{corollary}
\begin{corollary} Let $X$ be a compact space. If $AD(X)$ admits an $r$-skeleton, then the induced space $Y=\pi(\hat{Y}\cap X_0)$ of $X$ is unique. That is, if $Y'$ is an induced space by an arbitrary $r$-skeleton on $X$, then $Y'=Y$. \end{corollary}
\begin{proof} Let $Y'$ be a subset of $X$ induced by an $r$-skeleton on $X$. We suppose that $Y\neq Y'$. By Lemma 3.2 from \cite{cuth1}, we have that $Y\cap Y'$ cannot be dense in $X$. Hence, there is a nonempty open subset $V$ of $X$ such that $V\cap (Y\cap Y')=\emptyset$. Let $W$ a nonempty open subset such that $cl_X(W)\subseteq V$. By density of $Y'$, there is an infinite countable set $B\subseteq W \cap Y'$ . Since $Y'$ is countably closed, we must have $cl_{X}(B)\subseteq cl_X(W)\cap Y'\subseteq V\cap Y'$. On the other hand, Theorem \ref{principal} implies that $cl_{X}(B)\setminus B\subseteq Y$, but this is impossible since $V\cap (Y\cap Y')=\emptyset$. Therefore, $Y=Y'$. \end{proof}
As a consequence of the previous corollary, if $AD(X)$ is not Corson compact space and has an $r$-skeleton, then $X$ is not a super Valdivia\footnote{We say that a compact space $X$ is {\it super Valdivia} if for every $x\in X$ there is a dense $\Sigma$-subset $Y$ of $X$ such that $x\in Y$ (see \cite{kalenda1}).} space. In particular, Alexandroff duplicate of $[0,1]^{\kappa}$ does not admit an $r$-skeleton, for every $\kappa\geq \omega_1$.
In terms of $\omega$-monotone functions, we have the next result.
\begin{proposition}\label{novo2} Let $X$ be a compact space which admits an $r$-skeleton with induced space $Y$. Let us suppose that $\{r_A:A\in [Y]^{\leq \omega}\}$ is the $r$-skeleton obtained by Theorem \ref{numerables}. If the conditions $(*)$ and $(**)$ from above hold, and there is $\psi:[X\setminus Y]^{\leq\omega}\rightarrow [Y]^{\leq \omega}$ $\omega$-monotone such that for $B\in [X\setminus Y]^{\leq \omega}$, $cl_X(B)\setminus B \subseteq r_{\psi(B)}(X)$. Then $AD(X)$ has an $r$-skeleton. \end{proposition}
\begin{proof} Let $\Gamma'=[Y]^{\leq \omega}\times[X\setminus Y]^{\leq \omega}$. For each $(A,B)\in \Gamma'$, let $r_{(A,B)}:X\rightarrow X$ the function defined by $r_{(A,B)}=r_{A\cup \psi(B)}$. The family $\{r_{(A,B)}:(A,B)\in \Gamma'\}$ is an $r$-skeleton on $X$ which satisfies the conditions $(*)-(***)$. Therefore, using Theorem \ref{principal}, $AD(X)$ has an $r$-skeleton. \end{proof}
We do not know whether or not the monotony of Proposition \ref{novo2} is sufficient.
The next is an example of application of Theorem \ref{principal}. \begin{example} Let $\kappa$ an uncountable cardinal number. We considerer the $r$-skeleton $\{r'_A:A\in \mathbf{C}_\kappa\}$ given in the Example \ref{resqordinal}. For this $r$-skeleton, $Y=\bigcup\mathbf{C}_\kappa$ is the induced space. By properties of the ordinal space $[0,\kappa]$ it follows that $(*)$ and $(**)$ hold. By Theorem \ref{numerables}, from $\{r'_A:A\in \mathbf{C}_\kappa\}$ we obtain an $r$-skeleton $\{r_A:A\in [Y]^{\leq \omega}\}$ with induced space $Y$ that satisfies $(*)$ and $(**)$. Now, if $\psi:[X\setminus Y]^{\leq \omega}\rightarrow [Y]^{\leq \omega}$ is the function defined by $\psi(B)=\{\beta+1:\beta\in B\}$, for each $B\in [X\setminus Y]^{\leq \omega}$, then $\psi$ is $\omega$-monotone. We observe that for $B\in [X\setminus Y]^{\leq \omega}$, $cl_X(B)\setminus B\subseteq cl_X(\psi(B))\subseteq r_{\psi(B)}([0,\kappa])$. As a consequence of Proposition \ref{novo2}, we have that $AD([0,\kappa])$ has an $r$-skeleton. \end{example}
To finish this section we generalize Question 2.14 of the paper \cite{kalenda2} as follows.
\begin{question} If $X$ is a linearly ordered compact space with an $r$-skeleton, does $X$ admit an $r$-skeleton that satisfies the conditions of the Theorem \ref{principal}? \end{question}
\section{$\pi$-skeleton}
We remember that the Valdivia compact spaces are the compact spaces in a cube $[0,1]^\kappa$ with dense intersection with the $\Sigma$-product of $[0,1]^\kappa$. As the family of compact spaces with $r$-skeletons are the generalization of the Valdivia compact spaces, it is natural to ask about the possibility of describing the compact spaces with an $r$-skeleton in the vein of the definition of Valdivia compact spaces. To get an answer to this question we introduce the following notion.
\begin{definition} Let $X\subseteq [0,1]^{T}$ be a compact space.
A family $\{F_s:s\in \Gamma\}$ of cosmic spaces and an $ \omega$-monotone function $\psi:\Gamma\rightarrow [T]^{\leq\omega}$ generate a {\it $\pi$-skeleton} on $X$ if the next conditions are satisfied:
\begin{itemize}
\item[$(a)$] $F_s\subseteq F_{t}$ whenever $s\leq t$,
\item[$(b)$] $\bigcup_{s\in \Gamma}F_s$ is dense in $X$, and
\item[$(c)$] $\pi_{\psi(s)}\upharpoonright_{F_s}$ is an homeomorphism such that $\pi_{\psi(s)}(X)=\pi_{\psi(s)}(F_s)$, for each $s\in \Gamma$.
\end{itemize}
The set $\bigcup_{s\in \Gamma}F_s$ will be called {\it the induced space} of the $\pi$-skeleton. \end{definition} In what follows, when we will say that $X$ has $\pi$-skeleton generated by the pair $(\{F_s:s\in \Gamma\}, \psi:\Gamma\rightarrow [T]^{\leq\omega} )$ we shall understand that $X\subseteq [0,1]^T$.\\
Next, we shall describe two examples of spaces with $\pi$-skeletons.
\begin{example} If $\alpha$ is an infinite cardinal number, then there exists an embedding $h:[0,\alpha]\rightarrow [0,1]^{\alpha + 1}$ such that $h([0,\alpha])$ admits a $\pi$-skeleton. Indeed, let $h:[0,\alpha]\rightarrow [0,1]^{\alpha+1}$ be the function such that for every $\beta\leq \alpha$ we have
$$ \pi_\theta(h(\beta))=\left\{ \begin{array}{lcc} 0 & if &\theta< \beta\\ 1 & if &\theta\geq\beta, \end{array} \right. $$ for each $\theta<\alpha$. It is easy to verify that $h$ is an embedding. Let $L_\alpha$ be the set of limit points of $[0,\alpha]$ and consider the set $\Gamma=\{A\in [[0,\alpha]\setminus L_\alpha]^{\leq\omega}: 0\in A\}$. Let $X=h([0,\alpha])$ and, for each $A\in \Gamma$ let $F_A:=\overline{h(A\cup A+1)}$ and $\varphi(A):=A\cup\{\theta\leq\alpha:\theta+1\in A\mbox{ and } cof(\theta)\leq \omega\}$. It is clearly that the function $\varphi:\Gamma\rightarrow [\alpha+1]^{\leq\omega}$ is $\omega$-monotone and the family of cosmic subspaces $\{F_A:A\in [Y]^{\leq \omega}\}$ satisfies $(a)$ and $(b)$. To prove the last condition of being $\pi$-skeleton, we fix $A\in \Gamma$. First, we shall prove that $\pi_{\varphi(A)}(F_A)=\pi_{\varphi(A)}(X)$. Indeed, let $x\in X\setminus F_A$ and choose $\beta \leq \alpha$ so that $x=h(\beta)$. We point out that $\sup(\varphi(A))=\sup(A)$ and $\beta\notin h^{-1}(F_A)=\overline{A\cup A+1}$. Define $\beta_0:=\sup\{\theta\in \varphi(A):\theta<\beta\}$.\\
\begin{claim}{1} If $\beta_0\in \varphi(A)$, then $\beta_0+1<\beta$. \end{claim}
\begin{proofclaim} It is clearly from the choice of $\beta_0$ that $\beta_0<\beta$. Let us suppose that $\beta=\beta_0 + 1$. Since $\beta_0\in \varphi(A)$, $\beta_0\in A$ or $\beta_0+1\in A$. Hence, $\beta=\beta_0 +1\in A\cup A+1\subseteq h^{-1}(F_A) $. Since $\beta\notin \overline{A\cup A+1}$, we must have that $\beta_0+1<\beta$. \end{proofclaim}
To continue with the proof we need to consider two cases for $\beta_0$: For the first case, we suppose $\beta_0\in L_\alpha$. Clearly, $\beta_0\in \overline{\varphi(A)} \subseteq \overline{A\cup A+1}$. Since $\beta\notin \overline{A\cup A+1}$, we have that $\beta_0<\beta$ and $(\beta_0,\beta)\cap \varphi(A)=\emptyset$. Also, $\beta_0\notin \varphi(A)$, indeed, let us suppose $\beta_0\in \varphi(A)$. Since $\beta_0\in L_\alpha$, we have that $\beta_0+1\in A\subseteq \varphi(A)$. According to Claim 1, $\beta_0+1<\beta$, but this contradicts the equality $\beta_0=\sup\{\theta\in \varphi(A):\theta<\beta\}$. Hence, if $\theta\in \varphi(A)$, then $\theta<\beta_0$ or $\theta\geq\beta$. For $\theta\in \varphi(A)$ with $\theta< \beta_0$, we have $\pi_\theta(h(\beta_0))=0=\pi_\theta(h(\beta))$. And, for $\theta\in \varphi(A)$ with $\theta\geq \beta$, $\pi_\theta(h(\beta_0))=1=\pi_\theta(h(\beta))$. Thus, $\pi_{\varphi(A)}(h(\beta))=\pi_{\varphi(A)}(h(\beta_0))$ and $h(\beta_0)\in F_A$. The second case is when $\beta_0\notin L_\alpha$. In this case, we have that $\beta_0\in \varphi(A)$. It then follows that $\beta_0<\beta$. By the Claim 1, we have that $\beta_0\in A$. For $\theta\in \varphi(A)$ with $\theta\leq \beta_0$, we have that $\pi_\theta(h(\beta_0+1))=0=\pi_\theta(h(\beta))$. And, for $\theta\in \varphi(A)$ with $\theta\geq \beta$, $\pi_\theta(h(\beta_0+1))=1=\pi_\theta(h(\beta))$. Thus, $\pi_{\varphi(A)}(h(\beta))=\pi_{\varphi(A)}(h(\beta_0+1))$ and $h(\beta_0+1)\in F_A$. In both cases, we obtain $\pi_{\varphi(A)}(h(\beta))\in \pi_{\varphi(A)}(F_A)$. Hence, $\pi_{\varphi(A)}(F_A)=\pi_{\varphi(A)}(X)$.\\
Let us prove that $\pi_{\varphi(A)}(F_A)$ is homeomorphic to $F_A$. It is enough prove that the function $\pi_{\varphi(A)}\upharpoonright_{F_A}$ is one-to-one. Let $x,y\in F_A$ be distinct and $\theta,\beta\in h^{-1}(F_A)$ such that $h(\theta)=x$ and $h(\beta)=y$. Without loss of generality we may assume that $\theta<\beta$. Observe that $\pi_{\beta'}(x)\neq \pi_{\beta'}(y)$ for each $\beta'\in [\theta,\beta)$.
If $\beta$ is an isolated point, then there exist $\beta'<\alpha$ such that $\beta=\beta'+1$. So, we obtain that $\beta\in A\cup A+1$. If $\beta\in A$, then $\beta'\in \varphi(A)$. If $\beta\in A+1$, then $\beta'\in A\subseteq \varphi(A)$. If $\beta$ is not isolated point, by density, then $A\cap (\theta, \beta+1)\neq \emptyset$ and so there exist $\beta'\in A\cap [\theta,\beta)\subset \varphi(A)$. In all cases, $\varphi(A)\cap [\theta,\beta)\neq \emptyset$. Hence, we have $\pi_{\varphi(A)}(x)\neq \pi_{\varphi(A)}(y)$; that is, $\pi_{\varphi(A)}$ one-to-one on $F_A$. This shows that $\pi_{\varphi(A)}(F_A)$ is homeomorphic to $F_A$. Therefore, $(\{F_A:A\in \Gamma\},\varphi)$ is a $\pi$-skeleton on $X$. \end{example}
Our next example is the family of Valdivia compact spaces. To describe a $\pi$-skeleton on a Valdivia compact space we need prove, first, a useful technical lemma. We recall that for a power $[0,1]^T$ and a point $x\in X$, the support of $x$ is the set $supp(x):=\{t\in T: \pi_t(x)\neq 0\}$, if $A\subseteq [0,1]^T $, then $supp(A):=\bigcup_{x\in A} supp(x)$.
\begin{lemma}\label{lemmahomeo1} Let $Y\subseteq [0,1]^T$. If $A\in [Y]^{\leq\omega}$, then $\pi_{supp(A)}\upharpoonright_{\overline{A}}$ is a one-to-one function. \end{lemma}
\begin{proof} First, we shall prove that $supp(x)\subseteq supp(A)$ whenever $A\in [Y]^{\leq\omega}$ and $x\in \overline{A}$. If $x\in A$, we trivially have that $supp(x)\subseteq supp(A)$. Let $x\in \overline{A}\setminus A$ such that $ supp(x)\setminus supp(A)\not= \emptyset$. For a fix $t\in supp(x)\setminus supp(A)$ there is an open subset $U\subseteq [0,1]$ such that $0\notin U$ and $x\in \pi^{-1}_{t}(U)$. So, we have that $A\cap \pi^{-1}_{t}(U)=\emptyset$, which contradicts that $x\in \overline{A}$. Hence, for every $x\in \overline{A}$ we have $supp(x)\subseteq supp(A)$. Let $x,y\in \overline{A}$ with $x\neq y$. Then, there exists $t\in supp(x)\cup supp(y)$ such that $\pi_t(x)\neq \pi_t(y)$. Since $supp(x)\cup supp(y)\subseteq supp(A)$, we have that $\pi_{supp(A)}(x)\neq \pi_{supp(A)}(y)$. Thus, we have proved that $\pi_{supp(A)}\upharpoonright_{\overline{A}}$ is a one-to-one function. \end{proof}
\begin{proposition}\label{Valdiviapies} If $X$ is a Valdivia compact space, then $X$ admit a $\pi$-skeleton.
\end{proposition}
\begin{proof} Suppose $X$ is a compact space of $[0,1]^T$ and $Y=X\cap \Sigma[0,1]^T$ is dense in $X$. We shall prove that $X$ admits a $\pi$-skeleton.
\begin{claim}{1} The set $\Gamma=\{A\in [Y]^{\leq \omega}: \pi_{supp(A)}(X)=\pi_{supp(A)}(\overline{A})\}$ is cofinal in $[Y]^{\leq\omega}$. \end{claim}
\begin{proofclaim} Fix $A\in [Y]^{\leq\omega}$. We know that $\pi_{supp(A)}(X)$ is a separable space. Hence, it is possible to find $D'\in [Y]^{\leq\omega}$ such that $\pi_{supp(A)}(D')$ is dense in $\pi_{supp(A)}(X)$. We put $D_1=D'\cup A$ and for a positive $n<\omega$ suppose that we have established the set $D_n$. Choose $D'_{n+1}\in [Y]^{\leq\omega}$ so that $\pi_{supp(D_n)}(D'_{n+1})$ is dense in $\pi_{supp(D_n)}(X)$. We put $D_{n+1}=D'_{n+1}\cup D_n$. Thus, we have defined an increasing set $\{D_n:n<\omega\}\subseteq [Y]^{\leq\omega}$. We consider $D=\bigcup_{n<\omega}D_n$ and shall prove that $\pi_{supp(D)}(D)$ is dense on $\pi_{supp(D)}(X)$. Let $x\in X$, $t\in supp(D)$ and $U$ be an open subset of $[0,1]$ with $\pi_t(x)\in U$. Since $t\in supp(D)$ and $supp(D)=\bigcup_{n<\omega}supp(D_n)$, there is $n<\omega$ such that $t\in supp(D_n)$. As $\pi_{supp(D_n)}(D'_{n+1})$ is dense in $\pi_{supp(D_n)}(X)$, there is $d\in D'_{n+1}\subseteq D_{n+1}\subseteq D$ such that $\pi_t(d)\in U$. Hence, $\pi_{supp(D)}(D)$ is dense in $\pi_{supp(D)}(X)$. By Lemma \ref{lemmahomeo1}, $\pi_{supp(D)}\upharpoonright_{\overline{D}}$ is an homeomorphism and so we have that $\pi_{supp(D)}(\overline{D})=\pi_{supp(D)}(X)$. By construction, $A\subseteq D$ and thus we conclude that $\Gamma$ is cofinal in $[Y]^{\leq\omega}$. \end{proofclaim}
\begin{claim}{2} The set $\Gamma$ under the order given by the contention is a $\sigma$-complete directed partially ordered set. \end{claim}
\begin{proofclaim} It is immediate to see that $\Gamma$ is directed partially ordered set. To prove that $\Gamma$ is $\sigma$-complete we consider an increasing sequence $\{A_n\}_{n<\omega}$ in $\Gamma$, $A=\bigcup_{n<\omega}A_n$ and $x\in X$. For each $n<\omega$, using that $A_n\in \Gamma$, we pick $a_n\in \overline{A_n}$ such that $\pi_{supp(A_n)}(x)=\pi_{supp(A_n)}(a_n)$. Without loss of generality, we suppose $a=\lim_{n\rightarrow \infty}a_n$ since $Y$ is Frechet-Uryshon. As $\{a_n\}_{n<\omega}\subseteq \overline{\bigcup_{n<\omega}A_n}=\overline{A}$, $a\in \overline{A}$. By the choice of $\{a_n\}_{n<\omega}$ and the continuity of $\pi_{supp(A)}$, we have $$ \pi_{supp(A)}(x)=\lim_{n\rightarrow\infty}\pi_{supp(A)}(a_n)= \pi_{supp(A)}(\lim_{n\rightarrow\infty}a_n)=\pi_{supp(A)}(a). $$ \end{proofclaim}
Finally, for each $A\in \Gamma$ we define $F_A:=\overline{A}$. The family $\{F_A:A\in \Gamma\}$ consists of cosmic spaces which satisfy $(a)$ and $(b)$. Now, we define $\varphi:\Gamma\rightarrow [T]^{\leq \omega}$ by $\varphi(A)=supp(A)$, for each $A\in \Gamma$. Thus, we conclude that the pair $(\{F_A:A\in \Gamma\},\varphi)$ is a $\pi$-skeleton on $X$. \end{proof}
\begin{remark} We observe that if $\{F_s:s\in \Gamma\}$ is a family of cosmic spaces on a compact space $X$ which satisfies $(a)$, $(b)$ and $F_s\subseteq \Sigma_0$, for every $s\in \Gamma$, then $X$ is a Valdivia compact space. Hence and according to Proposition \ref{Valdiviapies}, we have that a compact space $X$ is Valdivia iff admits a $\pi$-skeleton $(\{F_s:s\in \Gamma\},\varphi)$ such that $F_s\subseteq \Sigma_0$, for every $s\in \Gamma$. \end{remark}
To prove the main result of this section, we require some technical lemmas. In the first of these lemmas, we will see that the notion of $\pi$-skeleton can be generalized when the base space is a subspace of a product of arbitrary cosmic compact spaces.
\begin{lemma}\label{obs12} Let $X$ be a compact subspace of a product of cosmic spaces $\Pi_{t\in T}Z_t$. Su\-ppose that there exists a pair $(\{F_s:s\in \Gamma\},\psi:\Gamma\rightarrow [T]^{\leq\omega})$ where $\{F_s:s\in \Gamma\}$ is a family which satisfies $(a)$ and $(b)$, $\psi$ is $\omega$-monotone and for each $s\in \Gamma$ we have that $\pi_{\psi(s)}\upharpoonright_{F_s}$ is an homeomorphism such that $\pi_{\psi(s)}(X)=\pi_{\psi(s)}(F_s)$. Then there is an embedding $g:X\rightarrow [0,1]^{T'}$ such that $g(X)$ admits a $\pi$-skeleton. \end{lemma} \begin{proof} For each $t\in T$, fix a countable set $T_t$ and an embedding $g_t:Z_t\rightarrow [0,1]^{T_t}$. We consider $g:\Pi_{t\in T}Z_t\rightarrow \Pi_{t\in T}[0,1]^{T_t}$ given by $g((x_t)_{t\in T})=(g_t(x_t))_{t\in T}$, for each $x\in \Pi_{t\in T}Z_t$. It is easy to verify that $g$ is an embedding. Without loss of generality, we suppose that $T_t\cap T_{t'}=\emptyset$, for distinct $t,t'\in T$. Let $T'=\bigcup_{t\in T}T_t$ and $\phi:[T]^{\leq\omega}\rightarrow [T']^{\leq\omega}$ be the mapping given by $\phi(A)=\bigcup_{t\in A}T_t$, for each $A\in [T]^{\leq \omega}$. It is immediate from its definition to see that $\phi$ is $\omega$-monotone. We shall prove that the pair $(\{g(F_s):s\in \Gamma\},\phi\circ\psi)$ is a $\pi$-skeleton on $g(X)$. Indeed, it is clearly that $\{g(F_s):s\in \Gamma\}$ is a family of cosmic spaces which satisfies $(a)$ and $(b)$, and since composition of $\omega$-monotone functions is $\omega$-monotone, the composition $\phi\circ \psi$ is an $\omega$-monotone function. It remains prove condition $(c)$ for the pair $(\{g(F_s):s\in \Gamma\},\phi\circ\psi)$. To prove that $\pi_{\phi\circ \psi(s)}(g(X))=\pi_{\phi\circ\psi(s)}(g(F_s))$, for each $s\in \Gamma$, we fix $s \in \Gamma$ and $x\in X$. By hypothesis there is $y\in F_s$ such that $\pi_{\psi(s)}(x)=\pi_{\psi(s)}(y)$.
So, $\pi_t(x)=\pi_t(y)$ and $g_t(\pi_t(x))=g_t(\pi_t(y))$, for each $t\in \psi(s)$. Hence, $\pi_{\phi\circ\psi(s)}(g(x))=\pi_{\phi\circ\psi}(g_t(\pi_t(x))_{t\in T})=(g_t(\pi_t(x)))_{t\in\psi(s)}= (g_t(\pi_t(y)))_{t\in\psi(s)}=\pi_{\phi\circ\psi(s)}(g(y))$. So, we conclude that $\pi_{\phi\circ\psi(s)}(g(X))=\pi_{\phi\circ\psi(s)}(g(F_s))$. Now, we shall prove that $\pi_{\phi\circ\psi(s)}\upharpoonright_{g(F_s)}$ is one-to-one. We pick $x, y\in F_s$ with $x\neq y$. Using that $\pi_{\psi(s)}\upharpoonright_{F_s}$ is an homeomorphism, we choose $t\in \psi(s)$ such that $\pi_t(x)\neq\pi_t(y)$. Since $g_t$ is an embedding, we have that $g_t(\pi_t(x))\neq g_t(\pi_t(y))$ and then we conclude that $\pi_{\phi\circ\psi(s)}(g(x))=(g_t(\pi_t(x)))_{t\in\psi(s)}\neq (g_t(\pi_t(y)))_{t\in\psi(s)}=\pi_{\phi\circ\psi(s)}(g(y))$. As a consequence of injectivity of $\pi_{\phi(\psi(s))}\upharpoonright_{g(F_s)}$, we have that $\pi_{\phi(\psi(s))}\upharpoonright_{g(F_s)}$ is an homeomorphism. Therefore, $(\{g(F_s):s\in \Gamma\},\phi\circ\psi)$ is a
$\pi$-skeleton on $g(X)$. \end{proof}
We observe that the reciprocal of Lemma \ref{obs12} is true. Hence, to prove that a space $X$ admits a $\pi$-skeleton it is enough to embed $X$ in an arbitrary product of cosmic spaces and to prove the existence of a pair as in the hypothesis of Lemma \ref{obs12}.\\
The next two lemmas are consequences of the characterization of spaces with $r$-skeletons using inverse limits. The first Lemma is an implicit result inside of the work of inverse limits and Valdivia compact spaces in the paper \cite{kubis1}.
\begin{lemma}\label{cen1} Let $X$ be a compact space. If $X$ admits an $r$-skeleton $\{r_s:s\in \Gamma\}$, then $\underleftarrow{lim}\langle r_s(X),r^{t}_s ,\Gamma\rangle =\{(r_s(x))_{s\in \Gamma}:x\in X\}$, where $r^t_s=r_s\upharpoonright_{r_t(X)}$, for each $t\geq s$. \end{lemma}
\begin{lemma}[\cite{kubis1}]\label{salceno} Let $X$ be a compact space. If $X$ admits an $r$-skeleton $\{r_s:s\in \Gamma\}$, then $X=\underleftarrow{lim}\langle r_s(X),r^{t}_s ,\Gamma\rangle$, where $r^t_s=r_s\upharpoonright_{r_t(X)}$, for each $t\geq s$. In particular, if $A$ is a countable directed subset of $\Gamma$, then $$r_{\sup(A)}(X)=\underleftarrow{lim}\langle r_{s}(X),r^t_s,A\rangle.$$
\end{lemma}
Next, we state the main result.
\begin{theorem}\label{teoprin} Let $X$ be a compact space. Then, $X$ admit an $r$-skeleton iff admit a $\pi$-skeleton. \end{theorem} \begin{proof} First, we shall prove the necessity. Let $\{r_s:s\in \Gamma\}$ be an $r$-skeleton on $X$ and ,by Corollary \ref{cen1} and Lemma \ref{salceno}, we may assume that $X=\underleftarrow{lim}\langle r_s(X),r^{t}_s ,\Gamma\rangle= \{(r_s(x))_{s\in \Gamma}:x\in X\}$, where $r^t_s=r_s\upharpoonright_{r_t(X)}$, for each $t\geq s$. We shall prove that $X$ admits a $\pi$-skeleton. Let $\Gamma'$ be the family of countable directed subsets of $\Gamma$. Observe that $\Gamma'$ is a $\sigma$-complete up-directed partially ordered set with the order given by set containment. For each $A\in \Gamma'$, let $F_A:=r_{sup(A)}(X)$ and define $\varphi:\Gamma'\rightarrow [\Gamma]^{\leq\omega}$ by $\varphi(A)=A$, for each $A\in \Gamma'$. It is easy to see that the function $\varphi$ is $\omega$-monotone and the family $\{F_A:A\in \Gamma'\}$ of cosmic subspaces of $X$ satisfies $(a)$. We note that $\bigcup_{A\in \Gamma'}r_{sup(A)}(X)$ is the induced set by the $r$-skeleton on $X$ because of $\{\{s\}:s\in \Gamma\}\subseteq\Gamma'$, and we deduce that $\{F_A:A\in \Gamma'\}$ satisfies property $(b)$. Let $A\in \Gamma'$. By using property $(ii)$ of the $r$-skeleton definition, we obtain that
$$\pi_A(F_A)=\{(r_s(r_{sup(A)}(x)))_{s\in A}:x\in X\}=\{(r_s(x))_{s\in A}:x\in X\}=\pi_A(X).$$
Hence, $\pi_{A}(X)=\pi_A(F_A)$. Now, by Corollary \ref{cen1} and Lemma \ref{salceno}, we know that $F_A=r_{sup(A)}(X)=\underleftarrow{lim}\langle r_s(X),r^t_s,A\rangle=\{(r_s(x))_{s\in A}:x\in F_A\}$, it then follows that $\pi_A\upharpoonright_{F_A}$ is one-to-one. Thus, we conclude that $\pi_A\upharpoonright_{F_A}$ induces an homeomorphism between $F_A$ and $\pi_A(X)$. Hence, the pair $(\{F_A:A\in \Gamma'\},\varphi)$ satisfies the hypothesis of Lemma \ref{obs12} and so $X$ admits a $\pi$-skeleton.
Now, we shall prove the sufficiency. Assume that $X\subseteq [0,1]^{T}$ and $(\{F_s:s\in \Gamma\},\varphi)$ is a $\pi$-skeleton on $X$ with induced space $Y$. For each $A\in [Y]^{\leq\omega}$, let $r_A:X\rightarrow F_A$ be the function defined by $r_A:=\pi^{-1}_{\varphi(A)}\upharpoonright_{F_A}\circ\pi_{\varphi(A)}$. As $\pi_{\varphi(A)}(F_A)$ is homeomorphic to $F_A$ and $\pi_{\varphi(A)}(X)\subseteq \pi_{\varphi(A)}(F_A)$, we have that $r_A$ is a retraction.
We assert that $\{r_A:A\in [Y]^{\leq\omega}\}$ is an $r$-skeleton on $X$. Indeed, condition $(i)$ is satisfied because $r_A(X)=F_A$ is homeomorphic to $\pi_{\varphi(A)}(F_A)$ and $\pi_{\varphi(A)}(F_A)$ is a cosmic space. For the conditions $(ii)$ and $(iii)$ we consider the next claim.\\
\begin{claim}{1} For every $A\in [Y]^{\leq \omega}$, $\pi_{\varphi(A)}(r_A(x))=\pi_{\varphi(A)}(x)$ for all $x\in X$. \end{claim}
\begin{proofclaim} Fix $A\in [Y]^{\leq \omega}$. By the definition of $r_A$, we have that $$\pi_{\varphi(A)}(r_A(x))=\pi_{\varphi(A)}(\pi^{-1}_{\varphi(A)}\upharpoonright_{F_A}(\pi_{\varphi(A)}(x)))=(\pi_{\varphi(A)}\circ\pi^{-1}_{\varphi(A)}\upharpoonright_{F_A})\circ\pi_{\varphi(A)}(x)=\pi_{\varphi(A)}(x),$$ for each $x\in X$. \end{proofclaim}
Let us prove condition $(ii)$. Fix $A, B\in [Y]^{\leq\omega}$ such that $A\subseteq B$ and fix $x\in X$. Since $r_A(x)\in F_A\subseteq F_B$ and $r_B$ is the identity on $F_B$, we have that $r_B(r_A(x))=r_A(x)$. Now, we shall prove that $r_A(x)=r_A(r_B(x))$. By Claim 1, $\pi_{\varphi(B)}(x)=\pi_{\varphi(B)}(r_B(x))$ and $\pi_{\varphi(A)}(r_A(r_B(x)))=\pi_{\varphi(A)}(r_B(x))$, for each $x\in X$. As $\varphi(A)\subseteq \varphi (B)$, we have that $\pi_{\varphi(A)}(r_A(r_B(x)))=\pi_{\varphi(A)}(r_B(x))=\pi_{\varphi(A)}(x)$. On the other hand, according to Claim 1, we have that $\pi_{\varphi(A)}(r_A(x))=\pi_{\varphi(A)}(x)$. Hence, $\pi_{\varphi(A)}(r_A(r_B(x)))=\pi_{\varphi(A)}(r_A(x))$. Since $\pi_{\varphi(A)}$ is injective on $F_A$, we conclude that $r_A(x)= r_A(r_B(x))$. Thus, $r_A=r_A\circ r_B= r_B\circ r_A$ and $(ii)$ hold. To prove condition $(iii)$ we consider the next claim.\\
In what follows, $\mathcal{B}$ will be a countable base of $[0,1]$.
\begin{claim}{2}\label{4obs} If $A\in [Y]^{\leq\omega}$, then
$$\{(\pi_{\varphi(A)}\upharpoonright_{F_A})^{-1}(\bigcap_{t\in G}[t,V_t]\cap \pi_{\varphi(A)}(F_A)): G\in [\varphi(A)]^{<\omega}\mbox{ and } V_t\in \mathcal{B}\}$$ is base for $F_A$. \end{claim}
\begin{proofclaim} It is evident that $\{\bigcap_{t\in G}[t,V_t]: G\in [\varphi(A)]^{<\omega}\mbox{ and } V_t\in \mathcal{B}\}$ is a base of the space $[0,1]^{\varphi(A)}$ and, so $\{\bigcap_{t\in G}[t,V_t]\cap \pi_{\varphi(A)}(F_A): G\in [\varphi(A)]^{<\omega}\mbox{ and } V_t\in \mathcal{B}\}$ is a base for the space $\pi_{\varphi(A)}(F_A)$. Since $\pi_{\varphi(A)}\upharpoonright_{F_A}$ is an homeomorphism, we obtain that $\{(\pi_{\varphi(A)}\upharpoonright_{F_A})^{-1}(\bigcap_{t\in G}[t,V_t]\cap \pi_{\varphi(A)}(F_A)): G\in [\varphi(A)]^{<\omega}\mbox{ and } V_t\in \mathcal{B}\}$ is a base for $F_A$.
\end{proofclaim}
For the condition $(iii)$, we consider an increasing sequence $\langle A_n\rangle_{n<\omega}\subseteq [Y]^{\leq\omega}$. Put $A=\bigcup_{n<\omega}A_n$ and fix $x\in X$. By Claim 2, we may choose $G\in [\varphi(A)]^{<\omega}$ and $\{V_t:t\in G\}\subseteq \mathcal{B} $ such that $r_A(x)\in (\pi_{\varphi(A)}\upharpoonright_{F_A})^{-1}(\bigcap_{t\in G}[t,V_t]\cap\pi_{\varphi(A)}(F_A))$. Since $\varphi(A)=\bigcup_{n<\omega}\varphi(A_n)$, there is $n_0<\omega$ such that $G\subseteq \varphi(A_n)$, for each $n\geq n_0$. According to Claim 1, we know that $\pi_{\varphi(A)}(r_A(x))=\pi_{\varphi(A)}(x)$ and $\pi_{\varphi(An)}(r_{A_n}(x))=\pi_{\varphi(An)}(x)$, for each $n\geq n_0$. In particular, as $G\subseteq \varphi(A_n)\subseteq \varphi(A)$, $\pi_{G}(r_A(x))=\pi_{G}(x)=\pi_{G}(r_{A_n}(x))$, for each $n\geq n_0$. Since $r_A(x)\in (\pi_{\varphi(A)}\upharpoonright_{F_A})^{-1}(\bigcap_{t\in G}[t,V_t]\cap\pi_{\varphi(A)}(F_A))$, it then follows that $\pi_t(r_A(x))\in V_t$, for each $t\in G$. Using that $\pi_{G}(r_A(x))=\pi_{G}(x)=\pi_{G}(r_{A_n}(x))$, we have that $\pi_t(r_{A_n}(x))\in V_t$, for each $t\in G$. Hence, for every $n\geq n_0$, $r_{A_n}(x)\in \bigcap_{t\in G}[t,V_t]$ and thus we have that condition $(iii)$ is satisfied.
For to establish the condition $(iv)$, we note that $r_A(X)=F_A$ and $\overline{\bigcup_{A\in [Y]^{\leq\omega}}r_A(X)}=\overline{\bigcup_{A\in [Y]^{\leq\omega}}F_A}=X$. Therefore, we have that $\{r_A:A\in [Y]^{\leq\omega}\}$ is an $r$-skeleton on $X$. \end{proof}
Following the definition of the $r$-skeletons, we say that a $\pi$-skeleton on a compact space $X$ is {\it full} if its induced space is the space $X$. Hence, we have the next consequence of the previous theorem. \begin{corollary} Let $X$ be a compact space. $X$ is a Corson space iff $X$ admits a full $\pi$-skeleton. \end{corollary}
\begin{proof} In \cite{cuth1}, the authors proved that $X$ is a Corson space iff $X$ admits a full $r$-skeleton. From the proof of Theorem \ref{teoprin} we have that the spaces induced by the $r$-skeleton and by a $\pi$-skeleton coincided. Hence, we deduce that $X$ admits a full $r$-skeleton iff admits a full $\pi$-skeleton. \end{proof}
Next, we will give a proof of the stability of $\pi$-skeletons under the product of spaces with a $\pi$-skeleton.
\begin{theorem}\label{preserprod} The product of spaces with $\pi$-skeleton admit a $\pi$-skeleton. \end{theorem}
\begin{proof} Let $\{X_i:i\in I\}$ be a family of compact spaces which admit $\pi$-skeletons. For each $i\in I$, let $T_i$ be such that $X_i\subseteq [0,1]^{T_i}$, $\mathcal{F}_i=\{F^{i}_s: s\in \Gamma_i\}$ be a family of cosmic closed subspaces of $X_i$ with induced space $Y_i$ and $\varphi_i:\Gamma_i\rightarrow [T_i]^{\leq\omega}$ be the $\omega$-monotone function such that $(\mathcal{F}_i,\varphi_i)$ is a $\pi$-skeleton on $X_i$. Without loss of generality, we suppose that the sets $T_i$'s are pairwise disjoints. For each $i\in I$, let $\leq_{i}$ be the order on $\Gamma_i$ and for every $A\subseteq I$ and $S_1, S_2\in \Pi_{i\in J}\Gamma_i$, $S_1\leq_{A} S_2$ will mean that $\pi_{i}(S_1)\leq_{i} \pi_{i}(S_2)$, for every $i\in A$.
To prove that $\Pi_{i\in I}X_i$ admit a $\pi$-skeleton we fix $y\in \Pi_{i\in I}Y_i$ which will help us in the process of the proof. Define $\Gamma:=\{(A,S):A\in [I]^{\leq \omega} \mbox{ and } S\in \Pi_{i\in A}\Gamma_i\}$. For $(A_1,S_1),(A_2,S_2)\in \Gamma$, we define $(A_1,S_1)\leq_{p}(A_2,S_2)$ if $A_1\subseteq A_2 $, $S_1\leq_{A_1}\pi_{A_1}(S_2)$ and $\pi_i(y)\in F^{i}_{\pi_{i}(S_2)}$, for each $i\in A_2\setminus A_1$.
First, we shall prove that $(\Gamma,\leq_{p})$ is a $\sigma$-complete up-directed partially ordered set. The reflexivity and antisymmetry of $\leq_p$ are immediate from the orders of the sets $\Gamma_i$. For the transitivity of $\leq_p$, let $(A_1,S_1),(A_2,S_2),(A_3,S_3)\in \Gamma$ such that $(A_1,S_1)\leq_p(A_2,S_2)$ and $(A_2,S_2)\leq_p (A_3,S_3)$. By definition of $\leq_p$, we have that $S_1\leq_{A_1}\pi_{A_1}(S_2)$ and $S_2\leq_{A_2}\pi_{A_2}(S_3)$. Using that $A_1\subseteq A_3$, $\pi_{A_1}(S_2)\leq_{A_1}\pi_{A_1}(S_3)$ and the transitivity of the orders of the sets $\Gamma_i$, for each $i\in A_1$, we obtain that $S_1\leq_{A_1}\pi_{A_1}(S_3)$. Now, let $i\in A_3\setminus A_1$, if $i\in A_3\setminus A_2$, then using that $(A_2,S_2)\leq_p (A_3,S_3)$ we have $\pi_{i}(y)\in F^{i}_{\pi_i(S_3)}$. Now, if $i\in A_2\setminus A_1$, then $\pi_{i}(y)\in F^{i}_{\pi_i(S_2)}$. Since $\pi_{i}(S_2)\leq_{i} \pi_{i}(S_3)$ and that $\mathcal{F}_i$ satisfies $(a)$, we have that $F^{i}_{\pi_{i}(S_2)}\subseteq F^{i}_{\pi_{i}(S_3)}$ and $\pi_{i}(y)\in F^{i}_{\pi_{i}(S_3)}$. Thus, for each $i\in A_3\setminus A_1$, $\pi_i(y)\in F^{i}_{\pi_i(S_3)}$. Hence, $(A_1,S_1)\leq_p (A_3,S_3)$. Now, we will prove that $\Gamma$ is an up-directed set. Let $(A_1,S_1),(A_2,S_2)\in \Gamma$. For each $i\in A_1\cap A_2$, using that $\Gamma_i$ is up-directed, fix $s_i\in \Gamma_i$ such that $s_i\geq_{i}\pi_i(S_1)$ and $s_i\geq_{i} \pi_i(S_2)$. For $i\in A_2\setminus A_1$, let $s_i$ be such that $s_i\geq_{i} \pi_{i}(S_2)$ and $\pi_i(y)\in F^{i}_{s_i}$. For $i\in A_1\setminus A_2$, let $s_i$ be such that $s_i\geq_{i} \pi_{i}(S_1)$ and $\pi_i(y)\in F^{i}_{s_i}$. Let $S\in \Pi_{i\in A_1\cup A_2}\Gamma_i$ such that $\pi_i(S)=s_i$, for each $i\in A_1\cup A_2$. We have that $(A_1\cup A_2,S)\in \Gamma$ and it is clear that $(A_1,S_1)\leq_p(A_1\cup A_2,S)$ and $(A_2,S_2)\leq_p(A_1\cup A_2,S)$. For the $\sigma$-completeness, let $\langle (A_n,S_n)\rangle_{n<\omega}\subseteq \Gamma$ be an increasing sequence and let $A=\bigcup_{n<\omega}A_n$. For each $i\in A$, let $n_i=\min\{n<\omega:i\in A_n\}$, $s_i=\sup_{n\geq n_i}\pi_i(S_n)$; and define $S\in \Pi_{i\in A}\Gamma_i$ by $\pi_i(S)=s_i$, for each $i\in A$. We have that $(A,S)\in \Gamma$ and claim that $(A,S)=\sup_{n<\omega}(A_n,S_n)$. Indeed, first we prove that $(A_n,S_n)\leq_p(A,S)$, for each $n<\omega$. Fix $n<\omega$. By definition, we have that $A_n\subseteq A$. By the choice of $S$, we have that $S_n\leq_{A_n}\pi_{A_n}(S)$. For $i\in A\setminus A_n$, there exists $n'\geq n$ such that $i\in A_{n'}$. Using that $(A_n,S_n)\leq_p(A_{n'},S_{n'})$, we have that $\pi_i(y)\in F^i_{\pi_i(S_{n'})}$.
Since $\mathcal{F}_i$ satisfies $(a)$ and $\pi_i(S_{n'})\leq_i \pi_i(S)$, we deduce that $F^i_{\pi_i(S_{n'})}\subseteq F^i_{\pi_i(S)}$. Hence, $\pi_{i}(y)\in F^i_{\pi_i(S)}$ and $(A_n,S_n)\leq_p (A,S)$. Now, let $(A',S')\in \Gamma$ be such that $(A_n,S_n)\leq_p (A',S')$, for each $n<\omega$. We shall prove that $(A,S)\leq_p (A',S')$. Since $A_n\subseteq A'$, for each $n<\omega$, $A\subseteq A'$. For each $i\in A$, we have that $\pi_i(S)=\sup_{n\geq n_i}\pi_i(S_n)$ and $\pi_i(S_n)\leq_i \pi_i(S')$, for all $n\geq n_i$. It then follows that $\pi_i(S)\leq_i \pi_i(S')$ and $S\leq_A \pi_A(S')$. For $i\in A'\setminus A$, is easy to see $\pi_(y)\in F^i_{\pi_i(S')}$. We conclude that $(A,S)\leq_p(A',S')$ and $(A,S)=\sup_{n<\omega}(A_n,S_n)$.
For each $(A,S)\in \Gamma$, let $F_{(A,S)}:=\Pi_{i\in A}F^{i}_{\pi_{i}(S)}\times \Pi_{i\in I\setminus A}\{\pi_i(y)\}$. The family $\mathcal{F}=\{F_{(A,S)}:(A,S)\in \Gamma\}$ of cosmic closed subspaces of $\Pi_{i\in I}X_i$ satisfies $(a)$. From the density of $Y_i$ in $X_i$, for each $i\in I$, we deduce that $\bigcup\mathcal{F}$ is dense in $\Pi_{i\in I}X_i$. Let $\varphi:\Gamma\rightarrow [\bigsqcup_{i\in I}T_i]^{\leq\omega}$ be defined by $\varphi((A,S))=\bigcup_{i\in A}\varphi_{i}(\pi_{i}(S))$, for each $(A,S)\in \Gamma$. We have that $\varphi$ is $\omega$-monotone. Indeed, let $(A_1,S_1),(A_2,S_2)\in \Gamma$, such that $(A_1,S_1)\leq_p (A_2,S_2)$. We have that $\pi_{A_1}(S_1)\leq_p \pi_{A_1}(S_2)$. Hence, $\varphi((A_1,S_1))=\bigcup_{i\in A_1}\varphi_i(\pi_i(S_1))\subseteq \bigcup_{i\in A_2}\varphi_i(\pi_i(S_2))=\varphi((A_2,S_2))$. Let $\langle(A_n,S_n)\rangle_{n<\omega}$ be an increasing sequence in $\Gamma$ and $(A,S)=\sup_{n<\omega}(A_n,S_n)$. It is easy to see that $\varphi((A_n,S_n))\subseteq \varphi((A,S))$, for each $n<\omega$, it then follows that $ \bigcup_{n<\omega}\varphi((A_n,S_n))\subseteq \varphi((A,S))$. Let $i\in A$ and $n_i=\min\{n<\omega:i\in A_n\}$, we know that $\pi_i(S)=\sup\{\pi_i(S_n): n\geq n_i\}$. Using that $\varphi_i$ is $\omega$-monotone, we have that $\varphi_i(\pi_i(S))=\bigcup_{n\geq n_i}\varphi_i(\pi_i(S_n))\subseteq \bigcup_{n\geq n_i}\varphi((A_n,S_n))$. Since $\varphi((A,S))=\bigcup_{i\in A}\varphi_{i}(\pi_{i}(S))$, we deduce that $\varphi((A,S))\subseteq\bigcup_{n<\omega}\varphi((A_n,S_n))$. Hence, $\varphi((A,S))=\bigcup_{n<\omega}\varphi((A_n,S_n))$; that is, $\varphi$ is $\omega$-monotone. Finally, for every $(A,S)\in \Gamma$, we shall prove that $\pi_{\varphi((A,S))}\upharpoonright_{F_{(A,S)}}$ is an homeomorphism. Note that
$$\pi_{\varphi((A,S))}(X)=\Pi_{i\in A}\pi_{\varphi_{i}(\pi_{i}(S))}(X_i)=\Pi_{i\in A}\pi_{\varphi_{i}(\pi_{i}(S))}(F^{i}_{\pi_{i}(S)})=\pi_{\varphi((A,S))}(F_{(A,S)}).
$$ Since $\pi_{\varphi_{i}(\pi_{i}(S))}\upharpoonright_{F^{i}_{\pi_{i}(S)}}$ is an homeomorphism, it is clear that $\pi_{\varphi((A,S))}\upharpoonright_{F_{(A,S)}}$ is an homeo\-morphism. Therefore, $(\mathcal{F}=\{F_{(A,S)}:(A,S)\in \Gamma\},\varphi)$ is a $\pi$-skeleton on $ \Pi_{i\in I}X_i$. \end{proof}
The previous theorem provides an alternative proof of the fact that the product of spaces with $r$-skeleton admits a $r$-skeleton, this proof does not use the theory of elementary submodels as it was done in \cite{cuth1}.
\begin{corollary}\label{preguntapi} The arbitrary product of compact spaces with $r$-skeleton admit an $r$-skeleton. \end{corollary} \begin{proof} This is a consequence of Theorem \ref{teoprin} and Theorem \ref{preserprod}. \end{proof}
Concerning the reciprocal of Theorem \ref{preserprod} we may ask the following.
\begin{question}
If $X_1$ and $X_2$ are compact spaces such that $X_1\times X_2$ admits a $\pi$-skeleton, must either $X_1$ or $X_2$ admit a $\pi$-skeleton?
\end{question}
The question when $X_1$ and $X_2$ are compact Valdivia spaces is posed in \cite[Q. 2.3]{kalenda2}.
Finally, we point out that the notion of $\pi$-skeleton somehow resembles the definition of Valdivia spaces using the notion of $\Sigma$-product.
\section{weak $c$-skeletons}
Our first task of this section is to weaken the notion of a $c$-skeleton:
For a family $\{F_s:s\in \Gamma\}$ of closed spaces on a compact space $(X,\tau)$ we recall the following conditions used in the previous section:
\begin{itemize}
\item[$(a)$] $F_s\subseteq F_{t}$, whenever $s\leq t$ and
\item[$(b)$] $\bigcup_{s\in \Gamma}F_s$ is dense in $X$.
\end{itemize} If $\mathcal{B}$ is a base for a topology on $X$ and $x \in X$, then define $\mathcal{N}(\mathcal{B},x):=\{U\in \mathcal{B}:x\in U\}$. A function $\psi:\Gamma \rightarrow [\tau]^{\leq\omega}$ is called an {\it assignment of bases for} $\{F_s:s\in \Gamma\}$ if $\psi(s)$ is a base for a topology on $X$, $\psi(s)\upharpoonright_{F_s}:=\{U\cap F_s:U\in \psi(s)\}$ is a base for $F_s$ and $\emptyset\notin \psi(s)\upharpoonright_{F_s}$.
\begin{definition}\label{wcesq} Let $(X,\tau)$ be a space. A pair $(\{F_s:s\in \Gamma\},\psi)$ is a \textit{weak $c$-skeleton} on $X$ if the following conditions hold: \begin{enumerate} \item\label{cond0} $\{F_s:s\in \Gamma\}$ is a family of cosmic subspaces of $X$ which satisfies $(a)$ and $(b)$;
\item\label{cond01} $\psi$ is an assignment of bases for $\{F_s:s\in \Gamma\}$; and
\item \label{condition1} if $x\in X$ and $z\in \bigcap_{U\in \mathcal{N}(\psi(s),x)}\overline{U\cap F_s}$, then for each $V\in \mathcal{N}(\psi(s),z)$ there is $U\in \mathcal{N}(\psi(s),x)$ such that $\overline{U\cap F_s}\subseteq V$. \end{enumerate}
The {\it induced space} of a weak $c$-skeleton is $\bigcup_{s\in \Gamma}F_s$. \end{definition}
Let us see how from a $c$-skeleton we can obtain a weak $c$-skeleton: Consider a $c$-skeleton $\{(F_s,\mathcal{B}_s):s\in \Gamma\}$ on a space $X$. If we define $\psi(s) = \mathcal{B}_s$ for every $s \in \Gamma$, then it is evident that $\psi$ is an assignment of bases for $\{F_s:s\in \Gamma\}$. Conditions (\ref{cond0}) and (\ref{cond01}) of a weak $c$-skeleton easily hold. The existence of a Tychonoff space $Z_s$ and continuous map $g_s:(X,\psi(s))\rightarrow Z_s$ which separates the points of $F_s$ implies the third Condition, for each $s\in\Gamma$.
The next technical lemma shows that a closed subset of a compact space is a retract whenever we have a base which satisfies the condition (\ref{condition1}). \begin{lemma}\label{lemaconti} Let $(X,\tau)$ be a compact space and $F\subseteq X$ be a cosmic closed subspace. Then the next conditions are equivalents:
\begin{itemize} \item There is a base $\mathcal{B}\in [\tau]^{\leq\omega}$ for a topology on $X$ such that $\mathcal{B}\upharpoonright_{F}$ is a base for $F$, $U\cap F\neq \emptyset$ for each $U\in \mathcal{B}$ and $F$ and $\mathcal{B}$ satisfies the condition (\ref{condition1}) of the definition of weak $c$-skeleton.
\item There exist a retraction $r:X\rightarrow F$. \end{itemize}
\end{lemma}
\begin{proof} Suppose that $\mathcal{B}\in [\tau]^{\leq\omega}$ is a base for a topology on $X$ such that $\mathcal{B}\upharpoonright_{F}$ is a base for $F$, $U\cap F\neq \emptyset$ for each $U\in \mathcal{B}$ and $F$ and $\mathcal{B}$ satisfies the condition (\ref{condition1}) of definition of weak $c$-skeleton. Let $x\in X$. First, we will prove that $\bigcap_{V\in \mathcal{N}(\mathcal{B},x)}\overline{V}\cap F\neq\emptyset$. Since $\mathcal{B}$ is a base for a topology on $X$, we have that $\mathcal{N}(\mathcal{B},x)\neq \emptyset$. As $U\cap F\neq \emptyset$, for every $U\in \mathcal{N}(\mathcal{B},x)$, we have that $\{\overline{U}\cap F: U\in \mathcal{N}(\mathcal{B},x)\}$ is a nonempty family of nonempty closed subsets. We claim that $\{\overline{U}\cap F: U\in \mathcal{N}(\mathcal{B},x)\}$ has the finite intersection property. In fact, let $U_1,U_2\in \mathcal{N}(\mathcal{B},x)$. Since $\mathcal{B}$ is a base for a topology on $X$, there is $U_3\in \mathcal{N}(\mathcal{B},x)$ such that $U_3\subseteq U_1\cap U_2$. Hence, $\emptyset\neq \overline{U_3}\cap F\subseteq \overline{U_1\cap U_2}\cap F$. As $\{\overline{U}\cap F: U\in \mathcal{N}(\mathcal{B},x)\}$ has the finite intersection property, by the compactness of $X$, we have that $\bigcap_{U\in \mathcal{N}(\mathcal{B},x)}\overline{U}\cap F\neq \emptyset$. Now, we will show that $|\bigcap_{U\in \mathcal{N}(\mathcal{B},x)}\overline{U}\cap F|=1$. Pick two distinct points $y_0,y_1\in \bigcap_{U\in \mathcal{N}(\mathcal{B},x)}\overline{U}\cap F$. Since $F$ is Hausdorff and $\mathcal{B}\upharpoonright_{F}$ is base for $F$, there are $W_0\in \mathcal{N}(\mathcal{B},y_0)$ and $W_1\in \mathcal{N}(\mathcal{B},y_1)$ such that $W_0\cap W_1\cap F=\emptyset$. Using condition (\ref{condition1}) of the Definition \ref{wcesq}, there are $V_0,V_1\in \mathcal{N}(\mathcal{B},x)$ such that $\overline{V_0\cap F}\subseteq W_0$ and $\overline{V_1}\cap F\subseteq W_1$. However, we have that $\emptyset\neq \bigcap_{U\in \mathcal{N}(\mathcal{B},x)}\overline{U}\cap F \subseteq \overline{V_0}\cap F\cap \overline{V_1}\cap F \subseteq W_0\cap W_1\cap F$, which contradicts that $W_0$ and $W_1$ are disjoints sets in $F$. Hence, $|\bigcap_{U\in \mathcal{N}(\mathcal{F},x)}\overline{U\cap F}|=1$. Now, consider the function $r:X\rightarrow F$ such that $r(x)$ is the unique point of $ \bigcap_{V\in \mathcal{N}(\mathcal{B},x)}\overline{V}\cap F$, for every $x\in X$. From the definition of $r$, it is easy to note that if $x\in F$ and $V\in\mathcal{B}$, then $r(x)=x$ and $r(V)\subseteq \overline{V}\cap F$. Let us show that $r$ is continuous. Fix $U\in \mathcal{B}$. Remember that $\mathcal{B}\upharpoonright_{F}$ is base for the subspace $F$. We assert that $r^{-1}(U\cap F)$ is an open subset of $X$. Indeed, fix $x\in X$ such that $r(x)\in U\cap F$. According to condition (\ref{condition1}) of Definition \ref{wcesq}, we may choose $V\in \mathcal{N}(\mathcal{B},x)$ with $\overline{V}\cap F\subseteq U$. Since $r(V)\subseteq \overline{V}\cap F$, we have that $V\subseteq r^{-1}(\overline{V}\cap F)\subseteq r^{-1}(U\cap F)$. Thus, $r$ is a continuous function. Finally, $r$ is a retraction because $r\upharpoonright_{F}$ is the identity.
Now, assume that $r:X\rightarrow F$ is a retraction. Let $\mathcal{B}'$ be a countable base for the space $F$ and $\mathcal{B}:=\{r^{-1}(V):V\in \mathcal{B}'\}$. We note that $\mathcal{B}\in [\tau]^{\leq\omega}$ is a base for a topology on $X$, $\mathcal{B}\upharpoonright_{F}$ is a base for $F$ and $U\cap F\neq \emptyset$, for each $U\in \mathcal{B}$. It remains to show that the third condition of the definition of weak $c$-skeleton is satisfied. Indeed, fix $ x\in X$. First, we will prove that $r(x)\in\bigcap_{V\in \mathcal{N}(\mathcal{B},x)}\overline{V\cap F}$. To have this done choose $V\in \mathcal{N}(\mathcal{B},x)$ and $U\in \mathcal{N}(\mathcal{B},r(x))$. We assert that $U\cap V\cap F\neq \emptyset$. Certainly, we can find $V',U'\in \mathcal{B}'$ such that $V=r^{-1}(V')$ and $U=r^{-1}(U')$. Then, we have that $r(x)\in V'$, and since $r$ is a retraction, $r(x)\in U'$. Hence, $r(x)\in U'\cap V'$. As $\mathcal{B}'$ is a base in $F$, there exist $W'$ such that $r(x)\in W'\subseteq U'\cap V'$. Thus, $x\in r^{-1}(W')\subseteq r^{-1}(U'\cap V')= U\cap V$. As $r^{-1}(W')\in\mathcal{B}$, we obtain that $\emptyset\neq r^{-1}(W')\cap F\subseteq U\cap V\cap F$. It then follows that $r(x)\in \overline{V}\cap F$ and so $r(x)\in\bigcap_{V\in \mathcal{N}(\mathcal{B},x)}\overline{V\cap F}$.
Next, we will show that $|\bigcap_{U\in \mathcal{N}(\mathcal{B},x)}\overline{U\cap F}|=1$. Suppose that $y\in \bigcap_{U\in \mathcal{N}(\mathcal{B},x)}\overline{U\cap F}$ and $y\neq r(x)$. Since $F$ is Hausdorff and $\mathcal{B}\upharpoonright_{F}$ is base for $F$, there are $W_0\in \mathcal{N}(\mathcal{B},y)$ and $W_1\in \mathcal{N}(\mathcal{B},r(x))$ such that $W_0\cap W_1\cap F=\emptyset$. As $r$ is a retraction, $r(x)\in r(W_1)=W_1'$. Hence, $x\in r^{-1}(W_1')=W_1\in\mathcal{B}$. Since $y\in \overline{W_1}\cap F$, we have that $W_0\cap W_1\cap F\neq \emptyset$, but this contradicts the choice of $W_0$ and $W_1$. Hence, $|\bigcap_{U\in \mathcal{N}(\mathcal{B},x)}\overline{U\cap F}|=1$ and $\{r(x)\}=\bigcap_{V\in \mathcal{N}(\mathcal{B},x)}\overline{V\cap F}$. Finally, to prove condition (\ref{condition1}). Consider $r^{-1}(W)\in \mathcal{N}(\mathcal{B},r(x))$ where $W\in \mathcal{B}$. As $r$ is a retraction, $r(x)=r(r(x))\in W$. Choose $W'\in \mathcal{B}$ with $r(x)\in W'\subseteq \overline{W'}\subseteq W$. Hence, $x\in r^{-1}(W')\subseteq r^{-1}(\overline{W'})\subseteq r^{-1}(W)$. Thus, $\overline{r^{-1}(W')\cap F}\subseteq r^{-1}(W)$ and condition (\ref{condition1}) is satisfied for the point $x$. \end{proof}
\begin{remark}\label{observacionlemaconti} Implicitly in the proof of Lemma \ref{lemaconti} there is a useful equality that we will use in the proofs of our next results: $\{r(x)\}=\bigcap_{V\in \mathcal{N}(\mathcal{B},x)}\overline{V}\cap F$, where $r: X \to F$ is the retraction obtained in the previous lemma. \end{remark}
In the next result, we show that the notion of $r$-skeleton implies the notion of weak $c$-skeleton.
\begin{theorem}\label{weakteo1}
If $X$ is a compact space with an $r$-skeleton $\{r_s:s\in \Gamma \}$, then $X$ admits a weak $c$-skeleton $(\{F_s:s\in \Gamma \},\psi)$ such that $\{r_s(x)\}=\bigcap_{U\in \mathcal{N}(\psi(s),x)}\overline{U}\cap F$, for each $x\in X$ and each $s\in \Gamma$.
\end{theorem}
\begin{proof}
Let $\{r_s:s\in \Gamma\}$ be an $r$-skeleton on $X$ where $Y$ is its induced space. For each $s\in \Gamma$, put $F_s=r_s(X)$, choose a countable base $\mathcal{B}_s$ of $F_s$ and define $\psi(s):=r^{-1}_s(\mathcal{B}_s)$. Consider this function $\psi:\Gamma\rightarrow [\tau]^{\leq\omega}$. From Lemma \ref{lemaconti} we have that $\psi:\Gamma\rightarrow [\tau]^{\leq\omega}$ satisfies conditions (\ref{cond01}) and (\ref{condition1}). As $\{r_s:s\in \Gamma\}$ is an $r$-skeleton, the conditions $(i)$, $(ii)$ and $(iv)$ of the $r$-skeleton definition imply that $\{F_s:s\in \Gamma\}$ is a family of cosmic spaces which satisfies $(a)$ and $(b)$. Therefore, $(\{F_s:s\in\Gamma\},\psi)$ is a weak $c$-skeleton.
\end{proof}
The next example is a compact space which admits a weak $c$-skeleton but does not admit an $r$-skeleton. This space is one of the examples of non-Valdivia compact spaces, given by O. Kalenda in \cite{kalenda1}, to show that the continuous images of Valdivia compact spaces are not necessarily Valdivia compact spaces.
\begin{example} Let $X$ the space obtained from $[0,\omega_1]\times\{0,1\}$ by identifying the points $(\omega_1,0)$ and $(\omega_1,1)$ and denote by $\tau_X$ the quotient topology on $X$. First, we shall prove that $X$ admits a weak $c$-skeleton. Let $\mathcal{B}$ a base for the space $[0,\omega_1]$. The set $\Gamma :=\omega_1\times \omega_1$ is a $\sigma$-complete up-directed partially ordered set with the order defined by $(\alpha,\beta)\leq_{\Gamma}(\alpha',\beta')$ if $\alpha\leq \alpha'$ and $\beta \leq \beta'$. For every $\alpha<\omega_1$, fix a subset $\mathcal{B}_\alpha\in [\mathcal{B}]^{\leq \omega}$ such that $\{U\cap [0,\alpha]:U\in \mathcal{B}_\alpha\}$ is a base for the subspace $[0,\alpha]$ and if $\alpha\notin U\in \mathcal{B}_\alpha$, then $U\subseteq [0,\alpha)$. For $\alpha<\omega_1$ and $i\in \{0,1\}$, let $\mathcal{B}_\alpha\times \{i\}:=\{U\times \{i\}:U\in \mathcal{B}_\alpha\}$. For each, $(\alpha,\beta)\in \Gamma$, define $F_{(\alpha,\beta)}:=([0,\alpha]\times\{0\})\cup ([0,\beta])\times\{1\}$ and $$ \psi((\alpha,\beta)):= (\mathcal{B}_\alpha\times\{0\}) \cup (\mathcal{B}_\beta\times\{1\}) \cup\{\big( (U\cup (\alpha,\omega_1])\times\{0\}\big)\cup \big ( (W\cup (\beta,\omega_1])\times\{1\}\big):
$$
$$ U\in \mathcal{B}_\alpha, \alpha\in U, W\in\mathcal{B}_\beta \mbox{ and } \beta\in W\}. $$ We assert that $(\{F_{(\alpha,\beta)}:(\alpha,\beta)\in \Gamma\},\psi)$ is a weak $c$-skeleton on $X$. It is easy to see that $\{F_{(\alpha,\beta)}:(\alpha,\beta)\in\Gamma\}$ satisfies the condition (\ref{cond0}). Now, we shall prove that $\psi$ satisfies condition (\ref{cond01}). Fix $(\alpha,\beta)\in \Gamma$. By the choice of the sets $\mathcal{B}_\alpha$ and $\mathcal{B}_\beta$, we have that $\psi((\alpha,\beta))\upharpoonright_{F_{(\alpha,\beta)}}$ is a base for the space $F_{(\alpha,\beta)}$ and $\emptyset\notin \psi((\alpha,\beta))\upharpoonright_{F_{(\alpha,\beta)}}$. We shall prove that $\psi((\alpha,\beta))$ is a base for a topology on $X$. Indeed, by assumption, $\psi((\alpha,\beta))$ covers $X$. Let $U, U'\in \mathcal{B}_\alpha$ and $W, W'\in \mathcal{B}_\beta$. Let us consider three cases: \begin{itemize}
\item[Case 1] If $x\in (U\times\{0\})\cap (U'\times\{0\})$ and $\alpha\notin U\cap U'$, since $\mathcal{B}_{\alpha}$ is base for $[0,\alpha]$, then we have that there exist $V\in \mathcal{B}_\alpha$ such that $x\in (V\times\{0\})\subseteq (U\times\{0\})\cap (U'\times\{0\})$ and $V\times\{0\}\in \psi((\alpha,\beta))$. \item[Case 2] When $x\in (W\times\{1\})\cap(W'\times\{1\})$ and $\beta\notin W\cap W'$ we proceed in a similar way as in the previous case. \item [Case 3]Let us suppose $x\in (((U\cup (\alpha,\omega_1])\times\{0\})\cup((W\cup (\beta,\omega_1])\times\{1\}))\cap (((U'\cup (\alpha,\omega_1])\times\{0\})\cup((W'\cup (\beta,\omega_1])\times\{1\}))$. As $\mathcal{B}_\alpha$ and $\mathcal{B}_\beta$ are bases for the spaces $[0,\alpha]$ and $[0,\beta]$, respectively, we can find $V\in \mathcal{B}_\alpha$ and $V'\in \mathcal{B}_\alpha$ such that either $x\in (V\cup(\alpha,\omega_1])\times\{0\}\subseteq ((U\cup(\alpha,\omega_1])\times\{0\})\cap ((U'\cup(\alpha,\omega_1])\times\{0\})$ or $x\in (V'\cup (\beta,\omega_1])\times\{1\}\subseteq ((W\cup(\beta,\omega_1])\times\{1\})\cap ((W'\cup(\beta,\omega_1])\times\{1\})$. Thus, $x\in ((V\cup(\alpha,\omega_1])\times\{0\})\cup ((V'\cup (\beta,\omega_1])\times\{1\}) \subseteq (((U\cup (\alpha,\omega_1])\times\{0\})\cup((W\cup (\beta,\omega_1])\times\{1\}))\cap (((U'\cup (\alpha,\omega_1])\times\{0\})\cup((W'\cup (\beta,\omega_1])\times\{1\}))$. \end{itemize} This shows that $\psi((\alpha,\beta))$ is a base for a topology on $X$ and so the condition (\ref{cond01}) holds.
It remains to show the condition (\ref{condition1}). Fix $(\alpha,\beta)\in \Gamma$ and $x\in X$. First, we suppose $x\in F_{(\alpha,\beta)}$. It is easy to see that $\bigcap_{U\in \mathcal{N}(\psi((\alpha,\beta)),x)} \overline{U\cap F_{(\alpha,\beta)}}=\{x\}$, and condition (\ref{condition1}) follows trivially. Now, let us suppose that $x\notin F_{(\alpha,\beta)}$. We have that $\bigcap_{U\in \mathcal{N}(\psi((\alpha,\beta)),x)} \overline{U\cap F_{(\alpha,\beta)}}=\{(\alpha,0),(\beta,1)\}$. Note that if $z\in \{(\alpha,0),(\beta,1)\}$ and $U\in \mathcal{N}(\psi((\alpha,\beta)),z)$, then $U\in \mathcal{N}(\psi((\alpha,\beta)),x)$ and so condition \ref{condition1} holds. Therefore, $(\{F_{(\alpha,\beta)}:(\alpha,\beta)\in \Gamma\},\psi)$ is a weak $c$-skeleton on $X$. We know that $X$ cannot be a Valdivia compact space and does not admit an $r$-skeleton, because every compact space with weight $\aleph_1$ and $r$-skeleton is a Valdivia compact space (see \cite{kubis1}). \end{example}
In the next theorem, we establish conditions on a weak $c$-skeleton in order to be equivalent to an $r$-skeleton. Thus we obtain a characterization of spaces that admit an $r$-skeleton in terms of a special weak $c$-skeleton.
\begin{theorem}\label{teoprincipalceqsresq} For a compact space $X$, the following statements are equivalent: \begin{itemize} \item There is a weak $c$-skeleton $(\{F_s:s\in\Gamma \}, \psi)$ on $X$ which satisfies \begin{itemize} \item[$(*)$] For every $s\leq t$, $x\in X$, $z\in \bigcap_{U\in \mathcal{N}(\psi(t),x)}\overline{U\cap F_t}$ and $V\in \mathcal{N}(\psi(s),x)$, there is $W\in \mathcal{N}(\psi(s),z)$ such that $\overline{W\cap F_s}\subseteq \overline{V\cap F_s}$; and
\item[$(**)$] if $\langle s_n \rangle_{n<\omega}$ is an increasing sequence in $\Gamma$ with $s =\sup\langle s_n\rangle_{n<\omega}$, then for each $x\in X$ and $U\in \mathcal{N}(\psi(s),x)$ there is $m<\omega$ such that $\overline{U\cap F_s}\cap {V\cap F_{s_n}}\neq \emptyset$ for every $n\geq m$ and for every $V\in \mathcal{N}(\psi(s_n),x)$. \end{itemize} \item There is an $r$-skeleton on $X$. \end{itemize} \end{theorem}
\begin{proof} Let $(\{F_s:s\in\Gamma \}, \psi)$ be a weak $c$-skeleton that satisfies $(*)$ and $(**)$. For each $s\in \Gamma$, consider the retraction $r_s$ given in Lemma \ref{lemaconti}. We shall prove that $\{r_s: s\in \Gamma\}$ is an $r$-skeleton on $X$. In fact, the condition $(i)$ is true because $r_s(X)=F_s$ and $F_s$ is a cosmic space, for each $s\in \Gamma$. For condition $(ii)$, fix $s, t \in \Gamma$ with $s\leq t$. On one hand, we know that $r_s(X)=F_s\subseteq F_t=r_t(X)$ and $r_t$ is the identity on $F_t$, hence, $r_t\circ r_s=r_s$. On the other hand, fix $x\in X$. Using condition $(*)$, for every $V\in \mathcal{N}(\psi(s),x)$ choose $W_V\in \mathcal{N}(\psi(s),r_{t}(x)) $ such that $\overline{W_V\cap F_s}\subseteq \overline{V\cap F_s}$. It then follows that $\bigcap_{V\in \mathcal{N}(\psi(s),x)}\overline{W_V\cap F_s}\subseteq \bigcap_{V\in \mathcal{N}(\psi(s),x)}\overline{V\cap F_s}$. So $\bigcap_{W\in \mathcal{N}(\psi(s),r_t(x))}\overline{W\cap F_s}\subseteq\bigcap_{V\in \mathcal{N}(\psi(s),x)}\overline{W_V\cap F_s}$ and this implies that $$ \{r_s(r_t(x))\}=\bigcap_{W\in \mathcal{N}(\psi(s),r_t(x))}\overline{W\cap F_s}\subseteq \bigcap_{V\in \mathcal{N}(\psi(s),x)}\overline{V\cap F_s}=\{r_s(x)\}.
$$
Thus, $r_s\circ r_t=r_s$ and $(ii)$ holds. For condition $(iii)$, let $\langle s_n\rangle_{n<\omega}$ be an increasing sequence in $\Gamma$ and $s=\sup\langle s_n\rangle_{n<\omega}$. Fix $x\in X$ and $U\in \mathcal{N}(\psi(s),r_s(x))$. Choose $V\in \mathcal{N}(\psi(s),r_s(x))$ such that $\overline{V}\cap F_s\subseteq U\cap F_s$. For condition $(**)$, pick $m<\omega$ such that for every $n\geq m$ we have that $\overline{V\cap F_s}\cap {W\cap F_{s_n}}\neq \emptyset$, for every $W\in \mathcal{N}(\psi(s_n),r_s(x))$. Fix $n\geq m$. By property $(**)$, it is easy to see that $\{\overline{V\cap F_s}\cap {W\cap F_{s_n}}: W\in \mathcal{N}(\psi(s_n),r_s(x)) \}$ has the finite intersection property, so $\{\overline{V\cap F_s}\cap \overline{W\cap F_{s_n}}: W\in \mathcal{N}(\psi(s_n),r_s(x)) \}$ has the finite intersection property. Since $F_s$ is a compact space, we have that $\overline{V\cap F_s}\cap \big(\bigcap_{W\in \mathcal{N}(\psi(s_n),r_s(x))}\overline{W\cap F_{s_n}}\big)\neq \emptyset$. As $\{r_{s_n}(x)\}=\{r_{s_n}(r_s(x))\}= \bigcap_{W\in \mathcal{N}(\psi(s_n),r_s(x))}\overline{W\cap F_{s_n}}$, we obtain that $r_{s_n}(x)\in \overline{V\cap F_s}\subseteq U$. Thus, for each $n\geq m$, $r_{s_n}(x)\in U$; that is, $r_{s_n}(x)\rightarrow r_s(x)$ in $F_s$. Hence, $r_{s_n}(x)\rightarrow r_s(x)$ in $X$ and $(iii)$ holds.
The last condition $(iv)$ follows from the fact that $\{F_s:s\in \Gamma\}$ satisfies condition $(b)$. Thus we have shown that $\{r_s: s\in \Gamma\}$ is an $r$-skeleton on $X$.
Assume that $\{r_s:s\in \Gamma\}$ is an $r$-skeleton on $X$ and $Y$ is the induced space. For each $s\in \Gamma$, define $F_s:=r_s(X)$, choose a countable base $\mathcal{B}_s$ of $F_s$ and put $\psi(s):=r^{-1}_s(\mathcal{B}_s)$. Theorem \ref{weakteo1} asserts that $(\{F_s:s\in\Gamma \},\psi)$ is a weak $c$-skeleton on $X$ with induced set $Y$. To establish property $(*)$, let $s\leq t$, $x\in X$ and $W\in \mathcal{B}_s$ such that $x\in r^{-1}_s(W)$. According to Theorem \ref{weakteo1}, we know that $\{r_t(x)\}=\bigcap_{U\in \mathcal{N}(\psi(t),x)}\overline{U\cap F_t}$. Since $\{r_s:s\in \Gamma\}$ is an $r$-skeleton, we have that $r_s(r_t(x))=r_s(x)\in W$. Hence, $r_t(x)\in r^{-1}_s(W)$ and $(*)$ follows trivially. For property $(**)$, let $\langle s_n\rangle_{n<\omega}\subseteq \Gamma$ be an increasing sequence and suppose that $s=\sup\langle s_n\rangle_{n<\omega}$, $x\in X$ and $W\in \mathcal{B}_s$ satisfy that $x\in r^{-1}_s(W)$. As $\{r_s:s\in\Gamma\}$ is an $r$-skeleton, $r_{s_n}(x)\rightarrow r_s(x)$. Since $r_s(x)\in W$, we have that there is $m<\omega$, such that for every $n\geq m$, $r_{s_n}(x)\in W$. Let $n\geq m$ and $r^{-1}_{s_n}(W')\in \mathcal{N}(\psi(s_n),x)$, where $W'\in \mathcal{B}_{s_n}$. Since $\{r_{s_n}(x)\}=\bigcap_{U\in \mathcal{N}(\psi(s_{n}),x)}\overline{U\cap F_{s_n}}$, we have that $r_{s_n}(x)\in \overline{r^{-1}_{s_n}(W')\cap F_{s_n}}$. As $r_{s}(r_{s_n}(x))=r_{s_n}(x)$ and $r_{s_n}(x)\in W$ , we have that $r_{s_n}(x)\in r^{-1}_{s}(W)$. Using that $r_{s_n}(x)\in \overline{r^{-1}_{s_n}(W')\cap F_{s_n}}$, we have that $r^{-1}_{s}(W)\cap r^{-1}_{s_n}(W')\cap F_{s_n}'\neq \emptyset$. Since $r^{-1}_{s}(W)\cap r^{-1}_{s_n}(W')\cap F_{s_n}=r^{-1}_{s}(W)\cap F_s\cap r^{-1}_{s_n}(W')\cap F_{s_n}$, we conclude that $\overline{r^{-1}_s(W)\cap F_s}\cap r^{-1}_{s_n}(W')\cap F_s \neq \emptyset$ and then $(**)$ holds. Therefore, $(\{F_s:s\in \Gamma\},\psi)$ is a weak $c$-skeleton on $X$ which satisfies $(*)$ and $(**)$. \end{proof}
The Corollary 5.6 of the paper \cite{casa1} shows that the notion of full $c$-skeleton is equivalent to the notion of full $r$-skeleton; that is, the Corson compact spaces are the spaces which admits a full $c$-skeleton. This result motivates the next characterization of a Valdivia compact space by mean of a special weak $c$-skeleton.
\begin{theorem} For a compact space $X$, the following statements are equivalent: \begin{itemize} \item There is a weak $c$-skeleton $(\{F_s:s\in\Gamma \}, \psi)$ on $X$ which satisfies \begin{itemize}
\item[$(**)$] if $\langle s_n \rangle_{n<\omega}$ is an increasing sequence in $\Gamma$ with $s =\sup\langle s_n\rangle_{n<\omega}$, then for each $x\in X$ and $U\in \mathcal{N}(\psi(s),x)$ there is $m<\omega$ such that $\overline{U\cap F_s}\cap {V\cap F_{s_n}}\neq \emptyset$ for every $n\geq m$ and for every $V\in \mathcal{N}(\psi(s_n),x)$; and
\item[$(***)$] for every $s,t\in \Gamma$, $x\in X$ and $V\in \mathcal{N}(\psi(s),r_t(x))$, there is $U_V\in \mathcal{N}(\psi(t),r_s(x))$ such that $\emptyset\neq \overline{U_V}\cap F_t\cap F_s\subseteq \overline{V}\cap F_s$; and $\{\overline{U_V}\cap F_t\cap F_s: V\in \mathcal{N}(\psi(s),r_t(x))\}$ has the finite intersection property. \end{itemize} \item There is a commutative $r$-skeleton on $X$. \end{itemize} \end{theorem}
\begin{proof} Let us suppose $(\{F_s:s\in\Gamma \}, \psi)$ is a weak $c$-skeleton on $X$ which satisfies $(**)$ and $(***)$. According the proof of the Theorem \ref{teoprincipalceqsresq}, $(\{F_s:s\in\Gamma \}, \psi)$ induces a family of retractions $\{r_s:s\in \Gamma\}$ on $X$ which satisfies $(i)$ and $(iv)$ of the $r$-skeleton definition, $\{r_s(x)\}=\bigcap_{V\in \mathcal{N}(\psi(s),x)}\overline{V\cap F_s}$ for each $s\in \Gamma$ and each $x\in X$; and $r_s=r_t\circ r_s$ provided that $s\leq t$. Next, to prove the commutativity of the retractions (also, this implies $(ii)$) fix $s,t\in \Gamma$. From $(***)$ we have that for each $V\in \mathcal{N}(\psi(s),r_t(x))$ we can find $U_V\in \mathcal{N}(\psi(t),r_s(x))$ so that $\overline{U_V}\cap F_t\cap F_s\subseteq \overline{V}\cap F_s$. As $\{\overline{U_V}\cap F_t\cap F_s:V\in \mathcal{N}(\psi(s),r_t(x))\}$ has the finite intersection property and $\bigcap_{V\in \mathcal{N}(\psi(t),r_s(x)) }\overline{U_V}\cap F_t\cap F_s\subseteq \bigcap_{V\in \mathcal{N}(\psi(t),r_s(x)) }\overline{V}\cap F_s$, we have that $\bigcap_{V\in \mathcal{N}(\psi(t),r_s(x)) }\overline{U_V}\cap F_t\cap F_s=\{r_t(r_x(x))\}$. Now, the equality $\bigcap_{U\in \mathcal{N}(\psi(s),r_t(x)) }\overline{U}\cap F_s = \bigcap_{V\in \mathcal{N}(\psi(t),r_s(x)) }\overline{U_V}\cap F_t\cap F_s $ implies that $\{r_s(r_t(x))\}=\bigcap_{V\in \mathcal{N}(\psi(t),r_s(x)) }\overline{U_V}\cap F_t\cap F_s $. Hence, $r_s(r_t(x))=r_t(r_s(x))$. This proves the commutativity of the retractions and we also obtain condition $(ii)$. Using the condition $(ii)$ and $(**)$ and some ideas from the proof in the Theorem \ref{teoprincipalceqsresq}, we obtain $(iii)$. Therefore, $\{r_s:s\in \Gamma\}$ is a commutative $r$-skeleton.
Now, let us suppose that $\{r_s:s\in \Gamma\}$ is a commutative $r$-skeleton. For each $s\in \Gamma$ define $F_s:=r_s(X)$, choose a countable base $\mathcal{B}_s$ of $F_s$ and define $\psi(s):=r^{-1}_s(\mathcal{B}_s)$. From Theorem \ref{teoprincipalceqsresq} we have that $(\{F_s:s\in\Gamma \},\psi)$ is a weak $c$-skeleton on $X$ with induced set $Y$ which satisfy conditions $(*)$ and $(**)$. It remains to prove condition $(***)$. Let $s,t\in \Gamma$ and $x\in X$. Fix $r^{-1}_s(W),r^{-1}_s(V)\in \mathcal{N}(\psi(s),r_t(x))$ so that $r_t(x)\in r^{-1}_s(W)\subseteq \overline{r^{-1}_s(W))}\subseteq r^{-1}_s(V)$. We have that $r_s(r_t(x))\in W$. Since $r_s$ is a retraction and $W\subseteq F_s$, we obtain that $r_s(r_s(r_t(x)))\in W$. Hence, $r_s(r_t(x))\in r^{-1}_s(W)$. By the commutativity of the $r$-skeleton, we obtain that $ \bigcap_{U\in \mathcal{N}(\psi(t),r_s(x)) }\overline{U}\cap F_t=\{r_t(r_s(x))\}=\{r_s(r_t(x))\}=\bigcap_{W'\in \mathcal{N}(\psi(s),r_t(x)) }\overline{W'}\cap F_s\subseteq r^{-1}_s(W)\cap F_s$. That is, $\bigcap_{U\in \mathcal{N}(\psi(t),r_s(x)) }\overline{U}\cap F_t\subseteq r^{-1}_s(W)\cap F_s$. Since $r_t(r_s(x))=r_s(r_t(x))$, we have that $\bigcap_{U\in \mathcal{N}(\psi(t),r_s(x)) }\overline{U}\cap F_t\cap F_s=\bigcap_{U\in \mathcal{N}(\psi(t),r_s(x)) }\overline{U}\cap F_t\subseteq r^{-1}_s(W)\cap F_s$. By the compactness of $X$, there is $k<\omega$ and $\{U_0,...,U_k\}\subseteq \mathcal{N}(\psi(t),r_s(x))$ such that $\bigcap^{k}_{i=0}\overline{U_i}\cap F_t\cap F_s\subseteq r^{-1}_s(W)\cap F_s$. Since $\psi(t)$ is a base on $X$, there is $U_V\in \mathcal{N}(\psi(t),r_s(x))$ such that $\overline{U_V}\cap F_t\cap F_s\subseteq \bigcap^{k}_{i=0}\overline{U_i}\cap F_t\cap F_s$. Hence, $\overline{U_V}\cap F_t\cap F_s\subseteq r^{-1}_s(W)\cap F_s$. As $\bigcap_{V\in \mathcal{N}(\psi(t),r_s(x)) }\overline{U_V}\cap F_t\cap F_s =\{r_s(r_t(x))\}$, we have that $\{\overline{U_V}\cap F_t\cap F_s: V\in \mathcal{N}(\psi(s),r_t(x))\}$ has the finite intersection property. \end{proof}
In the paper \cite{casa1}, the authors proved that for a compact space $X$, if $X$ has a (full) $c$-skeleton, then $C_p(X)$ has a (full) $q$-skeleton. And, if $X$ has a (full) $q$-skeleton, then $C_p(X)$ has a (full) $c$-skeleton. In this sense, we have the next problem.
\begin{question} Let $X$ be a compact space. Is there a notion weaker than the notion of a $q$-skeleton on $C_p(X)$ that is related to the notion of a weak $c$-skeleton on $X$? \end{question}
In the following two diagrams, we illustrate the connection among all the topological properties that we considered in this article.
The first diagram is about the general case: \begin{displaymath}
\xymatrix{ c\mbox{-skeleton} \ar[d] & \mbox{Corson} \ar[l]\ar[d] \\
r\mbox{-skeleton} \ar[d]\ar[r] & \pi\mbox{-skeleton}\ar[l]\\
\mbox{weak } c\mbox{-skeleton}& \mbox{Valdivia}\ar[l] \ar[u]} \end{displaymath}
This second diagram is a particular case when the skeletons are full. \begin{displaymath}
\xymatrix{\mbox{full } c\mbox{-skeleton} \ar[d]\ar[r] & \mbox{Corson} \ar[l]\ar[d] \\
\mbox{full } r\mbox{-skeleton}\ar[u] \ar[r] & \mbox{full } \pi\mbox{-skeleton}\ar[l]\ar[u] } \end{displaymath}
\end{document} | arXiv |
Orthogonal Bases
Let $V$ be a subspace in $\R^n$.
If a basis $B$ for $V$ is an orthogonal set, then $B$ is called an orthogonal basis.
If a basis $B$ for $V$ is an orthonormal set, then $B$ is called an orthonormal basis.
From any basis $B$ of $V$, the Gram-Schumidt orthogonalization produces an orthogonal basis $B'$ for $V$.
=solution
Using Gram-Schmidt orthogonalization, find an orthogonal basis for the span of the vectors $\mathbf{w}_{1},\mathbf{w}_{2}\in\R^{3}$ if
\mathbf{w}_{1}
\begin{bmatrix}
1 \\ 0 \\ 3
,\quad
2 \\ -1 \\ 0
\mathbf{v}_{1}
1 \\ 1
,\;
1 \\ -1
\] Let $V=\Span(\mathbf{v}_{1},\mathbf{v}_{2})$. Do $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ form an orthonormal basis for $V$? If not, then find an orthonormal basis for $V$.
Let $S=\{\mathbf{v}_1, \mathbf{v}_2\}$ be the set of the following vectors in $\R^4$.
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
\end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix}
\end{bmatrix}.\] Find an orthogonal basis of the subspace $\Span(S)$ of $\R^4$.
See (a)
Let $W$ be a subspace of $\R^4$ with a basis
\[\left\{\, \begin{bmatrix}
\end{bmatrix}, \begin{bmatrix}
\end{bmatrix} \,\right\}.\] Find an orthonormal basis of $W$.
(The Ohio State University)
1 & 0 & 1 \\
0 &1 &0
(a) Find an orthonormal basis of the null space of $A$.
(b) Find the rank of $A$.
(c) Find an orthonormal basis of the row space of $A$.
Let $\mathbf{v}_1=\begin{bmatrix}
2/3 \\ 2/3 \\ 1/3
\end{bmatrix}$ be a vector in $\R^3$. Find an orthonormal basis for $\R^3$ containing the vector $\mathbf{v}_1$.
Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ be a set of nonzero vectors in $\R^n$. Suppose that $S$ is an orthogonal set.
(a) Show that $S$ is linearly independent.
(b) If $k=n$, then prove that $S$ is a basis for $\R^n$.
Linear Algebra Version 0 (11/15/2017)
Introduction to Matrices
Elementary Row Operations
Gaussian-Jordan Elimination
Solutions of Systems of Linear Equations
Linear Combination and Linear Independence
Nonsingular Matrices
Inverse Matrices
Subspaces in $\R^n$
Bases and Dimension of Subspaces in $\R^n$
General Vector Spaces
Subspaces in General Vector Spaces
Linearly Independency of General Vectors
Bases and Coordinate Vectors
Dimensions of General Vector Spaces
Linear Transformation from $\R^n$ to $\R^m$
Linear Transformation Between Vector Spaces
Determinants of Matrices
Computations of Determinants
Introduction to Eigenvalues and Eigenvectors
Eigenvectors and Eigenspaces
Diagonalization of Matrices
The Cayley-Hamilton Theorem
Dot Products and Length of Vectors
Eigenvalues and Eigenvectors of Linear Transformations
Jordan Canonical Form | CommonCrawl |
\begin{document}
\newcommand{{\mathbb R}}{{\mathbb R}} \newcommand{{\mathbb H}}{{\mathbb H}} \newcommand{{\mathbb N}}{{\mathbb N}} \newcommand{{\R^n}}{{{\mathbb R}^n}} \newcommand{{\R_+^n}}{{{\mathbb R}_+^n}} \newcommand{\begin{equation}}{\begin{equation}} \newcommand{\end{equation}}{\end{equation}} \newcommand{\begin{equation*}}{\begin{equation*}} \newcommand{\end{equation*}}{\end{equation*}} \newcommand{\begin{eqnarray}}{\begin{eqnarray}} \newcommand{\end{eqnarray}}{\end{eqnarray}} \newcommand{\begin{eqnarray*}}{\begin{eqnarray*}} \newcommand{\end{eqnarray*}}{\end{eqnarray*}} \newcommand{\alpha}{\alpha} \newcommand{\lambda}{\lambda} \newcommand{\lambda^+_\al}{\lambda^+_\alpha} \newcommand{\lambda^-_\al}{\lambda^-_\alpha} \newcommand{u^+_\al}{u^+_\alpha} \newcommand{u^-_\al}{u^-_\alpha} \newcommand{\Omega}{\Omega} \newcommand{\overline{\Omega}}{\overline{\Omega}} \newcommand{\omega}{\omega} \newcommand{\partial}{\partial} \newcommand{\epsilon}{\epsilon} \newcommand{\overline{x}}{\overline{x}} \newcommand{\overline{y}}{\overline{y}} \newcommand{x_\al}{x_\alpha} \newcommand{\widetilde{g}}{\widetilde{g}} \renewcommand{\noindent\textbf{Proof.} }{\noindent\textbf{Proof.} } \newcommand{\noindent\textbf{Proof} }{\noindent\textbf{Proof} } \newcommand{\finedim}{{\unskip\nobreak\hfil\penalty50
\hskip2em\hbox{}\nobreak\hfil\mbox{$\Box$ \qquad}
\parfillskip=0pt \finalhyphendemerits=0\par
}}
\def{\mathcal A}{{\mathcal A}} \def{\mathcal S}{{\mathcal S}} \def{\mathcal H}{{\mathcal H}} \def{\mathcal{M}_{a,A}^+}{{\mathcal{M}_{a,A}^+}} \def{\mathcal T}{{\mathcal T}} \def{\mathcal I}{{\mathcal I}} \def{\mathcal F}{{\mathcal F}} \def{\mathcal J}{{\mathcal J}} \def{\mathcal E}{{\mathcal E}} \def{\mathcal P}{{\mathcal P}} \def{\mathcal H}{{\mathcal H}} \def{\mathcal V}{{\mathcal V}} \def{\mathcal B}{{\mathcal B}}
\font\teneufm=eufm10 \font\seveneufm=eufm7 \font\fiveeufm=eufm5 \newfam\eufmfam \textfont\eufmfam=\teneufm \scriptfont\eufmfam=\seveneufm \scriptscriptfont\eufmfam=\fiveeufm \def\eufm#1{{\fam\eufmfam\relax#1}} \def{\eufm D}{{\eufm D}} \def{\eufm M}{{\eufm M}}
\newcommand{\average}{{\mathchoice {\kern1ex\vcenter{\hrule height.4pt width 6pt depth0pt} \kern-11pt} {\kern1ex\vcenter{\hrule height.4pt width 4.3pt depth0pt} \kern-7pt} {} {} }} \newcommand{\average\int}{\average\int} \title{A Neumann eigenvalue problem for fully nonlinear operators.} \author{Isabeau Birindelli, Stefania Patrizi} \date{} \maketitle \begin{center} {\em Dedicated to prof. Louis Nirenberg for his $85^{\rm{th}}$ birthday.}\end{center} \section{Introduction} In this introduction and in the rest of the paper we quote some works of Louis Nirenberg that are used explicitly in order to give the right definitions and to prove the results; but the influence of his research, here and in all the papers both the authors have written, goes well beyond the citations. His mathematical ideas have been very important for us, specially for the first named author, but his teaching of how to approach mathematical problems has been as important. We are happy to have this opportunity to thank him for his generosity.
In this paper, for $\Omega$ a $C^2$ bounded domain of ${\mathbb R}^n$ and for any $\alpha>0$, we consider the eigenvalue problem: \begin{equation}\label{000} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2u)+\lambda u=0 & \text{in} \quad\Omega, \\
\frac{\partial u}{\partial \overrightarrow{n}}=\alpha u & \text{on} \quad\partial\Omega, \\
\end{cases} \end{equation} where ${\mathcal{M}_{a,A}^+}$ is the Pucci operator, i.e. $\displaystyle{\mathcal{M}_{a,A}^+}(M)=\sup_{0<aI\leq\sigma\leq AI}\rm{tr}(\sigma M) $.
It is useless to emphasize the importance of the concept of eigenvalue for the understanding of the structural properties of the solutions both for linear and non linear equations. The pioneering work of Berestycki, Nirenberg and Varadhan \cite{BNV} has open the way to enlarge this fundamental concept to non linear operators. Indeed, even if they treat linear equations, their theory is very well adapted to fully nonlinear operators and viscosity solutions being based primarily on the use of the maximum principle. This has been done by many and in many different contests, let us mention the works of Armstrong, Busca, Demengel, Juutinen, Ishii, Quaas, Sirakov, Yoshimura and the authors of this note (\cite{SA,bd,BEQ,IY,J,p2,QS1}). It should be mentioned that P.-L. Lions in \cite{PLL}, with a completely different approach, first introduces what he called demi-eigenvalues. Indeed when the operator is not odd with respect to the Hessian (as is the case of the Pucci operators), eigenvalues corresponding to positive eigenfunctions or to negative eigenfunctions may not coincide and one could interpret these two eigenvalues as a "splitting" of the eigenvalue.
The eigenvalue problem for Robin boundary conditions associated with a fully-nonlinear operator was already treated in \cite{p2}. The novelty here is that we consider $\alpha>0$ which is the "wrong sign" in the sense that the boundary conditions are not "proper", see e.g. \cite{cil}. The boundary source and the reaction-diffusion equation are somehow in competition.
In analogy to \cite{BNV} we define the eigenvalues in the following way:
\begin{equation*}\begin{split}\lambda^+_\al:=\sup\{&\lambda\in {\mathbb R}\;|\;\exists\,v>0 \text{ on $\overline{\Omega}$ bounded viscosity supersolution of }\\& {\mathcal{M}_{a,A}^+}(D^2v)+\lambda v=0 \text{ in } \Omega,\, \frac{\partial v}{\partial \overrightarrow{n}} =\alpha v\text{ on }\partial \Omega \}, \end{split}\end{equation*}
\begin{equation*}\begin{split}\lambda^-_\al:=\sup\{&\lambda\in {\mathbb R}\;|\;\exists\,v<0 \text{ on $\overline{\Omega}$ bounded viscosity subsolution of }\\& {\mathcal{M}_{a,A}^+}(D^2v)+\lambda v=0\text{ in } \Omega,\, \frac{\partial v}{\partial \overrightarrow{n}} =\alpha v\text{ on }\partial \Omega \}. \end{split}\end{equation*}
The first step is to prove that there exists $u^+_\al>0$ and $u^-_\al<0$ solutions of (\ref{000}) when respectively $\lambda=\lambda^+_\al$ and $\lambda=\lambda^-_\al$ (Proposition \ref{esistautof}). We shall also prove that below these eigenvalues there are solutions of the equation with a forcing term $f(x)$ as long as the $f$ has the right sign, i.e. $f\leq 0$ below $\lambda^+_\al$ and $f\geq0$ below $\lambda^-_\al$.
We are mainly interested in the asymptotic behavior with respect to $\alpha$ of the eigenvalues. When $\alpha\rightarrow 0$, $\lambda^+_\al$ and $\lambda^-_\al$ tend to $0$ which is the principal eigenvalue of the pure Neumann boundary problem \begin{equation*} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2u)+\lambda u =0& \text{in} \quad\Omega, \\
\frac{\partial u}{\partial \overrightarrow{n}}=0 & \text{on} \quad\partial\Omega, \\
\end{cases} \end{equation*}
But our main goal is to study the behavior when $\alpha\rightarrow+\infty$, this is done in our main
\begin{thm}\label{mainthm} The following limits hold: \begin{equation}\label{lamsbehavior}\lim_{\alpha\rightarrow+\infty}\frac{\lambda^+_\al}{-\alpha^2}=A,\end{equation} \begin{equation}\label{lamsobehavior}\lim_{\alpha\rightarrow+\infty}\frac{\lambda^-_\al}{-\alpha^2}=a.\end{equation} \end{thm} Interestingly this asymptotic behavior emphasizes the "splitting" of the eigenvalue. In the linear case, i.e. when $a=A=1$ and the Pucci operator is nothing else but the Laplacian, this problem was treated in \cite{lz} by Lou and Zhu with a variational approach. Very recently Daners and Kennedy \cite{dk} have proved that this asymptotic behavior is valid for the whole spectrum.
We also prove that for any $K\subset\subset\Omega$, the normalized eigenfunctions $u^+_\al$ and $u^-_\al$ satisfy
$$\|u^+_\al\|_{L^\infty(K)}\rightarrow 0\quad\mbox{ and }\quad \|u^-_\al\|_{L^\infty(K)}\rightarrow 0\quad\text{as }\alpha\rightarrow+\infty.$$ So that the eigenfunctions tend to concentrate on the point of the boundary where they reach the sup or the inf.
The idea of the proof of Theorem \ref{mainthm} which somehow follows the line adopted in \cite{lz}, is the following: first we establish that $u^+_\al$ reaches its maximum on the boundary and then we perform a blow up around this point.
Then a key tool will be a Liouville result in the half space (Theorem \ref{halfspacethm}). Precisely we prove that for $\gamma>A$ (respectively $\gamma>a$) there are no bounded subsolutions (respectively supersolutions) of $$ \left\{ \begin{array}{ll}
{\mathcal{M}_{a,A}^+}(D^2 u)- \gamma u=0 & \hbox{in }{\R_+^n}, \\
-\frac{\partial u}{\partial x_n}=u & \hbox{on } \partial{\R^n}.\\ \end{array} \right. $$
that are positive (repectively negative) somewhere. In \cite{lz} the analogous result for the Laplacian is proved using the construction of sub and super solutions in the flavor of what is done in \cite{bcn}.
Let us mention here that it would be interesting to extend the results of Berestycki, Caffarelli, Nirenberg \cite{bcn} in half spaces,
to this class of fully-nonlinear operators and to these boundary conditions.
Lipschitz estimates up to the boundary will be required in the proofs of both the existence results and the asymptotic behavior. These estimates which are interesting in their own right, are established here using an argument inspired by \cite{il} (see also Barles and Da Lio \cite{bdl} and Milakis and Silvestre \cite{ms} ).
In the whole paper the fully-nonlinear operator considered is the Pucci operator ${\mathcal{M}_{a,A}^+}$, but, mutatis mutandis, parallel results can be stated for the Pucci operator ${\mathcal{M}_{a,A}^-}$ defined by $\displaystyle{\mathcal{M}_{a,A}^-}(M)=\inf_{0<aI\leq\sigma\leq AI}\rm{tr}(\sigma M) $.
\section{Preliminary results} Let us recall the definition of viscosity sub and supersolution of the Neumann problem associated to a general elliptic operator $F:\overline{\Omega}\times{\mathbb R}\times{\mathbb R}^n \times \emph{S(n)}\rightarrow{\mathbb R}$. Here $\emph{S(n)}$ is the space of symmetric matrices on ${\mathbb R}^n$, equipped with the usual ordering. We denote by $USC(\overline{\Omega})$ (resp., $LSC(\overline{\Omega})$) the set of upper (resp., lower) semicontinuous functions on $\overline{\Omega}$. Let $f:\overline{\Omega}\rightarrow{\mathbb R}$, $g:\partial\Omega\times{\mathbb R}\rightarrow{\mathbb R}$. \begin{de} A function $u\in USC(\overline{\Omega})$ (resp., $u\in LSC(\overline{\Omega})$ ) is called \emph{viscosity subsolution} (resp., \emph{supersolution}) of \begin{equation*} \begin{cases}
F(x,u,Du,D^2u)=f(x) & \text{in} \quad\Omega, \\
\frac{\partial u}{\partial \overrightarrow{n}}=g(x,u) & \text{on} \quad\partial\Omega, \\
\end{cases}
\end{equation*}if the following conditions hold \begin{itemize} \item[(i)] For every $x_0\in \Omega$, for any $\varphi\in C^2(\overline{\Omega})$, such that $u-\varphi$ has a local maximum (resp., minimum) at $x_0$ then $$F(x_0,u(x_0),D\varphi(x_0),D^2\varphi(x_0))\geq \,(\text{resp., } \leq\,)\,f(x_0).$$ \item[(ii)]For every $x_0\in \partial\Omega$, for any $\varphi\in C^2(\overline{\Omega})$, such that $u-\varphi$ has a local maximum (resp., minimum) at $x_0$ then $$-(F(x_0,u(x_0),D\varphi(x_0),D^2\varphi(x_0))-f(x_0))\wedge \left(\frac{\partial \varphi}{\partial \overrightarrow{n}}(x_0)-g(x_0,u(x_0))\right)\leq 0$$(resp., $$-(F(x_0,u(x_0),D\varphi(x_0),D^2\varphi(x_0))-f(x_0))\vee \left(\frac{\partial \varphi}{\partial \overrightarrow{n}}(x_0)-g(x_0,u(x_0))\right)\geq 0).$$\end{itemize} A \emph{viscosity solution} is a continuous function which is both a subsolution and a supersolution. \end{de} One of the motivation for these relaxed boundary conditions is the stability under uniform convergence. Actually, if the domain $\Omega$ satisfies the exterior sphere condition and $F$ is uniformly elliptic, viscosity subsolutions (resp., supersolutions) satisfy in the viscosity sense $\frac{\partial u}{\partial \overrightarrow{n}}\leq $ (resp., $\geq$ )$g(x,u)$ for any $x\in\partial\Omega$, see e.g. Proposition 2.1 in \cite{p2}.
We assume throughout the paper that $\Omega$ is a bounded domain of ${\R^n}$ of class $C^2$. \begin{thm}[Strong Comparison Principle, \cite{p2} Theorem 3.1]\label{stcompneu} Assume that $c$ and $f$ are continuous on $\overline{\Omega}$. Let $u\in USC(\overline{\Omega})$ and $v\in LSC(\overline{\Omega})$ be respectively a sub and a supersolution of \begin{equation*} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2u)+c(x)u= f(x) & \text{in} \quad\Omega, \\
\frac{\partial u}{\partial \overrightarrow{n}} =\alpha u & \text{on} \quad\partial\Omega. \\
\end{cases}
\end{equation*} If $u\leq v$ on $\overline{\Omega}$ then either $u<v$ on $\overline{\Omega}$ or $u\equiv v$ on $\overline{\Omega}$. \end{thm}
\begin{prop}[Maximum Principle for $\lambda<\lambda^+_\al$, \cite{p2} Theorem 4.5]\label{maxprinc} Assume $\lambda<\lambda^+_\al$. Let $v\in USC(\overline{\Omega})$ be a viscosity subsolution of \begin{equation}\label{sysmaxp} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2v)+\lambda v=0 & \text{in} \quad\Omega, \\
\frac{\partial v}{\partial \overrightarrow{n}} =\alpha v & \text{on} \quad\partial\Omega, \\
\end{cases}
\end{equation} then $v\leq 0$ on $\overline{\Omega}$. \end{prop} \begin{prop}[Minimum Principle for $\lambda<\lambda^-_\al$, \cite{p2} Remark 4.6]\label{minprinc} Assume $\lambda<\lambda^-_\al$. Let $v\in LSC(\overline{\Omega})$ be a viscosity supersolution of \begin{equation*} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2v)+\lambda v=0 & \text{in} \quad\Omega, \\
\frac{\partial v}{\partial \overrightarrow{n}} =\alpha v & \text{on} \quad\partial\Omega, \\
\end{cases}
\end{equation*} then $v\geq 0$ on $\overline{\Omega}$. \end{prop}
\section{Lipschitz estimates} In this section we shall prove a local Lipschitz regularity result for solutions of the Neumann problem associated to general uniformly elliptic operators, that we will use in the next sections.
Let us consider the Neumann problem \begin{equation}\label{lipgensys} \begin{cases}
F(x,u,Du,D^2u)= f(x) & \text{in} \quad\Omega, \\
\frac{\partial u}{\partial\overrightarrow{n}}= g(x) & \text{on} \quad\partial\Omega, \\
\end{cases}
\end{equation} where the operator $F$ is supposed to be continuous on $\overline{\Omega}\times{\mathbb R}\times{\mathbb R}^n \times \emph{S(n)}$ and satisfying the following assumptions: \renewcommand{(F\arabic{enumi})}{(F\arabic{enumi})} \begin{enumerate}
\item There exist $b,c>0$ such that for $x\in\overline{\Omega},\,r,s\in{\mathbb R},\, p,q\in {\mathbb R}^n,\, X, Y\in \emph{S(n)}$
\begin{equation*}\begin{split} \mathcal{M}_{a,A}^-(Y-X)-b|p-q|-c|r-s|&\leq
F(x,r,p,Y)-F(x,s,q,X)\\&
\leq \mathcal{M}_{a,A}^+(Y-X)+ b|p-q|+c|r-s|.\end{split}\end{equation*}
\item There exists $C_1>0$ such that for all $x,y\in\overline{\Omega}$ and $X\in\emph{S(n)}$
$$|F(x,0,0,X)-F(y,0,0,X)|\leq C_1|x-y|^\frac{1}{2}\|X\|.$$ \end{enumerate}
\begin{prop}\label{regolaritaloc} Assume that (F1) and (F2)
hold. Let $f:\overline{\Omega}\rightarrow{\mathbb R}$ be bounded, $g:\partial\Omega\rightarrow{\mathbb R}$ be Lipschitz continuous. Let $u\in C(\overline{\Omega})$ be a viscosity solution of \eqref{lipgensys}, then, for any $x_0\in\overline{\Omega}$ and for any $\rho>0$, there exists $K>0$ such that
\begin{equation}\label{uliploc}|u(x)-u(y)|\leq (MK+|g|_{L^\infty(\partial\Omega)})|x-y|\quad \forall x,y\in B_\rho(x_0)\cap\overline{\Omega}\end{equation} and
\begin{equation}\label{klipestlocal}\begin{split} K^2-b K&
\leq C\left[c|u|_{L^\infty(\overline{B}_{3\rho}(x_0)\cap\overline{\Omega})}
+|f|_{L^\infty(\overline{\Omega})}
+\left(1+b\right)|g|_{C^{0,1}(\partial\Omega)}+\frac{b}{\rho}+\frac{1}{\rho^2}+1\right],\end{split}\end{equation} where $M\leq C(|u|_{L^\infty(\overline{B}_{3\rho}(x_0)\cap\overline{\Omega})}+|g|_{L^\infty(\partial\Omega)}+1)$
and $C$ depends on $a,\,A,\,C_1,\,n$ and $\Omega$.
\end{prop}
\begin{cor}\label{corregu}Assume $\lambda\in{\mathbb R}$ and $\alpha\geq0$. Let $u\in C(\overline{\Omega})$ be a viscosity solution of \begin{equation}\label{generallameq} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2u)+\lambda u=f(x) & \text{in} \quad\Omega, \\
\frac{\partial u}{\partial \overrightarrow{n}} =\alpha u & \text{on} \quad\partial\Omega. \\
\end{cases}
\end{equation}
Then, for any $\rho>0$, there exists $K>0$ such that for any $x,y\in\Omega_\rho:=\{x\in\overline{\Omega}\,|\,d(x)\leq\rho\}$
$$|u(x)-u(y)|\leq (\alpha|e^{\alpha
d(x)}u|_{L^\infty(\Omega_\rho)}+MK)|x-y|$$ and
\begin{equation}\label{kvlocal}K^2-C\alpha K\leq C\left[\left(\alpha+\alpha^2+|\lambda|\right)|e^{\alpha
d(x)}u|_{L^\infty(\Omega_{3\rho})}+ |e^{\alpha
d(x)}f|_{L^\infty(\overline{\Omega})}+\frac{\alpha}{\rho}+\frac{1}{\rho^2}+1\right],\end{equation} where $M\leq C(|e^{\alpha
d(x)}u|_{L^\infty(\Omega_{3\rho})}+1)$ and $C$ depends on $a,A,n$ and $\Omega$.
\end{cor}
\noindent\textbf{Proof} {\bf of Proposition \ref{regolaritaloc}}
We follow the proof of Proposition III.1 of \cite{il}, that we modify taking test functions which depend on the distance function and that are suitable
for the Neumann boundary conditions. Moreover, as in \cite{bdl}, we are going to use a regularization of
$g$. In order to do so, it is convenient to introduce the
following classical lemma.
\begin{lem}\label{gextension}Assume $\rho\in C^\infty({\mathbb R}^n)$, $\rho>0$,
supp$(\rho)\subset B_1(0)$ and $\int_{{\mathbb R}^n}\rho(y)dy=1$. If $g\in
C^{0,1}({\mathbb R}^n)$ and $g$ is bounded, then the function
$\widetilde{g}:{\mathbb R}^n\times[0,+\infty)\rightarrow{\mathbb R}$ defined by
\begin{equation*}\widetilde{g}(x,\epsilon):=\int_{{\mathbb R}^n}g(z)\rho\left(\frac{x-z}{\epsilon}\right)\frac{1}{\epsilon^n}dz,\,\epsilon>0,\end{equation*}
\begin{equation*} \widetilde{g}(x,0)=g(x)\quad\text{for }x\in{\mathbb R}^n,\end{equation*} is in
$C^{0,1}({\mathbb R}^n\times[0,+\infty))$. Moreover, the function
$\widetilde{g}$ is in $C^2({\mathbb R}^n\times(0,+\infty))$ with
\begin{equation*}|D_x\widetilde{g}(x,\epsilon)|,\,|D_\epsilon\widetilde{g}(x,\epsilon)|\leq
C_0,\end{equation*}
\begin{equation*}
|D^2_{xx}\widetilde{g}(x,\epsilon)|,\,|D^2_{x\epsilon}\widetilde{g}(x,\epsilon)|,\,|D^2_{\epsilon\epsilon}\widetilde{g}(x,\epsilon)|\leq
\frac{C_0}{\epsilon}\quad\text{in }{\mathbb R}^n\times(0,+\infty)\end{equation*} for some
positive constant $C_0\leq C|g|_{C^{0,1}({\R^n})}$, with $C$ depending only on $\rho$ and $n$.
\end{lem}
We first extend $g$ to a $C^{0,1}$ function of ${\mathbb R}^n$ and we still denote by $g$ this extension. Then, we consider the function $\widetilde{g}$ associated to $g$ as in Lemma \ref{gextension}.
Since $\Omega$ is a domain of class $C^2$, it satisfies the uniform exterior sphere condition, i.e., there exists $r>0$ such that $B(x+r\overrightarrow{n}(x),r)\cap \Omega =\emptyset$ for any $x\in \partial\Omega.$ From this property it follows that \begin{equation}\label{sferaest} \langle
\overrightarrow{n}(x),y-x\rangle\leq\frac{1}{2r}|y-x|^2\quad\text{for }x\in\partial\Omega \text{ and } y\in\overline{\Omega}.\end{equation} Moreover, the $C^2$-regularity of $\Omega$ implies the existence of a neighborhood of $\partial\Omega$ in $\overline{\Omega}$ on which the distance from the boundary
$$d(x):=\inf\{|x-y|, y\in\partial\Omega\},\quad x\in\overline{\Omega}$$ is of class $C^2$. We still denote by $d$ a
$C^2$ extension of the distance function to the whole $\overline{\Omega}$. Without loss of generality we can assume that $|Dd(x)|\leq1$ on $\overline{\Omega}$.
Let $x_0\in\overline{\Omega}$ and $\rho>0$. Let us denote $\overline{B}_{\overline{\Omega}}(x_0,\rho):=\overline{B}_\rho(x_0)\cap\overline{\Omega}$ and $B_{\overline{\Omega}}(x_0,\rho):=B_\rho(x_0)\cap\overline{\Omega}$.
We are going to show that $u$ is Lipschitz continuous on $\overline{B}_{\overline{\Omega}}(x_0,\rho)$. For this aim, let us introduce the functions
$$\Phi(x)=MK|x|-M(K|x|)^2,$$ $$\Psi_1(x,y)=e^{-L(d(x)+d(y))}\Phi(x-y),$$ $$\Psi_2(x,y)=\widetilde{g}\left(\frac{x+y}{2},
(\delta^2+|x-y|^2)^\frac{1}{2}\right)(d(x)-d(y)),$$ and $$\varphi (x,y)=\Psi_1(x,y)-\Psi_2(x,y),$$ where $L$ is a fixed number greater than $\frac{1}{r}$ with $r$ the radius in \eqref{sferaest}, $K$ and $M$ are positive constants to be chosen later and $\delta$ is a small parameter. We also use the notation $$\widetilde{g}(Z,T)=\widetilde{g}\left(\frac{x+y}{2},
(\delta^2+|x-y|^2)^\frac{1}{2}\right).$$ If $K|x|\leq \frac{1}{4}$, then \begin{equation}\label{phimagg}\Phi(x)\geq
\frac{3}{4}MK|x|.\end{equation} We define
$$\Delta _K:=\left\{(x,y)\in {\mathbb R}^n\times{\mathbb R}^n:\, |x-y|\leq\frac{1}{4K}\right\}.$$ We fix $M>1$ and $j>0$ such that
\begin{equation}\label{M}\max_{\overline{B}_{\overline{\Omega}}(x_0,\rho)^{\,2}}|u(x)-u(y)|+2d_0(|g|_\infty+C_0\delta)\leq e^{-2Ld_0}\frac{M}{8},\end{equation} $$j=\frac{M}{\rho^2},$$ where $d_0=\max_{x\in\overline{\Omega}}d(x),$ and we claim that taking $K$ large enough, for any small $\delta$ one has \begin{equation}\label{u-vindelta}
u(x)-u(y)-\varphi(x,y)-je^{-Ld(x)}|x-x_0|^2\leq 0\quad\text{for }(x,y)\in \Delta_K\cap\overline{B}_{\overline{\Omega}}(x_0,\rho)^2. \end{equation}
To show \eqref{u-vindelta} we suppose by contradiction that the maximum of $u(x)-u(y)-\varphi(x,y)-je^{-Ld(x)}|x-x_0|^2$ on $\Delta_K\cap\overline{B}_{\overline{\Omega}}(x_0,\rho)^2$ is positive. Then, for $\delta$ small enough, there is $(\overline{x},\overline{y})\in\Delta_K\cap \overline{B}_{\overline{\Omega}}(x_0,\rho)^2$ such that $\overline{x}\neq \overline{y}$ and \begin{equation}\label{u-vcontr}\begin{split} u(\overline{x})- u(\overline{y})-\widetilde{\varphi}(\overline{x},\overline{y}) =\max_{\Delta_K\cap \overline{B}_{\overline{\Omega}}(x_0,\rho)\,^2}( u(x)- u(y)-\widetilde{\varphi}(x,y))>0,\end{split}\end{equation} where
$$\widetilde{\varphi}(x,y)=\varphi(x,y)+je^{-Ld(x)}|x-x_0|^2-C_0\delta(d(x)+d(y)),$$with $C_0$ the constant defined as in Lemma \ref{gextension}.
The point $(\overline{x},\overline{y})$ belongs to $\text{int}(\Delta_K)\cap B_{\overline{\Omega}}(x_0,\rho)^2$. Indeed, if $|x-y|=\frac{1}{4K}$, by \eqref{M} and \eqref{phimagg}, we have \begin{equation*}\begin{split}
u(x)- u(y)&\leq e^{-2Ld_0}
\frac{M}{8}-2d_0(|g|_\infty+C_0\delta)\leq e^{-L(d(x)+d(y))}\frac{1}{2}MK|x-y|-\Psi_2(x,y)\\&-C_0\delta(d(x)+d(y))\leq\widetilde{\varphi}(x,y).\end{split}\end{equation*}
On the other hand, if $|x-x_0|=\rho$, then \begin{equation*}\begin{split}
u(x)- u(y)&\leq e^{-Ld_0}M-2d_0(|g|_\infty+C_0\delta)\leq
e^{-Ld(x)}\frac{M}{\rho^2}|x-x_0|^2+\Psi_1(x,y)-\Psi_2(x,y)\\&-C_0\delta(d(x)+d(y))=\widetilde{\varphi}(x,y).\end{split}\end{equation*}
Similarly, if $|y-x_0|=\rho$ and $K>K_0/\rho$, for some constant $K_0>0$, then $ u(x)- u(y)\leq \widetilde{\varphi}(x,y)$.
Hence, $(\overline{x},\overline{y})\in\text{int}(\Delta_K)\cap B_{\overline{\Omega}}(x_0,\rho)^2$.
Since $\overline{x}\neq \overline{y}$ we can compute the derivatives of $\widetilde{\varphi}$ at $(\overline{x},\overline{y})$ obtaining
\begin{equation*}\begin{split}D_x\widetilde{\varphi}(\overline{x},\overline{y})=&-Le^{-L(d(\overline{x})+d(\overline{y}))}MK|\overline{x}-\overline{y}|(1-K|\overline{x}-\overline{y}|)Dd(\overline{x})\\&+
e^{-L(d(\overline{x})+d(\overline{y}))}MK(1-2K|\overline{x}-\overline{y}|)\frac{(\overline{x}-\overline{y})}{|\overline{x}-\overline{y}|}-C_0\delta Dd(\overline{x})\\&-jLe^{-Ld(\overline{x})}|\overline{x}-x_0|^2Dd(\overline{x})+2je^{-Ld(\overline{x})}(\overline{x}-x_0)-D_x\Psi_2(\overline{x},\overline{y}),\end{split}\end{equation*}
\begin{equation*}\begin{split}D_y\widetilde{\varphi}(\overline{x},\overline{y})=&-Le^{-L(d(\overline{x})+d(\overline{y}))}MK|\overline{x}-\overline{y}|(1-K|\overline{x}-\overline{y}|)Dd(\overline{y})\\&-e^{-L(d(\overline{x})+d(\overline{y}))}MK
(1-2K|\overline{x}-\overline{y}|)\frac{(\overline{x}-\overline{y})}{|\overline{x}-\overline{y}|}-C_0\delta Dd(\overline{y})\\&-D_y\Psi_2(\overline{x},\overline{y}),\end{split}\end{equation*}where \begin{equation*}\begin{split}D_x\Psi_2(\overline{x},\overline{y})&=\frac{d(\overline{x})-d(\overline{y})}{2}D_Z\widetilde{g}\left(\frac{\overline{x}+\overline{y}}{2},
(\delta^2+|\overline{x}-\overline{y}|^2)^\frac{1}{2}\right)\\&+(d(\overline{x})-d(\overline{y}))\frac{(\overline{x}-\overline{y})}{(\delta^2+|\overline{x}-\overline{y}|^2)^\frac{1}{2}}D_T\widetilde{g}\left(\frac{\overline{x}+\overline{y}}{2},
(\delta^2+|\overline{x}-\overline{y}|^2)^\frac{1}{2}\right) \\&+\widetilde{g}\left(\frac{\overline{x}+\overline{y}}{2},
(\delta^2+|\overline{x}-\overline{y}|^2)^\frac{1}{2}\right)Dd(\overline{x})\end{split}\end{equation*}and \begin{equation*}\begin{split}D_y\Psi_2(\overline{x},\overline{y})&=\frac{d(\overline{x})-d(\overline{y})}{2}D_Z\widetilde{g}\left(\frac{\overline{x}+\overline{y}}{2},
(\delta^2+|\overline{x}-\overline{y}|^2)^\frac{1}{2}\right)\\&-(d(\overline{x})-d(\overline{y}))\frac{(\overline{x}-\overline{y})}{(\delta^2+|\overline{x}-\overline{y}|^2)^\frac{1}{2}}D_T\widetilde{g}\left(\frac{\overline{x}+\overline{y}}{2},
(\delta^2+|\overline{x}-\overline{y}|^2)^\frac{1}{2}\right) \\&-\widetilde{g}\left(\frac{\overline{x}+\overline{y}}{2},
(\delta^2+|\overline{x}-\overline{y}|^2)^\frac{1}{2}\right)Dd(\overline{y}).\end{split}\end{equation*}
Observe that \begin{equation}\label{dxphistima}
|D_x\widetilde{\varphi}(\overline{x},\overline{y})|,|D_y\widetilde{\varphi}(\overline{x},\overline{y})|\leq C(MK+C_0+j\rho).\end{equation} Here and henceforth C denotes various positive constants independent of $K,b,c,f,g$ and $u$.
By Lemma \ref{gextension} \begin{equation*}
\left|\widetilde{g}\left(\frac{\overline{x}+\overline{y}}{2},
(\delta^2+|\overline{x}-\overline{y}|^2)^\frac{1}{2}\right)-g(\overline{x})\right|\leq C_0(2|\overline{x}-\overline{y}|+\delta),\end{equation*} then, if $\overline{x}\in\partial \Omega$ we have \begin{equation*}\begin{split}-\langle\overrightarrow{n}(\overline{x}),D_x\Psi_2(\overline{x},\overline{y})\rangle-g(\overline{x})\geq
-C_0(4|\overline{x}-\overline{y}|+\delta).\end{split}\end{equation*}
Hence, using \eqref{sferaest}, if $\overline{x}\in\partial \Omega$ we get \begin{equation}\label{condbordo}\begin{split} \langle \overrightarrow{n}(\overline{x}),D_x\widetilde{\varphi}(\overline{x},\overline{y})\rangle
-g(\overline{x})&=Le^{-Ld(\overline{y})}MK|\overline{x}-\overline{y}|(1-K|\overline{x}-\overline{y}|)
\\&+e^{-Ld(\overline{y})}MK(1-2K|\overline{x}-\overline{y}|)\langle
\overrightarrow{n}(\overline{x}),\frac{(\overline{x}-\overline{y})}{|\overline{x}-\overline{y}|}\rangle \\&
+jL|\overline{x}-x_0|^2+2j\langle \overrightarrow{n}(\overline{x}),\overline{x}-x_0\rangle\\& -\langle\overrightarrow{n}(\overline{x}),D_x\Psi_2(\overline{x},\overline{y})\rangle-g(\overline{x})+C_0\delta\\&\geq
\frac{1}{2}e^{-Ld(\overline{y})}MK|\overline{x}-\overline{y}|\left(\frac{3}{2}L-\frac{1}{r}\right)+j|\overline{x}-x_0|^2\left(L-\frac{1}{r}\right)\\&-4C_0|\overline{x}-\overline{y}|>0, \end{split}\end{equation} for $MK>\frac{16rC_0e^{Ld_0}}{(3rL-2)}$, since $\overline{x}\neq\overline{y}$ and $L>\frac{1}{r}$. Similarly, if $\overline{y}\in\partial\Omega$ then \begin{equation*} \langle \overrightarrow{n}(\overline{y}),-D_y\widetilde{\varphi}(\overline{x},\overline{y})\rangle -g(\overline{y})\leq
\frac{1}{2}e^{-Ld(\overline{x})}MK|\overline{x}-\overline{y}|\left(-\frac{3}{2}L+\frac{1}{r}\right)+4C_0|\overline{x}-\overline{y}|<0.\end{equation*} Then, by definition of sub and supersolution $$F(\overline{x}, u(\overline{x}),D_x\widetilde{\varphi}(\overline{x},\overline{y}),X)\geq f(\overline{x}),\quad\text{if }(D_x\widetilde{\varphi}(\overline{x},\overline{y}),X)\in \overline{J}^{2,+} u(\overline{x}),$$ $$F(\overline{y}, u(\overline{y}),-D_y\widetilde{\varphi}(\overline{x},\overline{y}),Y)\leq
f(\overline{y})\quad\text{if }(-D_y\widetilde{\varphi}(\overline{x},\overline{y}),Y)\in \overline{J}^{2,-} u(\overline{y}).$$ Since $(\overline{x},\overline{y})\in\text{int}\Delta_K\cap B_{\overline{\Omega}}(x_0,\rho)\,^2$, it is a local maximum point of $ u(x)- u(y)-\widetilde{\varphi}(x,y)$ in $\overline{\Omega}\,^2$. Then applying Theorem 3.2 in \cite{cil}, for every $\epsilon>0$ there exist $X,Y\in \emph{S(n)}$ such that $$
(D_x\widetilde{\varphi}(\overline{x},\overline{y}),X-C_0\delta D^2d(\overline{x})+D^2(je^{-Ld(x)}|x-x_0|^2))\in \overline{J}\,^{2,+} u(\overline{x}),$$ $$(-D_y\widetilde{\varphi}(\overline{x},\overline{y}),Y+C_0\delta D^2d(\overline{y}))\in \overline{J}\,^{2,-} u(\overline{y})$$ and \begin{equation}\label{tm2ishii}\begin{split}\left( \begin{array}{cc}
X & 0 \\
0 & -Y \\ \end{array} \right)&\leq D^2\varphi(\overline{x},\overline{y})+\epsilon (D^2\varphi(\overline{x},\overline{y}))^2\\&\leq D^2\Psi_1(\overline{x},\overline{y})-D^2\Psi_2(\overline{x},\overline{y})+2\epsilon(D^2\Psi_1(\overline{x},\overline{y}))^2+2\epsilon(D^2\Psi_2(\overline{x},\overline{y}))^2. \end{split}\end{equation} Now we want to estimate the matrix on the right-hand side of the last inequality.
Using Lemma \ref{gextension}, it is easy to check that \begin{equation}\label{psi2hessian}-CC_0\left(\begin{array}{cc}
I & 0 \\
0 & I \\ \end{array} \right)\leq D^2\Psi_2(\overline{x},\overline{y})\leq CC_0\left(\begin{array}{cc}
I & 0 \\
0 & I \\ \end{array} \right).\end{equation} Next, let us estimate $D^2\Psi_1(\overline{x},\overline{y})$. \begin{equation*}\begin{split}D^2\Psi_1(\overline{x},\overline{y})&=\Phi(\overline{x}-\overline{y})D^2(e^{-L(d(\overline{x})+d(\overline{y}))})+D(e^{-L(d(\overline{x})+d(\overline{y}))})\otimes D(\Phi(\overline{x}-\overline{y}))\\&+D(\Phi(\overline{x}-\overline{y}))\otimes D(e^{-L(d(\overline{x})+d(\overline{y}))})+e^{-L(d(\overline{x})+d(\overline{y}))}D^2(\Phi(\overline{x}-\overline{y})).\end{split}\end{equation*}We set $$A_1:=\Phi(\overline{x}-\overline{y})D^2(e^{-L(d(\overline{x})+d(\overline{y}))}),$$ $$A_2:=D(e^{-L(d(\overline{x})+d(\overline{y}))})\otimes D(\Phi(\overline{x}-\overline{y}))+D(\Phi(\overline{x}-\overline{y}))\otimes D(e^{-L(d(\overline{x})+d(\overline{y}))}),$$ $$A_3:=e^{-L(d(\overline{x})+d(\overline{y}))}D^2(\Phi(\overline{x}-\overline{y})).$$Observe that
\begin{equation}\label{a1}A_1\leq CMK|\overline{x}-\overline{y}|\left(\begin{array}{cc}
I & 0 \\
0 & I \\ \end{array} \right).\end{equation}
For $A_2$ we have the following estimate \begin{equation}\label{a2}A_2\leq CMK\left( \begin{array}{cc}
I & 0 \\
0 & I \\ \end{array} \right) +CMK\left( \begin{array}{cc}
I & -I \\
-I & I \\ \end{array} \right)\leq CMK\left( \begin{array}{cc}
I & 0 \\
0 & I \\ \end{array} \right).\end{equation} Indeed for $\xi,\,\eta\in{\mathbb R}^n$ we compute
\begin{equation*}\begin{split}\langle A_2(\xi,\eta),(\xi,\eta)\rangle&=2Le^{-L(d(\overline{x})+d(\overline{y}))}\{\langle Dd(\overline{x})\otimes D\Phi(\overline{x}-\overline{y})(\eta-\xi),\xi\rangle\\&+\langle Dd(\overline{y})\otimes D\Phi(\overline{x}-\overline{y})(\eta-\xi),\eta\rangle\}\\&\leq CMK(|\xi|+|\eta|)|\eta-\xi|\\&\leq CMK(|\xi|^2+|\eta|^2)+CMK|\eta-\xi|^2.\end{split}\end{equation*} Now we consider $A_3$. The matrix $D^2(\Phi(\overline{x}-\overline{y}))$ has the form $$D^2(\Phi(\overline{x}-\overline{y}))=\left( \begin{array}{cc}
D^2\Phi(\overline{x}-\overline{y}) & - D^2\Phi(\overline{x}-\overline{y}) \\
- D^2\Phi(\overline{x}-\overline{y}) & D^2\Phi(\overline{x}-\overline{y}) \\ \end{array} \right),$$and the Hessian matrix of $\Phi(x)$ is
\begin{equation}\label{hessianphi}D^2\Phi(x)=\frac{MK}{|x|}\left(I-\frac{x\otimes x}{|x|^2}\right)-2MK^2I.\end{equation} If we choose
\begin{equation}\label{epsilon}\epsilon=\frac{|\overline{x}-\overline{y}|}{4MKe^{-L(d(\overline{x})+d(\overline{y}))}},\end{equation} then we have the following estimates
$$\epsilon A_1^2\leq CMK|\overline{x}-\overline{y}|^3I_{2n},\quad \epsilon A_2^2\leq CMK|\overline{x}-\overline{y}|I_{2n},$$
\begin{equation}\label{aprodotti}\begin{split} \epsilon (A_1A_2+A_2A_1)\leq CMK|\overline{x}-\overline{y}|^2I_{2n},\end{split}\end{equation}
$$\epsilon (A_1A_3+A_3A_1)\leq CMK|\overline{x}-\overline{y}|I_{2n},\quad \epsilon (A_2A_3+A_3A_2)\leq CMKI_{2n},$$
where $I_{2n}:=\left( \begin{array}{cc}
I & 0 \\
0 & I \\ \end{array} \right)$. Then using \eqref{psi2hessian}, \eqref{a1}, \eqref{a2}, \eqref{aprodotti} and observing that $$(D^2(\Phi(\overline{x}-\overline{y})))^2=\left( \begin{array}{cc}
2(D^2\Phi(\overline{x}-\overline{y}))^2 & - 2(D^2\Phi(\overline{x}-\overline{y}))^2 \\
- 2(D^2\Phi(\overline{x}-\overline{y}))^2 & 2(D^2\Phi(\overline{x}-\overline{y}))^2 \\ \end{array} \right),$$from \eqref{tm2ishii} we can conclude that $$\left( \begin{array}{cc}
X & 0 \\
0 & -Y \\ \end{array} \right)\leq (MO(K)+CC_0)\left( \begin{array}{cc}
I & 0 \\
0 & I \\ \end{array} \right)+\left( \begin{array}{cc}
B & -B \\
-B & B \\ \end{array} \right),$$ where
\begin{equation}\label{matriceB}B=e^{-L(d(\overline{x})+d(\overline{y}))}\left[D^2\Phi(\overline{x}-\overline{y})+\frac{|\overline{x}-\overline{y}|}{MK} (D^2\Phi(\overline{x}-\overline{y}))^2\right].\end{equation}The last inequality can be rewritten as follows $$\left( \begin{array}{cc}
\widetilde{X} & 0 \\
0 & -\widetilde{Y} \\ \end{array} \right)\leq\left( \begin{array}{cc}
B & -B \\
-B & B \\ \end{array} \right),$$ with $\widetilde{X}=X-(MO(K)+CC_0)I$ and $\widetilde{Y}=Y+(MO(K)+CC_0)I.$
Now we want to get a good estimate for tr($\widetilde{X}-\widetilde{Y}$), as in \cite{il}. For that aim let
$$0\leq P:=\frac{(\overline{x}-\overline{y})\otimes (\overline{x}-\overline{y})}{|\overline{x}-\overline{y}|^2}\leq I.$$ Since $\widetilde{X}-\widetilde{Y}\leq 0$ and $\widetilde{X}-\widetilde{Y}\leq 4B,$ we have
$$\text{tr}(\widetilde{X}-\widetilde{Y})\leq \text{tr}(P(\widetilde{X}-\widetilde{Y}))\leq 4 \text{tr}(PB).$$ We have to compute tr($PB$). From \eqref{hessianphi}, observing that the matrix $(1/|x|^2)x\otimes x$ is idempotent, i.e.,
$[(1/|x|^2)x\otimes x]^2=(1/|x|^2)x\otimes x$, we compute
$$(D^2\Phi(x))^2=\frac{M^2K^2}{|x|^2}(1-4K|x|)\left(I-\frac{x\otimes x}{|x|^2}\right)+4M^2K^4I.$$ Then, since $\text{tr}P=1$ and
$4K|\overline{x}-\overline{y}|\leq1$, we have
\begin{equation*}\begin{split}\text{tr}(PB)&=e^{-L(d(\overline{x})+d(\overline{y}))}MK^2(-2+4K|\overline{x}-\overline{y}|) \leq -e^{-L(d(\overline{x})+d(\overline{y}))}MK^2<0. \end{split}\end{equation*}This gives
\begin{equation}\label{estimatex-y}|\text{tr}(\widetilde{X}-\widetilde{Y})|=-\text{tr}(\widetilde{X}-\widetilde{Y})\geq 4e^{-L(d(\overline{x})+d(\overline{y}))}MK^2\geq CMK^2.\end{equation}
Since $\|B\|\leq \frac{CMK}{|\overline{x}-\overline{y}|},$ we have
\begin{equation*}\begin{split}\|B\|^{\frac{1}{2}}|\text{tr}(\widetilde{X}-\widetilde{Y})|^{\frac{1}{2}}&\leq
\left(\frac{CMK}{|\overline{x}-\overline{y}|}\right)^{\frac{1}{2}}|\text{tr}(\widetilde{X}-\widetilde{Y})|^{\frac{1}{2}} \leq
\frac{C}{K^{\frac{1}{2}}|\overline{x}-\overline{y}|^{\frac{1}{2}}}|\text{tr}(\widetilde{X}-\widetilde{Y})|.
\end{split}\end{equation*}The Lemma III.I in \cite{il} ensures the existence of a universal constant $C$ depending only on $n$ such that $$\|\widetilde{X}\|, \|\widetilde{Y}\|\leq C\{|\text{tr}(\widetilde{X}-\widetilde{Y})|+\|B\|^{\frac{1}{2}}|\text{tr}(\widetilde{X}-\widetilde{Y})|^{\frac{1}{2}}\}.$$ Thanks to the above estimates we can conclude that
\begin{equation}\label{normaxy}\|\widetilde{X}\|,\,\|\widetilde{Y}\|\leq C|\text{tr}(\widetilde{X}-\widetilde{Y})|\left(1+\frac{1}{K^{\frac{1}{2}}|\overline{x}-\overline{y}|^{\frac{1}{2}}}\right).\end{equation}
Now, using assumptions (F1) and (F2) concerning $F$, the definition of $\widetilde{X}$ and $\widetilde{Y}$ and the fact that $ u$ is sub and supersolution we compute
\begin{equation*}\begin{split}f(\overline{x})&\leq F(\overline{x},u(\overline{x}),D_x\widetilde{\varphi}(\overline{x},\overline{y}),X-C_0\delta D^2d(\overline{x})+D^2(je^{-Ld(x)}|x-x_0|^2))\\&\leq F(\overline{x},u(\overline{x}),D_x\widetilde{\varphi}(\overline{x},\overline{y}),\widetilde{X})+MO(K)+CC_0+Cj\\& \leq F(\overline{x},u(\overline{y}),-D_y\widetilde{\varphi}(\overline{x},\overline{y}),\widetilde{Y})+
c|u(\overline{x})-u(\overline{y})|
+b|D_x\widetilde{\varphi}(\overline{x},\overline{y})+D_y\widetilde{\varphi}(\overline{x},\overline{y})|\\&+a\text{tr}(\widetilde{X}-\widetilde{Y})+MO(K)+CC_0+Cj\\&
\leq F(\overline{y},u(\overline{y}),-D_y\widetilde{\varphi}(\overline{x},\overline{y}),\widetilde{Y})+C_1|\overline{x}-\overline{y}|^\frac{1}{2}\|\widetilde{Y}\|+2c|u(\overline{y})|+2b|D_y\widetilde{\varphi}(\overline{x},\overline{y})|
\\&+c|u(\overline{x})-u(\overline{y})|
+b|D_x\widetilde{\varphi}(\overline{x},\overline{y})+D_y\widetilde{\varphi}(\overline{x},\overline{y})|+a\text{tr}(\widetilde{X}-\widetilde{Y})+MO(K)+CC_0+Cj\\&
\leq f(\overline{y})+C_1|\overline{x}-\overline{y}|^\frac{1}{2}\|\widetilde{Y}\|+2c|u(\overline{y})|+2b|D_y\widetilde{\varphi}(\overline{x},\overline{y})|+c|u(\overline{x})-u(\overline{y})|
\\&+b|D_x\widetilde{\varphi}(\overline{x},\overline{y})+D_y\widetilde{\varphi}(\overline{x},\overline{y})|+a\text{tr}(\widetilde{X}-\widetilde{Y})+MO(K)+CC_0+Cj. \end{split}\end{equation*}
From these inequalities, using \eqref{dxphistima}, \eqref{normaxy} and \eqref{estimatex-y}, for $K>\overline{K}$, where $\overline{K}$ is a constant depending only on $a,A,C_1,n$ and $\Omega$, we get \begin{equation}\label{ultimalemm}\begin{split}
&-2|f|_{L^\infty(\overline{\Omega})}-4c|u|_{L^\infty(\overline{B}_{\overline{\Omega}}(x_0,\rho))}-C|g|_{C^{0,1}(\partial\Omega)}\\&\leq Cb|D\widetilde{\varphi}|_\infty+MO(K)+Cj
+C|\text{tr}(\widetilde{X}-\widetilde{Y})|(|\overline{x}-\overline{y}|^{\frac{1}{2}}+K^{-\frac{1}{2}}) +a\text{tr}(\widetilde{X}-\widetilde{Y})
\\&\leq CM\left(-K^2+bK+\frac{b}{\rho}+\frac{1}{\rho^2}\right)+Cb|g|_{C^{0,1}(\partial\Omega)}. \end{split}\end{equation}Then, since we have chosen $M>1$, for
$K>\overline{K}$ we obtain \begin{equation}\label{stimKneu}K^2-bK\leq C\left(|f|_{L^\infty(\overline{\Omega})}+c|u|_{L^\infty(\overline{B}_{\overline{\Omega}}(x_0,\rho))}
+(1+b)|g|_{C^{0,1}(\partial\Omega)}+\frac{b}{\rho}+\frac{1}{\rho^2}\right),\end{equation} and this is a contradiction for $K$ large enough. This implies that there exists $K$ satisfying \eqref{stimKneu}, such that \eqref{u-vindelta} holds true. Next, choosing $x=x_0$, \eqref{u-vindelta} gives $$u(x_0)-u(y)\leq \varphi(x_0,y)\quad \forall y\in \overline{B}_{\overline{\Omega}}(x_0,\rho)\cap\overline{\Omega}.$$ Repeating the proof in $\overline{B}_{\overline{\Omega}}(x,2\rho)$ for any $x\in \overline{B}_{\overline{\Omega}}(x_0,\rho)$, we finally find the $u$ satisfies \eqref{uliploc} and \eqref{klipestlocal}.\finedim
\noindent\textbf{Proof} {\bf of Corollary \ref{corregu}}
Let us define $$v(x):=e^{\alpha d(x)}u(x).$$ Then, $v$ is a solution of \begin{equation*} \begin{cases}
F(x,v,Dv,D^2v)= e^{\alpha d(x)}f(x)& \text{in} \quad\Omega, \\
\frac{\partial v}{\partial \overrightarrow{n}} =0& \text{on} \quad\partial\Omega, \\
\end{cases}
\end{equation*} where
$$F(x,r,p,X)={\mathcal{M}_{a,A}^+}\{X-\alpha(Dd\otimes p+p\otimes Dd)+\alpha^2 r(Dd\otimes
Dd)-\alpha rD^2d\}+\lambda r.$$
It is easy to check that $F$ satisfies assumptions (F1) and (F2)
with $C_1=0$, and
$$c=C(\alpha^2+\alpha+|\lambda|),\quad b=C\alpha,$$ where $C$ depends on
$a,\,A,\,n$ and $\Omega$. Then, by Proposition \ref{regolaritaloc}, the
Lipschitz constant of $v$ on $\Omega_\rho$ is bounded from above by $M_vK_v$, where $M_v\leq C(|e^{\alpha
d(x)}u|_{L^\infty(\Omega_{3\rho})}+1)$ and $K_v$ satisfies
\eqref{kvlocal}. Hence, for any $x,y\in\Omega_\rho$, we have
\begin{equation*} |u(x)-u(y)|\leq|e^{-\alpha d(x)}-e^{-\alpha d(y)}||v(x)|+e^{-\alpha
d(y)}|v(x)-v(y)|\leq (\alpha|e^{\alpha
d(x)}u|_{L^\infty(\Omega_\rho)}+M_vK_v)|x-y|,\end{equation*} and this concludes
the proof.\finedim
\section{Properties of the principal eigenvalues}
\begin{prop}[Existence of principal eigenfunctions]\label{esistautof}
There exists $u_\alpha^+>0$ and $u_\alpha^-<0$ on $\overline{\Omega}$ respectively viscosity solution of \begin{equation}\label{poseigen} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2u^+_\al)+\lambda^+_\al u^+_\al=0 & \text{in} \quad\Omega, \\
\frac{\partial u^+_\al}{\partial \overrightarrow{n}} =\alpha u^+_\al & \text{on} \quad\partial\Omega, \\
\end{cases}
\end{equation} \begin{equation}\label{negeigen} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2u^-_\al)+\lambda^-_\alu^-_\al=0 & \text{in} \quad\Omega, \\
\frac{\partial u^-_\al}{\partial \overrightarrow{n}} =\alpha u^-_\al & \text{on} \quad\partial\Omega. \\
\end{cases}
\end{equation} \end{prop} \noindent\textbf{Proof.} We follow the arguments of \cite{bd}. To show the existence of positive eigenfunctions, the first step is to prove that if $f$ is a continuous function such that $f\leq 0$, $f\not\equiv0$, then for any $\lambda<\lambda^+_\al$ there exists a positive solution of \begin{equation}\label{syslam<lams} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2u)+\lambda u=f(x) & \text{in} \quad\Omega, \\
\frac{\partial u}{\partial \overrightarrow{n}} =\alpha u & \text{on} \quad\partial\Omega. \\
\end{cases}
\end{equation} Observe that $v\equiv1$ is a positive subsolution of \eqref{sysmaxp} for $\lambda\geq0$. This implies, by Proposition \ref{maxprinc}, that if $\lambda<\lambda^+_\al$ then $\lambda<0$. Let $(v_{n})_n$ be the sequence defined by $v_1=0$
and $v_{n+1}$ be the solution of \begin{equation*} \begin{cases} F(x,v_{n+1},Dv_{n+1},D^2v_{n+1})-(c-\lambda) v_{n+1}=e^{\alpha d(x)}f(x)-cv_n & \text{in} \quad\Omega, \\
\frac{\partial v_{n+1}}{\partial \overrightarrow{n}} =0 & \text{on} \quad\partial\Omega, \\
\end{cases}
\end{equation*} where $$F(x,r,p,X)={\mathcal{M}_{a,A}^+}\{X-\alpha(Dd\otimes p+p\otimes Dd)+\alpha^2 r(Dd\otimes
Dd)-\alpha rD^2d\}$$ and $c=C(\alpha^2+\alpha)$. By comparison, the sequence is positive and increasing. Let $(u_n)_n$ be the sequence defined by
$u_n(x):=e^{-\alpha d(x)}v_n(x)$, then $u_{n+1}$ is solution of \begin{equation*} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2u_{n+1})-(c-\lambda)u_{n+1}=f(x)-cu_n & \text{in} \quad\Omega, \\
\frac{\partial u_{n+1}}{\partial \overrightarrow{n}} =\alpha u_{n+1} & \text{on} \quad\partial\Omega. \\
\end{cases}
\end{equation*}
We claim that $(u_n)_n$ is bounded. Suppose that it is not, then defining
$w_n:=\frac{u_n}{|u_n|_\infty}$ one gets that $w_{n+1}$ is a solution of \begin{equation*} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2w_{n+1})-(c-\lambda)w_{n+1}=\frac{f(x)}{|u_{n+1}|_\infty}-c\frac{|u_n|_\infty}{|u_{n+1}|_\infty}w_n & \text{in} \quad\Omega, \\
\frac{\partial w_{n+1}}{\partial \overrightarrow{n}} =\alpha w_{n+1} & \text{on} \quad\partial\Omega. \\
\end{cases}
\end{equation*} By Corollary \ref{corregu}, $(w_n)_n$ converges along a subsequence to a positive function $w$ which
satisfies \begin{equation*} \begin{cases} {\mathcal{M}_{a,A}^+}(D^2w)+\lambda w=c(1-k)w\geq0 & \text{in} \quad\Omega, \\
\frac{\partial w}{\partial \overrightarrow{n}}=\alpha w & \text{on} \quad\partial\Omega, \\
\end{cases}
\end{equation*} where
$k:=\limsup_{n\rightarrow+\infty}\frac{|u_n|_\infty}{|u_{n+1}|_\infty}\leq
1$. This contradicts the Maximum Principle, Proposition
\ref{maxprinc}. Then $(u_n)_n$ is bounded and letting $n$ go to infinity, by the compactness result, the sequence converges uniformly to a function $u$ which is a solution of \eqref{syslam<lams}. Moreover, $u$ is positive by the Strong Comparison Principle, Theorem \ref{stcompneu}.
We are now in position to construct a sequence $(u_n)_n$ of positive solutions of \begin{equation*} \begin{cases} {\mathcal{M}_{a,A}^+}(D^2u_{n})+\lambda_n u_{n}=-1 & \text{in} \quad\Omega, \\
\frac{\partial u_{n}}{\partial \overrightarrow{n}} =\alpha u_n & \text{on} \quad\partial\Omega, \\
\end{cases}
\end{equation*} where $(\lambda_n)_n$ is an increasing sequence which converges to $\lambda^+_\al$. The sequence $(u_n)_n$ is unbounded, otherwise one would
contradict the definition of $\lambda^+_\al$ (see Theorem 8 of
\cite{bd}).
Then, up to subsequence, $|u_n|_\infty\rightarrow+\infty$ as
$n\rightarrow+\infty$ and defining $\phi_n:=\frac{u_n}{|u_n|_\infty}$
one gets that $\phi_n$ satisfies \begin{equation*} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2\phi_{n})+\lambda_n \phi_{n}=-\frac{1}{|u_n|_\infty} & \text{in} \quad\Omega, \\
\frac{\partial \phi_{n}}{\partial \overrightarrow{n}} =\alpha \phi_n & \text{on} \quad\partial\Omega. \\
\end{cases}
\end{equation*}
By Corollary \ref{corregu}, an extracted subsequence converges to a function $u^+_\al$ with $|u^+_\al|_\infty=1$, which is a solution of
\eqref{poseigen}. Moreover, by Theorem \ref{stcompneu}, $u^+_\al>0$ on $\overline{\Omega}$.
Similar arguments show the existence of negative solutions of \eqref{negeigen}.
\finedim
\begin{prop}[Simplicity of the first eigenvalues, \cite{p2} Proposition 7.1]\label{simplicityprop} Let $v\in C(\overline{\Omega})$ be a viscosity subsolution (resp. supersolution) of \eqref{poseigen} (resp. \eqref{negeigen}), then there exists $t\in{\mathbb R}$ such that $v\equiv t u^+_\al$ (resp. $v\equiv t u^-_\al$). \end{prop}
\begin{rem}\label{boundprop}{\em Remark that \begin{equation}\label{lamsbound} \lambda^+_\al< -A\alpha^2,\end{equation} \begin{equation}\label{lamsobound} \lambda^-_\al< -a\alpha^2.\end{equation} Indeed, the function $v(x):=e^{\alpha x_1}$, where $x_1$ is the first coordinate of $x\in{\mathbb R}^n$, is a positive subsolution of \begin{equation}\label{propboundeq} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2v)-A\alpha^2 v=0 & \text{in} \quad\Omega,\\
\frac{\partial v}{\partial \overrightarrow{n}} =\alpha v & \text{on} \quad\partial\Omega. \\
\end{cases}
\end{equation} Then the Maximum Principle, Proposition \ref{maxprinc}, implies that $\lambda^+_\al\leq -A\alpha^2$. If $\lambda^+_\al=-A\alpha^2$, then by Proposition \ref{simplicityprop}, $v(x)$ is a solution of \eqref{poseigen} and this implies that $\Omega={\R^n}$. Hence \eqref{lamsbound} holds true. Similarly, inequality \eqref{lamsobound} is a consequence of the Minimum Principle, Proposition \ref{minprinc}, of Proposition \ref{simplicityprop} and the fact that $-v(x)$ is a negative supersolution of \eqref{propboundeq} with $A$ replaced by $a$.}\end{rem} \begin{rem}{\em Since $\lambda^+_\al,\lambda^-_\al<0$ the operator ${\mathcal{M}_{a,A}^+}(D^2u)+\lambda u$, with $\lambda=\lambda^+_\al$ or $\lambda=\lambda^-_\al$ satisfies the Dirichlet Comparison Principle.}\end{rem} \begin{prop}\label{monot}The sequences $(\lambda^+_\al)_\alpha$ and $(\lambda^-_\al)_\alpha$ are decreasing. \end{prop} \noindent\textbf{Proof.} Let us prove that $(\lambda^+_\al)_\alpha$ is decreasing. Consider $0<\alpha_1<\alpha_2$ and let $u_{\alpha_1}^+$ be a solution of \eqref{poseigen} with $\alpha=\alpha_1$. Then $u_{\alpha_1}^+$ is a positive subsolution of \begin{equation*} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2u)+\lambda_{\alpha_1}^+u=0 & \text{in} \quad\Omega, \\
\frac{\partial u}{\partial \overrightarrow{n}} =\alpha_2 u & \text{on} \quad\partial\Omega, \\
\end{cases}
\end{equation*} and the Maximum Principle, Proposition \ref{maxprinc}, implies $\lambda_{\alpha_1}^+\geq \lambda_{\alpha_2}^+$. The strict inequality $\lambda_{\alpha_1}^+> \lambda_{\alpha_2}^+$ follows from Proposition \ref{simplicityprop}. \finedim
\begin{lem}\label{maxboundlem} Let $u^+_\al$ and $u^-_\al$ be respectively a positive solution of \eqref{poseigen} and a negative solution of \eqref{negeigen}, then \begin{equation*} u^+_\al(x)<\max_{\partial\Omega} u^+_\al\quad\forall x\in\Omega,\end{equation*} \begin{equation*}u^-_\al(x)>\min_{\partial\Omega} u^-_\al\quad\forall x\in\Omega.\end{equation*} \end{lem} \noindent\textbf{Proof.} Let us show the result for $u^+_\al$. Suppose by contradiction that the maximum of $u^+_\al$ is attained at some point $x_0\in\Omega$ and let $v(x):=u^+_\al(x)-u^+_\al(x_0)$. Since $u^+_\al(x_0)>0$ and $\lambda^+_\al<0$, $v$ satisfies $${\mathcal{M}_{a,A}^+}(D^2 v)+\lambda^+_\al v\geq 0\quad\text{in }\Omega$$ and $v\leq 0$ in $\Omega$, $v(x_0)=0$. Then the Strong Maximum Principle implies $u^+_\al\equiv u^+_\al(x_0)$ in $\Omega$ and this contradicts the fact that $u^+_\al$ solves \eqref{poseigen}. \finedim
\section{Liouville type results} For $\gamma>0$ let us introduce the system \begin{equation}\label{pbhalfspace}\left\{ \begin{array}{ll}
{\mathcal{M}_{a,A}^+}(D^2 u)- \gamma u=0 & \hbox{in }{\R_+^n}, \\
-\frac{\partial u}{\partial x_n}=u & \hbox{on } \partial{\R^n}.\\ \end{array} \right.\end{equation}
\begin{thm}\label{halfspacethm}If $\gamma >A$, any bounded subsolution of \eqref{pbhalfspace} is non-positive in ${\R_+^n}$.
If $\gamma >a$, any bounded supersolution of \eqref{pbhalfspace} is non-negative in ${\R_+^n}$.
Hence, if $\gamma>A$ there are no, non trivial bounded solutions of \eqref{pbhalfspace}. \end{thm} \begin{rem}{\em It turns out that Theorem \ref{halfspacethm} is sharp: $u(x)=e^{-x_n}$ (resp., $u(x)=-e^{-x_n}$) is a positive bounded subsolution (resp., negative bounded supersolution) of \eqref{pbhalfspace} for every $\gamma\leq A$ (resp., $\gamma\leq a$).}
{\em Theorem \ref{halfspacethm} also fails without the boundedness condition. Indeed, $u(x)=e^{\nu\cdot x}$ (resp., $u(x)=-e^{\nu\cdot x}$), with $\nu=(\nu_1,...,\nu_{n-1},-1)$,
$|\nu|>1$, is an unbounded subsolution (resp., supersolution) of
\eqref{pbhalfspace} for $A<\gamma\leq A|\nu|^2$ (resp.,
$a<\gamma\leq |\nu|^2a$).} \end{rem}
We assume that $u(x)$ is a bounded subsolution of \eqref{pbhalfspace} with $\gamma>0$, which is positive somewhere.
We normalize $u$ so that \begin{equation}\label{supequal1}\sup_{{\R_+^n}} u=1.\end{equation} Then $u$ is a viscosity subsolution of \begin{equation}\label{neumhalfspace}\left\{ \begin{array}{ll}
{\mathcal{M}_{a,A}^+}(D^2 u)- \gamma u=0 & \hbox{in }{\R_+^n}, \\
-\frac{\partial u}{\partial x_n}=1 & \hbox{on } \partial{\R^n}.\\ \end{array} \right.\end{equation}
\begin{prop}\label{comparisonhalfspace} Assume $\gamma>0$ and $k\in{\mathbb R}$. Let $u\in USC({\mathbb R}_+^n)$ and $v\in LSC({\mathbb R}_+^n)$ be respectively bounded viscosity sub and supersolution of \begin{equation}\label{neumhalfspacek}\left\{ \begin{array}{ll}
{\mathcal{M}_{a,A}^+}(D^2 u)- \gamma u=0 & \hbox{in }{\R_+^n},\\
-\frac{\partial u}{\partial x_n}=k & \hbox{on } \partial{\R^n}.\\ \end{array} \right.\end{equation} Then $u\leq v$ in ${\R_+^n}$. \end{prop} \noindent\textbf{Proof.}
Suppose by contradiction that $\sup_{{\R_+^n}}(u-v)=M>0$. Let $\psi$ be a smooth positive function with bounded derivatives and such that $\psi(x)\rightarrow+\infty$
as $|x|\rightarrow+\infty$. Let $\chi(x)=\chi(x_n)$ be a smooth function such that $\chi(x_n)=x_n$ for $|x_n|\leq1$ and
$\chi(x_n)\equiv 0$ for $|x_n|>2$. Let
$$\varphi(x,y)=\frac{j}{2}|x-y|^2-k(x_n-y_n)+\beta\psi(x)-\epsilon(\chi(x)+\chi(y)).$$ Then, for $\beta$ and $\epsilon$ small enough and $j>0$, the supremum of the function $u(x)-v(y)-\varphi(x,y)$
is greater than $\frac{M}{2}$ and it is reached at some point $(\overline{x},\overline{y})\in \overline{{\R_+^n}}\times\overline{{\R_+^n}}$.
If $\overline{x}\in\partial{\R_+^n}$ then, for $\overline{y}\in{\R_+^n}$, \begin{equation*}\begin{split}-\partial_{x_n}\varphi(\overline{x},\overline{y})-k=-j(\overline{x}_n-\overline{y}_n)+k
-\beta\partial_{x_n}\psi(\overline{x})+\epsilon-k=j\overline{y}_n-\beta\partial_{x_n}\psi(\overline{x})+\epsilon>0\end{split}\end{equation*} for $\epsilon>\beta|D\psi|_\infty$.
If $\overline{y}\in\partial{\R_+^n}$ then, for $\overline{x}\in{\R_+^n}$ \begin{equation*}\partial_{y_n}\varphi(\overline{x},\overline{y})-k=-j(\overline{x}_n-\overline{y}_n)-\epsilon=-j\overline{x}_n-\epsilon< 0.\end{equation*} Both inequalities contradict the definition of sub and supersolution, therefore $\overline{x},\overline{y}\in{\R_+^n}$.
Applying Theorem 3.2 of \cite{cil}, there exist $X,Y\in\emph{S(n)}$ such that $(D_x\varphi(\overline{x},\overline{y}),X+\beta D^2\psi(\overline{x})-\epsilon D^2\chi(\overline{x}))\in \overline{J}^{2,+}u(\overline{x})$, $(-D_y\varphi(\overline{x},\overline{y}),Y+\epsilon D^2\chi(\overline{y}))\in \overline{J}^{2,-}v(\overline{y})$ and
\begin{equation*}-3j\left( \begin{array}{cc}
I & 0 \\
0 & I \\ \end{array} \right)\leq \left( \begin{array}{cc}
X & 0 \\
0 & -Y \\ \end{array} \right)\leq 3j\left( \begin{array}{cc}
I & -I \\
-I & I \\ \end{array} \right). \end{equation*} Since $u$ and $v$ are respectively sub and supersolution, we have \begin{equation*}{\mathcal{M}_{a,A}^+}(X+\beta D^2\psi(\overline{x})-\epsilon D^2\chi(\overline{x}))\geq \gamma u(\overline{x}),\end{equation*} \begin{equation*} {\mathcal{M}_{a,A}^+}(Y+\epsilon D^2\chi(\overline{y}))\leq \gamma v(\overline{y}).\end{equation*} Subtracting the two previous inequalities, using the properties of Pucci's operators and that
$$u(\overline{x})-v(\overline{y})>\frac{M}{2}+\frac{j}{2}|\overline{x}_n-\overline{y}_n|^2-k(\overline{x}_n-\overline{y}_n)-\epsilon(\chi(\overline{x})+\chi(\overline{y}))\geq \frac{M}{2}-\frac{k^2}{2j}-C\epsilon,$$ we finally get \begin{equation*}\begin{split} \frac{\gamma}{2}\left(M-\frac{k^2}{j}-C\epsilon\right)&<\gamma(u(\overline{x})-v(\overline{y}))\leq {\mathcal{M}_{a,A}^+}(X+\beta D^2\psi(\overline{x})-\epsilon D^2\chi(\overline{x}))\\&-{\mathcal{M}_{a,A}^+}(Y+\epsilon D^2\chi(\overline{y}))\\&\leq {\mathcal{M}_{a,A}^+}(X-Y)+\beta {\mathcal{M}_{a,A}^+}(D^2\psi(\overline{x}))-\epsilon\mathcal{M}_{a,A}^-( D^2\chi(\overline{x})+D^2\chi(\overline{y}))\\&\leq \beta {\mathcal{M}_{a,A}^+}(D^2\psi(\overline{x}))-\epsilon\mathcal{M}_{a,A}^-( D^2\chi(\overline{x})+D^2\chi(\overline{y})).\end{split}\end{equation*} This is a contradiction for $\beta$ and $\epsilon$ small enough and $j$ large. Then $u\leq v$ in ${\R_+^n}$. \finedim
\noindent\textbf{Proof} {\bf of Theorem \ref{halfspacethm}} The function $$v(x)=\sqrt{\frac{A}{\gamma}}e^{-\sqrt{\frac{\gamma}{A}}x_n}$$ is the bounded viscosity solution of \eqref{neumhalfspace}. Then by
Proposition \ref{comparisonhalfspace} $$u(x)\leq \sqrt{\frac{A}{\gamma}}e^{-\sqrt{\frac{\gamma}{A}}x_n}, \quad\text{for any }x\in{\R_+^n}.$$ It follows from \eqref{supequal1} that $$1=\sup_{{\R_+^n}}u\leq \sqrt{\frac{A}{\gamma}},$$ i.e. $\gamma\leq A$.
Similarly, if $u$ is a negative supersolution of \eqref{pbhalfspace}, normalized so that $\min_{{\R_+^n}}u=-1$, then $u$ is a supersolution of \begin{equation*}\left\{ \begin{array}{ll}
{\mathcal{M}_{a,A}^+}(D^2 u)- \gamma u=0 & \hbox{in }{\R_+^n}, \\
-\frac{\partial u}{\partial x_n}=-1 & \hbox{on } \partial{\R^n},\\ \end{array} \right.\end{equation*} and by comparison $$u(x)\geq -\sqrt{\frac{a}{\gamma}}e^{-\sqrt{\frac{\gamma}{a}}x_n}.$$ This implies $\gamma\leq a$ and Theorem \ref{halfspacethm} is proved.\finedim
\section{Asymptotic behavior and Proof of Theorem \ref{mainthm}} We start by the following simple result: \begin{prop} $\displaystyle \lim_{\alpha\rightarrow 0}\lambda^{\pm}_{\alpha}=0$. \end{prop} \noindent\textbf{Proof.} By Proposition \ref{monot}, $\lambda^+_\al$ increases to some value $\lambda_0\leq 0$. On the other hand, the sequence of normalized solutions $(u^+_\al)_\alpha$, by the Lipschitz estimates Corollary \ref{corregu}, converges to $u_0$ a positive solution of \begin{equation*} \begin{cases}
{\mathcal{M}_{a,A}^+}(D^2u)+\lambda_0 u=0 & \text{in} \quad\Omega, \\
\frac{\partial u}{\partial \overrightarrow{n}}=0 & \text{on} \quad\partial\Omega, \\
\end{cases} \end{equation*}
which satisfies $|u_0|=1$. Recall that $0$ is the principal eigenvalue for the Neumann problem. If $\lambda_0<0$, the Maximum Principle below the first eigenvalue, i.e. Proposition \ref{maxprinc}, implies that $u_0\leq 0$ a contradiction. \finedim
We consider now the asymptotic behavior at infinity. By Remark \ref{boundprop}, it is enough to show that \begin{equation}\label{limsuplams}\limsup_{\alpha\rightarrow+\infty}\frac{\lambda^+_\al}{-\alpha^2}\leq A,\end{equation} and \begin{equation}\label{limsuplamso}\limsup_{\alpha\rightarrow+\infty}\frac{\lambda^-_\al}{-\alpha^2}\leq a.\end{equation}
We are going to show \eqref{limsuplams}. For $\alpha>0$, let $u^+_\al$ be a positive solution of \eqref{poseigen}. By Lemma \ref{maxboundlem}, we know that $u^+_\al$ attains its maximum at $x_\alpha\in\partial\Omega$. After normalization, we can assume that $\max_{\overline{\Omega}}u^+_\al=1$ and $x_\al\rightarrow0$ as $\alpha\rightarrow+\infty$. Furthermore, we can assume that there is a $C^2$ function $\phi$ and $r>0$ such that \begin{eqnarray*}
&&x_n=\phi(x'),\quad \forall (x',x_n)\in \partial\Omega\cap B_r(0) \\&&
x_n>\phi(x'),\quad \forall (x',x_n)\in \Omega\cap B_r(0) \\&&
\phi(0)=0, \\&& \partial_{x_i}\phi(0)=0,\quad\text{for }i=1,...,n-1.\\&& \end{eqnarray*}
We flatten $\partial\Omega$ near the origin. Let $\Phi(x):\Omega\cap B_r(0)\rightarrow \Omega_{\Phi}:=\Phi(\Omega\cap B_r(0))$, be such that \begin{equation}\label{Phi}\begin{split}&\Phi_i(x)=x_i,\quad i=1,...,n-1,\\&
\Phi_n(x)=x_n-\phi(x'). \end{split}\end{equation}
Denote by $x=\Psi(y)$ the inverse of $y=\Phi(x)$. The function $$v_\alpha(y)=u^+_\al(\Psi(y))$$ is solution of \begin{equation}\label{systemflatbound}\left\{ \begin{array}{ll}
{\mathcal{M}_{a,A}^+}\left[ \left(\sum_{l,k=1}^n
\partial^2_{y_ly_k}v_\alpha\partial_{x_j}\Phi_k(\Psi(y))\partial_{x_i}\Phi_l(\Psi(y))\right)_{ij}\right.\\
\quad\left.+\left(\sum_{k=1}^n\partial_{y_k}v_\alpha\partial^2_{x_ix_j}\Phi_k(\Psi(y))\right)_{ij}\right]+\lambda^+_\al v_\alpha=0 & y\in \Omega_{\Phi}, \\
\sum_{k,j=1}^n\partial_{y_k}v_\alpha\partial_{x_j}\Phi_k(\Psi(y))\overrightarrow{n}_j(\Psi(y))=\alpha v_\alpha & y\in\partial\Omega_{\Phi}.\\ \end{array} \right.\end{equation}
Since the exterior normal $\overrightarrow{n}(x)$ at $x\in \partial\Omega\cap B_r(0)$ is
$$\overrightarrow{n}(x)=\frac{(D\phi(x'),-1)}{\sqrt{|D\phi(x')|^2+1}},$$ by \eqref{Phi}, the boundary condition in \eqref{systemflatbound} can be rewritten as follows
\begin{equation*}\frac{1}{\sqrt{|D\phi(y')|^2+1}}\sum_{k=1}^{n-1}\partial_{y_k}v_\alpha\partial_{x_k}\phi(y')-\left(\sqrt{|D\phi(y')|^2+1}\right)\partial_{y_n}v_\alpha=\alpha v_\alpha,\quad y\in\partial\Omega_{\Phi}.\end{equation*} Notice that, since $D\phi(x')\rightarrow 0$ as $x'\rightarrow0$, $D\Phi(\Psi(y))\rightarrow I$ as $y\rightarrow0$, where $I$ is the identity matrix of $\emph{S(n)}$.
We now consider two different cases.
\noindent\emph{Case 1.} $$\limsup_{\alpha\rightarrow+\infty}\frac{\lambda^+_\al}{-\alpha^2}=\gamma<+\infty.$$ Without loss of generality, we may assume that $\frac{\lambda^+_\al}{-\alpha^2}\rightarrow\gamma$ as $\alpha\rightarrow+\infty$, and $u^+_\al(x_\al)=\max_{\overline{\Omega}}u^+_\al=1$, $x_\al\rightarrow0$ as $\alpha\rightarrow+\infty$. We let $$z=\alpha(y-y_\alpha),$$ where $y_\alpha=\Phi(x_\al)=(x_\al',0)$. We set $$w_\alpha(z)=v_\alpha(y)=u^+_\al(x),$$ then for any $R>0$, as $\alpha$ becomes sufficiently large, $w_\alpha$ is solution of \begin{equation}\label{systemblowup}\left\{ \begin{array}{ll}
{\mathcal{M}_{a,A}^+}\left[ \left(\sum_{l,k=1}^n
\partial^2_{z_lz_k}w_\alpha\partial_{x_j}\Phi_k(\Psi\left(y\right))\partial_{x_i}\Phi_l(\Psi\left(y\right))\right)_{ij}\right.\\
\quad\left.+\frac{1}{\alpha}\left(\sum_{k=1}^n\partial_{z_k}w_\alpha\partial^2_{x_ix_j}\Phi_k(\Psi\left(y\right))\right)_{ij}\right]
+\frac{\lambda^+_\al}{\alpha^2} w_\alpha=0 & z\in B_{R}^+, \\
\frac{1}{\sqrt{\left|D\phi\left(y'\right)\right|^2+1}}\sum_{k=1}^{n-1}\partial_{z_k}w_\alpha\partial_{x_k}\phi\left(y'\right)
-\left(\sqrt{\left|D\phi\left(y'\right)\right|^2+1}\right)\partial_{z_n}w_\alpha=w_\alpha & z\in \Gamma_{R},\\ \end{array} \right.\end{equation} where $$y=y(z)=\frac{z}{\alpha}+y_\alpha$$ and $$B_R^+:=B_R(0)\cap{\R_+^n},\quad \Gamma_R:=B_R(0)\cap\partial{\R_+^n}.$$ Since for $z\in B_{R}^+$, $z/\alpha+y_\alpha\rightarrow0$ as $\alpha\rightarrow+\infty$ and $\partial_{x_i}\phi(0)=0$ for $i=1,...,n-1$, for $\alpha$ sufficiently large, $I/2\leq D\Psi\left(z/\alpha+y_\alpha\right)\leq 2I$. Hence, if $L_\alpha$ is the Lipschitz constant of $u^+_\al$ in the set
$\{x=\Psi(z/\alpha+y_\alpha),\,|z|\leq R\}$, we have \begin{equation*}
|w_\alpha(z_1)-w_\alpha(z_2)|=\left|u^+_\al\left(\Psi\left(\frac{z_1}{\alpha}+y_\alpha\right)\right)-u^+_\al\left(\Psi\left(\frac{z_2}{\alpha}+y_\alpha\right)\right)\right|
\leq \frac{2L_\alpha}{\alpha}|z_1-z_2|.\end{equation*}
Remark that if $|z|\leq R$, then $d(x)\leq CR/\alpha$ for
$x=\Psi(z/\alpha+y_\alpha)$, where $C$ depends on $\phi$. Hence, since for $\rho=CR/\alpha$, $|e^{\alpha d(x)}u^+_\al|_{L^\infty(\Omega_{3\rho})}\leq e^{3CR}$, Corollary \ref{corregu} gives $$L_\alpha\leq C e^{3CR}(\alpha+K_\alpha),$$ where $K_\alpha$ satisfies
$$K_\alpha^2-C\alpha K_\alpha\leq C\left[(\alpha+\alpha^2+|\lambda^+_\al|)e^{3CR}+\frac{\alpha^2}{CR^2}+1\right].$$ This implies that the sequence $(w_\alpha)_\alpha$ is bounded in the space of Lipschitz continuous functions of $\overline{B}_R^+$ for any fixed $R>0$, and then, up to subsequence, $w_\alpha\rightarrow w_0$ uniformly on $\overline{B}_R^+$, with $\sup_{{\R_+^n}}w=1$, viscosity solution of \eqref{pbhalfspace}. Moreover by the Strong Comparison Principle, $w>0$ on $\overline{{\R_+^n}}$. Then, by Theorem \ref{halfspacethm}, $\gamma\leq A$ and this proves \eqref{limsuplams}.
\noindent\emph{Case 2.} $$\limsup_{\alpha\rightarrow+\infty}\frac{\lambda^+_\al}{-\alpha^2}=+\infty.$$
Let $u^+_\al$ be the sequence of positive solutions of \eqref{poseigen} such that $$\frac{\lambda^+_\al}{-\alpha^2}=:l_\alpha\rightarrow+\infty\quad\text{as }\alpha\rightarrow+\infty,$$ and $u^+_\al(x_\al)=\max_{\overline{\Omega}}u^+_\al=1$. Define $$z=\sqrt{l_\alpha}\alpha(y-y_\alpha)\quad\text{and}\quad w_\alpha(z)=u_\alpha(x),$$ where $y=\Phi(x)$ and $y_\alpha=\Phi(x_\alpha)$. Then, for any $R>0$, as $\alpha$ becomes sufficiently large, $w_\alpha$ satisfies \begin{equation*}\left\{ \begin{array}{ll}
{\mathcal{M}_{a,A}^+}\left[ \left(\sum_{l,k=1}^n
\partial^2_{z_lz_k}w_\alpha\partial_{x_j}\Phi_k(\Psi(y))\partial_{x_i}\Phi_l(\Psi\left(y\right))\right)_{ij}\right.\\
\quad\left.+\frac{1}{\sqrt{-\lambda^+_\al}}\left(\sum_{k=1}^n\partial_{z_k}w_\alpha\partial^2_{x_ix_j}\Phi_k(\Psi\left(y\right))\right)_{ij}\right]
- w_\alpha=0 & z\in B_{R}^+, \\
\frac{1}{\sqrt{\left|D\phi\left(y'\right)\right|^2+1}}\sum_{k=1}^{n-1}\partial_{z_k}w_\alpha\partial_{x_k}\phi\left(y'\right)
-\left(\sqrt{\left|D\phi\left(y'\right)\right|^2+1}\right)\partial_{z_n}w_\alpha=\frac{1}{\sqrt{l_\alpha}}w_\alpha & z\in \Gamma_{R},\\ \end{array} \right.\end{equation*} where $y=y(z)=\frac{z}{\alpha}+y_\alpha$. As in Case 1, we can show that $w_\alpha\rightarrow w_0$ which is a bounded positive viscosity solution of \begin{equation}\label{pureneumann}\left\{ \begin{array}{ll}
{\mathcal{M}_{a,A}^+}(D^2 w_0)- w_0=0 & \hbox{in }{\R_+^n}, \\
-\frac{\partial w_0}{\partial x_n}=0 & \hbox{on } \partial{\R^n}.\\ \end{array} \right.\end{equation} On the other hand, by Proposition \ref{comparisonhalfspace}, the only bounded viscosity solution of \eqref{pureneumann} is $u\equiv 0$ and we reach a contradiction. \finedim
\begin{prop}\label{convcompacts}Let $u^+_\al$ and $u^-_\al$ be respectively the normalized solution of \eqref{poseigen} and \eqref{negeigen}, i.e.
$\|u_\alpha^{\pm}\|_\infty=1$, then for any compact set $K\subset\Omega$
\begin{equation*} \|u_\alpha^{\pm}\|_{L^\infty(K)}\rightarrow0\quad\text{as }\alpha\rightarrow+\infty.\end{equation*} \end{prop} \noindent\textbf{Proof.} Let $u^+_\al$ be the normalized solution of \eqref{poseigen} and let $K$ be a compact set contained in $\Omega$. Let $x_\al\in K$ be such that $\max_K u^+_\al=u^+_\al(x_\al)$. Define $z=\alpha(x-x_\al)$ and
$w_\alpha(z)=u^+_\al(x)$ for $|z|< \alpha r$ where $r=$dist$(K,\partial\Omega)$. Then for any $R>0$, as $\alpha$ becomes large, $w_\alpha(z)$ satisfies \begin{equation*}{\mathcal{M}_{a,A}^+}(D^2 w_\alpha)+\frac{\lambda^+_\al}{\alpha^2} w_\alpha=0\quad\text{in
}B_{2R},\end{equation*} and $\|w_\alpha\|_\infty\leq 1$. By standard elliptic estimates, see e.g. \cite{cc} and Theorem \ref{mainthm}, $w_\alpha\rightarrow w_0$ non-negative solution of \begin{equation*} {\mathcal{M}_{a,A}^+}(D^2 w_0)- Aw_0=0\quad\text{in }{\R^n}.\end{equation*} It is well-know that there are no nontrivial bounded solutions of the above equation, see e.g. \cite{cil}, hence $w_\alpha(0)=\max_K u^+_\al\rightarrow0$ as $\alpha\rightarrow+\infty$ and Proposition \ref{convcompacts} is proved.\finedim
\end{document} | arXiv |
Abstract: Halevi, Lindell, and Pinkas (CRYPTO 2011) recently proposed a model for secure computation that captures communication patterns that arise in many practical settings, such as secure computation on the web. In their model, each party interacts only once, with a single centralized server. Parties do not interact with each other; in fact, the parties need not even be online simultaneously.
In this work we present a suite of new, simple and efficient protocols for secure computation in this "one-pass" model. We give protocols that obtain optimal privacy for the following general tasks: -- Evaluating any multivariate polynomial $F(x_1, \ldots ,x_n)$ (modulo a large RSA modulus N), where the parties each hold an input $x_i$. -- Evaluating any read once branching program over the parties' inputs.
As a special case, these function classes include all previous functions for which an optimally private, one-pass computation was known, as well as many new functions, including variance and other statistical functions, string matching, second-price auctions, classification algorithms and some classes of finite automata and decision trees. | CommonCrawl |
\begin{definition}[Definition:Operation/Binary Operation/Product/Right]
Let $x$ and $y$ be elements which are operated on by a given operation $\circ$.
The '''right-hand product of $x$ by $y$''' is the product $x \circ y$.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Best Rational Approximation]
Let $x \in \R$ be an (irrational) real number.
The rational number $a = \dfrac p q$ is a '''best rational approximation''' to $x$ {{iff}}:
:$(1): \quad a$ is in canonical form, that is $p$ is coprime to $q$: $p \perp q$
:$(2): \quad \left\vert{x - \dfrac p q}\right\vert = \min \left\{ {\left\vert{x - \dfrac {p'} {q'} }\right\vert: q' \le q}\right\}$
That is:
:$\left\vert{x - \dfrac p q}\right\vert$ is smaller than for any $\dfrac {p'} {q'}$ where $q' \le q$
where $\left\vert{x}\right\vert$ denotes the absolute value of $x$.
\end{definition} | ProofWiki |
This article describes the extension of recent methods for a posteriori error estimation such as dual-weighted residual methods to node-centered finite volume discretizations of second order elliptic boundary value problems including upwind discretizations. It is shown how different sources of errors, in particular modeling errors and discretization errors, can be estimated with respect to a user-defined output functional.
We prove the $L^\infty(L^\infty)$-boundedness of a higher-order shock-capturing streamline-diffusion DG-method based on polynomials of degree $p\geq 0$ for general scalar conservation laws. The estimate is given for the case of several space dimensions and for conservation laws with initial and boundary conditions.
The paper presents results on piecewise polynomial approximations of tensor product type in Sobolev-Slobodecki spaces by various interpolation and projection techniques, on error estimates for quadrature rules and projection operators based on hierarchical bases, and on inverse inequalities. The main focus is directed to applications to discrete conservation laws.
Vol 6: Azacitidine in patients with WHO-defined AML - Results of 155 patients from the Austrian Azacitidine Registry of the AGMT-Study Group.
Vol 12: Position paper on the importance of psychosocial factors in cardiology: Update 2013. | CommonCrawl |
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Johansson Mikrokator Comparator: Construction And Working Principle.
Like sigma comparator, Johansson Mikrokator is also a measuring instrument and it was developed and introduced by C.F. Johansson.Ltd. Johansson Mikrokator is a type of mechanical comparator.
1 What is Johansson Mikrokator?
2 Construction (or) parts of Johansson Mikrokator
2.1 Plunger
2.2 Slit washer
2.3 Elbow
2.4 Bellcrank lever
2.5 Twisted strip
2.6 Cantilever strip
2.7 Pointer
2.8 Reading scale
3 Working principle of Johansson Mikrokator
4 Magnification of Johansson Mikrokator
5 Advantages of Johansson Mikrokator
6 Disadvantages of Johansson Mikrokator
7 Applications of Johansson Mikrokator
7.1 Latest articles 👇
This article is about Johansson Mikrokator along with its construction, working principle, applications, advantages, and disadvantages.
What is Johansson Mikrokator?
Johansson Mikrokator is a type of mechanical comparator invented by a Swedish engineer named H.Abramson, which was later introduced and manufactured by a Sweden company called C.H. Johansson.Ltd. This comparator is used as measuring and comparing device. It is one of the most important mechanical comparators due to its simple working principle and its construction.
Construction (or) parts of Johansson Mikrokator
(Diagram of Johansson Mikrokator)
This comparator is constructed by the following components or parts,
Slit washer
Bellcrank lever (or) spring elbow
Twisted strip
Cantilever strip
Let's discuss them in detail.
Plunger act as a detecting component in the comparator. It is arranged in a vertical setup.
This circular slit washer helps in mounting the plunger to give a frictionless motion. It is made up of polymer material. The slit washer is held by the elbow. It is a flexible diaphragm. It prevents axial rotation of the plunger.
Elbow holds the slit washer at the bottom where the measuring plunger is mounted. Both sides of the elbow hold the bell crank lever and the cantilever strip.
Bellcrank lever
The base of the bell crank lever is fixed with the plunger and one arm of the spring elbow. It acts as a kinematic link.
The twisted strip is fastened between the bell crank lever and the cantilever strip. It is flexible in nature. It is made up of phosphor bronze.
The cantilever strip is fastened with the twisted strip and the base is fixed with the arm. This cantilever strip is an adjustable type.
The pointer is lightweight in nature and it is made up of glass materials. It is attached to the twisted strip.
Reading scale
The reading scale is placed parallel to the glass pointer to estimate the readings.
Working principle of Johansson Mikrokator
Johansson Mikrokator works on the simple principle of button spinning on a loop of string. Since H. Abramson developed this instrument, this simple basic principle was also known as the Abramson movement.
When there is vertical movement either upward or downward in the plunger is transmitted through the elbow to the bell crank lever.
The bell crank lever moves either left or right side respective to the vertical movement of the plunger.
The displacement of the bell crank lever makes the strip to twist or untwist (stretching) which is fastened to the cantilever strip at the opposite end. Stretching subjects it into the tensile force.
Perforated stripes are used in this comparator to prevent excessive stress.
The glass pointer placed in the center of the twisted strip start rotates either in the clockwise or anti-clockwise direction. This rotation is directly proportional to change in length of strip due to stretching caused by the plunger movement.
The deflection of the glass pointer is recorded by the calibrated scale parallel to the pointer.
Magnification of Johansson Mikrokator
The magnification of the Johansson MikroKator is expressed in the ratio shown in the below equation.$$ \displaystyle \text{Magnification = }\frac{{d\theta }}{{dl}}$$ $$ \displaystyle \frac{{d\theta }}{{dl}}\propto \frac{l}{{n{{w}^{2}}}}$$where,
l = Length of the metal strip
n = Number of turns on the metal strip
w = Width of the metal strip.
Magnification of this instrument depends upon the length of the strip and the number of twists in the twisted strip.
To obtain better magnification the twisted strip must be smaller in dimension(width). The average dimension of the twisted stip may up to 0.06 mm to 0.0025 mm.
By adjusting the length of the cantilever strip by the screws fitted on it can result in the variation of magnification.
Johansson Mikrokator has an accuracy of $ \displaystyle \pm 1\%$.
Advantages of Johansson Mikrokator
It is small and compact in size.
It is easy to handle and operate.
It has a simple basic working principle compared to other comparators.
Since it is a portable device, it can be easily carried from one place to another place.
It is cheaper in cost.
It is robust and magnification up to 5000 times can be obtained.
Higher sensitivity can be obtained under certain controlled laboratory environmental conditions.
No external power source is required to perform the operation.
Disadvantages of Johansson Mikrokator
Due to improper observation, there may be a chance of getting a parallax error.
Wear can occur in the internal components such as in slit washer.
Accuracy and sensitivity can be varied to the variation of environmental conditions.
Applications of Johansson Mikrokator
It is used as a tool in the inspection field.
It is used as a measuring device in mass production and in automation.
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Why are imaginary numbers important
In order to understand and appreciate the need for imaginary numbers, you should start from the very basics of math, the numbers. Even though this is a high school level topic, we will. Imaginary numbers are important since it exceeds the limitation of what we thought mathematics is for a long time. A square root of a negative number.. First lets review all sets of different number types, where they fail, and where in their list is the set of complex numbers. (Complex numbers include imaginary numbers as a special case.) Each set in the list will be shown to arise from the inade..
Part 1: What Are Imaginary Numbers? Why Do We Need Them
The square root of any negative number is not a real number. denoted as i for imaginary because it does not exist, in the normal concept of numbers.Complex numbers (which include real and.
Learn about the imaginary unit i, about the imaginary numbers, and about square roots of negative numbers. Google Classroom Facebook Twitter. Email. The imaginary unit i. Intro to the imaginary numbers. Intro to the imaginary numbers. This is the currently selected item
An imaginary number is a real number that has been multiplied by i, an imaginary unit that is equivalent to the square root of -1. This means that imaginary numbers are essentially negative perfect squares. Since it is otherwise impossible to achieve a negative square root through standard multiplication, imaginary numbers become necessary to.
An imaginary number is a number that, when squared, has a negative result. Essentially, an imaginary number is the square root of a negative number and does not have a tangible value Imaginary numbers, like real numbers, are simply ideas without any physical existence. They are both very useful (though with real numbers, it is much more obvious why that is so). But it is hard to see how one would (convincingly) argue that real numbers actually exist while imaginary numbers do not Why Do Electrical Engineers Use Imaginary Numbers - Visit us today for local electrician training programs, including campus locations, start dates an {Electricians are an important part of all our daily lives,The use of electricians within our daily lives are very important,Electricians are a necessity in our
Why are negative numbers important? I can't hold -4 fingers up, I can't put -2 cookies in a jar...so in a sense, they don't exist. Same with imaginary numbers. In a lot of contexts, they aren't relevant. But if you are trying to do something very specific, like compute the impedance of an AC circuit, they are invaluable Here comes an important point- IMAGINARY NUMBERS are Imaginary but their existance is not Imaginary they really exist. it was imaginary in the sense as it was left to the people's imagination to imagine a solution to the square root of negative numbers and use the letter i this was fancy and impressive The word imaginary can be a bit misleading in the sense that it implies imaginary numbers don't exist or that they aren't important. A better way to think about it is that normal (real) numbers can directly refer to actual quantities, for example the number 3 can refer to 3 loaves of bread Imaginary numbers run contra to common sense on a basic level, but you must accept them as a system, and then they make sense: remember that nothing makes 2+2=4 except the fact that we SAY SO. Imaginary numbers are just another class of number, exactly like the two new classes of numbers we've seen so far. Let's see why and how imaginary numbers came about. Let's see.
Imaginary Numbers are not Imaginary. Imaginary Numbers were once thought to be impossible, and so they were called Imaginary (to make fun of them).. But then people researched them more and discovered they were actually useful and important because they filled a gap in mathematics but the imaginary name has stuck.. And that is also how the name Real Numbers came about (real is not. Argand was also a pioneer in relating imaginary numbers to geometry via the concept of complex numbers. Complex numbers are numbers with a real part and an imaginary part. For instance, 4 + 2i is a complex number with a real part equal to 4 and an imaginary part equal to 2i. It turns out that both real numbers and imaginary numbers are also.
Why are imaginary numbers important? Study
Imaginary Numbers Definition. Imaginary numbers are the numbers when squared it gives the negative result. In other words, imaginary numbers are defined as the square root of the negative numbers where it does not have a definite value
Complex numbers are broadly used in physics, normally as a calculation tool that makes things easier due to Euler's formula. In the end, it is only the real component that has physical meaning or the two parts (real and imaginary) are treated separately as real quantities
An international research team has proven that the imaginary part of quantum mechanics can be observed in action in the real world. For almost a century, physicists have been intrigued by the fundamental question: why are complex numbers so important in quantum mechanics, that is, numbers containing.
The Story of Imaginary Numbers and why they are not imaginary 1. Prelude, and a Challenge 2. The Oath, the Sign, and the Discovery 3. The Dilemma 4. Bombelli's Breakthrough 5. Geometric Progress: John Wallis 6. Completing the Jigsaw Girolamo Cardano (Pavia, Bologna) The Story of Imaginary Numbers and why they are not imaginary
Why is imaginary number s important for Electrical Engineering?Without it, it's impossible to analyze the electric circuit. We tell you why.https://www.iklea..
These are numbers we can't picture, numbers that normal human consciousness cannot comprehend. And when we add the imaginary numbers to the real numbers, we have the complex number system. The first number system in which it's possible to explain satisfactorily the crystal formation of ice. It's like a vast, open landscape. The horizons
Using complex numbers means you are trying to describe a value in a different domain and in complex number systems, the Imaginary number doesn't mean that the value of capacitor is imaginary. The imaginary number helps to signify the vector rotation when voltage is applied across it or when current flows through it
Why are imaginary numbers important? - Quor
In a general manner, imaginary numbers are used where beyond real numbers a category is needed which is less real. One important example are quantum observables which are treating with complex (and thus with imaginary) numbers. Only the measurable values of an observable, called eigenvalues, are real numbers, and inversely only real numbers.
Imaginary numbers are extremely useful in all areas of physics, because you can use the natural exponent and imaginary powers (exp(a+bi)) to represent sinusoids that grow or decay over time. That's everything from a weight on a spring to current through a wire (which is my field) in a single general representation
-- but having to learn about imaginary numbers -- numbers that didn't even exist -- numbers that were made up by some guy back in the 16th century -- I just didn't understand that at all -- it may be important for some fields like quantum mechanics and electrical engineering, but for a farmer or for a wildlife biologist -- which is what I.
An imaginary number is the square root of a negative number. That is why they are called imaginary, what René Descartes called them, because he thought such a number could not exist. In this paper, I will discuss how complex numbers and imaginary numbers were discovered, the interesting math of complex numbers, and how they are used in other.
A short and sweet video explaining why Complex Numbers are so interesting!
-The word imaginary was meant to be downgrading for these numbers, because at a certain point in time they were deemed useless. -Even after imaginary numbers were concluded as important and useful, mathematicians decided that it would be best to keep that name
Why is the unit when you are dealing with imaginary numbers important: When you're dealing with the theoretical concept of imaginary numbers, the term unit is used to describe first term and is equivalent to how the numeral one, is the first number which exists
Why are imaginary numbers important? - Answer
But imaginary numbers, and the complex numbers they help define, turn out to be incredibly useful. They have a far-reaching impact in physics, engineering, number theory and geometry . And they are the first step into a world of strange number systems, some of which are being proposed as models of the mysterious relationships underlying our.
the imaginary unit, i, is defined as. i= sqrt(-1). Obviously there is no real number that is the square root of an imaginary number. However, in doing math problems you inevitably need to take square roots of negative numbers, which you can't do
us sign: Uh oh. This question makes most people cringe the first time they see it. You want the square root of a number less than zero? That's absurd
Complex numbers are made up of two components, real and imaginary. They have the form a + bi, where the numbers a and b are real. The bi component is responsible for the specific features of complex numbers. The key role here is played by the imaginary number i, i.e. the square root of -1
Either one of them can be termed real or imaginary. Since complex numbers provide ready means of describing numbers with two dimensions, they come in handy to describe the wave function. But there is no reason why an alternate mathematical structure that provides two dimensions to represent a number can not be employed to describe wave function Imaginary numbers live in a world of their own; the numbers are counted on an entirely different plane or axis that is solely devised for them. However, imaginary numbers have acquired a somewhat nefarious reputation, considering that their discovery has compounded the difficulty of problems that math was already replete with. I mean, as if the numbers we already had weren't enough An international research team has proven that the imaginary part of quantum mechanics can be observed in action in the real world. For almost a century, physicists have been intrigued by the fundamental question: why are complex numbers so important in quantum mechanics, that is, numbers containing a component with the imaginary number i Representation of Waves via Complex Numbers In mathematics, the symbol is conventionally used to represent the square-root of minus one: that is, the solution of (Riley 1974). A real number, (say), can take any value in a continuum of values lying between and . On the other hand, an imaginary number takes the general form , where is a real number The name of imaginary numbers. The name of the imaginary numbers includes the impression of the numbers are in the imagination and they do not actually exist. However, the imaginary number is rotational transformation as I explained it in this page. The imaginary number exists almost as same as rotational transformation existing well
An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i 2 = −1. The square of an imaginary number bi is −b 2.For example, 5i is an imaginary number, and its square is −25.By definition, zero is considered to be both real and imaginary. Originally coined in the 17th century by René Descartes as a. For real numbers, a horizontal number line is used, with numbers increasing in value as you move to the left. John Wallis added a vertical line to represent the imaginary numbers. This is called the complex number plane where the x-axis is named the real axis and the y-axis is named the imaginary axis By the way, imaginary and complex numbers really became important with Cardano's formula for solving cubic equation. For some cubic equations having only real solutions, Cardano's formula would require working with complex numbers (the imaginary parts cancelled out at the end) I've been reading up on imaginary numbers and how they work. I even have a rudimentary understanding of how the complex plane works. But one thing that has eluded me is just why they were invented in the first place.why they were invented in the first place
Imaginary numbers belong to the complex number system. All numbers of the equation a + bi, where a and b are real numbers are a part of the complex number system. Imaginary Numbers at Work Imaginary numbers are used in a variety of fields and holds many uses. Without imaginary numbers you wouldn't be able to listen to the radio or talk on. If you become a mathematician, engineer or physicist, imaginary numbers become very important. Imaginary numbers are mainly used in mathematical modeling. They can affect values in models where the state of a model at a particular moment in time is affected by the state of a model at an earlier time Imaginary Number. The number is the basis of any imaginary number, which, in general, is any real number times i. For example, 5i is an imaginary number and is equivalent to - 1 ÷ 5. The real numbers are those numbers that can be expressed as terminating, repeating, or nonrepeating decimals; they include positive and negative numbers
The most fundamental of the imaginary numbers, so called because, in reality, no number can be multiplied by itself to produce a negative number (and, therefore, negative numbers have no real. A complex number is a number comprising area land imaginary part. It can be written in the form a+ib, where a and b are real numbers, and i is the standard imaginary unit with the property i2=-1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of. The imaginary number unlike real numbers cannot be represented on a number line but are real in the sense that it is used in Mathematics. Imaginary numbers are also known as complex numbers. Imaginary numbers also show up in equations of quadratic planes where the imaginary numbers don't touch the x-axis
Imaginary numbers are not called real numbers - but this meaning of real is the mathematic definition, pertaining to cauchy sequences... and does not at all refer to the generic meaning of real, which you seem to be implying. we can easily see why imaginary and complex numbers are so very important to things like electronics and. An imaginary number is a complex number that can be defined as a real number multiplied by the imaginary number i. i is defined as the square root of negative one. If you've taken basic math, you know that the square of every real number is a positive number, and that the square root of every real number is, therefore, a positive number.
A complex number is a mathematical tool, and it is widely used in mechanics, electrodynamics, optics and other related fields of physics to provide an elegant formulation of the corresponding. The Nikola Tesla Numbers. This idea of a slowly increasing snowball is vitally important when we explore the 3 6 9 numerology. It relates to an idea known as vortex mathematics. In this form of math's, the number 1, 2, 4, 5, 7 and 8 are the numbers that represent the physical world while the 3 6 9 numerology belongs t We use imaginary numbers to represent time delays in circuits. That's all. There is a long story about what imaginary numbers mean in pure math and why they are called imaginary Why Imaginary Numbers are as Real as Real Numbers Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being. ~ Gottfried Leibniz. For any positive number, one can find it's square root. For example, 2 2 = 4 and so the square root of 4 is 2. But what about negative numbers
Intro to the imaginary numbers (article) Khan Academ
Complex numbers are the sums and differences of real and imaginary numbers. In order to work with complex numbers, we must first understand imaginary numbers. Real numbers are the numbers that we are most familiar with such as: 1, 0.67, -5, etc. The square root of any real number has two roots, a positive and a negative
Well, i^2 is a negative number. Therefore, i is not a positive number, and therefore the field of complex numbers is not ordered. Also, a negative number squared is a positive number, and i^2 is a negative number. Therefore, i is not a negative number. Imaginary numbers are neither positive nor negative. Now for the 1 thru 4 bit. 1. -1 = (sqrt.
ed through imaginary polynomial equations by engineers
The imaginary part is an imaginary number , that is, the square-root of a negative number. To keep things standardized, the imaginary part is usually reduced to an ordinary number multiplied by the square-root of negative one. As an example, the complex number: t '1 % &1.041 , is first reduced to: t '1 % 1.041 &1 , and then t
Complex Numbers can also have zero real or imaginary parts such as: Z = 6 + j0 or Z = 0 + j4.In this case the points are plotted directly onto the real or imaginary axis. Also, the angle of a complex number can be calculated using simple trigonometry to calculate the angles of right-angled triangles, or measured anti-clockwise around the Argand diagram starting from the positive real axis
What Are Real-Life Uses of Imaginary Numbers
The imaginary number i: i p 1 i2 = 1: (1) Every imaginary number is expressed as a real-valued multiple of i: p 9 = p 9 p 1 = p 9i= 3i: A complex number: z= a+ bi; (2) where a;bare real, is the sum of a real and an imaginary number. The real part of z: Refzg= ais a real number. The imaginary part of z: Imfzg= bis a also a real number. Imaginary Numbers displays the fruits of this cross-fertilization by collecting the best creative writing about mathematical topics from the past hundred years. In this engaging anthology, we can explore the many ways writers have played with mathematical ideas Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century.Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number. It is known, for example, that I. Newton did not include imaginary quantities within the notion of number, and that G. Leibniz said that complex numbers are a fine and wonderful refuge of the divine spirit, as if it were an amphibian of existence and non-existence
What Are Imaginary Numbers? Live Scienc
The course starts with the basics. You will get an in depth understanding of the fundamentals of complex numbers. Fundamentals are the most important part of building expert knowledge and skills. You will learn everything from what is number axis all the way up to different representation forms of complex numbers and conversions this video is going to be a quick review of complex numbers if you studied complex numbers in the past this will knock off some of the rust and it'll help explain why we use complex numbers in electrical engineering if complex numbers are new to you I highly recommend you go look on the Khan Academy videos that Sal's done on complex numbers and those are in the algebra 2 section so let's get. This seemingly toy theorem has very significant corollaries and I would say that almost all applications of complex numbers is linked to it. For example, it's known that complex numbers make a great aid in solving some important kinds of differential equations. The simplest case where they are needed, is: a*y'' + b*y' + c*y = 0
Q: What the heck are imaginary numbers, how are they
So the complex conjugate is just the act of multiplying the imaginary part of a complex number by negative one. Why is this important? The complex conjugate has a nice property that if you multiply a complex number by its conjugate, the imaginary parts will cancel out 1 Complex Numbers in Quantum Mechanics Complex numbers and variables can be useful in classical physics. However, they are not essential. To emphasize this, recall that forces, positions, momenta, potentials, electric and magnetic fields are all real quantities, and the equations describing them One: It's an important placeholder digit in our number system. Two: It's a useful number in its own right. The first uses of zero in human history can be traced back to around 5,000 years ago. Here giving a longer than normal introduction to imaginary and complex numbers because as a student I couldn't see why some lecturers and professors wanted to use complex numbers instead of tangents, sines and cosines which I knew from school and also because the explanations in Maths books are not always very helpful
An important concept with complex or imaginary numbers is the complex conjugate. If y = (2 + 4i), then y* = (2 - 4i) is the complex conjugate.The multiplication of y by y* yields a real rather than imaginary number (2 + 8i - 8i + 16) = (18). This operation will be performed throughout the text to generate real from imaginary numbers A complex number is expressed in standard form when written a + bi where a is the real part and bi is the imaginary part. For example, [latex]5+2i[/latex] is a complex number. So, too, is [latex]3+4i\sqrt{3}[/latex]. Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number
Why Do Electrical Engineers Use Imaginary Numbers
There are some important things to note about this. Firstly, technically speaking, all real and imaginary numbers are complex. For real numbers, b happens to be 0, and for imaginary numbers, a is 0. However, All complex numbers are not imaginary or real, 3+6i sits on neither line but in the middle of the plane
A number whose square is less than or equal to zero is termed as an imaginary number. Let's take an example, √-5 is an imaginary number and its square is -5. An imaginary number can be written as a real number but multiplied by the imaginary unit.in a+bi complex number i is called the imaginary unit,in given expression a is the real part.
Imaginary Numbers were originally laughed at, and so got the name imaginary. That's all of the most important number types in mathematics. From the Counting Numbers through to the Complex Numbers. There are other types of numbers, because mathematics is a broad subject, but that should do you for now
ELI5: Why imaginary numbers are important? : explainlikeimfiv
The complex number, $6i$, only contains an imaginary part, so its position is expected to lie along the imaginary axis - in fact, $6$ units on the positive imaginary axis. From inspection alone, we can see that the distance of $6i$ from the origin is $6$
which has the form [2.53]. We call the set of numbers of the form [2.53] the complex numbers and denote this set .Given a complex number z = a + bi, we call the real number a the real part of z.We call the real number b the imaginary part of z.This motivates the Re and Im functions that map a complex number z = a + bi to its real and imaginary parts a and b, respectively
At first, imaginary numbers were considered useless (an imaginary number is a number that, when squared, gives a negative result; e.g. 5i = -25). But by the Enlightenment Era, thinkers began to.
Why are imaginary numbers called imaginary? If they
Math has many important constants that give the discipline structure, like pi and i, the imaginary number equal to the square root of -1.But one constant that's equally important, though perhaps. Complex numbers of the form , or just , are called Imaginary Numbers. This somewhat problematic nomenclature is perhaps one reason why they can be viewed as mystical by some people. Having defined , we can state that , where is any positive Real Number. A Complex Number, , can be seen to have a Real part, , and an Imaginary part, . We write and
Imaginary Numbers - Maths Career
Complex numbers are made up of two components, real and imaginary. They have the form a + bi, where the numbers a and b are real. The bi component is responsible for the specific features of.
The most important thing to remember about imaginary numbers is the pattern of exponents. For the most part, dealing with imaginary numbers is pretty similar to dealing with polynomials (though do not mistake i for just another variable-it hates that). Just think of complex numbers as polynomials with a new set of rules to follow, and you.
As we've discussed, every complex number is made by adding a real number to an imaginary number: a + b•i, where a is the real part and b is the imaginary part. We can plot a complex number on the complex plane—the position along the x-axis of this plane represents the real part of the complex number and the position along the y-axis.
A common visualisation of complex numbers is the use of Argand Diagrams. To construct this, picture a Cartesian grid with the x-axis being real numbers and the y-axis being imaginary numbers
Remarks on the History of Complex Numbers. The study of numbers comes usually in succession. Children start with the counting numbers. Move to the negative integers and fractions. Dig into the decimal fractions and sometimes continue to the real numbers. The complex numbers come last, if at all. Every expansion of the notion of numbers has a valid practical explanatio
a and b are real while i is imaginary. A new paper says that rather than complex numbers being a purely mathematical invention to facilitate calculations for physicists, quantum states and complex numbers are instead ironically and inextricably linked. They even can show it experimentally
this is just part of the code that matters and needs to be fixed. I don't know what i'm doing wrong here. all the variables are simple numbers, it's true that one is needed for that other, but there shouldn't be anything wrong with that. the answer for which I'm getting imaginary numbers is supposed to be part of a loop, so it's important I get it right. please ignore the variables that are.
There are many important numbers that have made this world what it currently is. But the following 10 are the most important numbers, or constants, in the entire world. Imaginary Unit: i. First of all, complex numbers (and imaginary numbers) do appear in real-world phenomena; they have lots of practical applications. But now, on to the philosophical portion of the problem. Numbers are abstractions. They don't exist in the same way that, say, physical objects exist. You can give me two apples, but you can't just give me a two Numbers are really *two dimensional*; and just like the integer 1 is the unit distance on the axis of the real numbers, i is the unit distance on the axis of the imaginary numbers
What use are imaginary numbers in the real world? Do they
Let's looks at some of the important features of complex numbers using math module function. Phase of complex number The phase of a complex number is the angle between the real axis and the vector representing the imaginary part Complex numbers are an important and useful extension of the real numbers. In particular, they can be thought of as an extension which allows us to take the square root of a negative number. We define the imaginary unit as the number which squares to \( -1 \), \[ \begin{aligned} i^2 = -1. \end{aligned} \ Complex numbers and imaginary numbers surround us all the time and, as any mathematician will tell you, they are no less real (or less important) than numbers like 1, 2 and 3. Uses of complex numbers in our daily life almost always go unnoticed, but they surround you whenever you turn on a light, pick up a guitar or even watch a tree swaying in.
A Mathematical History: Imaginary Numbers
Imaginary Numbers are defined in Mathematics as numbers so big, you can't even think about how big they are. However, a parallel school of thought claims that the concept of an imaginary number of based on the ancient Indian war game I am thinking of a number from one to ten.A fair guess in this case would be seven (7)(VII), as the Indians have had the number placed in their minds for time. I am confused as why do we need to represent the complex numbers with the imaginary y-axis if we can simply represent them as (x,y) ? I've read that Multiplication by i is an anti-clockwise rotation of a quarter-circle over y-axis.. Multiplying 1 by i gives i The Imaginary Numbers are a codename for artificial, genetically engineered humans that appears as enemies in the later stages of Front Mission 3. Along with the Real Numbers, the Imaginary Numbers are the result of a research project initiated by the Ravnui National Laboratory under the oversight of Bal Gorbovsky, with the primary motive is to create a perfect human being in every aspect.
Imaginary Numbers - MAT
In India, negative numbers did not appear until about 620 CE in the work of Brahmagupta (598 - 670) who used the ideas of 'fortunes' and 'debts' for positive and negative.By this time a system based on place-value was established in India, with zero being used in the Indian number sytem. Brahmagupta used a special sign for negatives and stated the rules for dealing with positive and negative. These are much better described by complex numbers. Rather than the circuit element's state having to be described by two different real numbers V and I, it can be described by a single complex number z = V + i I. Similarly, inductance and capacitance can be thought of as the real and imaginary parts of another single complex number w = C + i L. The terms 'real' and 'imaginary' are not meant to refer to the legitimacy of the numbers involved. If you like, you could just as easily refer to the real numbers as 'happy numbers' and the imaginary numbers as 'super happy numbers.' The important point is, that the names we give to these numbers are just labels Its simple, apart from the magnitudes of current and voltage in AC circuits the relative phase of current and voltage is also very important, therefore the impedance is given in complex form In statistics, the average and the median are two different representations of the center of a data set and can often give two very different stories about the data, especially when the data set contains outliers. The mean, also referred to by statisticians as the average, is the most common statistic used to measure the [
This article provides insight into the importance of complex conjugates in electrical engineering. Complex Numbers. Complex numbers are numbers which are represented in the form $$ z = x + i y $$, where x and y are the real and imaginary parts (respectively) and $$ i =\sqrt{-1} $$.. Complex numbers can also be represented in polar form, which has a magnitude term and an angular term As a conclusion, the most important thing you should know is that they are composed of a real part and an imaginary part, that the number i is equal to the root of -1 and therefore, the number i squared is equal to -1 and how to obtain the conjugate of a complex number Or, one can expand this number system to include additional concepts, such as negative numbers, fractions, even the so-called imaginary numbers (which are not really imaginary at all). Each of these concepts exists provided we look for it in the context of a large enough number system
Imaginary numbers: a brief history to these complex
Plus, many consider the math involved to be a lot more elegant using complex arithmetic (where, for strictly real input, the cosine correlation or even component of an FFT result is put in the real component, and the sine correlation or odd component of the FFT result is put in the imaginary component of a complex number. whether the sinusoidal real and imaginary components are periodic. In addition to the basic signals discussed in this lecture, a number of ad-ditional signals play an important role as building blocks. These are intro-duced in Lecture 3. Suggested Reading Section 2.2, Transformations of the Independent Variable, pages 12-1 Other authors have already discussed how important complex numbers can be for object rotation. Here I am adding a couple of other examples where we can see the use of complex/ imaginary numbers. A really cool application of complex numbers is Fractals which is used in procedural generations i
Imaginary Numbers (Definition, Rules, Operations, & Examples
Complex numbers include everyday real numbers like 3, -8, and 7/13, but in addition, we have to include all of the imaginary numbers, like i, 3i, and -πi, as well as combinations of real and imaginary.You see, complex numbers are what you get when you mix real and imaginary numbers together — a very complicated relationship indeed You may be wondering why it is even necessary to raise the numbers to the second power if we're just going to solve for the square root, anyway. As noted earlier, raising the real and imaginary numbers to the second power pulls them out of the complex number by eliminating all imaginary parts (j squared is -1)
Numbers for the greater part of history have been viewed alternately as concepts and as quantities. Now, this raises problems about many types of numbers, which include negative numbers and imaginary numbers, because these cannot be viewed as quantities although there are compelling theories that can treat them logically as concepts Simplifying Radicals with Imaginary Numbers Maze Activity Students will simplify 13 radicals which include negative coefficients, negative radicands and imaginary numbers. This resource allows for student self-checking and works well as independent work, homework assignment, or even to leave wit Why quadratic equation may have complex solutions? Anywhere you read you will learn that when you calculate the discriminant (the expression inside the square root) and if it is greater than 0 then you have two solutions, when it is equal to 0 than you have two equal solutions, but if it is less than 0 then there are no solutions among real numbers So you see, in its basic form, you need two numbers to represent the ratio of two sine waves: amplitude and phase. A complex number is a mathematical convenience to carry over those two values, although not directly amplitude and phase as such, but the x-y components of the related phasor or vector
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Uplifting sympathy card messages. | CommonCrawl |
\begin{definition}[Definition:Emirp]
An '''emirp''' is a prime number whose reversal is a different prime number.
\end{definition} | ProofWiki |
Heine–Borel theorem
In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:
For a subset S of Euclidean space Rn, the following two statements are equivalent:
• S is closed and bounded
• S is compact, that is, every open cover of S has a finite subcover.
History and motivation
The history of what today is called the Heine–Borel theorem starts in the 19th century, with the search for solid foundations of real analysis. Central to the theory was the concept of uniform continuity and the theorem stating that every continuous function on a closed and bounded interval is uniformly continuous. Peter Gustav Lejeune Dirichlet was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof.[1] He used this proof in his 1852 lectures, which were published only in 1904.[1] Later Eduard Heine, Karl Weierstrass and Salvatore Pincherle used similar techniques. Émile Borel in 1895 was the first to state and prove a form of what is now called the Heine–Borel theorem. His formulation was restricted to countable covers. Pierre Cousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized it to arbitrary covers.[2]
Proof
If a set is compact, then it must be closed.
Let S be a subset of Rn. Observe first the following: if a is a limit point of S, then any finite collection C of open sets, such that each open set U ∈ C is disjoint from some neighborhood VU of a, fails to be a cover of S. Indeed, the intersection of the finite family of sets VU is a neighborhood W of a in Rn. Since a is a limit point of S, W must contain a point x in S. This x ∈ S is not covered by the family C, because every U in C is disjoint from VU and hence disjoint from W, which contains x.
If S is compact but not closed, then it has a limit point a not in S. Consider a collection C ′ consisting of an open neighborhood N(x) for each x ∈ S, chosen small enough to not intersect some neighborhood Vx of a. Then C ′ is an open cover of S, but any finite subcollection of C ′ has the form of C discussed previously, and thus cannot be an open subcover of S. This contradicts the compactness of S. Hence, every limit point of S is in S, so S is closed.
The proof above applies with almost no change to showing that any compact subset S of a Hausdorff topological space X is closed in X.
If a set is compact, then it is bounded.
Let $S$ be a compact set in $\mathbf {R} ^{n}$, and $U_{x}$ a ball of radius 1 centered at $x\in \mathbf {R} ^{n}$. Then the set of all such balls centered at $x\in S$ is clearly an open cover of $S$, since $\cup _{x\in S}U_{x}$ contains all of $S$. Since $S$ is compact, take a finite subcover of this cover. This subcover is the finite union of balls of radius 1. Consider all pairs of centers of these (finitely many) balls (of radius 1) and let $M$ be the maximum of the distances between them. Then if $C_{p}$ and $C_{q}$ are the centers (respectively) of unit balls containing arbitrary $p,q\in S$, the triangle inequality says:
$d(p,q)\leq d(p,C_{p})+d(C_{p},C_{q})+d(C_{q},q)\leq 1+M+1=M+2.$
So the diameter of $S$ is bounded by $M+2$.
Lemma: A closed subset of a compact set is compact.
Let K be a closed subset of a compact set T in Rn and let CK be an open cover of K. Then U = Rn \ K is an open set and
$C_{T}=C_{K}\cup \{U\}$
is an open cover of T. Since T is compact, then CT has a finite subcover $C_{T}',$ that also covers the smaller set K. Since U does not contain any point of K, the set K is already covered by $C_{K}'=C_{T}'\setminus \{U\},$ that is a finite subcollection of the original collection CK. It is thus possible to extract from any open cover CK of K a finite subcover.
If a set is closed and bounded, then it is compact.
If a set S in Rn is bounded, then it can be enclosed within an n-box
$T_{0}=[-a,a]^{n}$
where a > 0. By the lemma above, it is enough to show that T0 is compact.
Assume, by way of contradiction, that T0 is not compact. Then there exists an infinite open cover C of T0 that does not admit any finite subcover. Through bisection of each of the sides of T0, the box T0 can be broken up into 2n sub n-boxes, each of which has diameter equal to half the diameter of T0. Then at least one of the 2n sections of T0 must require an infinite subcover of C, otherwise C itself would have a finite subcover, by uniting together the finite covers of the sections. Call this section T1.
Likewise, the sides of T1 can be bisected, yielding 2n sections of T1, at least one of which must require an infinite subcover of C. Continuing in like manner yields a decreasing sequence of nested n-boxes:
$T_{0}\supset T_{1}\supset T_{2}\supset \ldots \supset T_{k}\supset \ldots $
where the side length of Tk is (2 a) / 2k, which tends to 0 as k tends to infinity. Let us define a sequence (xk) such that each xk is in Tk. This sequence is Cauchy, so it must converge to some limit L. Since each Tk is closed, and for each k the sequence (xk) is eventually always inside Tk, we see that L ∈ Tk for each k.
Since C covers T0, then it has some member U ∈ C such that L ∈ U. Since U is open, there is an n-ball B(L) ⊆ U. For large enough k, one has Tk ⊆ B(L) ⊆ U, but then the infinite number of members of C needed to cover Tk can be replaced by just one: U, a contradiction.
Thus, T0 is compact. Since S is closed and a subset of the compact set T0, then S is also compact (see the lemma above).
Heine–Borel property
The Heine–Borel theorem does not hold as stated for general metric and topological vector spaces, and this gives rise to the necessity to consider special classes of spaces where this proposition is true. They are called the spaces with the Heine–Borel property.
In the theory of metric spaces
A metric space $(X,d)$ is said to have the Heine–Borel property if each closed bounded[3] set in $X$ is compact.
Many metric spaces fail to have the Heine–Borel property, such as the metric space of rational numbers (or indeed any incomplete metric space). Complete metric spaces may also fail to have the property; for instance, no infinite-dimensional Banach spaces have the Heine–Borel property (as metric spaces). Even more trivially, if the real line is not endowed with the usual metric, it may fail to have the Heine–Borel property.
A metric space $(X,d)$ has a Heine–Borel metric which is Cauchy locally identical to $d$ if and only if it is complete, $\sigma $-compact, and locally compact.[4]
In the theory of topological vector spaces
A topological vector space $X$ is said to have the Heine–Borel property[5] (R.E. Edwards uses the term boundedly compact space[6]) if each closed bounded[7] set in $X$ is compact.[8] No infinite-dimensional Banach spaces have the Heine–Borel property (as topological vector spaces). But some infinite-dimensional Fréchet spaces do have, for instance, the space $C^{\infty }(\Omega )$ of smooth functions on an open set $\Omega \subset \mathbb {R} ^{n}$[6] and the space $H(\Omega )$ of holomorphic functions on an open set $\Omega \subset \mathbb {C} ^{n}$.[6] More generally, any quasi-complete nuclear space has the Heine–Borel property. All Montel spaces have the Heine–Borel property as well.
See also
• Bolzano–Weierstrass theorem
Notes
1. Raman-Sundström, Manya (August–September 2015). "A Pedagogical History of Compactness". American Mathematical Monthly. 122 (7): 619–635. arXiv:1006.4131. doi:10.4169/amer.math.monthly.122.7.619. JSTOR 10.4169/amer.math.monthly.122.7.619. S2CID 119936587.
2. Sundström, Manya Raman (2010). "A pedagogical history of compactness". arXiv:1006.4131v1 [math.HO].
3. A set $B$ in a metric space $(X,d)$ is said to be bounded if it is contained in a ball of a finite radius, i.e. there exists $a\in X$ and $r>0$ such that $B\subseteq \{x\in X:\ d(x,a)\leq r\}$.
4. Williamson & Janos 1987.
5. Kirillov & Gvishiani 1982, Theorem 28.
6. Edwards 1965, 8.4.7.
7. A set $B$ in a topological vector space $X$ is said to be bounded if for each neighborhood of zero $U$ in $X$ there exists a scalar $\lambda $ such that $B\subseteq \lambda \cdot U$.
8. In the case when the topology of a topological vector space $X$ is generated by some metric $d$ this definition is not equivalent to the definition of the Heine–Borel property of $X$ as a metric space, since the notion of bounded set in $X$ as a metric space is different from the notion of bounded set in $X$ as a topological vector space. For instance, the space ${\mathcal {C}}^{\infty }[0,1]$ of smooth functions on the interval $[0,1]$ with the metric $d(x,y)=\sum _{k=0}^{\infty }{\frac {1}{2^{k}}}\cdot {\frac {\max _{t\in [0,1]}|x^{(k)}(t)-y^{(k)}(t)|}{1+\max _{t\in [0,1]}|x^{(k)}(t)-y^{(k)}(t)|}}$ (here $x^{(k)}$ is the $k$-th derivative of the function $x\in {\mathcal {C}}^{\infty }[0,1]$) has the Heine–Borel property as a topological vector space but not as a metric space.
References
• P. Dugac (1989). "Sur la correspondance de Borel et le théorème de Dirichlet–Heine–Weierstrass–Borel–Schoenflies–Lebesgue". Arch. Int. Hist. Sci. 39: 69–110.
• BookOfProofs: Heine-Borel Property
• Jeffreys, H.; Jeffreys, B.S. (1988). Methods of Mathematical Physics. Cambridge University Press. ISBN 978-0521097239.
• Williamson, R.; Janos, L. (1987). "Construction metrics with the Heine-Borel property". Proc. AMS. 100 (3): 567–573. doi:10.1090/S0002-9939-1987-0891165-X.
• Kirillov, A.A.; Gvishiani, A.D. (1982). Theorems and Problems in Functional Analysis. Springer-Verlag New York. ISBN 978-1-4613-8155-6.
• Edwards, R.E. (1965). Functional analysis. Holt, Rinehart and Winston. ISBN 0030505356.
External links
• Ivan Kenig, Dr. Prof. Hans-Christian Graf v. Botthmer, Dmitrij Tiessen, Andreas Timm, Viktor Wittman (2004). The Heine–Borel Theorem. Hannover: Leibniz Universität. Archived from the original (avi • mp4 • mov • swf • streamed video) on 2011-07-19.
• "Borel-Lebesgue covering theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Mathworld "Heine-Borel Theorem"
• "An Analysis of the First Proofs of the Heine-Borel Theorem - Lebesgue's Proof"
| Wikipedia |
Astronomy Meta
Astronomy Stack Exchange is a question and answer site for astronomers and astrophysicists. It only takes a minute to sign up.
Astronomy Beta
Is there a upper limit to the number of planets orbiting a star?
Our sun has 8 planets orbiting as well as a number of dwarf planets. Are there any calculations that hint as to whether this number is close to some theoretical maximum value or are we simply an average solar system in this particular way?
I could imagine that if you have many planets, they will likely interact with each other. Can you calculate any theoretical value for the maximum number of planets which have long-term stable orbits around their own star?
orbit planet
astromax
bogenbogen
$\begingroup$ I imagine this will vary greatly depending on the size and mass of the star too if such a limit does exist $\endgroup$ – RhysW Nov 5 '13 at 22:38
There do exist somewhat trivial configurations, which are stable in the long term and which include arbitrarily many bodies. Consider, for example, a set of $N$ circularly moving bodies of the same mass $m$, which obeys the constraint $mN\ll M$, where $M$ is the mass of the star. So long as $mN\ll M$, the bodies move dominantly in the gravitational field of the star and are hence moving stably over long term period. However, as $N$ is arbitrary, one concludes that there is no upper limit on the number of planets, provided that their total mass is small.
A more physical example would be a protoplanetary disc, or an accretion disc, which is a limit $N\rightarrow \infty$ of an arbitrary planetary system (not not necessarily circular) of a given mass. A yet more physical example is an asteroid belt, consisting of a large number of bodies on, roughly, stable orbits. Finally, during planet formation process the star goes through stages, when it is surrounded by sets of pebbles and asteroids, which keep their structure constant over a large number of orbits (roughly, of order $10^5$). And these all are real physical examples of planetary-like systems.
The answer to your question would start to alter, though, if you start imposing additional conditions apart from $N\rightarrow \infty$. For example, if you require that bodies do not collide in the long term, some of the above named systems would not work (for example, accretion disc model), but some other would (sets of concentric particles). If you additionally require that the object should obey the definition of a planet, that is have some range of masses, then interesting things will start happening when the total mass of the planets will start being comparable to the mass of the star. So the limit would certainly exist. Finally, you might be more strict about what do you really mean by stability here, and that could also have a bearing on the answer.
To summarize, unless you impose any constraints, there do exist N-body systems orbiting a star in a stable fashion and having arbitrarily large $N$.
Alexey BobrickAlexey Bobrick
The limit would depend on the size of the central star as well as the location and sizes of the planets in the system.
Really the limit would be the number of planets that you can fit within the area of which the orbital velocity is >0. Once you reach that distance, you can't orbit anymore. Though adding a planet would move this further out due to the added mass itself. So in theory you could keep pushing this limit and stick more planets in for forever (depending on what you consider to be a planet).
The problem comes more with having stable orbits. Each planet that you add to the system would affect the rest of the system and could cause the orbits to not be stable anymore. Also adding planets would allow more planets further out due to the additional mass but it does make figuring out if you have a stable orbit more complicated (https://en.wikipedia.org/wiki/N-body_problem).
SchleisSchleis
I am not feeling completely satisfied with Alexey Bobrick's argument: "interesting things will start happening when the total mass of the planets will start being comparable to the mass of the star. So the limit would certainly exist.".
Let us consider a self-similar hierarchical planetary system, where planet number $p$ has semi-major axis $a_p$ and where $a_{p+1}>>a_p$ (say, as in a geometric progression). For planet $p$, all the mass inside its orbit is that "of the star". In other words, the effective mass of the star depends on the planet we are considering and it has no limits.
I don't see any argument going against the stability of such a system.
$\begingroup$ This argument has an obvious flaw: We know that planets perturb each other, so your $a_{p+1}$ has to be much greater than $a_{p}$ to an extent that we quickly leave distances where we comfortably can form and bind planets. $\endgroup$ – AtmosphericPrisonEscape May 5 '17 at 11:40
Lets start with some basics and, before I continue, this is a criteria based answer.
Short answer: 30. (OK that sounds nuts, but hear me out). That's about the upper, upper, gonzo, bananas limit for planet definition and long-term stable orbits. I'm tempted to say 25 as an upper limit just because 30 seems too improbable.
The gist of the problem, is that a star and protoplanetary disk is unlikely to form the maximum possible number of planets. Gravity tends to clump around the larger objects. Planetary perturbations and migration make the maximum possible stable number unlikely to be reached, but with luck of a "just right" formation and some planet capture, I reached a ballpark estimate of about 30.
Long answer: let's assume we're talking about only stable planetary orbits by the definition of having cleared out their orbital path and don't cross each other's orbits. This eliminates any Trojan planets and doesn't eliminate, but makes highly elliptical orbits problematic because they span a greater orbital range.
And lets dismiss any large planetesimals that might be planet sized and any planet sized dwarf planets that cross other planet's orbits. We're only counting orbit dominating planet definition planets.
Lets also eliminate any binary or trinary systems, and only use single star systems, but the star could have some very massive planets that are borderline brown dwarf stars if you like.
Using our solar-system as a guideline and quoting from the planetesimals article above:
It is generally thought that about 3.8 billion years ago, after a period known as the Late Heavy Bombardment, most of the planetesimals within the Solar System had either been ejected from the Solar System entirely, into distant eccentric orbits such as the Oort cloud, or had collided with larger objects due to the regular gravitational nudges from the giant planets
I'd also like to set some kind of time-limit because young solar-systems can have hundreds of large planitesimals. By about 700 million years of age, our solar-system had, for the most part, settled down into the 8, maybe soon to be 9, planets that are currently known.
A larger star probably has the potential for a good deal more than 9. But if it takes 700 million years (give or take) for a protoplanetary disk to work itself out into planets with stable, semi-permanent orbits, that puts a limit on the size of the star.
A 40 solar-mass star has a lifespan of only a million years or so before it goes Supernova. That's far too short a lifespan for planetary systems to form. Even a 10 solar mass star lasts just 30 million years or so. Again, too short.
A 4 solar mass star has a lifespan some 30 times shorter than our sun (using the 2.5 power rule, which I've also seen as a 3 power rule, but all this is pretty ballpark. Point is, a star with 4 solar masses has less than 400 million years for it's planetary system. 5 solar masses, as little as 200 million years. That's pretty close to what I'd call the minimum amount of time for a planetary system to have relevance, so I'm going to go with a 4 solar mass upper limit. The romantic notion of a star 20 times the mass of our sun, with 100 planets might make good science fiction, but it's not realistic.
A 2nd factor to consider is the mass and size of the planetary debris field. Our sun is about 99.8% of the mass of the solar-system, leaving 0.2% of the solar-system's mass to form all the planets and other stuff. There was probably more mass in the debris field originally, some of which was lost as rogue planets, rogue comets and asteroids, so the original planetary debris field might have been higher, but not all that much higher. Larger objects can cast out smaller ones. The ratio of lost debris to remaining debris shouldn't be all that high. (if anyone knows, feel free to post a comment).
The highest percentage of mass in a forming solar-system is difficult to calculate and it depends on the total angular momentum of the debris field that collapses into the spiraling disk of matter, but it's improbable that the % of mass gets too high. 1%-3% might be on the upper limit. If we go with 3% of the mass of a 4 solar mass star in the planetary disk that's about 40,000 Earth masses or about 125 Jupiter masses. That's obviously ballpark, perhaps too ballpark, but it helps to have a sense of how much stuff we have to work with.
The size of a debris field is important too. By this article, the largest debris field ever observed is about 1,000 AU in diameter (500 AU in radius) with a debris field mass of about 3.1 += .6 Jupiter masses and a central star perhaps less massive than our sun. Whether such a system could form planets as far out as 500 AU is hard to say, but I'm inclined to think that the outermost planet would form comfortably inside that debris field, not at the observed edge.
It's worth pointing out that planetary formation is a chaotic mess. A young protoplanetary disk, especially one with some 125 jupiter masses worth of material could easily form over 100 planet sized objects early in formation, but it wouldn't retain that many.
Planets perturb each other's orbits and they need space. You'd get collisions like the collection that formed our Moon and larger planets can send smaller planets any which way. No system could keep 100 planets. It's too many and would be much too unstable. There would be far fewer when a mostly stable formation is reached.
Jupiter, for example, is believed to have migrated towards the sun when our solar system was young, them migrated back outwards, called type II migration. Migrating Jupiters are both good and bad if you want a lot of planets. Jupiter's migration is believed to be the reason why there's no planets and so much empty space between Mars and Jupiter and why Mars is so small. Jupiter's migration may have also sent Uranus, Neptune out to their current distant orbits, so gas giant migration can move planets around, but it can also cast them completely out of a solar-system. The larger the gas giant, the greater a kick it can give to smaller planets.
Very massive planets are bad if you want the highest number of planets because they cause greater perturbations and demand the greatest space around them. With a lot of debris in a planetary disk, very large planets are likely to form so more debris isn't always better. What you probably want is a larger, more spread out disk, where you don't get any super massive planets, but some massive enough to push some young forming planets outwards to create more planets at greater distances. Planets are unlikely to form at very great distances, but they can be tossed out there by larger planets to very distant orbits. By tossing a number of fledgling planets outwards early in formation, the total number of planets in a solar-system could increase.
How close can the planets be to each other?
Planets don't like to be too close to each other. While we can't see small planets very well, Kepler observations seem to confirm this that very close planets are rare. When they're too close, there's orbital instability. Earth and Venus are the closest planets by multiple, where Earth is 1.38 times the distance from the sun as Venus. By this short article, a multiple of 1.4 to 1.8 times the distance between planets is suggested. Observations of exo-solar-systems find very few planets closer than 1.4 times their nearest observed neighbor, so for an entire system, a 1.4 to 1.8 multiple seems about right on average.
Planets around small stars, like Trappist 1 can get very close to each other, close enough that they can appear about moon sized from their closest neighbors, but those systems are almost entirely around small red-dwarf stars with very tight orbits, often with orbital resonance and even with very close orbiting planets, they still average out to about the 1.4 multiple or greater. Planets in a 3/2 orbital resonance that corresponds to a 1.31 distance multiple, and such resonances depend on the interactive tidal force that are only possible at close distances around smaller stars.
Kepler 36 is an oddball with two very close planets with a 7:6 orbital resonance, but building an entire solar system from planets that close seems enormously improbable. So, a key criteria to my estimate is the 1.4 distance multiple, and that's probably conservative over an entire system.
How close can the closest planets be to the star?
The heat of a 4 solar-mass star is a problem for very close planets. A 4 solar mass star, (while the luminosity changes over it's lifetime), is over 100 times more luminous than our sun, so the innermost rocky planet should probably begin at roughly about 10 times the distance Mercury is from our sun. Much closer than that and the planet would be in danger of being vaporized. So for a 4 solar mass star, 3 AU might be a good starting point. Applying the 1.4 multiple to a 3 AU starting point. A hot Jupiter might survive closer than that, but a hot Jupiter couldn't form that close, so that would probably require too much migration for our goal of highest number of planets.
so, if we start at 3 AU, and we do a 1.4 distance multiple, then our 4 solar mass star can have up to 30 planets within an orbits less than one light year, and just 32 within 2 light years, so you don't add much by doubling the distance, at least, using the 1.4 multiple.
An obvious question that follows might be, well, maybe the 1.4 multiple no longer applies at larger distances, but planets would need to grow fairly large to effectively clear out their orbit and have an effect on near-by asteroids and comets, like Neptune does and Planet 9 is believed to, so as the distance grows, you can't have mercury sized planets and define them as planets, and as the distance grows, the planets gravitational effect on each other remains consistent, so the 1.4 multiple rule should still apply even at very distant orbits.
Mercury for example, is massive enough to be a planet where it is, but if it was out past Neptune, it would be perhaps too small to clear out it's orbit. Here's a question that discusses this in more detail and it raises the problem that if Pluto was some 15-20 times more massive, the minimum mass it would need and assuming it wasn't crossing Neptune's orbit, That theoretical object would still need a billion years to clear it's orbit and that's over twice our star's lifetime and the necessary minimum size grows a larger at greater distances.
So, if we go with our one-light year proposal, an object orbiting around a 4 solar mass star at 1 light year distance has an orbital period of about 8 million years and an orbital velocity of about .23 km/s and it would have a required minimum mass to clear out it's orbit of at least several Earths. Planet 9, for comparison, is thought to have an orbital period between 10,000 and 20,000 years and an orbital velocity in the .5-.7 km/s range and a semi-major axis of about 600-800 AU or about 1/90th of a lightyear. Those numbers are all ballpark and just posted for comparison. But it points out the difficulty in recognizing a planet in a very distant orbit.
And for a planet to get that distant, it would need to be thrown out there by a larger planet, presumably undergoing type II migration or, perhaps captured from a passing star. I think you'd probably want some of both to maximize the number of planets. A star with a very large very distant planet could be effective in helping capture planets and/or debris from nearby stars that pass too close.
In both cases, the planet cast out very far or captured planets would initially have a very eccentric orbit and it would take some time for any such planets to circularize and you'd need the orbits to circularize, because a handful of eccentric orbits don't meet the planet criteria if they cross other planets.
Again, using our solar-system as a model, the outer planets, Uranus, Neptune and Planet 9 (if it exists) are all thought to have formed quite a bit closer to the sun than where they are now and migrated outwards, presumably by Jupiter.
A large star could have upwards of 100 Mercury or maybe even Earth sized objects in it's orbit, but no where close to that many that would meet the planet criteria. 30 is pushing it.
A large star capturing planets whether rogue, or capturing planets off a smaller star is certainly possible. 3 body dynamics does make planet capture possible, but there's still the problem of eccentricity and orbits crossing other orbits not meeting the criteria of a planet. If you dismiss that standard orbital criteria or a planet, then the number goes up.
So, using the criteria for a large star (4 solar masses) an innermost planet (3 AU) an outermost (1 light year - a bit of a stretch), and distance multiple (1.4 - also probably on the low side), a 4 solar mass star could have a maximum of 30 planets. If you run different criteria, you get different numbers, but I think that's a pretty good upper benchmark, perhaps on the generous side. Such a system could have a lot more objects that meet the dwarf-planet criteria, some of them even what we think of as planet sized, but meeting the complete planet criteria, 30 seems a pretty good gonzo upper limit.
Something interesting happens if you make the star smaller. If we make the star 2 solar masses instead of 4 and put the outermost planet at the inverse square law or .707 light years, not 1 light year. A 2 solar mass planet is about 12-16 times as luminous as our sun and 12-16 times less luminous than a 4 solar mass star, so the outermost planet that wouldn't get vaporized is now about 1 AU, not 3 AU. So the inner part of the planet region is 3 times closer and just 1.4 times close on the outside, so curiously a 2 solar mass star could perhaps maybe hold more planets than the 4 solar mass star. It wouldn't capture as many, on average, but the upper limit still goes up, using the same criteria to 32 or 33 for a 2 solar mass star and continues to grow as the star gets smaller.
At the same time, as stars get smaller, the upper end mass of the planetary debris field grows smaller too and the ability to capture planets drops, so I don't small stars are good candidates for the most planets, but interestingly, smaller stars with smaller protoplanetary disks could still, on average have as many planets as their larger neighbors. When James Webb starts to take a look, maybe we'll get an answer on this.
Obviously if you had all no criteria, and a star a few million light years from the nearest galaxy or massive object, you could design something with many more planets, but I'm thinking formation within a galaxy and I'm thinking that both planet capture and the right set of circumstances during formation would both play a role in maximizing the number of planets. A star that far from other stars would be unlikely to capture any planets.
Hope that's not too world-building an an answer or too long. I'll try to check it for typos tomorrow. (kinda late now).
userLTKuserLTK
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Leonid Khachiyan
Leonid Genrikhovich Khachiyan[1][lower-alpha 1] (/kɑːtʃiːən/;[4] Russian: Леони́д Ге́нрихович Хачия́н; May 3, 1952 – April 29, 2005) was a Soviet and American mathematician and computer scientist.
Leonid Khachiyan
Born
Leonid Genrikhovich Khachiyan
(1952-05-03)May 3, 1952
Leningrad, Russian SFSR, Soviet Union
DiedApril 29, 2005(2005-04-29) (aged 52)
South Brunswick, New Jersey, U.S.
CitizenshipSoviet Union, United States
Children2, including Anna
AwardsFulkerson Prize (1982)
Scientific career
InstitutionsComputer Center of the Soviet Academy of Sciences
Rutgers University
He was most famous for his ellipsoid algorithm (1979) for linear programming,[5] which was the first such algorithm known to have a polynomial running time. Even though this algorithm was shown to be impractical, it has inspired other randomized algorithms for convex programming and is considered a significant theoretical breakthrough.
Early life and education
Khachiyan was born on May 3, 1952, in Leningrad to Armenian parents Genrikh Borisovich Khachiyan, a mathematician and professor of theoretical mechanics, and Zhanna Saakovna Khachiyan, a civil engineer.[6][1] His grandparents were Karabakh Armenians.[7][8] He had two brothers: Boris and Yevgeniy (Eugene).[6][4] His family moved to Moscow in 1961, when he was nine.[1][6] He received a master's degree from the Moscow Institute of Physics and Technology.[4] In 1978 he earned his Ph.D. in computational mathematics/theoretical mathematics from the Computer Center of the Soviet Academy of Sciences and in 1984 a D.Sc. in computer science from the same institution.[6][4][1]
Career
Khachiyan began his career at the Soviet Academy of Sciences,[4] working as a researcher at the academy's Computer Center in Moscow.[1] He also worked as an adjunct professor at the Moscow Institute of Physics and Technology.[9] In 1979 he stated: "I am a theoretical mathematician and I'm just working on a class of very difficult mathematical problems."[1] Khachiyan immigrated to the United States in 1989.[10][6] He first taught at Cornell University as a visiting professor. In 1990 he joined Rutgers University as a visiting professor.[4][6][9] He became professor[11] of computer science at Rutgers in 1992.[4][6] By 2005, he held the position of Professor II at Rutgers, reserved for those faculty who have achieved scholarly eminence in their discipline.[6]
Work on linear programming
Ellipsoid method
Khachiyan is best known for his four-page February 1979 paper[12] that indicated how an ellipsoid method for linear programming can be implemented in polynomial time.[13][9] The paper was translated into several languages and spread around the world unusually quickly. Authors of a 1981 survey of his work noted that it "has caused great excitement and stimulated a flood of technical papers" and was covered by major newspapers.[13] It was originally published without proofs, which were provided by Khachiyan in a later paper published in 1980[14] and by Peter Gács and Laszlo Lovász in 1981.[15][9][13] It was Gács and Lovász who first brought attention to Khachiyan's paper at the International Symposium on Mathematical Programming in Montreal in August 1979.[13][6] It was further popularized when Gina Kolata reported it in Science Magazine on November 2, 1979.[16][11]
Khachiyan's theory is considered a groundbreaking one that "helped advance the field of linear programming."[11] Giorgio Ausiello noted that the method was not practical, "but it was a real breakthrough for the world of operations research and computer science, since it proved that the design of polynomial time algorithms for linear programming was possible and in fact opened the way to other, more practical, algorithms that were designed in the following years."[17]
Personal life and death
Khachiyan spoke Russian and English, but not Armenian.[7] Bahman Kalantari noted that "For some, his English accent wasn’t always easy to understand."[18] A 1979 New York Times profile of him described Khachiyan as "a relaxed, friendly young man in a sweater who speaks a little English, which he learned in high school."[1]
He was known as "Leo"[7][19] and "Lenya" to his friends and colleagues.[20] Václav Chvátal described him as "selfless, open, patient, sympathetic, understanding, considerate."[19] Michael Todd, another colleague, described him as "cynical about politics," "very modest and kind to his friends," and "intolerant of condescension and pomposity."[9]
Khachiyan married Olga Pischikova Reynberg, of Russian-Jewish origin,[21] in 1985.[6][9] They had two daughters, Anna and Nina,[6][4] who were teenagers at the time of his death.[9] He became a naturalized U.S. citizen in 2000.[4][11] He died of a heart attack in South Brunswick, New Jersey on April 29, 2005, at the age of 52.[4][6][11]
Recognition
In 1982 he was awarded the prestigious Fulkerson Prize by the Mathematical Programming Society and the American Mathematical Society[10] for outstanding papers in the area of discrete mathematics,[6] particularly his 1979 article "A polynomial algorithm in linear programming."[22]
Khachiyan was considered a "noted expert in computer science whose work helped computers process extremely complex problems."[10] He was called one of the world's most famous computer scientists at the time of his death by Haym Hirsh, chair of the computer science department at Rutgers.[6][23] "Computer scientists and mathematicians say his work helped revolutionize his field," noted his New York Times obituary.[4] Bahman Kalantari, a friend and colleague at Rutgers, wrote: "Surely, Khachiyan shall always remain to be among the greatest and most legendary figures in the field of mathematical programming."[18]
References
Notes
1. His last name was often spelled in English as Khachian.[2][3] Anglicized as Leonid Henry Khachiyan.[4]
Citations
1. Whitney, Craig R. (November 27, 1979). "Soviet Mathematician Is Obscure No More". The New York Times.
2. Boas, Harold P. (30 November 1979). "Linear Programming Discovery". Science. 206 (4422): 1022. Bibcode:1979Sci...206.1022B. doi:10.1126/science.206.4422.1022-c.
3. Browne, Malcolm W. (November 7, 1979). "A Soviet Discovery Rocks World of Mathematics". The New York Times.
4. Pearce, Jeremy (May 22, 2005). "Leonid Khachiyan Is Dead at 52; Advanced Computer Math". The New York Times.
5. Lawler, Eugene L. (1980). "The Great Mathematical Sputnik of 1979". The Sciences. 20 (7): 12–15. doi:10.1002/j.2326-1951.1980.tb01345.x. S2CID 56588045.
6. "World Renowned Computer Scientist Leonid G. Khachiyan Dies at 52". Rutgers University Department of Computer Science. Archived from the original on 2016-09-11. (archived PDF)
7. Gurvich, Vladimir (6 June 2008). "Recalling Leo". Discrete Applied Mathematics. 156 (11): 1957–1960. doi:10.1016/j.dam.2008.04.013.
8. Khachiyan, Anna (April 25, 2020). "Family portrait of Armenian ancestors, Nagorno-Karabakh, 1920s (great great grandparents in the center, grandmother little girl on the left with the pigtails)". Twitter. Archived from the original on 17 August 2020. Retrieved 17 August 2020.
9. Todd, Michael (October 2005). "Leonid Khachiyan, 1952–2005: An Appreciation". SIAG/OPT Views-and-News. SIAM Activity Group on Optimization. 16 (1–2): 4–6. CiteSeerX 10.1.1.131.3938.
10. "Leonid Khachiyan, 52; Computer Science Expert at Rutgers". Los Angeles Times. May 5, 2005.
11. Madden, Andrew P. (September 1, 2005). "Obituary: Mystery Man". MIT Technology Review. Massachusetts Institute of Technology. (archived PDF)
12. Khachiyan, L. G. 1979. "A Polynomial Algorithm in Linear Programming". Doklady Akademii Nauk SSSR 244, 1093-1096 (translated in Soviet Mathematics Doklady 20, 191-194, 1979).
13. Bland, Robert G.; Goldfarb, Donald; Todd, Michael J. (1981). "The Ellipsoid Method: A Survey" (PDF). Operations Research. 29 (6): 1039–1091. doi:10.1287/opre.29.6.1039. JSTOR 170362. Archived from the original (PDF) on 2015-07-01.
14. Khachiyan, L. G. 1980. "Polynomial Algorithms in Linear Programming". Zhurnal Vychisditel'noi Matematiki i Matematicheskoi Fiziki (USSR Computational Mathematics and Mathematical Physics) 20, 51-68.
15. Gács, Peter; Lovász, Laszlo (1981). "Khachiyan's algorithm for linear programming". In König, H.; Korte, B.; Ritter, K. (eds.). Mathematical Programming at Oberwolfach. Mathematical Programming Studies. Vol. 14. pp. 61–68. doi:10.1007/BFb0120921. ISBN 978-3-642-00805-4.
16. Kolata, Gina Bari (November 2, 1979). "Mathematicians Amazed by Russian's Discovery". Science. 206 (4418): 545–546. Bibcode:1979Sci...206..545B. doi:10.1126/science.206.4418.545. JSTOR 1749236. PMID 17759415.
17. Ausiello, Giorgio (2018). The Making of a New Science: A Personal Journey Through the Early Years of Theoretical Computer Science. Springer. p. 174. ISBN 9783319626802.
18. Kalantari, Bahman (2005). "My Memories of Leonid Khachiyan and a Personal Tribute for His Contributions in Linear Programming" (PDF). Allen Institute for AI. S2CID 15568389. Archived from the original (PDF) on 2020-01-13. {{cite journal}}: Cite journal requires |journal= (help)
19. Chvátal, Václav (6 June 2008). "Remembering Leo Khachiyan". Discrete Applied Mathematics. 156 (11): 1961–1962. doi:10.1016/j.dam.2007.08.001.
20. Todd, Michael J. (1 December 2005). "SIAM: Leonid Khachiyan, 1952 - 2005: An Appreciation". archive.siam.org. Philadelphia: Society for Industrial and Applied Mathematics. Archived from the original on 21 January 2021. Retrieved 27 June 2021.
21. Khachiyan, Anna (December 4, 2019). "I had such a shambolic, dysfunctional upbringing my parents didn't even bother to teach me chess — unheard of and frankly shameful for Russian family of Armenian and Ashkenazi origins lol!". Twitter. Archived from the original on 17 August 2020. Retrieved 17 August 2020.
22. "The Fulkerson Prize". mathopt.org. Mathematical Optimization Society. Archived from the original on 12 February 2019.
23. "Leonid Khachiyan, professor, leading computer scientist". The Boston Globe. (via Associated Press). May 5, 2005. Archived from the original on 4 September 2017.
External links
• DBLP: Leonid Khachiyan.
• In Memoriam: Leonid Khachiyan from the Computer Science Department, Rutgers University.
• SIAM news: Leonid Khachiyan, 1952–2005: An Appreciation.
• Discrete Applied Mathematics: Remembering Leo Khachiyan
• The Mathematics Genealogy Project: Leonid Khachiyan.
• New York Times : Obituary.
Authority control
International
• VIAF
Academics
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Last edited by Shakazuru
Saturday, April 18, 2020 | History
2 edition of Analysis of make-to-stock queues with general production times. found in the catalog.
Analysis of make-to-stock queues with general production times.
Nima Sanajian
Published 2006 .
In this thesis, we obtain the exact steady-state distribution of the number of customers in a GI/G/1 make-to-stock queue, therefore we observe the impact of production time variability on optimal inventory control policies. We also shed light on how system performance deteriorates more because of production stoppages and variability in repair times.We also analyze a production/inventory producing different products. We compare the performances of different scheduling policies in regard of the inventory control parameters and average system costs which are obtained by using the exact steady-state distribution of the number of customers for each static policy (first-come-first-served, preemptive and non-preemptive priority disciplines). We provide the inequalities indicating when it is optimal to switch from make-to-stock to make-to-order policy. We extend the application area of a well-known dynamic scheduling heuristic, Myopic( T), for systems with non-exponential service times. We additionally analyze inventory pooling problem in the M/G/1 make-to-stock queue.
Pagination 66 leaves.
Purpose of the Operations Management Body of Knowledge (OMBOK) – The Operations Management Body of Knowledge provides an outline of the areas needed to successfully manage the processes for producing and delivering common products or services. The descriptions give an overview of each area and when taken together define a generally accepted.
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Analysis of make-to-stock queues with general production times. by Nima Sanajian Download PDF EPUB FB2
Analyzing make-to-stock queues. In particular, the asymptotic analysis of the inventory aspect of the system is novel.
Other than appearing in two classical textbooks Morse () and Zipkin (), the multi-server make-to-stock queue as a tool for supply chain analysis has received no attention in. Motivated by make-to-stock production systems, we consider a scheduling problem for a single server queue that can process a variety of different job classes.
After jobs are processed, they enter a Cited by: In this paper, we analyse a production/inventory system modelled as an M/G/1 make-to-stock queue producing different products requiring different and general production menards.club: Apurva Jain.
This study is the first treatment of rationing caused by limited capacity in a periodic review setting. In the context of a make-to- stock queue, Ha [14, 16] studied two comparable cases but the. May 25, · Performance analysis of a decoupling stock in a Make-to-Order system both types of demand arrive according to a Poisson process and production times of the workstation are exponentially distributed.
van Donk D., Gaalman menards.clubed make-to-order and make-to-stock in a food production system. International Journal of Production Economics Cited by: 2.
Two queues are fed by independent, time-homogeneous Poisson arrival processes. One server is available to handle both. All service durations, in both queues, are drawn independently from the same distribution. A setup time is incurred whenever the server moves (switches) from one queue to the other.
We prove that in order to minimize the sum of discounted setup charges and holdings costs Cited by: Waiting Lines and Queuing Theory Models Introduction Queuing theory is the study of waiting lines.
It is one of the oldest and most widely used quantitative analysis techniques. Waiting lines are an everyday occurrence for most people. Queues form in business process as well. Mar 11, · We consider joint capacity–inventory management for multi-server make-to-stock queues operating under a base stock policy.
The number of servers corresponds to the capacity decision, and the base stock level is the inventory decision. Our goal is to minimize a combination of capacity, inventory, and backordering costs. We develop a square-root rule for the joint decision and justify the rule Cited by: 2.
Operations management is an area of management concerned with designing and controlling the process of production and redesigning business operations in the production of goods or services. It involves the responsibility of ensuring that business operations are efficient in terms of using as few resources as needed and effective in terms of meeting customer requirements.
Downloadable (with restrictions). Abstract We consider joint capacity–inventory management for multi-server make-to-stock queues operating under a base stock policy.
The number of servers corresponds to the capacity decision, and the base stock level is the inventory decision. Our goal is to minimize a combination of capacity, inventory, and backordering costs. In this paper, we study the production time allocation issue for a multi-purpose manufacturing facility.
This production facility can produce different types of make-to-order and make-to-stock products. Using a vacation queueing model, we develop a set of quantitative performance measures for a two-parameter time allocation policy. Based on the renewal cycle analysis, we derive an average cost Cited by: 2.
For make-to-stock products, it is the length of time between the release of an order to the production process and receipt into inventory. Included here are order preparation time, queue time, setup time, run time, move time, inspection time, and put-away time.
Start studying Business process productivity. Learn vocabulary, Analysis of make-to-stock queues with general production times. book, and more with flashcards, games, and other study tools.
In a make-to-stock environment, total production required for a chase production plan when the firm wants to increase inventory is given by: One of the objectives of facility location analysis is to select a.
Downloadable. We model an isolated portion of a competitive supply chain as a M/M/1 make-to-stock queue. The retailer carries finished goods inventory to service a Poisson demand process, and specifies a policy for replenishing his inventory from an upstream supplier.
The supplier chooses the service rate, i.e., the capacity of his manufacturing facility, which behaves as a single-server queue. Jul 12, · We analyze the double queue that arises when arriving customers simultaneously place two demands handled independently by two servers.
It is assumed that the customer arrivals form a Poisson process with mean 1, the servers have exponential service times with rates $\alpha,\beta $ and $1. We consider bi-criteria scheduling problems for three queueing systems (a single queue, a two-station closed network, and a two-station network with controllable inputs) populated by various customer Cited by: 1.
capacitated systems where replenishment lead times are not constant and depend on the congestion in the system. For capacitated systems under continuous review, Buzacott and Shanthikumar () consider an M/M/1 make-to-stock queue and investigate the effects of ADI with constant customer order lead times.
Jan 15, · JIT is a stock replenishment policy that aims to reduce final product stocks and work-in-process (WIP); it coordinates requirements and replenishments in order to minimize stock-buffer needs, and it has reversed the old make-to-stock production approach, leading most companies to adopt "pull" instead of "push" policies to manage material.
The theory of constraints (TOC) is an overall management philosophy introduced by Eliyahu M. Goldratt in his book titled The Goal, that is geared to help organizations continually achieve their goals.
Goldratt adapted the concept to project management with his book Critical Chain, published in J. George Shanthikumar Richard E. Dauch Chair of Manufacturing and Operations Management and Distinguished Professor of Management Multi-Class Production Systems with Setup Times, Operations Research, 46 (May-June ), S -S On Level Crossing Analysis of Queues, The Australian Journal of Statis¬tics, 23, (December ), This is a first-rate and authoritative book that will be invaluable to anyone involved with systems analysis and evaluation.
The first chapter defines renewal theory, discusses Poisson processes, and, as in the rest of the book, gives more Cited by: You can write a book review and share your experiences.
Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.
inventory problem with advance demand information and a restricted production Inderfurth () extends the analysis of Simpson to more general supply-chain structures. Most of these models treat uncapacitated multi-echelon systems and assume type make-to-stock queues, using analytical results and approximations.
Liberopoulos. This is a very technical book that contains a thorough treatment of the various queueing models used in the analysis of polling systems. The purpose of the book is to present the essential results obtained so far and to give the derivations of t more Cited by: In such systems, the order lead times are load-dependent.
Make-to-stock systems with production modeled by servers will be referred to as make-to-stock queues. An M/M/1 make-to-stock queue is a make-to-stock system with one server, an exponentially distributed manufacture time and demand arriving according to a Poisson menards.club by: 9.
A compilation of the practice questions from the cd and the practice questions in the book and maybe even some vocab. Which of the following is used to manage queues and lead times.
Forward scheduling. Rough-cut capacity planning. Under which of the following circumstances will firms generally make-to-stock.
Demand is 4/5. This paper considers a two-station tandem production system consisting of make-to-stock and make-to-order facilities. The make-to-stock facility produces components which are served for external demands as well as internal make-to-order operations while the make-to-order facility processes customer orders with the option to accept or menards.club by: 4.
Mass production, in one hand, has the advantage that IKEA can achieve economic of scale, as already mentioned in order to meet the performance objective cost, in the other hand, it has the disadvantage that the inventory level is high which in particular means that the products of IKEA are make-to-stock (MTS).
Mar 23, · Ever wanted the short version of 'What's New in Dynamics AX ' from the functional / application side. Here it is. The following is a summarization for Microsoft's What's New documentation around the new functionality in AX So, if you want a high level summary (again not technical changes), here you go.
User Interface. May 06, · A fully revised and updated edition of the landmark work on material requirements planning (MRP), Orlicky's Material Requirements Planning, Third Edition focuses on the new rules required to effectively support a manufacturing operation using MRP systems in the twenty-first century.
An analytic approach to a general class of G/G/s queuing systems, Operations Research, 38, 1,Transient and busy period analysis of the GI/G/1 queue as a Hilbert factorization problem, (with J. Keilson, D. Nakazato, H. Zhang), Journal of Applied Probability, 28,This book develops a modeling framework to analyze the problem of inventory management with alternative delivery times.
The general context considered here is that a seller replenishes its inventory in fixed intervals and, between replenishments, allocates the limited inventory to satisfy customers who are both price and delivery-time sensitive. Today's manufacturing system operations are becoming increasingly complex.
Advanced knowledge of best practices for treating these problems is not always well known. The purpose of the book is to create a foundation for the development of stochastic models and their analysis in.
Operations management explained. Operations management is an area of management concerned with designing and controlling the process of production and redesigning business operations in the production of goods or services.
It involves the responsibility of ensuring that business operations are efficient in terms of using as few resources as needed and effective in terms of meeting customer.
Our research interests encompass a broad range of topics in the fields of logistics and supply chain management, production, service operations managment, and menards.club these fields, we have been and will be working with various corporations on real-world practical problems.
Mar 19, · Production & operation management 1. Southern Illinois University Carbondale OpenSIUC Honors Theses University Honors Program Production & Operations Management: Study Guide for Management David J.
Bolling Southern Illinois University Carbondale This Dissertation/Thesis is brought to you for free and open access by the University Honors Program at. May 10, · He figures out what the goal of his organisation is, and then involves his team of Bob Donovan, the Production Manager, Lou, the Plant Controller, who measures efficiency, Stacey Potazenik, the Materials Manager, and Ralph Nakamura, the data processor, at different times, sometimes together, sometimes on a one- to- one basis, to find answers to.
IEMS Production Planning and Scheduling to Undergraduate Industrial Engineering Students IEMS Production and Logistics II to Graduate Industrial Engineering Students IEMS Reliability and Maintenance in Production Systems to Graduate Industrial Engineering Students. First prize by the Ministry of Culture and Science for his first poetry book "Starting at Nafplion," Athens Harry Kurnitz Literary Award at UCLA twice in and for poems in English.
Best Book Award by the Greek Writers Association for his poetry book. Oct 04, · Operations management is an area of management concerned with designing and controlling the process of production and redesigning business operations in the production of goods or services.
It involves the responsibility of ensuring that business operations are efficient in terms of using as few resources as needed and effective in terms of meeting customer requirements. Make-to-Stock: • Pull systems do replenish inventory voids. • But jobs can be associated with customer orders.
Forecast Free: • Toyota's classic system made cars to forecasts. • Use of takt times or production smoothing often involves production without firm orders (and hence forecasts).
5.inGoogle announces a plan to take every page of every book and permit everyone in the world to access it on the Internet for free-Step 1: digitize the text – turn each page into a .Production activity control must balance the flow of work to and from different work centers. This is to ensure that queue, work-in-process, and lead times are controlled.
The input/output control system is a method of managing queues and work-in-process lead times by monitoring and controlling the input to, and output from, a facility.
menards.club - Analysis of make-to-stock queues with general production times. book © 2020 | CommonCrawl |
What is the smallest prime number that is the sum of two other distinct prime numbers?
The two smallest prime numbers are 2 and 3, and $2+3=5$ is also prime. Therefore, $\boxed{5}$ is the smallest prime number that is the sum of two other distinct primes.
Note: If $p$ and $q$ are odd primes, then $p+q$ is an even number greater than 7 and is therefore composite. So the only sets of three primes for which two primes sum to the third are of the form $\{2,p,p+2\}$. | Math Dataset |
npj materials degradation
Article | Open | Published: 21 March 2019
Inhibiting hydrogen embrittlement in ultra-strong steels for automotive applications by Ni-alloying
Sung Jin Kim1,
Eun Hye Hwang1,
Jin Sung Park1,
Seung Min Ryu1,
Dae Won Yun2 &
Hwan Goo Seong3
npj Materials Degradationvolume 3, Article number: 12 (2019) | Download Citation
With the stricter international regulations on CO2 emissions, fuel economy, and auto-safety, the application of novel materials with both higher strength and lower weight is becoming a major technical issue in automotive industries. Among the various lightweight concepts, ultra-strong GIGA STEEL with a tensile strength of more than 2 GPa is a major breakthrough in light of the remarkable weight reduction of vehicle without a decrease in auto-safety. Despite the outstanding mechanical performance, hydrogen embrittlement induced by aqueous and/or atmospheric corrosion is a serious problem that has restricted the application of steel to auto-parts. This study reports that such a critical challenge can be overcome by Ni-alloying, which leads to a lower cathodic reduction rate on the steel surface and slower H-infusion kinetics in the steel matrix. In contrast to the beneficial effects of Ni-alloying, conflicting results can be obtained when steel with a higher Ni content (≥1 wt.%) is exposed to neutral-corrosive environments, but the results have not been verified using conventional metallurgical approaches. This paper proposes a mechanism for these conflicting results, and provides a new and economic strategy for superior resistance to corrosion-induced hydrogen embrittlement, by making optimal use of Ni-alloying of ultra-strong steel.
Considering that the material cost of a car accounts for ~50% of the total car price (Manufacturing: 30%, Balance: R&D and others),1 there is increasing demand for materials with much higher strength and lower weight. Ultra-strong GIGA STEEL developed recently is three times stronger than Al, a typical non-ferrous metal used in auto-parts. Therefore, much thinner parts would be needed to produce lighter but safer and more economical cars.2,3 In general, increasing the tensile strength of ferrous alloys can be achieved by metallurgical strategies, such as grain refinement and/or precipitation hardening. In addition, a much higher tensile strength (over 2 GPa) can be achieved practically and economically using a higher C content in ferrous alloys, leading to the precipitation of iron carbides (ε-Fe2.4C/Fe3C) with a low H overvoltage.4,5 Furthermore, carbides are metallic conductors5,6 and act as a cathode,5,7 resulting in selective dissolution of the matrix. For this reason, it is generally accepted that an increase in the strength of steels is accompanied essentially by a substantial decrease in the resistance to aqueous/atmospheric corrosion5,8 as well as HE.4,9,10,11
HE of a variety of ferrous alloys has been investigated extensively,11,12,13,14,15 but most experimental approaches in these studies focused primarily on the mechanical degradation of specimens charged fully with H by applying an extremely high cathodic current in an aqueous solution.14,15,16 Although a number of researchers have provided significant insights into H-trapping and the resulting cracking behaviors of ferrous alloys, their experimental approaches using electrochemical H-charging in aqueous solutions exclude corrosion reactions on the alloy surface. On the other hand, the surface properties related to aqueous corrosion become predominant over the internal metallurgical effects under mild and near-neutral environments with low H-concentrations, to which automotive steels are normally exposed. From these perspectives, conventional experimental methods, involving severe H-charging and subsequent simple mechanical testing, are inadequate practically. Therefore, alloy design for ultra-strong automotive steel with superior resistance to HE has not been optimized, and the precise role of alloying elements in this resistance is not completely understood. Among the elements, the precise role of Ni in the H-absorption/embrittlement of steels is controversial.17,18,19,20,21 The beneficial effects of Ni-addition have been reported17,18 and the mechanism is based primarily on the microstructural modifications induced by Ni. On the other hand, adverse effects19,20 or no effects21 have also been reported. Nevertheless, the effects of Ni-alloying on the H-reduction and uptake kinetics on the surface have not been clarified. The aim of this study was to gain deeper insights into corrosion-induced HE from two perspectives: surface properties and internal H-diffusion behaviors. Based on this, an effective strategy for the production of much stronger steels with superior resistance to HE was developed by the optimal use of Ni-alloying.
Microstructure
Figure 1 presents the microstructure of the medium C (0.43 wt.%) ultra-strong steel with a tensile strength of 2 GPa. Electron backscatter diffraction (EBSD) and transmission electron microscopy (TEM) showed that the representative microstructure was composed of lath-type martensite with fine need-shaped carbides (ε-Fe2.4C is considered a transition carbide that precipitates during low temperature tempering22,23) along the lath. From TEM observations using the extraction replica technique, two types of precipitate, which are characterized as coarse TiN and fine (Ti,Mo)C, respectively, were distributed uniformly throughout the matrix. It is noteworthy that the presence of fine carbides with a nanometer size may not only give rise to significant strengthening24,25 by hindering dislocation motion or pinning effect, but also act as H-traps,26,27,28 leading to slower kinetics of H-diffusion.
a Microstructure with tempered martensite obtained by an image quality (IQ) map constructed from electron backscatter diffraction (EBSD). b TEM image showing distribution of ε-Fe2.4C carbides in a martensite lath in higher magnification. c TEM image of the precipitates obtained using the extraction replica technique
Surface characteristics of the bare steel samples, and their effects on the electrochemical behaviors
As listed in Table 1, three different levels of Ni (0.1–0.3, 0.6–0.8, and 1.0–1.2 wt%) were added to the tested ultra-strong steel. The specimens are simply referred to as N1, N2, and N3, respectively.
Table 1 Chemical composition of the tested samples (in weight-%)
Figure 2a presents the auger electron spectroscopy (AES) depth profile of the surface of N3, showing a metallic Ni enriched layer, approximately 4 nm in thickness, beneath the surface oxide film. The two controlling factors, H-generation by a cathodic reduction reaction (H+ + e− → H) on the surface and H-penetration through the surface, were analyzed separately to determine the surface barrier effect by the preferential enrichment of Ni. As shown in Fig. 2b, N3 exhibited a higher polarization resistance (Rp) than N1, as measured under a cathodic bias potential in the electrochemical impedance spectroscopy (EIS) test, suggesting that H-generation by a cathodic reduction reaction is suppressed on the surface enriched by Ni. For the other controlling factor, Fig. 2c presents the electrochemical permeation test results showing the H-penetration behaviors obtained from H-permeation through the thin steel membrane. H-penetration through the Ni enriched layer was more restricted, as illustrated in the lower H-diffusivity (Dapp) and steady-state permeation current (Jss) of N3. Faster H-diffusion kinetics is normally expected in N3 considering that Ni-addition leads to a decrease in the amount of iron carbides acting as H-traps29,30 in the steel matrix (refer to Supplementary Fig. 1, in accordance with previous results17,31). Nevertheless, the much slower diffusion kinetics of N3 means that the infusion of nascent H into the steel is prevented by the thin Ni enriched layer on the surface. These results are in accordance with previous studies,32 which reported the beneficial effect of Ni on the resistance to HE of high-strength steels with a tensile strength of 1.5 GPa.
a Compositional depth-profile, as analyzed by AES, of the as-polished N3. b EIS Nyquist plots of N1 and N3 measured immediately after immersion in a Walpole solution of 0.2 M CH3COONa + 0.185 M HCl. c Electrochemical hydrogen permeation transients with apparent H-diffusivity values (Dapp), measured under an applied cathodic current density of 1 mA cm−2 in a 3.5% NaCl + 0.3% NH4SCN solution in the H-charging side, and applied anodic potential of 270 mVSCE in a 0.1 M NaOH solution in the H-detection side
Corrosion tests and an analysis of the corrosion products formed on the surfaces of tested samples
The surface inhibiting effects imparted by Ni, which were observed in bare steel under electrochemical treatments, persisted in the neutral aqueous environment of 3.5% NaCl even after longer exposure times. Under long-term immersion conditions, a high corrosion resistance could be attributed primarily to the corrosion products with protective characteristics, formed on the surface. A previous study33 reported that the corrosion scale formed on high-strength carbon steel exposed to a neutral solution containing chloride ions has a bi-layer structure composed mainly of an adhesive inner-layer of Fe3O4 and a non-adhesive outer- layer of γ-FeOOH. Although 1.2 wt.% Ni was added to the ultra-strong steel, the nature of the corrosion scale formed on the surface was not changed and the difference in the structure of the inner scale could not be identified by X-ray diffraction (XRD), as shown in Supplementary Fig. 2. As shown in Fig. 3a, however, the weight loss per unit area decreased linearly with increasing Ni-addition to the steel. Several mechanisms for the beneficial effects of Ni-addition in minute quantities to the steel have been proposed: partial substitution of Ni atoms into the Fe-sites of Fe3O4 to form Fe3-xNixO4,34 formation of a protective amorphous layer in the rust,35 and the refinement of rust particles by increasing the atomic level heterogeneity in rust.36 Although it may not be possible to elucidate the precise mechanism, the results suggest that the beneficial effect of Ni-addition on the long-term corrosion resistance is closely associated with the denser and finer rusts (Fe3O4) enriched with Ni formed on the steel surface, as shown in the cross-sectional observations and EDS spectrum in Fig. 3b, c, respectively. Supplementary Fig. 3 shows the results of depth profile analysis obtained by glow discharge spectroscopy (GDS); Ni is enriched in the thinner scale formed on N3 compared to N1. A higher open circuit potential (OCP) and lower anodic dissolution rate on the N3 surface, as shown in the potentiodynamic polarization measured after 3 h pre-exposure to a 3.5% NaCl solution (Fig. 3d), can also be understood in the same context.
a Weight loss measurements after the long-term immersion test for four weeks. b Cross-sectional morphologies of N1 and N3 observed after the long-term immersion test. c EDS-mapping analyses on the cross-sectional regions shown in (b). d Linear polarization resistance tests of N1 and N3, measured after 3 h pre-exposure in a 3.5% NaCl solution
Slow strain rate test (SSRT) and surface morphology observation to evaluate the resistance to corrosion-induced HE
One of the important findings in this study was the conflicting results of the Ni-alloying effect on corrosion-induced HE. In contrast to expectation based on the pre-described beneficial effects of Ni-alloying, a significant decrease in elongation with increasing Ni content was observed from the SSRT conducted in a neutral aqueous solution, as shown in Fig. 4a. More pits as a type of localized corrosion, which form normally at the 2nd-phase particles characterized primarily as (Al,Ca)-based oxide, were observed on the N3 surface in a neutral solution (Fig. 4b), and vice versa in an acidic solution. Considering these facts, the adverse effects of Ni-alloying are closely related to localized corrosion induced by Ni-depletion around the local heterogeneities (2nd-phase particles) acting as stress intensifiers under an applied stress, which increase the susceptibility of N3 to corrosion-induced HE. The localized corrosion behavior presented in this study could be understood in a similar context to the discontinuities of the passive film around the 2nd-phase particles, and preferential pitting corrosion in the particles on the surface of stainless steel.37 The Rp value obtained by EIS shortly before fracture of the specimen can be an electrochemical index indicating the surface state. The Rp values measured under neutral condition were marked in Fig. 4b, and their Nyquist plots can be found in Supplementary Fig. 4. N3 showed a lower Rp in a neutral solution, which suggests that N3 has a higher pit density on the surface. In contrast to localized corrosion in the neutral solution, local pits or corrosion products were not observed preferentially in the Walpole solution. This may be caused by the corrosion process in an acidic solution being controlled primarily by the strong anodic dissolution rate and resulting uniform corrosion. Because the diffusible H-content is regarded as one of the factors for HE, the contents in the tested steels were measured by thermal desorption spectroscopy (TDS), and they can be found in Supplementary Fig. 5. The results showed that N1 has a much higher diffusible H-content than N3 when measured in an acidic environment. This is understandable from the SSRT result (i.e. a comparatively lower elongation level of N1) in Fig. 4a. On the other hand, the H-contents were similar in N1 and N3 and extremely low, which were measured under neutral conditions, suggesting that the significant decrease in the total elongation of N3 under neutral conditions may be due primarily to the local stress concentration in the pits formed by localized corrosion around the 2nd-phase particles with Ni-depletion. Although the H-entry mechanism changes according to the environment to which the specimen is exposed, and H can be infused in the steel by a hydrolysis reaction38,39 following local acidification when exposed to a neutral environment, H infuses in extremely small amounts and may not be the predominant factor for HE, at least in this case.
$${\mathrm{Fe}}^{2 + } + 2{\mathrm{H}}_2{\mathrm{O}} \to {\mathrm{Fe}}\left( {{\mathrm{OH}}} \right)_2 \,+\, 2{\mathrm{H}}^ +$$
a SSRT curves of N1 and N3 evaluated in a neutral sol. (3.5% NaCl + 0.3% NH4SCN) and acidic sol. (0.2 M CH3COONa + 0.185 M HCl) after 6 h pre-exposure in each solution. b Surface morphologies of N1 and N3 observed after immersion in the two solutions for 15 h
Therefore, the decrease in HE resistance of N3 evaluated by SSRT can be estimated more accurately by the extent of local corrosion that appears in the form of Rp. Figure 5 presents a schematic illustration of the proposed corrosion mechanism in the two corrosive media. Anodic and cathodic sites were distributed throughout the surface of specimen exposed to the acidic solution. On the other hand, when the specimens were exposed to a neutral solution, anodic steel dissolution occurred preferentially around the 2nd-phase particle with Ni-depletion, leading to localized corrosion.
Schematic diagram illustrating the proposed mechanism of the effects of Ni-alloying (1–1.2 wt.% Ni) on the corrosion behaviors on the surfaces in the two corrosive media
Considering the diagram (Supplementary Fig. 6) showing the strain loss rate of N1, N2, and N3 in a neutral solution, the threshold Ni-content in which the detrimental effect of HE had been reached was more than 0.8 wt.%.
SSRT and fracture surface observation to evaluate the effects of ε-Fe2.4C on HE
Another possible mechanism explaining the conflicting results of Ni alloying is based on the slower H-effusion kinetics from ε-Fe2.4C distributed in the microstructure. ε-Fe2.4C is considered to be a strong H-trap with a high H-binding energy that partially immobilizes diffusible-H by the strong trapping capacity,12,23 suggesting that it can alleviate HE by suppressing the local H-concentrations around potential crack sites, such as surface defects or local-pits. As mentioned above, Ni-addition leads to a lower fraction of ε-Fe2.4C in the microstructure. Faster H-diffusion to the triaxial stress field40,41,42 around surface defects is expected in steel with a higher Ni content. In contrast to the permeation experiment showing the slower diffusion of H reduced cathodically from the surface to the internal matrix of steel with a higher Ni-content, the H mentioned in this case had been infused already in the steel matrix by local corrosion involving the hydrolysis reaction38,39 described in Eq. (1) and local acidification in the pits. Therefore, the faster diffusion of H can be expected in the matrix of steel with a higher Ni-content. To examine the effects of ε-Fe2.4C on HE, a mechanistic study using SSRT after the two H-charging processes was conducted, as summarized in Fig. 6. Figure 6a, b shows the engineering stress-strain curves conducted on the two samples tempered for 30 and 45 min, respectively, under different charging conditions. The HE-indices in the figures means the degree of mechanical degradation of pre-charged samples, which are quantified as follows:
$$HE\,{\rm{index}} = \left( {\omega _{{\rm{uncharged}}} - \omega _{{\rm{charged}}}} \right)/\omega _{{\rm{uncharged}}},$$
where ωcharged and ωuncharged are the elongation reduction measured after H-pre-charging and uncharging, respectively.
SSRT curves of pre-charged N1 specimens which had been tempered for 30 min (T30m), and for 45 min (T45m), with a charging current density of −1 mA cm−2 for 3 and 10 min, respectively
It is considered that an increase in tempering duration from 30 to 45 min results in a higher fraction of ε-Fe2.4C. From Supplementary Fig. 7, however, there was negligible variation in the residual strain and dislocation density with increasing tempering duration, as analyzed by EBSD kernel average misorientation (KAM). The distribution of KAM suggests that there was no significant variation in the dislocation density until the tempering duration was 60 min. A significant decrease in elongation and typical intergranular cracking along the prior-γ austenite grain boundaries under long H-charging conditions (10 min) was observed clearly in the samples regardless of the tempering duration (Fig. 7a, b). On the other hand, it is interesting to note that a lower fraction of ε-Fe2.4C by the shorter tempering duration (30 min) results in a comparatively high HE index (0.45) and typical intergranular cracking (Fig. 7c in contrast to Fig. 7d) under short H-charging conditions (3 min), which is in accordance with the results reported by Zhu et al.23 Considering that once the ε-Fe2.4C was filled with H-atoms under longer H-charging period of 10 min, it cannot be an effective H-trap leading to slower diffusion kinetics of H-atoms. A beneficial effect of ε-Fe2.4C can only be highlighted under shorter H-charging conditions. This suggests that N3 with a lower fraction of ε-Fe2.4C acting as a strong H-trap can be comparatively susceptible to HE when it is exposed to near-neutral environments of a low H-concentration under applied stress conditions.
Fracture surfaces after SSRT of pre-charged N1 which had been tempered (a) for 30 min, with a charging current density of −1 mA cm−2 for 10 min, (b) for 45 min, with a charging current density of −1 mA cm−2 for 10 min, (c) for 30 min, with a charging current density of −1 mA cm−2 for 3 min, (d) for 45 min, with a charging current density of −1 mA cm−2 for 3 min, respectively
These findings pave new ways for the development of much stronger automotive steels with superior resistance to corrosion-induced HE. For novel steel alloys, it is essential to optimize the use of Ni with an economical alloy design based on extremely minute quantities of Ni (0.6–0.8 wt%) because Ni has both beneficial and adverse effects on HE by controlling the kinetics of H-uptake, formation of localized pitting, and fraction of Fe2.4C in the steel. In particular, the adverse effects of Ni-alloying were determined by exposing steel with a Ni content greater than 1 wt.% to neutral corrosive environments under an applied tensile stress.
This study also highlights the need to consider the corrosion behavior on the surface when evaluating HE, which is in contrast to conventional test methods involving severe cathodic H-charging and subsequent mechanical testing.
The steel under investigation was reheated to 1200 °C for 2 h, and hot-rolled and cold-rolled to 2 mm in thickness. The samples were then austenitized by heating to 930 °C for 7 min and quenched in a mixture of oil and water. The quenched specimens were additionally tempered at 200 °C for 45 min.
After mechanical polishing, the elemental distribution on the surfaces of the bare steels (N1 and N3) was analyzed by AES. EIS was conducted over the frequency range, 0.01 Hz to 100.000 Hz, with a 10 mV amplitude sinusoidal voltage applied at −50 mV vs. OCP. The test electrolyte for the EIS test was a Walpole solution of 0.2 M CH3COONa + 0.185 M HCl with an initial pH of 3.5. The H-uptake and diffusion kinetics in the specimens were examined using an electrochemical permeation technique (EPT).43,44 This technique involves the diffusion of H-atoms generated on one side of a steel membrane by galvanostatic polarization with −1 mA cm−2 in 3.5% NaCl + 0.3% NH4SCN (hereinafter called a neutral charging sol.), and the H-permeation flux was evaluated by measuring the H-oxidation current on the other side of a steel membrane electroplated with palladium to which a constant anodic potential of 270 mVSCE was applied. The detailed experimental procedure is reported elsewhere.40,45 After the experiment, the apparent H-diffusivity (Dapp) was determined using the breakthrough method, which is expressed as follows:
$$D_{{\rm{app}}} = L^2/\left( {15.3\,\times\,t_{{\rm{bt}}}} \right),$$
where tbt is the time needed for H to begin arriving at the detection side, and L is the thickness of the steel membrane.
An immersion test was performed to evaluate the long-term corrosion behaviors on the surfaces, and to measure the weight loss. For the weight loss measurement, the specimens, 50 mm in width, 60 mm in length, and 1 mm in thickness, were cleaned with acetone and weighed in an analytical balance to a precision of 0.0001 g. After immersion in a 3.5% NaCl solution for four weeks, the specimens were cleaned ultrasonically in ethanol for 1–2 min to remove the corrosion products completely, and dried for 1 min. The specimens were weighed again in the same analytical balance. The difference between the initial and final weights divided by the initial areas was the weight loss. The cross-sectional images of the scale formed on the steel surfaces after the immersion test were also obtained. Prior to the observation, the samples were prepared using a focused ion beam. The resulting cross-sectional parts were then analyzed by TEM and energy dispersive spectroscopy (EDS). In addition, linear polarization resistance measurements were conducted in a 3.5% NaCl solution after pre-immersion for 3 h in solution. For these measurements, the samples were polarized dynamically from approximately −20 mV to 20 mV with respect to the OCP at a scan rate of 0.2 mV s−1.
The resistance of the samples to corrosion-induced HE was evaluated by providing corrosive environments and dynamic tensile loading condition, simultaneously. For this, a slow strain rate test (SSRT) at a strain rate of 10−6 s−1 was performed in two types of aqueous solution: Walpole solution and neutral charging sol. after 6 h pre-exposure to the solution at the OCP. The Rp values were obtained by EIS shortly before fracture of each specimen by the SSRT. The other experimental conditions were the same as those of the EIS measurements for bare steels described above. In addition, thermal desorption analysis of hydrogen was conducted to measure the diffusible hydrogen content in the steel shortly before fracture by the SSRT. The specimens were heated from 25 to 270 °C at a heating rate of 200 °C/h.
For a mechanistic study, the surface morphologies of N1 and N3 obtained after immersion in the two types of solution for 15 h were observed by FE-SEM. For another mechanistic study of the effects of Fe2.4C on HE, a SSRT at a strain rate of 10−5 s−1 was also conducted on two pre-charged specimens that had been tempered at 200 °C for 30 and 45 min, respectively, with a charging current density of −1 mA cm−2 for 3 and 10 min, respectively. After the SSRT, the fracture surfaces were observed by FE-SEM.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
The authors declare no competing interests.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Department of Advanced Materials Engineering, Sunchon National University Jungang-ro, Suncheon, Jeonnam, 540-742, Republic of Korea
Sung Jin Kim
, Eun Hye Hwang
, Jin Sung Park
& Seung Min Ryu
Korea Institute of Materials Science (KIMS), 797 Changwondaero, Seongsan-gu, Changwon, 642-831, Republic of Korea
Dae Won Yun
POSCO Technical Research Laboratories, Kumho-dong, Gwangyang, Jeonnam, 545-090, Republic of Korea
Hwan Goo Seong
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S.J.K. designed the study and wrote the paper; E.H.W., J.S.P. and S.M.R. performed the electrochemical experiments; S.J.K. and D.W.Y. analyzed the data; H.G.S. designed and produced the samples. All authors discussed the results and commented on the manuscript.
Correspondence to Sung Jin Kim.
Supplementary File
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
Baking Effect on Desorption of Diffusible Hydrogen and Hydrogen Embrittlement on Hot-Stamped Boron Martensitic Steel
Hye-Jin Kim
, Hyeong-Kwon Park
, Chang-Wook Lee
, Byung-Gil Yoo
& Hyun-Yeong Jung
Metals (2019)
npj Materials Degradation menu | CommonCrawl |
1 52 honeycomb
In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. It contains 142 and 151 facets, in a birectified 8-simplex vertex figure. It is the final figure in the 1k2 polytope family.
152 honeycomb
(No image)
TypeUniform tessellation
Family1k2 polytope
Schläfli symbol{3,35,2}
Coxeter symbol152
Coxeter-Dynkin diagram
8-face types142
151
7-face types132
141
6-face types122
{31,3,1}
{35}
5-face types121
{34}
4-face type111
{33}
Cells{32}
Faces{3}
Vertex figurebirectified 8-simplex:
t2{37}
Coxeter group${\tilde {E}}_{8}$, [35,2,1]
Construction
It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the end of the 2-length branch leaves the 8-demicube, 151.
Removing the node on the end of the 5-length branch leaves the 142.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 8-simplex, 052.
Related polytopes and honeycombs
1k2 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = ${\tilde {E}}_{8}$ = E8+ E10 = ${\bar {T}}_{8}$ = E8++
Coxeter
diagram
Symmetry
(order)
[3−1,2,1] [30,2,1] [31,2,1] [[32,2,1]] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 1,920 103,680 2,903,040 696,729,600 ∞
Graph - -
Name 1−1,2 102 112 122 132 142 152 162
See also
• 521 honeycomb
• 251 honeycomb
References
• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• Coxeter Regular Polytopes (1963), Macmillan Company
• Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter 5: The Kaleidoscope)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 GoogleBook
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
Methodology article | Open | Published: 21 July 2017
A deep convolutional neural network approach to single-particle recognition in cryo-electron microscopy
Yanan Zhu1,
Qi Ouyang1,2,3 &
Youdong Mao ORCID: orcid.org/0000-0001-9302-22571,2,4
Single-particle cryo-electron microscopy (cryo-EM) has become a mainstream tool for the structural determination of biological macromolecular complexes. However, high-resolution cryo-EM reconstruction often requires hundreds of thousands of single-particle images. Particle extraction from experimental micrographs thus can be laborious and presents a major practical bottleneck in cryo-EM structural determination. Existing computational methods for particle picking often use low-resolution templates for particle matching, making them susceptible to reference-dependent bias. It is critical to develop a highly efficient template-free method for the automatic recognition of particle images from cryo-EM micrographs.
We developed a deep learning-based algorithmic framework, DeepEM, for single-particle recognition from noisy cryo-EM micrographs, enabling automated particle picking, selection and verification in an integrated fashion. The kernel of DeepEM is built upon a convolutional neural network (CNN) composed of eight layers, which can be recursively trained to be highly "knowledgeable". Our approach exhibits an improved performance and accuracy when tested on the standard KLH dataset. Application of DeepEM to several challenging experimental cryo-EM datasets demonstrated its ability to avoid the selection of un-wanted particles and non-particles even when true particles contain fewer features.
The DeepEM methodology, derived from a deep CNN, allows automated particle extraction from raw cryo-EM micrographs in the absence of a template. It demonstrates an improved performance, objectivity and accuracy. Application of this novel method is expected to free the labor involved in single-particle verification, significantly improving the efficiency of cryo-EM data processing.
Single-particle cryo-EM images suffer from heavy background noise and low contrast, due to the limited electron dose used in imaging in order to reduce radiation damage to the biomolecules of interest [1]. Hence, a large number of single-particle images, extracted from cryo-EM micrographs, is required to perform a reliable 3D reconstruction of the underlying structure. Particle recognition thus represents the first bottleneck in the practice of cryo-EM structure determination. During the past decades, many computational methods have been proposed for automated particle recognition, mostly based on template matching, edge detection, feature extraction or neural networks [2,3,4,5,6,7,8,9,10,11,12,13,14,15]. The template matching methods depend on a local cross-correlation that is sensitive to noise, and a substantial fraction of false positives may result from false correlation peaks [2,3,4,5,6,7,8]. Similarly, both the edge-based [9, 10] and feature-based methods [11,12,13] suffer from a dramatical reduction of performance with lower contrast of the micrographs. In a different approach, a method based on a three-layer pyramidal-type artificial neural network was developed [14, 15]. However, there is only one hidden layer in the designed neutral network, which is insufficient to extract rich features from single-particle images. A common problem for these automated particle recognition algorithms lies in the fact that they cannot distinguish "good particles" from "bad" ones, including overlapped particles, local aggregates, background noise fluctuations, ice contamination and carbon-rich areas. Thus, additional steps comprising unsupervised image classification or manual verification and selection are necessary to sort out "good particles" after initial automated particle picking. For example, TMaCS uses the support vector machine (SVM) algorithm to classify the particles initially picked by a template-matching method to remove false positives [16].
Deep learning is a type of machine learning that focuses on learning from multiple levels of feature representation, and can be used to make sense of multi-dimensional data such as images, sound and text [17,18,19,20,22]. It is a process of layered feature extraction. In other words, features in greater detail can be extracted by moving the hidden layer down to a deeper level using multiple non-linear transformations [22]. Convolutional neural network (CNN) is a biologically inspired deep, feed-forward neural network that has demonstrated an outstanding performance in speech recognition [23] and image processing, such as handwriting recognition [24], facial detection [25] and cellular image classification [26]. Its unique advantage lies in the fact that the special structure of shared local weights reduces the complexity of the network [27, 28]. Multidimensional images can be directly used as inputs of the network, which avoids the complexities of feature extraction in the reconstructed data [17, 27].
The particle recognition problem in cryo-EM is fundamentally a binary classification problem, and is based on the features of single-particle images. We devised a novel automated particle recognition approach based on deep CNN learning [27]. Our algorithm, named DeepEM, is built upon an eight-layer CNN, including an input layer, three convolutional layers, three subsampling layers, and an output layer (Fig. 1). In this study, we applied this deep-learning approach to tackle the problem of automated template-free particle recognition. The DeepEM algorithm was examined through the task of detecting "good particles" from cryo-EM micrographs taken in a variety of situations, and demonstrated improved accuracy over other template-matching methods.
The architecture of the convolutional neural network used in DeepEM. The convolutional layer and the subsampling layer are abbreviated as C and S, respectively. C1:6@222×222 means that it is a convolutional layer and is the first layer of the network. This layer is comprised of six feature maps, each of which has a size of 222 × 222 pixels. The symbols and numbers above the feature maps of other layers have the equivalent corresponding meaning
Design of the DeepEM algorithm
The DeepEM algorithm is based on a convolutional neural network, a multilayered neural network with local connections. It contains convolutional layers, subsampling layers and fully connected layers, in addition to the input and output layers (Fig. 1). The convolutional and subsampling layers produce feature maps through repeated application of the activation function across sub-regions of the images, which represent low-frequency features extracted from the previous layer (Additional file 1: Figure S1).
In the convolutional layer, which is the core building block of a CNN, the connections are local, but expand throughout the entire input image. Such a network architecture ensures that the outputs of the convolutional layer are effectively activated in response to the detection of meaningful input spatial features. The feature maps from the previous layer are convoluted by a learnable kernel. All convolution operation outputs are then transformed by a nonlinear activation function. We used the sigmoid function (1) as the nonlinear activation function.
$$ sigmoid(x)=1/\left(1+{e}^{- x}\right) $$
The convolution operations in the same convolutional layer share the same connectivity weights with the previous layer, so that:
$$ {X}_j^{\left[ l\right]}= sigmoid\left(\sum_{i\in {M}_j}{X_i^{\left[ l-1\right]}}^{\ast }{W}_{i j}^{\left[ l\right]}+{B}^{\left[ l\right]}\right), $$
where l represents the convolutional layer; W represents the shared weights; M represents different feature maps from the previous layer; j represents one of the output feature maps; B represents the bias in the layer; and the star symbol (*) represents the convolution operation.
Subsampling is another important concept in CNNs. A subsampling layer is designed to subsample the input data to progressively decrease the spatial size of the representation and reduce the number of parameters and computational cost in the network, thus reducing potential over-fitting [29]. We computed the subsampling averages after each convolutional layer using the following expression:
$$ {X}_{ij}^{\left[ l\right]}=\frac{1}{ M N}{\sum}_m^M{\sum}_n^N{X}_{iM+ m, jN+ n}^{\left[ l-1\right]} $$
where i and j represent the position of the output map; M and N represent the subsampling size in two orthogonal dimensions.
The basic network architecture of DeepEM contains three convolutional layers (the first, third, and fifth layers) and three subsampling layers (the second, fourth and sixth layers). The last layer is fully connected to the previous layer, which outputs a prediction for the classification of the input image by the weight matrix and the activation function (Fig. 1).
Training of the DeepEM network
Prior to the application of DeepEM for automated particle recognition, the CNN needs to be trained with a manually assembled dataset, sampling both true particle images (positive training data) and non-particle images (negative training data) (Examples in Fig. 3a, b). Only a well-trained CNN should be used to recognize particles from raw micrographs. We used the error back-propagation method [30] to train the network, which produces an output of "1" for the true particle images and "0" for the non-particle images. The weights and biases in the CNN model are initialized with a random number between 0 and 1, and are then updated in the training process. We used the squared-error loss function [30] as the objective function in our model. For a training dataset with the number of N, it is defined as:
$$ {E}_N=\frac{1}{2 N}{\sum}_{n=1}^N{\left\Vert {t}_n-{y}_n\right\Vert}^2, $$
where t n is the target of the nth training image, and y n is the value of the output layer in response to the nth input training image. During the process of training, the objective function is minimized using an error back-propagation algorithm [30], which performs a gradient-based update as follows:
$$ \omega \left( t+1\right)=\omega (t)-\frac{\eta}{N}{\sum}_{k=1}^N{\varepsilon}_n\frac{\partial {\varepsilon}_n}{\partial \omega} $$
where ε n = ‖t n − y n ‖; ω(t) and ω(t + 1) represent the parameters before and after the update of an iteration, respectively; η is the learning rate and was set to 1 in this study.
The data augmentation technique has shown a certain improvement in the accuracy of CNN training with a large number of parameters [14, 26]. During our DeepEM training, each original particle image in the training dataset was rotated by 90°, 180° and 270°, in order to augment the size of data sampling by a factor of four. The intensity of each pixel from an original or rotated image was then used as the input of a neuron of the input layer. The desired output was set to 1 for the positive data and 0 for the negative data in the error back-propagation procedure.
The experimental cryo-EM micrographs may contain heterogeneous objects, such as protein impurities, ice contamination, carbon-rich areas, overlapping particles and local aggregates. Moreover, since the molecules in the single-particle images assume random orientations, significantly different projection structures of the same macromolecule may coexist in a micrograph. These factors make it difficult to assemble a relatively balanced training dataset at the beginning, which must include representative positive and negative particle images. The initially trained CNN is prone to missing some target particles in certain views or recognizing some unwanted particles whose appearances are similar to the target. The training dataset can be optimized by adding a greater number of representative particle images to the original training dataset after testing on a separate set of micrographs that are independent of the ones used for assembling the original training dataset, and then re-training the network following the workflow chart shown in Fig. 2. After a sufficient number of iterations of training, the CNN becomes more "knowledgeable" in differentiating positive particles from negative ones.
The workflow diagram of the DeepEM algorithm. The dashed box on the left represents the learning process; the dashed box on the right represents the recognition process
Since the input particle images size may vary in different datasets, one can set different hyper-parameters for each case, including the number of feature maps, the kernel size of the convolutional layers and the pooling region size of the pooling layers. We empirically initialized these hyper-parameters and fine-tuned them during the training process (Fig. 2). The details of the hyper-parameters used in this study are shown in Table 1. In general, the output dimension of the convolutional layer is chosen as 70–90% of its input dimension, and the output dimension of the subsampling layer is scaled to about half its input dimension. We implemented the DeepEM algorithm based on the DeepLearnToolbox [31], a toolbox for the development of deep learning algorithms, in conjunction with Matlab.
Table 1 Hyper-parameters used in different datasets
Particle recognition and selection in the DeepEM model
When a well-trained CNN is used to recognize particles, a square box of pixels is taken as the CNN input. Each input image boxed out of a testing micrograph is rotated incrementally, to generate three additional copies of the input image with rotations of 90°, 180° and 270°, relative to the original. Each copy is used as a separate input to generate a CNN output. The final expectation value of each input image is taken as the average of its four output values from the non-rotated and rotated copies. The boxed area is initially placed into a corner of the testing micrograph, and is raster-scanned across the whole micrograph to generate an array of CNN outputs.
We used two criteria to select particles. First, a threshold score must be defined. The boxed image is identified as a candidate if the CNN output score of the particle is above the threshold score. Those particles whose CNN scores are below the threshold are rejected. We used the F-measure [32], which is a measure of the accuracy of a test that combines both precision and recall for binary classification problems, to determine the threshold score in our approach, which is defined as.
$$ {F}_{\beta}={\left(1+{\beta}^2\right)}^{\ast}\frac{precision^{\ast } recall}{\left({\beta^2}^{\ast } precision+ recall\right)}, $$
where β is a coefficient weighting the importance of precision and recall. In our method, we used the F 2 score, which weights the recall higher than the precision. The F 2-score reaches its best value at 1 and its worst at 0. We defined the cutoff threshold at the highest value of the F 2-score.
Secondly, candidate images were further selected based on the standard deviation of the pixel intensities. There are often carbon-rich areas or contaminants in raw micrographs where the initially detected particles may not be good choices for downstream single-particle analysis. The pixels belonging to the "particles" in these areas usually have higher or lower standard deviations compared with those in other areas with clean amorphous ice. We therefore set a narrow range of the pixel standard deviation to remove the candidate particles that are initially picked from these unwanted areas [6, 16] (Additional file 1: Figure S2).
DeepEM algorithm workflow
Input: Training dataset.
Output: Trained CNN parameters (weights and biases)
Rotate each input particle image three times, each with a 90° increment;
Set the output of the positive data as 1, and the output of the negative data as 0;
Initialize the hyper-parameters;
Randomly initialize the weights and biases in each convolutional layer;
While (Learning error > Defined error), do
Tune the hyper-parameters or optimize the training dataset by adding more representative positive and negative particles from a new set of micrographs, which are independent of those used in the previous iterations, to the training dataset;
Train weights and biases via the error back-propagation algorithm;
Apply the trained CNN to an independent testing dataset to measure the learning error
Input: Micrographs and trained CNN.
Output: Box files of selected particles in the EMAN2 [33] format for each micrograph
Iterate the following steps (a-c) until the whole micrograph has been raster-scanned;
Extract a square the size of a particle, starting from a corner of the input micrograph;
Rotate the boxed image three times, each with a 90-degree increment;
Use the trained CNN to process four copies of the boxed image, including the non-rotated and rotated copies, and average the resulting output scores of the four images;
Pick the particle candidates based on scores that are not only local maxima but also above the threshold score;
Select particle images based on their standard deviations;
Write the coordinates of the selected particle images into the box file.
We evaluated the performance of the method based on the precision-recall curve [34], which is one of the most popular metrics for the performance evaluation of various particle-selection algorithms. The precision and recall are defined by Eqs. (7) and (8), respectively.
$$ \mathrm{Precision}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}} $$
$$ \mathrm{Recall}=\frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}} $$
The precision represents the fraction of true positives (TP) among the total particle images selected (TP + FP), and the recall represents the fraction of true particle images selected among all the true particle images (TP + FN) contained in the micrographs. The precision-recall curve is generated from the algorithm by varying the threshold score used in the particle recognition procedure. When the threshold increases, the precision would increase and the recall would decrease accordingly. Thus, the threshold is manifested as a balance between the precision and the recall. For a good performance in particle selection, both the precision and the recall are expected to achieve higher values at a certain threshold.
DeepEM training on the keyhole limpet Hemocyanin (KLH) dataset
The KLH dataset was acquired from the US National Resource for Automated Molecular Microscopy (nramm.scripps.edu). KLH is ~8 MDa protein particle with a size of ~40 nm. It consists of 82 micrographs at 2.2 Å/pixel that were acquired on a Philips CM200 microscope at 120 kV. The size of the micrograph is 2048 by 2048 pixels. There are two main types of projection views of the KLH complex, the side view and the top view. We boxed the particle images with a dimension of 272 pixels. 800 particle images were manually selected for the positive training dataset. The same number of randomly selected non-particle images from the first fifty micrographs was used as a negative dataset (Fig. 3a). Each original image in the training dataset was rotated at 90° increments to create three additional images to augment the training data. We also selected some particle images as a testing dataset containing positive and negative data that were not used in the prior training step. The testing dataset was used to test the intermediately trained CNN model (Fig. 2). The accuracy or error of the CNN learning output from the testing dataset was used as a feedback parameter to tune the hyper-parameters, including the number of feature maps, kernel size of the convolutional layers, and subsampling size of the subsampling layers in the network. Throughout the training-testing cycles, we tuned the hyper-parameters and updated the training dataset until the accuracy of the CNN learning reached a satisfactory level. The acceptable value was often set as ~95% at the threshold of 0.5 (Fig. 2).
The DeepEM results for the KLH and 19S regulatory particle datasets. a and b Examples of positive and negative particle images selected for the CNN training in conjunction with the KLH and 19S datasets, respectively. c and d Typical micrographs from the KLH and 19S datasets, respectively. The white square boxes indicate the positive particle images selected by DeepEM. The boxes with a triangle inside indicate that a false-positive particle image was picked. The star marks one example of a false negative, a true particle missed by the recognition program. e The F 2-score curves provide different thresholds for particle recognition in the KLH and 19S datasets, the arrows indicate the peaks of each curve, where the cutoff threshold value is defined. f The precision-recall curves plotted against a manually selected list of particle images
Application to experimental cryo-EM data
The original sizes of the micrographs of the inflammasome, 19S regulatory particle and 26S proteasome were 7420 by 7676, 3710 by 3838 and 7420 by 7676 pixels, respectively. The pixel sizes of the inflammasome, 19S regulatory particle and proteasome holoenzyme were 0.86, 0.98 and 0.86 Å/pixel, respectively. For the inflammasome and 26S proteasome, the micrographs were binned 4 times. Therefore, the pixel size used for the inflammasome and proteasome holoenzyme was 3.44 Å/pixel. For the 19S regulatory particle, the micrographs were binned 2 times, resulting in a pixel size of 1.96 Å/pixel. Thus, the resulting sizes of the micrographs used in our tests were all 1855 by 1919 pixels; the dimension of the particle images of the inflammasome, 19S and 26S complexes were 112, 160 and 150 pixels, respectively. These experimental cryo-EM datasets were acquired using a FEI Tecnai Arctica microscope (FEI, USA) at 200 kV, equipped with a Gatan K2 Summit direct electron detector. Finally, we applied the DeepEM algorithm to these cryo-EM datasets. The hyper-parameters tuned for these datasets are shown in Table 1. Different from the training for the KLH dataset, we added true positive and false positive data, which were manually verified on a separate set of micrographs independent of the testing dataset used for tuning the hyper-parameters, to optimize the training dataset and to train the network recursively for the low-contrast datasets (Additional file 1: Figure S3).
Experiments on the KLH dataset
We first tested our DeepEM algorithm on the Keyhole Limpet Hemocyanin (KLH) dataset [35] that was previously used as a standard testing dataset to benchmark various particle selection methods [3, 4, 6, 8, 11,12,13, 16]. For the KLH dataset, the recall and the precision both reached ~90% at the same time in the precision-recall curve (Fig. 3f) plotted against a manually selected set of particle images from 32 micrographs that did not include any particle images used in the training dataset. Our approach achieved a higher precision over all the particle images selected, whereas the recall was kept at a high value, indicating that fewer false-negative particle images were missed among the micrographs. In a typical KLH micrograph (Fig. 3c), all true particle images were automatically recognized by our method with a threshold of 0.84, as determined by the F 2-score (see Methods and Eq. 6) (Fig. 3e). A comparison of the precision-recall curves between DeepEM, RELION [36] and TMACS [16] suggests that DeepEM outperforms these two template-matching based methods (Additional file 1: Figure S4).
To understand the impact of the number of training particles on algorithm performance, we varied the particle number in the KLH training dataset from 100 to 1200, and plotted the corresponding precision-recall curves (Fig. 4). In each testing case, the number of positive particles was kept equal to that of the negative particles. Although there was clear improvement in the precision-call curve when the training particle number was increased from 100 to 400, there was little improvement with a further increase of the training dataset size. The best result was obtained in the training run with 800 positive particle images.
Impact of the training image number on the precision-recall curve. The black, blue, red and green curves were obtained with the training datasets including 100, 400, 800 and 1200 positive or negative images, respectively
Experiments on cryo-EM datasets
We also applied our method to several challenging cryo-EM datasets collected using a direct electron detector, including the 19S regulatory particle, 26S proteasome and NLRC4/NAIP2 inflammasome [37]. Figure 3d shows a typical micrograph of the 19S regulatory particle, in which DeepEM selected almost all true particle images contained in the micrograph. At the same time, it avoided selecting non-particles from areas containing aggregates and carbon film. The precision-recall curve resulting from the test on the 19S dataset is shown in Fig. 3f. The precision and recall both reach ~80% at the same time. The picked particles were approximately as well-centered as the manually boxed ones. To further verify that the selected particle images are correct, we performed unsupervised 2D classification. The resulting reference-free class averages from about 100 micrographs were consistent with different views of the protein samples (Additional file 1: Figure S5).
Two difficult cases from the inflammasome dataset were examined. Figure 5a shows a micrograph with a high particle density that contains excessively overlapped particles and ice contamination. Most methods based on template matching were incapable of avoiding particle picking from overlapped particles and ice contaminants in this case. Figure 5b presents another difficult situation, in which the side views of the inflammasome display a lower SNR, lack low-frequency features, and are dispersed with a very low spatial density. In both cases, DeepEM still performed quite well in particle recognition, while avoiding the selection of overlapping particles and non-particles. Further tests on similar cases from other protein samples suggested that this observation had a good reproducibility (Additional file 1: Figure S6). Most importantly, DeepEM was able to determine the structure of the human 26S proteasome [38].
Two challenging examples of automated particle recognition. a A typical micrograph showing high-density top views of the inflammasome complex. Considerable ice contaminants and overlapping particles are present. b A typical micrograph of the side views of the inflammasome showing both a paucity of features and a low density of objects. The white square boxes indicate the positive particle images selected by DeepEM. The boxes with a triangle inside indicate that false-positive particle images were picked. The boxes with a star inside indicate the omitted particle images. c The precision-recall curves corresponding to the cases shown in (a) and (b)
Computational efficiency
The DeepEM algorithm was first tested on a Macintosh with a 3.3 GHz Intel Core i5 and 32 GB memory, running Matlab 2014b. When the size of the particle images increases, the parameter space increases substantially, so that it costs more computational time for each micrograph. We usually binned the original micrographs 2 or 4 times to reduce the size of the particle images. For the KLH dataset, it took about 7300 s per micrograph with a micrograph size of 2048 by 2048 pixels and particle image size of 272 by 272 pixels. For the 19S regulatory particle, inflammasome and 26S proteasome datasets, it took about 790, 560, and 1160 s per micrograph with a binned micrograph size of 1855 by 1919 pixels and particle image sizes of 112 by 112, 160 by 160, and 150 by 150 pixels, respectively. To speed up the calculations, multiple instances of the code were run in parallel. We also implemented a Graphic Processing Unit (GPU)-accelerated version of DeepEM in Matlab. We tested it on a desktop computer with 4.0 GHz Intel Core i7-6700 k, 64GB memory and Geforce GTX 970, running Matlab 2016a and CUDA 8.0. It only took about 190, 50, 40, and 60 s per micrograph for the KLH, 19S regulatory particle, inflammasome and 26S proteasome datasets, respectively. The GPU-accelerated DeepEM version therefore speeds up the computation by at least an order of magnitude.
Based on the principles of deep CNN, we have developed the DeepEM algorithm for single-particle recognition in cryo-EM. The method allows automated particle extraction from raw cryo-EM micrographs, thus improving the efficiency of cryo-EM data processing. In our current scheme, a new dataset containing particles of significantly different features may render the previously trained hyper-parameters suboptimal. Readers are directed to Table 1 as references for the hyper-parameter tuning for specific cases. Indeed, finding a set of fine-tuned hyper-parameters leading to optimized learning results on new datasets therefore demands additional user intervention in CNN training. In the above-described examples, we screened several combinations of hyper-parameters to empirically pinpoint an optimal setting. This procedure may be inefficient and can be laborious in certain cases. An automated method for the systemic tuning of hyper-parameters could be developed in the future to address this issue.
The execution of the DeepEM algorithm requires users to first label several hundreds of 'good particles' and 'bad particles' for CNN training purpose, which can be readily assembled from several micrographs. Further processing of these raw particle images is not needed. By contrast, in the traditional template-matching methods [2,3,4,5,6,7,8, 36], users need to first obtain many high-quality class averages or an initial 3D model, which involves multiple steps of single-particle analysis significantly more laborious than the single step of manual particle labeling required by our DeepEM approach. If the template is based on a 3D model, it is usually not trivial to determine a high-quality initial model from new samples, which involves a complete procedure of the ab initio 3D structure determination at low resolution [1]. If the template is based on a set of 2D class averages, users still have to first manually pick thousands of particles and then perform 2D image clustering to generate high-quality 2D classes. Moreover, the number of the reference images are often very limited and hardly include all kinds of orientations, potentially introducing orientation bias in particle picking through template matching. Thus, the preparation step of DeepEM is considerably easier than those of template-matching methods.
Although there are unlimited possibilities for the design of deep CNNs, we made some explorations that helped us understand the optimal use of CNNs for our single-particle recognition problem. First, we examined the noise tolerance of the algorithm with simulated datasets. When the SNR is decreased to 0.005, the DeepEM can still recognize particle images after proper training (Fig. 6). Second, we replaced the sigmoid activation function with a rectified linear unit (ReLU) function. Our results indicate that the ReLU function gives rise to a slightly inferior accuracy in particle recognition than the sigmoid function (Additional file 1: Figure S7). Third, we attempted to design a six-layer CNN, but found that it failed to produce a better or equivalent performance (data not shown). Thus, it is likely that the eight-layer CNN we designed possesses the minimum depth suited to our problem. A deeper CNN might enable greater capacities in these tasks and awaits further investigation. Finally, from the experiments on the inflammasome dataset, we noticed that DeepEM is more effective for feature-rich data. It exhibits a reduced performance when tested on the side views as compared to the top views of the inflammasome (Fig. 4c), because the side views exhibit significantly less low-frequency features than the top views. Thus, the richness of low-frequency particle features is positively correlated with the achievable performance of CNNs.
Effect of the signal-to-noise ratio (SNR) on the precision-recall curves. Three synthetic datasets were generated through computational simulation of micrographs containing single-particle images with SNRs of 0.01, 0.008, 0.005, 0.003, 0.002 and 0.001. For each case, the CNN was first trained on the synthetic dataset of a given SNR and then used to examine the precision-recall relationship using another synthetic dataset with the same SNR. All synthetic datasets used the 70S ribosome as the single-particle model
Our DeepEM algorithm framework exhibits several advantages. First, with sufficient training, DeepEM can select true particles without picking non-particles in a single, integrative step of particle recognition. In fact, it performs as well as a human worker. Similar performance was previously only made possible by combining several steps, encompassing automated particle picking, unsupervised classification and manual curation. Second, DeepEM features traits representative of other artificial intelligence (AI) or machine learning systems. The more it is trained or learned, the better it performs. We found that with iterative updating or optimization of the training dataset, the particle recognition performance of DeepEM can be further improved, which was not possible for conventional particle-recognition algorithms developed so far. Therefore, the performance of earlier algorithms was intricately bound by their mathematics and control parameters, and DeepEM overcomes these limitations.
DeepEM, which is derived from deep CNNs, has proved to be a very useful tool for particle extraction from noisy micrographs in the absence of templates. This approach gives rise to improved "precision-recall" performance in particle recognition, and demonstrates a higher tolerance to much lower SNRs in the micrographs than was possible with older methods based on template-matching. Thus, it enables automated particle picking, selection and verification in an integrated fashion, with a quality comparable to that of a human worker. We expect that this development will broaden the applications of modern AI technology in expediting cryo-EM structure determination. Related AI technologies may be developed in the near future to address key challenges in this area, such as deep classification of highly heterogeneous cryo-EM datasets.
AI:
Convolutional neural network
Cryo-EM:
KLH:
Keyhole Limpet Hemocyanin
SNR:
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The authors thank H. Liu, Y. Xu, M. Lin, D. Yu, Y. Wang, J. Wu and S. Chen for helpful discussions, as well as S. Zhang for assistance in the code adaptation for GPU-based acceleration. The computation was performed in part using the high-performance computational platform at the Peking-Tsinghua Center for Life Science at Peking University, Beijing, China.
The cryo-EM experiments were performed in part at the Center for Nanoscale Systems at Harvard University, Cambridge, MA, USA, a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation of the USA, under NSF award no. 1541959. This work was funded by a grant of the Thousand Talents Plan of China (Y.M.), by grants from the National Natural Science Foundation of China No. 11434001 and No. 91530321 (Y.M., Q.O.), and by the Intel Parallel Computing Center program (Y.M.).
Our software implementation in Matlab is freely available at http://ipccsb.dfci.harvard.edu/deepem. The experimental micrograph data are freely available at the Electron Microscopy Pilot Image Archive (https://www.ebi.ac.uk/pdbe/emdb/empiar/) under the accession codes EMPIAR-10063 and EMPIAR-10072.
Center for Quantitative Biology, Peking University, Beijing, 100871, China
, Qi Ouyang
& Youdong Mao
State Key Laboratory for Artificial Microstructure and Mesoscopic Physics, Peking University, Institute of Condensed Matter Physics, School of Physics, Beijing, 100871, China
Qi Ouyang
Peking-Tsinghua Center for Life Sciences, Peking University, Beijing, 100871, China
Intel Parallel Computing Center for Structural Biology, Department of Microbiology and Immunobiology, Dana-Farber Cancer Institute, Harvard Medical School, Boston, MA, 02115, USA
Youdong Mao
Search for Yanan Zhu in:
Search for Qi Ouyang in:
Search for Youdong Mao in:
Conceived and designed the experiments: YZ QO YM. Performed the experiments: YZ. Analyzed the data: YZ YM. Contributed reagents/materials/analysis tools: QO YM. Wrote the manuscript: ZY YM. All authors have read and approved the final manuscript.
Correspondence to Youdong Mao.
Additional file 1: Figure S1.
The feature maps of the convolutional and subsampling layers from a typical particle image of KLH learned by our CNN. Figure S2. (a) and (b) show a comparison of the results obtained before and after additional selection using standard deviation of the KLH dataset, respectively. (c) and (d) show a comparison of the results obtained before and after additional selection using standard deviation of the 19S, respectively. Figure S3. (a) and (b) show a comparison of the results obtained before and after optimization of the training dataset, respectively. Figure S4. Comparison of DeepEM with TMACS and RELION using the KLH dataset as benchmark. The curves of TMACS [16] and RELION [36] were directly obtained from published data. Figure S5. Reference-free 2D classification of 19S proteasomes recognized by DeepEM. Figure S6. Results of the recognition of the side view of the 26S proteasome by DeepEM. Figure S7. A comparison of the results of different activation functions tested on the KLH dataset (PDF 477 kb)
Particle recognition
Single-particle reconstruction | CommonCrawl |
Mary Pugh
Mary Claire Pugh is an applied mathematician known for her research on thin films, including the thin-film equation and Hele-Shaw flow. She is a professor of mathematics at the University of Toronto.[1]
Pugh completed her Ph.D. in 1993 at the University of Chicago. Her dissertation, Dynamics of Interfaces of Incompressible Fluids: The Hele-Shaw Problem, was supervised by Peter Constantin.[2] Before moving to Toronto, she worked at the Courant Institute of Mathematical Sciences at New York University[3] and then as a faculty member at the University of Pennsylvania, where she won a Sloan Research Fellowship in 1999.[4]
References
1. "Mary Pugh, Professor", Faculty, Department of Mathematics, University of Toronto, retrieved 2019-09-09
2. Mary Pugh at the Mathematics Genealogy Project
3. Applied Mathematics Colloquium (1996-1997), Courant Institute of Mathematical Sciences, New York University, retrieved 2019-09-09
4. Past Fellows, Sloan Foundation, retrieved 2019-09-09
External links
• Home page
• Mary Pugh publications indexed by Google Scholar
Authority control: Academics
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
| Wikipedia |
\begin{document}
\baselineskip=17pt
\title[Is the quartic Diophantine equation $A^4+hB^4=C^4+hD^4$ solvable for any integer $h$? ]{Is the quartic Diophantine equation $A^4+hB^4=C^4+hD^4$ solvable for any integer $h$?}
\author[F. Izadi]{Farzali Izadi} \address{Farzali Izadi \\ Department of Mathematics \\ Faculty of Science \\ Urmia University \\ Urmia 165-57153, Iran} \email{[email protected]}
\author[M. Baghalaghdam]{Mehdi Baghalaghdam} \address{Mehdi Baghalaghdam \\ Department of Mathematics\\ Faculty of Science \\ Azarbaijan Shahid Madani University\\Tabriz 53751-71379, Iran} \email{[email protected]}
\date{}
\begin{abstract} The Diophantine equation $A^4+hB^4=C^4+hD^4$, where $h$ is a fixed arbitrary positive integer, has been investigated by some authors. Currently, by computer search, the integer solutions of this equation are known for all positive integer values of $h \le 5000$ and $A, B, C, D \le 100000$, except for some numbers, while a solution of this Diophantine equation is not known for arbitrary positive integer values of $h$. Gerardin and Piezas found solutions of this equation when $h$ is given by polynomials of degrees $5$ and $2$ respectively. Also Choudhry presented some new solutions of this equation when $h$ is given by polynomials of degrees $2$, $3$, and $4$.
In this paper, by using the elliptic curves theory, we study this Diophantine equation, where $h$ is a fixed arbitrary rational number. We work out some solutions of the Diophantine equation for certain values of $h$, in particular for the values which has not already been found a solution in the range where $A, B, C, D \le 100000$ by computer search. Also we present some new parametric solutions for the Diophantine equation when $h$ is given by polynomials of degrees $3$, $4$. Finally We present two conjectures such that if one of them is correct, then we may solve the above Diophantine equation for arbitrary rational number $h$. \end{abstract}
\subjclass[2010]{11D45, 11D72, 11D25, 11G05 \and 14H52}
\keywords{ Quaratic Diophantine equation, Biquadratics, Elliptic curves}
\maketitle
\section{Introduction} \noindent The Diophantine equation (DE) \begin{equation}\label{e8}
A^4+hB^4=C^4+hD^4,
\end{equation}
where $h$ is a fixed arbitrary positive integer, has been investigated by some authors. \\
\noindent The numerical solutions for $75$ integer values of $h\leq 101$ was given by Choudhry \cite{A.C}. Then these solutions were first extended by Piezas \cite{T.P} for all positive integer values of $h \leq 101$, and finally by Tomita \cite{S.T} for all positive integer values of $h \le 1000$ except $h = 967$. The lost solution for $h = 967$ was supplied by Bremner. Currently, by computer search, the small solutions of this DE are known for all positive integers $h \le 5000$, and $A, B, C, D \le 100000$ except for \\
\noindent $h= 1198, 1787, 1987$,\\
\noindent $2459, 2572, 2711, 2797, 2971$,\\
\noindent $3086, 3193, 3307, 3319, 3334, 3347, 3571, 3622, 3623, 3628, 3644$,\\
$3646, 3742, 3814, 3818, 3851, 3868, 3907, 3943, 3980$,\\
\noindent $4003, 4006, 4007, 4051, 4054, 4099, 4231, 4252, 4358, 4406, 4414$,\\
$4418, 4478, 4519, 4574, 4583, 4630, 4643, 4684, 4870, 4955, 4999.$ \\
\noindent We will work out some of these cases. \\
\noindent Gerardin and Piezas found solutions of this equation when $h$ is given by polynomials of degrees $5$, and $2$, respectively, see \cite{T.P} and \cite{S.T}. Also Choudhry presented several new solutions of this equation when $h$ is given by polynomials of degrees $2$, $3$, and $4$, see \cite{A.C2}. \\
\noindent In this paper, we used elliptic curves theory to study the DE \eqref{e8}. \\
\section{ The first method for solving the DE $A^4+hB^4=C^4+hD^4$}
\noindent Our main result in this section is the following:
\begin{theorem} Consider the DE \eqref{e8}, where $h$ is a fixed arbitrary rational number.\\ Then there exists a cubic elliptic curve of the form
\noindent $E(h):Y^2=X^3+FX^2+GX+H$, where the coefficients $F$, $G$, and $H$, are all functions of $h$. If the elliptic curve $E(h)$ or its counterpart $E(h)_t$ resulting from $E(h)$ by switching $h$ to $ht^4$ has positive rank, depending on the value of $h$ and an appropriate rational number $t$, then the DE \eqref{e8} has infinitely many integers solutions. By taking $h=\frac{v}{u}$, this also solves DE of the form $uA^4+vB^4=uC^4+vD^4$ for appropriate integer values of $u$ and $v$. \end{theorem}
\begin{pro} Let: $A=m-q$, $B=m+p$, $C=m+q$, and $D=m-p$, where all variables are rational numbers. By substituting these variables in the DE \eqref{e8} we get \begin{equation} -8m^3q-8mq^3+8hm^3p+8hmp^3=0. \end{equation}
\noindent Then after some simplifications and clearing the case $m=0$ we obtain \begin{equation}\label{e7} m^2(hp-q)=-hp^3+q^3. \end{equation}
\noindent We may assume that $hp-q=1$ and $m^2=-hp^3+q^3$.
\noindent By plugging $q=hp-1$
into the equation \eqref{e7} and some simplifications we obtain the equation
\begin{equation}\label{e1} m^2=(h^3-h)p^3-(3h^2)p^2+(3h)p-1. \end{equation} \\ \noindent By multiplying both sides of this equation in
$(h^3-h)^2$ and letting
\begin{equation} \label{e2}
X=(h^3-h)p \hspace{1cm} Y=(h^3-h)m, \end{equation} \\ \noindent we get the elliptic curve
\begin{equation}\label{e4}
Y^2=X^3-(3h^2)X^2+(3h(h^3-h))X-(h^3-h)^2.
\end{equation} \\ \noindent By Letting $X=Z+h^2$ in \eqref{e4}, we get the simple elliptic curve
\begin{equation}\label{e78} E(h): Y^2=Z^3-(3h^2)Z-(h^4+h^2). \end{equation}
\noindent If for a given $h$, the above elliptic curve $E(h)$ or its counterpart $E(h)_t$ resulting from $E(h)$ by switching $h$ to $ht^4$ has positive rank, then by calculating $m$, $p$, $q$, $A$, $B$, $C$, $D$, from the relations \eqref{e2}, $q=hp-1$, $A=m-q$, $B=m+p$, $C=m+q$, $D=m-p$, after some simplifications and canceling the denominators of $A$, $B$, $C$, $D$, we obtain infinitely many integer solutions for the DE \eqref{e8}. Now the proof of the main theorem is completed. \end{pro}
\noindent Although, we were able to find an appropriate $t$ such that $E(h)_t$ has positive rank in the case of rank zero $E(h)$ for many values of $h$, the proof for arbitrary $h$ seems to be difficult at this point. For this reason, we state it as a conjecture. \\
\begin{conj} Let $h$ be an arbitrary fixed rational number. Then there exists at least a rational number $t$ such that the rank of the elliptic curve \begin{equation} E(h)_t: Y^2=Z^3-(3h^2t^8)Z-(h^4t^{16}+h^2t^8),
\end{equation}
\noindent is positive. \end{conj}
\begin{rem}
If $h$ is a large number, for example $h=7000$, we may divide $h$ by $10^4$ to get an elliptic curve of positive rank with small coefficients, then solve the DE \eqref{e8} for $h'=\frac{7}{10}$ and finally get a solution for the main case of $h=7000$, by multiplying both sides of the DE \eqref{e8} by an appropriate number. As another example, if $h=9317=7.11^3$, we may first solve the DE \eqref{e8} for $h'=\frac{7}{11}$. \end{rem}
\begin{rem} Note that by substituting the relation $p=\frac{q+1}{h}$, in the equation
$m^2=-hp^3+q^3$, multiplying both side of the equation by $(\frac{h^2-1}{h^2})^2$, and letting
\begin{equation}\label{e18}
X'=(\frac{h^2-1}{h^2})q \hspace{1cm} Y'=(\frac{h^2-1}{h^2})m, \end{equation} \\ \noindent we get another elliptic curve
\begin{equation}\label{e21} Y'^2=X'^3-\frac{3}{h^2}X'^2-\frac{3(h^2-1)}{h^4}X'-(\frac{h^2-1}{h^3})^2. \end{equation} \\
\noindent Now, if we set $Y'=\frac{Y}{h^3}$, $X'=\frac{X+1-h^2}{h^2}$, then the elliptic curve \eqref{e21}, transforms to the elliptic curve \eqref{e4}. This means that two elliptic curves \eqref{e21} and \eqref{e4} are isomorphic. \end{rem}
\section{ Application to examples}
\subsection{Example: $A^4+B^4=C^4+D^4$}\noindent
\noindent i.e., sums of two biquadrates in two different ways. \\
\noindent $h=16$, here $h=1$ replaced by $h=2^4$.\\
\noindent $E(16)$: $Y^2=X^3-768X^2+195840X-16646400$.\\
\noindent Rank=1.\\
\noindent Generator: $P=(X,Y)=(340,680)$.\\
\noindent Points: $2P=(313,-275)$, $3P=(\frac{995860}{729},\frac{-727724440}{19683})$,
$4P=(\frac{123577441}{302500},\frac{305200800239}{166375000})$.\\
\noindent $(p',m',q')=(\frac{313}{4080},\frac{-55}{816},\frac{58}{255})$, \\
$(p'',m'',q'')=(\frac{2929}{8748},\frac{-1070183}{118098},\frac{9529}{2187})$,\\ $(p''',m''',q''')=(\frac{123577441}{1234200000},\frac{305200800239}{678810000000},\frac{46439941}{77137500})$.\\
\noindent Solutions:\\
$1203^4+76^4=653^4+1176^4$,\\
$1584749^4+2061283^4=555617^4+2219449^4$,\\
$103470680561^4+746336785578^4=713872281039^4+474466415378^4$.\\
\subsection{Example: $A^4+206B^4=C^4+206D^4$}\noindent \\
\noindent $h=\frac{103}{8}$.\\
\noindent $E(\frac{103}{8})$: $Y^2=X^3-\frac{31827}{64}X^2+\frac{335615715}{4096}X-\frac{1179689238225}{262144}$.\\
\noindent Rank=1.\\
\noindent Generator: $P=(X,Y)=(\frac{2131205}{32},\frac{8767168835}{512})$.\\
\noindent $(p,m,q)=(\frac{6819856}{217227},\frac{1753433767}{217227},\frac{850373}{2109})$.\\
\noindent Solution:\\
$3331690696^4+206.(1760253623)^4=3682044372^4+206.(1746613911)^4$.\\
\begin{rem}By searching, Noam Elkies found the smallest solution to this DE as $A, B, C, D=3923, 1084, 4747, 506.$ \end{rem}
\begin{rem} No computer research has come up with a solution for the following equations in the range of $A, B, C, D \le 100000$, see \cite{S.T}. \end{rem}
\subsection{Example: $A^4+2572B^4=C^4+2572D^4$}
\noindent $h=2572$.\\
\noindent $E(2572)$: $Y^2=X^3-(3.2572^2)X^2+(3.2572.(2572^3-2572))X-(2572^3-2572)^2$.\\
\noindent Rank=2.\\ \noindent Generators: $P_1=(X,Y)=(\frac{60035809}{9},\frac{302757191}{27})$,\\
\noindent $P_2=(X',Y')=(\frac{3435573760731933430513}{381659437643236},\frac{27488556048550361767336062809879}{7456139229698648679016})$. \\
\noindent $(p,m,q)=(\frac{23333}{59513508},\frac{117667}{178540524},\frac{194}{23139})$,
\\
\noindent $(p',m',q')=(\frac{1558040235953533}{2944884220855208976},\frac{12466120460409195830562539}{57531570296354773207287456},\frac{413061923023825}{1144978312929708})$. \\
\noindent Solutions:\\
$1379237^4+2572.(187666)^4=1614571^4+2572.(47668)^4$, \\
$8288946070402543055294861^4+2572.(12496558499611049062325037)^4=\\ 33221186991220934716419939^4+2572.(12435682421207342598800041)^4$. \\
\subsection{Example: $A^4+967B^4=C^4+967D^4$}\noindent
\noindent $h=967$.\\
\noindent $E(967)$: $Y^2=X^3-(3.967^2)X^2+(3.967.(967^3-967))X-(967^3-967)^2$.\\
\noindent Rank=1.\\
\noindent Generator: $P=(X,Y)=(\frac{238501273696}{245025},\frac{900632541139856}{121287375} )$.\\
\noindent $(p,m,q)=(\frac{2129475658}{1978205172075},\frac{8041361974463}{979211560177125},\frac{83761933}{2045713725})$.\\
\noindent Solution:\\
$32052543684982^4+967.(9095452425173)^4=\\ 48135267633908^4+967.(6987271523753)^4$. \\
\subsection{Example: $A^4+2797B^4=C^4+2797D^4$}
\noindent $h=2797$.\\
\noindent $E(2797)$: $Y^2=X^3-(3.2797^2)X^2+(3.2797.(2797^3-2797))X-(2797^3-2797)^2$.\\
\noindent Rank=1.\\
\noindent Generator: $P=(X,Y)=(\frac{18256234369}{2304},\frac{3411597220289}{110592})$.\\
\noindent $(p,m,q)=(\frac{6527077}{18024671232},\frac{1219734437}{865184219136},\frac{231563137}{18024671232})$.\\
\noindent Solution:\\
$9895296139^4+2797.(1533034133)^4=12334765013^4+2797.(906434741)^4$.\\
\subsection{Example: $A^4+4999B^4=C^4+4999D^4$}
\noindent $h=4999$.\\
\noindent $E(4999)$: $Y^2=X^3-(3.4999^2)X^2+(3.4999.(4999^3-4999))X-(4999^3-4999)^2$.\\
\noindent Rank=1.\\
\noindent Generator:$P=(X,Y)=(\frac{38932053386017900293094583125}{1502165941669975655844},\frac{51963991529347119364735376770810745620625}{58220625445784716642962064124328})$.\\
\noindent $p=\frac{62291285417628640468951333}{300252952455649800809709213504}$,\\
\noindent $m=\frac{83142386446955390983576602833297192993}{11637139545633456295150723664563706629248},$\\
\noindent $q=\frac{2228682405896333845684837}{60062603011732306623266496}$.\\
\noindent Solution:\\
\noindent $A=348665208625932834629908938859838853613$,\\ $B=85556658729553179445421813716725247139$,\\ $C=514949981519843616597062144526433239599$,\\ $D=80728114164357602521731391949869138847$.\\ \\
\subsection{Example: $A^4+2459B^4=C^4+2459D^4$}
\noindent $h=2459$.\\
\noindent $E(2459)$: $Y^2=X^3-(3.2459^2)X^2+(3.2459.(2459^3-2459))X-(2459^3-2459)^2$.\\
\noindent Rank=1.\\
\noindent Generator: $P=(X,Y)$, where \\
\noindent $X=\frac{2455940168334175449299068876662469864}{403764781843031846693075441721}$,\\
\noindent $Y=\frac{775339319798703416232888693955985044070341765700696}{256562189232730518019448407676170227655852269}$.\\
\noindent $p=\frac{249790497186144777186642481352977}{610607423109489416276237594383042485}$,\\
\noindent $m=\frac{78858759133309948762498850076890260788277234103}{387995150343819871443041535202557092730530548624665},$\\
\noindent $q=\frac{1475156352680191470401084694562}{248315340833464585716241396658415}$.
\\
\noindent Solution:\\
\noindent $A=2226087479458719030508635008690035436036778215959$,\\ $B=237581856564140327136761830581698727005176625756$,\\ $C=2383804997725338928033632708843815957613332684165$,\\ $D=79864338297520429611764130427918205428622157550$.\\
\section{The second method for solving the DE $A^4+hB^4=C^4+hD^4$}
\noindent In this section, we wish to look at the equation from a different perspective. To this end we take $X=Z^2+h^2$ \eqref{e4} to get the six degree curve
\begin{equation} Y^2=Z^6-3h^2Z^2-(h^4+h^2). \end{equation}
\noindent This curve can be considered as an quatic elliptic curve of $(h,Y)$ letting $Z$ as a parameter, i.e.,
\begin{equation}\label{e100} Y^2=-h^4-(3Z^2+1)h^2+Z^6. \end{equation}
\noindent Next we use theorem $2.17.$ of \cite{L.W} to transform this quartic to a cubic elliptic curve of the form
\begin{equation}\label{e500} E'(Z): Y'^2=X'^3-(3Z^2+1)X'^2+(4Z^6)X'-(12Z^8+4Z^6). \end{equation}
\noindent with the inverse transformation
\begin{equation}\label{e600} h=\frac{2Z^3(X'-(3Z^2+1))}{Y'} \hspace{1cm} Y=-Z^3+\frac{h^2X'}{2Z^3}. \end{equation}
\noindent Now by taking an appropriate rational value for $Z$ such that the rank of the elliptic curve \eqref{e500} is positive, we obtain an infinitely many rational points on \eqref{e500} and consequently an infinite set of rational values for $h$ (also for $Y$)), which is denoted by $H(Z)$ . Then for every $h$ obtained in this way, we can find a solution for the \eqref{e4} and finally a solution for the main DE by using all the necessary transformations namely
\noindent $X=Z^2+h^2$, $m=\frac{Y}{h^3-h}$, $p=\frac{X}{h^3-h}$, $h=\frac{q+1}{p}$, $A=m-q$, $B=m+p$, $C=m+q$, $D=m-p$. \\
\noindent To get infinitely many solutions one can use the Richmond method \cite{H.R}. The following examples clarify this idea better
\noindent $Z_1=3$.
\noindent $E'(3)$: $Y'^2=X'^3-28X'^2+2916X'-81648$.
\noindent Rank=1.
\noindent Generator: $P=(X',Y')=(108,1080)$.
\noindent $H(Z_1)=\{4,\frac{2^3.3^3}{197},\frac{-2^2.251.395449}{11.13.61.653}, \cdots \}$.\\
\noindent $Z_2=4$.
\noindent $E'(4)$: $Y'^2=X'^3-49X'^2+16384X'-802816$.
\noindent Rank=1.
\noindent Generator: $P=(X',Y')=(202,2958)$.
\noindent $H(Z_2)=\{\frac{2^6.3}{29},\frac{2^9.3.29}{61121},\frac{-2^6.3^2.19.3571.18131}{5.29.97.7746413},\cdots \}$.\\
\noindent $Z_3=6$.
\noindent $E'(6)$: $Y'^2=X'^3-109X'^2+186624X'-20342016$.
\noindent Rank=1.
\noindent Generator: $P=(X',Y')=(\frac{621}{4},\frac{-24975}{8})$.
\noindent $H(Z_3)=\{\frac{-2^2.5.317}{3^3.37},\frac{2^3.5^4.408841}{17.37^2.12757}$,$\frac{-2^2.5.10193.249587558933}{3^3.7.37.5101.181680953},\cdots \}$.\\
\noindent Having seen the examples, the natural question arises:
\noindent Does the set of natural numbers $\mathbb{N}$ contained in $\bigcup_ { t \in \mathbb{Q^*}}t^4 (\bigcup_{Z \in \Omega}H(Z))$?,
\noindent where
\noindent $\Omega=\{Z \in \mathbb{Q} \mid E'(Z) \hspace{.1cm}has \hspace{.1cm} positive \hspace{.1cm}rank \}.$
\noindent We state this as the second conjecture:
\begin{conj} With the above notations one has $\mathbb{N} \subset \bigcup_ { t \in \mathbb{Q^*}}t^4 (\bigcup_{Z \in \Omega}H(Z)) . $ \end{conj}
\section{Application to examples}\noindent \\
\noindent Now we are going to work out some examples. \subsection {Example: $h=108$} \noindent
\noindent $Z=\frac{5}{3}$.
\noindent $E'(\frac{5}{3})$: $Y'^2=X'^3-\frac{-28}{3}X'^2+\frac{62500}{729}X'-\frac{1750000}{2187}$.
\noindent Rank=1.
\noindent Generator: $P=(X',Y')=(\frac{2500}{81},\frac{109000}{729})$.
\noindent $(m,p,q)=(\frac{5}{4},\frac{123}{28},\frac{34}{7})$.
\noindent $h=\frac{4}{3}$.
\noindent Solution:
$303^4+108(158)^4=513^4+108(88)^4$.\\
\noindent Note: $108=4.27$.
\subsection {Example: $h=492$} \noindent
\noindent $Z=\frac{4}{3}.$
\noindent $E'(\frac{4}{3})$: $Y'^2=X'^3-\frac{19}{3}X'^2+\frac{16384}{729}X'-\frac{311296}{2187}$.
\noindent Rank=1.
\noindent Generator: $P=(X',Y')=(\frac{586}{81},\frac{5986}{729})$.
\noindent $(m,p,q)=(\frac{56908}{11033},\frac{-238251}{44132},\frac{-42025}{11033})$.
\noindent $h=\frac{64}{123}$.
\noindent Solution:
$42476^4+492(395732)^4=1863532^4+492(59532)^4$.\\
\noindent Note: $492=123.4$.
\subsection {Example $h=12256974$} \noindent \\ \noindent $Z=\frac{3}{2}$.
\noindent $E'(\frac{3}{2})$: $Y'^2=X'^3-\frac{31}{4}X'^2+\frac{729}{16}X'-\frac{22599}{64}$.
\noindent Rank=1.
\noindent Generator: $P=(X',Y')=(\frac{135}{4},\frac{351}{2})$.
\noindent $2P=(\frac{665}{64},\frac{10309}{512})$.
\noindent $h=\frac{54}{61}$.
\noindent $(m,p,q)=(\frac{145851}{12880},\frac{-306037}{19320},\frac{-48373}{3220})$.
\noindent Solutions:
$62099769^4+12256974(174521)^4=8718303^4+12256974(1049627)^4$.\\
\noindent Note: $54.61^3=12256974$.
\noindent For the point 3P, we obtain $h=\frac{-805}{3977}$, then as two more examples, we obtain solutions for the two cases $h'=3977.805^3$, $h''=3977^3.805$.
\section{Parametric solutions of $A^4+hB^4=C^4+hD^4$} \noindent Let: $A=m-q$, $B=m+p$, $C=m+q$, and, $D=m-p$, where all variables are rational numbers. By substituting these variables in the DE \eqref{e8}, and some simplification, we get
\begin{equation}\label{a} p(hm^2+hp^2)=q(m^2+q^2). \end{equation}
\noindent We may assume that
\begin{equation}\label{b} p=m^2+q^2, \end{equation} and \begin{equation}\label{c} q=h(m^2+p^2). \end{equation}
\noindent By substituting $p=m^2+q^2$, in the relation $h=\frac{q}{m^2+p^2}$, we get:
\begin{equation}\label{b}
h=\frac{q}{m^2+(m^2+q^2)^2}=\frac{q}{m^2+m^4+q^4+2m^2q^2}.
\end{equation} \\
\noindent Then using the reciprocal of $h$, we conclude that \begin{equation}
h=\frac{m^2+(m^2+q^2)^2}{q}=\frac{m^2+m^4+q^4+2m^2q^2}{q}, \end{equation} \noindent the DE \eqref{e8} has a parametric solution:
\noindent $A=m+m^2+q^2$,\\ $B=m-q$,\\ $C=m-m^2-q^2$,\\ $D=m+q$.\\
\begin{exa} $m=kq$;\\ $h=k^2q+(k^2+1)^2q^3;$\\ $A=k+k^2q+q$;\\ $B=k-1$;\\ $C=k-k^2q-q$;\\ $D=k+1$;\\
\begin{rem} The above example provides parametric solutions for the DE \eqref{e8} when $h$ is given by polynomials of the degrees $3$, and $4$ as follows. \end{rem}
\noindent $m=q$;\\ $h=q+4q^3;$\\ $A=1+2q$;\\ $B=0$;\\ $C=1-2q$;\\ $D=2$;\\
\noindent If we let $2q=p$, this recovers the third and the second parametric solutions of the table $1$ in \cite{A.C2}. Of course by replacing $p$ with $2p$ in the second parametric solution of the table $1$ in \cite{A.C2}, and dividing both sides of this equation by $16$, we get the third parametric solution of the table $1$.!\\ \\ $h=8p(p^2+1);$\\ $A=p+1$;\\ $B=0$;\\ $C=p-1$;\\ $D=1$;\\
\noindent $m=2q$;\\ $h=4q+25q^3;$\\ $A=2+5q$;\\ $B=1$;\\ $C=2-5q$;\\ $D=3$;\\
\noindent $m=3q$;\\ $h=9q+100q^3;$\\ $A=3+10q$;\\ $B=2$;\\ $C=3-10q$;\\ $D=4$;\\
\noindent $q=2$, $m=2k$;\\ $h=8k^4+18k^2+8$;\\ $A=2k^2+k+2$;\\ $B=k-1$;\\ $C=2k^2-k+2$;\\ $D=k+1$;\\
\noindent $q=3$, $m=3k$;\\ $h=27k^4+57k^2+27$;\\ $A=3k^2+k+3$;\\ $B=k-1$;\\ $C=3k^2-k+3$;\\ $D=k+1$;\\ \end{exa}
\begin{exa}\noindent
\noindent $h=8k^3p+512k^7p^3+512k^3p^3+1024k^5p^3$;\\ $A=8k^3p+8kp-k$;\\ $B=k+1$;\\ $C=8k^3p+8kp+k$;\\ $D=k-1$;\\
\noindent $k=\frac{1}{2}$;\\ $h=100p^3+p$;\\ $A=10p-1$;\\ $B=3$;\\ $C=10p+1$;\\ $D=1$;\\ \end{exa}
\begin{exa}\noindent \\ $h=n(p^4+(n^2+2)p^2+1)$;\\ $A=p^2+np+1$;\\ $B=p+1$;\\ $C=p^2-np+1$;\\ $D=p-1$;\\ \end{exa} \begin{rem} \noindent The case $n=1$, recovers the last parametric solution of the table 1 in \cite{A.C2}. \end{rem}
\begin{exa} \noindent $q=1$;\\ $h=m^4+3m^2+1;$\\ $A=m+m^2+1$;\\ $B=m-1$;\\ $C=m-m^2-1$;\\ $D=m+1$;\\ \end{exa} \noindent Again this recovers the last parametric solution of the table 1 in \cite{A.C2}.
\begin{exa}\noindent \\ $h=(p^2+2)(p^2+4)$;\\ $A=p^2+p+2$;\\ $B=p+1$;\\ $C=p^2-p+2$;\\ $D=p-1$;\\ \end{exa} \begin{exa}\noindent
\noindent$h=512m^4+1032m^2+512$;\\ $A=8m^2-m+8;$\\ $B=m+1$;\\ $C=8m^2+m+8$;\\ $D=m-1$;\\ \end{exa}
\begin{exa} \noindent $q=m^2$;\\ $h=1+m^2+m^6+2m^4;$\\ $A=1+m+m^3$;\\ $B=1-m$;\\ $C=1-m-m^3$;\\ $D=1+m$;\\ \end{exa}
\noindent\begin{exa} By taking $Z=h^2+1$, in the elliptic curve \eqref{e78} (or $X=2h^2+1$ in the \eqref{e4}), we get \begin{equation} Y^2=h^6-h^4-h^2+1=(h^2-1)(h^4-1)=(h^2-1)^2(h^2+1). \end{equation}
\noindent letting $h^2+1=t^2$, yields $h=\frac{r^2-1}{2r}$ and $t=\frac{-(r^2+1)}{2r}$, then we get \\
\noindent $Y=(h^2-1)(-t)=\frac{(r^6-5r^4-5r^2+1)}{8r^3}$, and $X=\frac{r^4+1}{2r^2}$.\\
\noindent Finally by calculating $m, p, q, A, B, C, D$, from the above relations, we obtain the parametric solution as follows:\\ $h=8r^3(r^2-1)$,\\ $A=4r(5r^4-1)$,\\ $B=(r^4+6r^3+6r^2+6r+1)(r-1)^2$,\\ $C=4r^3(r^4-5)$,\\ $D=(r^4-6r^3+6r^2-6r+1)(r+1)^2$.\\
\noindent Note that $h$ is given by polynomial of degree $5$.
\end{exa}
\noindent The Sage software and Denis Simon's ellrank code were used for calculating the rank of the elliptic curves, (see \cite{S.A}). \\
\begin{center}\textbf{Acknowledgements}
\noindent We are very grateful to Professor Allen MacLeod for the rank and generator computations of the elliptic curves with big generators. Finally the second author would like to present this work to his parents and his wife. \end{center}
\end{document} | arXiv |
\begin{document}
\title{A dynamical model for quantum memory channels} \author{Vittorio Giovannetti} \affiliation{NEST-INFM \& Scuola Normale Superiore, I-56126 Pisa, Italy.} \date{\today} \begin{abstract} A dynamical model for quantum channel is introduced which allows one to pass continuously from the memoryless case to the case in which memory effects are present. The quantum and classical communication rates of the model are defined and explicit expression are provided in some limiting case. In this context we introduce noise attenuation strategies where part of the signals are sacrificed to modify the channel environment. The case of qubit channel with phase damping noise is analyzed in details. \end{abstract} \pacs{03.67.Hk, 03.65.Ud, 89.70.+c} \maketitle \section{Introduction}\label{SEC1} In memoryless quantum channels successive signals (channel uses) are affected by independent, uniform sources of noise~\cite{CHUANG,SHOR,HSW,SETH,BEN}. On the other hand, memory channels are characterized by the presence of correlated source of noise where each channel use is directly or indirectly affected by the previous ones. Preliminary results in the study of such systems has been obtained in Ref.~\cite{MEMO} where it was pointed out that entangled codes can be useful in achieving optimal channel performances. Subsequently some of these results have been generalized to the continuous variable case in Refs.~\cite{GAUS,CERF}, while a systematic analysis of the problem has been proposed in Refs.~\cite{BOWEN,KRETS}. In this paper we present a ``dynamical'' model for studying memory effects in quantum communication where the noise correlations are derived from the interactions between the transmitted signals and the channel environment. By varying the time intervals at which signals are produced by the sender of the message, the model simulates different communication scenarios. Memoryless configurations for instance are recovered as a limiting case in which the signals are transmitted at a frequency much lower than the inverse of the characteristic time of the channel environment relaxation. In this context we introduce also {\em noise attenuation} protocols where the sender alternates sequences of carrying-messages signals with sequences of signals which are employed to modify the environment response but which do not carry any messages to the receiver. Since timescales are fundamental in our model, we characterize its efficiency by introducing the {\em transmission rates} of the communication line. These are dimensional quantities (of dimension equal to an inverse time) which measure the maximal number of qubits or bits
of information that can be transferred reliably (i.e. with unit fidelity) through the channel {\em per unit of transmitting time}. Transmission rates are peculiar of our model as previous works~\cite{BOWEN,GAUS,MEMO,KRETS,CERF} were concerned in characterizing memory channels in terms of information capacities, i.e. the maximum number of qubits (or bits) that can be reliably transferred through the channel {\em per channel uses}. These figures of merit (i.e. rates and capacities) are in general distinct, but are proportional to each other when the sender of the message encodes her/his messages in regular sequence of signals (see Sec.~\ref{s:updown}).
In Sec.~\ref{SEC2} we introduce the channel model by focusing on the physical assumption which underline its definition. In Sec.~\ref{s:memory} we discuss the memory effects present in the system and we introduce the noise attenuation protocols. In Sec.~\ref{SEC3} and Sec.~\ref{s:ratechannel} we define the transmission rates of the channel and we compute their values in some extremal case. Finally in Sec.~\ref{SEC4} an
example of a dephasing qubit channel with memory is discussed.
\section{The model}\label{SEC2}
Consider a communication line where messages are encoded into some internal degree of freedom (e.g. polarization, spin etc.) of a collection of identical physical objects C$_1$, C$_2$, $\cdots$ which propagate through the medium E that separates the sender (say Alice) from the receiver (Bob). The C$_j$ are the information carriers of the system: they are locally produced by Alice and organized in a time-ordered sequence ${s}= \{\tau_1, \tau_2, \cdots \}$ with $\tau_j>0$ being the time interval between the instants $t_{j+1}$ and $t_{j}$ at which C$_{j+1}$ and C$_j$ enter E respectively. We will assume the effective
transit time ${\cal T}_{tr}$ it takes for the carriers for reaching Bob to be constant and shorter than the intervals $\tau_j$ at which they are injected into the medium ({\em fast propagation condition}). The first condition guarantees that the time-ordering of ${s}$ is preserved in the propagation (i.e. Bob will receive the $({j+1})$-th carrier only after a time $\tau_j$ from the arrival of the $j$-th carrier). The second condition instead guarantees that E interacts only with one carrier at a time. Therefore, if $R$ is the density matrix of the carriers at Alice location, after a time ${\cal T}_{tr}$ Bob will receive the state \begin{eqnarray} R^\prime = \mbox{Tr}_E \; \big\{ W \; (R \otimes \rho_0 ) \; W^\dag \big\} \label{prima}\;, \end{eqnarray} where $\rho_0$ is the initial state of E, and where \begin{eqnarray} W = \cdots V_{j} U_{j}\; \cdots \; V_2 U_{2} \; V_1 U_{1} \;, \label{terza} \end{eqnarray} is the unitary operator which describes the coupling between the internal degree of freedom of the carriers and~E. In Eq.~(\ref{terza}) the terms
\begin{eqnarray} U_{j} \equiv
T \exp\left\{ -\frac{i}{\hbar} \int_{t_{j}}^{t_{j}+{\cal T}_{tr}} dt \; \left[ H_{C_j E} (t) + H_E \right] \right\} \;, \label{seconda3} \end{eqnarray} describe the interaction between C$_j$ and E (here $H_{C_j E}(t)$ is the effective time dependent Hamiltonian that couples C$_j$ and E, while $H_E$ is the free Hamiltonian of the medium). Working in a {\em strong coupling regime} we will neglect the contribution of $H_E$ in Eq.~(\ref{seconda3}) and we will assume the $U_j$ to be uniform with respect to the label~$j$. On the other hand, the terms $V_j$ of Eq.~(\ref{terza}) describe the free evolution of E in the time interval between the instant $t_j+{\cal T}_{tr}$ when C$_j$ leaves the environment and the instant $t_{j+1}$ when C$_{j+1}$ enters it, i.e. \begin{eqnarray} V_{j} \equiv \exp\left\{ -\frac{i}{\hbar} H_E (\tau_j- {\cal T}_{tr}) \right\} \simeq \exp\left\{ -\frac{i}{\hbar} H_E \tau_j \right\} \;. \label{seconda4} \end{eqnarray}
\begin{figure}
\caption{Schematic of the communication scenario. Alice encodes her messages in the internal degree of freedom of the carriers C$_1$, C$_2$, $\cdots$, which propagates in a time-ordered sequence toward Bob. The carriers interact one at a time with the local environment LE,
while LE undergoes a dissipative evolution through its interaction with the reservoir R.}
\label{f:figu0}
\end{figure}
In the following we identify two distinct components of the medium E: a finite dimensional Local Environment (LE) component which is directly coupled with the carriers through the $U_{j}$, and a huge Reservoir (R) component which is coupled with LE but not with the carriers (see Fig.~\ref{f:figu0}). The free evolution~(\ref{seconda4}) is supposed to induce a dissipative dynamics which transforms any initial states of LE into a stationary configuration $\sigma_0$, with $\tau_E$ being the characteristic time of the process. This is equivalent~\cite{PETRU} to introducing a one-parameter family ${\cal F}\equiv \{{\cal E}_\tau\}_{\tau\geqslant 0}$ of Completely Positive, Trace preserving (CPT) which, given $\sigma$ the initial state of LE at some time $t_0$, represents its evolution at time $t_0+\tau$ with the density matrix ${\cal E}_\tau(\sigma)$. In this formalism ${\cal E}_0$ coincides with identity map on ${\cal H}_{LE}$. On the other hand the stationary state $\sigma_0$ of LE is defined by the property \begin{eqnarray} {\cal E}_\tau(\sigma_0) &=& \sigma_0 \quad \mbox{for all $\tau\geqslant 0$}\;, \label{relax} \end{eqnarray} while the characteristic time $\tau_E$ by the property \begin{eqnarray} {\cal E}_{\tau\geqslant \tau_E} (\Theta) = \sigma_0\; \mbox{Tr}\; \Theta \label{mappaTAUE}\;, \end{eqnarray} for all bounded operator $\Theta$ of ${\cal H}_{LE}$. An example of $\cal F$ satisfying the above conditions will be presented in Sec.~\ref{SEC4}.
Under the above approximations Eq.~(\ref{prima}) provides a {\em bouncing ball} description of the carrier-environment interactions where the carriers-balls move toward the LE-wall according to the time-ordered sequence ${s} = \{\tau_1, \tau_2, \cdots \}$ chosen by the ``pitcher'' Alice and ``hit'' instantaneously the local environment LE one at a time (see Fig.~\ref{f:figu0}). The resulting transformation is a time ordered product of interactions $U_j$ and relaxation processes ${\cal E}_{\tau_j}$ (see Fig.~\ref{f:figu1}). Assuming LE to be initially in the stationary state $\sigma_0$ this gives \begin{eqnarray} R^\prime &=& \mbox{Tr}_{LE} \; \big\{ \cdots \circ {\cal E}_{\tau_{j}} \circ {\cal U}_j \circ \cdots \nonumber \\ &&\qquad \qquad \circ {\cal E}_{\tau_{2}} \circ {\cal U}_2
\circ {\cal E}_{\tau_{1}} \circ {\cal U}_{1} \; (R \otimes \sigma_0 ) \big\} \;, \label{mappaENNE} \end{eqnarray} where the partial trace is performed on ${\cal H}_{LE}$, ${\cal U}_j(\cdots)$ stands for the unitary mapping $U_j (\cdots ) U_j^\dag$ on ${\cal H}_{C_j}\otimes{\cal H}_{LE}$, and ``$\circ$'' indicates the composition of super-operators. It is important to note that in our model each sequence ${s}=\{\tau_1,\tau_2, \cdots\}$ is characterized by a distinct input-output relation~(\ref{mappaENNE}).
\begin{figure}
\caption{Circuit representation of Eq.~(\ref{mappaENNE}). The local environment LE interacts through the unitary couplings $U_j$ (represented by the small red circles in the figure) with one carriers at a time. Between two consecutive interactions with the carriers instead LE undergoes the dissipative evolution described by the transformations ${\cal E}_{\tau_j}$ (open circles).}
\label{f:figu1}
\end{figure}
\section{Memory effects}\label{s:memory}
Here we give an overview of the memory effects which are accounted for by the model introduced in Sec.~\ref{SEC2}.
Because of the time ordering of Eq.~(\ref{mappaENNE}) the output state of a carrier might depend on the input state of the carriers which precedes it in $s$ but it is always independent from the input state of the carriers which follows it in the sequence. As a matter of fact Eq.~(\ref{mappaENNE}) closely resembles the memory channels analyzed by Kretschmann and Werner in Ref.~\cite{KRETS}. To make this more explicit we rewrite this equation in terms of of the discrete family of CPT maps $\{\Phi_{s}^{(n)}\}_n$ where \begin{eqnarray} \Phi_{s}^{(n)} (R) \nonumber &\equiv&\mbox{Tr}_{LE} \; \big\{ {\cal U}_{n} \circ {\cal E}_{\tau_{n-1}} \circ {\cal U}_{n-1}\\ && \qquad \circ \cdots \circ {\cal E}_{\tau_{1}} \circ {\cal U}_{1} \; (R \otimes \sigma_0 ) \big\} \label{APmappaENNE}
\;, \end{eqnarray} is the output state~(\ref{mappaENNE}) corresponding to the
density matrix $R$ of $\otimes_{j=1}^{n} {\cal H}_{C_j}$ associated with the first $n$ carriers of the sequence ${s}$ (here ${\cal H}_{C_j}$ is the Hilbert space associated with the internal degree of freedom of the $j$-th carrier). Therefore the model of Sec.~\ref{SEC1} originates proper memory effects analogous to those of Refs.~\cite{BOWEN,KRETS,MEMO,CERF} but avoids the feed-forward correlations of Ref.~\cite{GAUS}. For instance Markovian correlated noise can be recovered by properly choosing the transformations ${\cal E}_{\tau_j}$ (see Appendix \ref{appendixA}).
\subsection{Memoryless configuration}\label{SEX}
Assume Alice is producing a sequence ${s}$ with intervals $\tau_j$ greater than or equal to the characteristic relaxation time $\tau_E$ of the dissipation process $\cal F$-- see part a) of Fig.~\ref{f:figu5}. In this case, after each interaction, the local environment LE has enough time to relax into the stationary configuration $\sigma_0$ before a new carrier begins interacting with it. Under this hypothesis Eqs.~(\ref{mappaTAUE}) and~(\ref{APmappaENNE}) yield
\begin{eqnarray} \Phi_{s}^{(n)} = {\cal N}^{\otimes n} \label{mappaENNE1} \end{eqnarray} where ${\cal N}$ is the CPT map which transforms the density matrices $\rho$ of a single carrier into
\begin{eqnarray} {\cal N}(\rho) = \label{mappaENNE2} \mbox{Tr}_{LE} \; \big\{ {\cal U} (\rho \otimes \sigma_0 ) \big\} \;. \end{eqnarray} Equation~(\ref{mappaENNE1}) describes a memoryless configuration where the noise acts on the C$_j$ independently.
\begin{figure}
\caption{Some relevant configurations. Part~{\em a)}: memoryless configuration~(\ref{mappaENNE1}). The carriers (represented by the green circles) are separated by time intervals $\tau_j$ which are greater than the dissipation time $\tau_E$ of the local environment. Part~{\em b)}: generalized memoryless configuration~(\ref{mappaENNE30}). Here the carriers are divided in groups labeled by the index~$g$. The groups are separated by time intervals $\Delta T_g$ which are greater than the dissipation time $\tau_E$. Part~{\em c)}: perfect memory channel~(\ref{mappaENNE10}). Here the distance between two consecutive carriers is negligible with respect to $\tau_E$ inhibiting the relaxation of LE. Part~{\em d)}: example of a noise attenuation protocol. Alice sends uniform sequences of signals composed by~$n$ carriers (the B carriers of the protocol represented by yellow circles in the picture) which have been prepared in the same input state $\rho_0$ and which are separated by time intervals $\tau$. These carriers do not convey any message to Bob and are employed only to ``program'' the environment response. The information is instead encoded into the $(n+1)$-th carrier (the A carriers of the protocol represented by the green circles). The sequence repeats after a time interval $\tau_E$ to allow LE to return to the stationary configuration.}
\label{f:figu5}
\end{figure}
\subsection{Generalized memoryless configuration} \label{SEX1} A generalization of~(\ref{mappaENNE1}) is obtained when the carriers are organized in identical independent groups of $m$ elements each. Here it is convenient to express the elements of $s$ as $\tau_{g,\ell}$ where $g=1,2,\cdots$ is the group index, while $\ell\in\{1, \cdots, m\}$ labels the carriers within a given group. In this notation the time interval \begin{eqnarray} T_g = \sum_{\ell=1}^{m-1} \tau_{g,\ell}\;, \label{imp} \end{eqnarray} gives the ``length'' of the $g$-th group while $\Delta T_g=\tau_{g,m}$ is the interval which separates the last element of the $g$-th group from the first element of the $(g+1)$-th group. We do not assume any restrictions on the time intervals $\{\tau_{g,\ell}\}_{\ell = 1, \cdots, m-1}$ which separates carriers belonging to the same group but we require carriers of distinct subgroups to be separated by time intervals larger than $\tau_E$, i.e. $\Delta T_{g} \geqslant \tau_E$ -- see Fig.~\ref{f:figu5} part~{\em b)}. In this case from Eq.~(\ref{APmappaENNE}) follows that the transformation of the carriers of the first $G$ groups can be expressed as \begin{eqnarray} \Phi_{s}^{(n)} = \label{mappaENNE30} \otimes_{g=1}^{G} {\cal M}_s^{(g)} \;, \end{eqnarray} where $n=mG$ and
\begin{eqnarray} &&{\cal M}_{s}^{(g)} (\rho) \equiv \label{mappaENNE40} \\ &&\quad \mbox{Tr}_{LE} \big\{ {\cal U}_{g,m} \circ {\cal E}_{\tau_{g,{m-1}}} \circ \cdots \circ {\cal E}_{\tau_{g,1}} \circ {\cal U}_{g,1} (\rho \otimes \sigma_0 ) \big\} \;,\nonumber \end{eqnarray} is the CPT map associated with the $m$ carriers C$_{g,1}$, $\cdots$, C$_{g,m}$ of the $g$-th group. By comparison with Eq.~(\ref{mappaENNE1}),
Eq.~(\ref{mappaENNE30}) describes a memoryless channel where the groups are the effective information carriers of the model. In particular if the sets
$\{ \tau_{g,\ell}\}_{\ell = 1,\cdots, m}$ are uniform with respect to the
group label $g$, one has ${\cal M}_s^{(g)}={\cal M}_s^{(g^\prime)}$ for all $g$ and $g^\prime$ and the transformation~(\ref{mappaENNE30}) has once again the standard tensor structure ${\cal M}_s^{\otimes G}$.
\subsection{Perfect memory channel} \label{PERF} Consider the case where $\tau_j \ll \tau_E$ for all $j$. In this limit the local environment relaxation process is inhibited by the frequent interactions with the carriers. Consequently the ${\cal E}_{\tau_j}$ are replaced by the identity transformation on ${\cal H}_{LE}$ and Eq.~(\ref{APmappaENNE}) yields \begin{eqnarray} \Phi_s^{(n)}(R) \label{mappaENNE10} = \mbox{Tr}_{LE} \; \big\{ {\cal U}_n \circ \cdots \circ {\cal U}_2 \circ {\cal U}_{1} \; (R \otimes \sigma_0 ) \big\} \;. \end{eqnarray} This expression describes a perfect memory channel~\cite{BOWEN,KRETS} where the information transferred from the carriers to the finite dimensional local environment LE is not dissipated into the reservoir R of Fig.~\ref{f:figu0}. These maps are asymptotically equivalent~\cite{KRETS} to noiseless channel where each carriers can transfer $\log_2 D$ qubits of quantum information reliably (here $D$ is the dimension of the Hilbert space ${\cal H}_C$ of a single carrier).
\subsection{Noise attenuation protocols}\label{s:noiseatt}
Here we present a communication strategy which explicitly exploits the fact that in our model the environment is effected by the signaling process. In this protocols only a subset A of the transmitted carriers is used to encode messages to Bob. The remaining carriers (subset B) are instead employed for perturbing LE in such a way that the C$_j$ on which the messages are encoded have a better chance to reach Bob without being corrupted. In other words the B carriers are used by the sender as control parameters to program the environment response to the A carriers. A simple implementation of a noise attenuation scheme is shown in Fig.~\ref{f:figu5}~part {\em d)}. Here the B carriers are composed by uniform strings of $n$ states $\rho_0$ (represented by the yellow circles) separated by equal time intervals $\tau$. The information is instead encoded a single carrier (green circles) and the whole structure repeats after a relaxation time $\tau_E$ -- this last assumption is not fundamental but allows us to treat the input-output relations of the A carriers as a memoryless channel of the form~(\ref{mappaENNE1}). In this configuration the transformation of the A carriers which comes from solving Eq.~(\ref{mappaENNE}) can be computed as follows. First we determine the modified state $\sigma_n$ of LE which arises from the interactions with the B carriers. This is accomplished by solving the set of coupled equations analogous to those of Ref.~\cite{SCARANI}, \begin{eqnarray} \left\{ \begin{array}{l} \sigma_j^\prime = \mbox{Tr}_C \;\{ {\cal U}(\rho_0 \otimes \sigma_j) \} \;, \\ \sigma_{j+1} = {\cal E}_{\tau} (\sigma_{j}^\prime) \;, \end{array} \right. \label{MinputoutputNEW} \end{eqnarray} where the trace is performed over the carrier degree of freedom, ${\cal U}$ is the usual carrier-LE coupling super-operator and $j=0,1, \cdots, n-1$. The density matrix $\sigma_n$ which results from~(\ref{MinputoutputNEW}) is then used to determine the output state of the A carriers according to the equation \begin{eqnarray} \overline{\cal N}(\rho) \equiv \mbox{Tr}_{LE} \{ {\cal U} (\rho \otimes \sigma_{n}) \} \;. \label{MinputoutputNEW1} \end{eqnarray} The transformation (\ref{MinputoutputNEW1}) is in general different from Eq.~(\ref{mappaENNE2}) and depends explicitly on the parameters $n$, $\tau$ and $\rho_0$ that are controlled by Alice. The basic idea of a noise attenuation scheme is to appropriately select such parameters in order to get a transformed mapping $\overline{\cal N}$ which is less noisy than the original mapping $\cal N$. An example of this effect will be presented in Sec.~\ref{SEC4}.
\section{Transmission rate of a sequence}\label{SEC3}
Timescales play a fundamental role in the model presented in Sec.~\ref{SEC2}. Therefore a proper way to characterize it, is by introducing its quantum and classical transmission rates. In simple terms these quantities measure, respectively, the maximum number of qubits and bits per second that Alice can encode into the carriers sequence $s$ without compromising the readability of the transmitted messages. The formal definition of the rate of the sequence $s$ is constructed as
follows.
First of all we introduce the discrete value function $n_s(T)$ which, given the sequence $s$, counts the number of carriers which fit~\cite{NOTAZERO} in the time interval $[0,T[$. Furthermore, for any $\epsilon >0$ and $T >0$ we define $q_{s}(\epsilon,T)$ to be the dimension --in qubits units-- of the largest Hilbert sub-space of ${\cal H}(T) \equiv \otimes_{j=1}^{n_s(T)} {\cal H}_{C_j}$ which allows for a fidelity of the transmitted state greater than $1-\epsilon$. This is \begin{eqnarray} q_s({\epsilon},T) = \max_{d}
\Big\{ \log_2 d : \exists {\cal H}_{code} \; \dim {\cal H}_{code}=d , \; \exists \; {\cal A}, {\cal D} \nonumber \\
\forall |\Psi\rangle \in {\cal H}_{code} \;\; F(\Psi,{\cal D}\circ \Phi_{s}^{(T)}\circ{\cal A})>1-\epsilon \; \Big\},\quad \; \label{formularate2} \end{eqnarray} where ${\cal H}_{code}$ are Hilbert sub-spaces of ${\cal H}(T)$, ${\cal A}$ and ${\cal D}$ are {\em encoding} and {\em decoding} CPT maps on ${\cal H}(T)$ applied, respecitively, by Alice and Bob to the carriers, and \begin{eqnarray}
F(\Psi,{\cal D}\circ\Phi_{s}^{(T)}\circ{\cal A} )\equiv \langle \Psi |{\cal D}
\circ \Phi_{s}^{(T)}\circ{\cal A}(|\Psi\rangle\langle
\Psi|) | \Psi\rangle \label{fidelity}\;, \end{eqnarray}
is the fidelity between the input state $|\Psi\rangle \in {\cal H}_{code}$ and the {\em decoded} output state ${\cal D}
\circ \Phi_{s}^{(T)}\circ {\cal A} (|\Psi\rangle\langle
\Psi|)$ (for easy of notation $\Phi_{s}^{(T)}$ indicates the map $\Phi_{s}^{(n_s(T))}$ of Eq.~(\ref{APmappaENNE}) that acts on the $n_s(T)$ carriers of $s$ which lie on $[0,T[$).
The quantum transmission rate $r_q(s)$ of $s$ is thus given by the ratio $q_s({\epsilon},T)/T$ in the the limits $\epsilon\rightarrow 0$, $T\rightarrow \infty$ , i.e. \cite{NOTASUP} \begin{eqnarray} r_q(s) = \lim_{\epsilon \rightarrow 0} \; \limsup_{T\rightarrow \infty} \; \frac{ q_{s}(\epsilon,T)}{T}\;. \label{formalrate1}\end{eqnarray} Analogously we define the {\em classical} transmission rate $r_c(s)$ of $s$ by substituting the function $q_s({\epsilon},T)$ with the largest number of classical distinguishable messages $c_s({\epsilon},T)$ that can be transmitted to Bob with fidelity greater than $1-\epsilon$, i.e. \begin{eqnarray} r_c(s) = \lim_{\epsilon \rightarrow 0} \; \limsup_{T\rightarrow \infty} \; \frac{c_s({\epsilon},T)}{T}\;, \label{formalrate10}\end{eqnarray} where as in Eq.~(\ref{formularate2}) one has \begin{eqnarray} c_s({\epsilon},T) = \max_{d}
\Big\{ \log_2 d : \exists {\cal H}_{code} \; \dim {\cal H}_{code}=d , \; \exists \; {\cal A}, {\cal D} \nonumber \\ \forall k\in\{1, \cdots, d\} \;\; F(\Psi_k,{\cal D}\circ \Phi_{s}^{(T)}\circ{\cal A} )>1-\epsilon \; \Big\}, \; \label{formularate3} \end{eqnarray}
with $|\Psi_1\rangle,|\Psi_2\rangle, \cdots, |\Psi_d\rangle$ being an orthonormal basis of ${\cal H}_{code}$.
\subsection{Upper and lower bounds}\label{s:updown} A simple upper bound for the quantum rate $r_q({s})$ of $s$ can be derived from Eq.~(\ref{formalrate1}) as follows,~\cite{NOTALIM} \begin{eqnarray} r_q({s}) &=& \lim_{\epsilon \rightarrow 0} \; \limsup_{T\rightarrow \infty} \; \frac{n_s(T)}{T} \frac{q_s({\epsilon},T)}{n_s(T)} \nonumber \\ &\leqslant &\Big[ \lim_{\epsilon \rightarrow 0} \;
\limsup_{T\rightarrow \infty} \frac{q_s({\epsilon},T)}{n_s(T)}\Big]\; \limsup_{T^\prime \rightarrow \infty} \; \frac{n_s(T^\prime)}{T^\prime}\nonumber \\ &=& Q_s/ {\tau}^\prime_s \;, \label{upper1} \end{eqnarray} where ${\tau}^\prime_s$ is the {\em minimum average first-neighbors distance} among the carriers of ${s}$ defined by \begin{eqnarray} 1/{\tau}^\prime_s = \limsup_{T^\prime\rightarrow \infty} \; \frac{n_s(T^\prime)}{T^\prime}= \lim_{T^\prime \rightarrow \infty} \sup_{t\geqslant T^\prime} \frac{n_s(t)}{t}\;. \label{unosutauprimo} \end{eqnarray} On the other hand \begin{eqnarray} Q_s = \lim_{\epsilon \rightarrow 0} \limsup_{T\rightarrow \infty} \; \frac{q_s({\epsilon},T)}{n_s(T)}= \lim_{\epsilon\rightarrow 0} \limsup_{n \rightarrow \infty} \frac{q_s({\epsilon},n)}{n} \label{qdis} \end{eqnarray} defines the quantum capacity~\cite{BARNUM,KRETS,KRETS1,SHOR} associated with the maps $\{\Phi_{s}^{(n)}\}_n$ of Eq.~(\ref{APmappaENNE}) (in this expression $q_s({\epsilon,n})$ is given by (\ref{formularate2}) with $n_s(T)$ replaced by $n$).
A lower bound for $r_q(s)$ is instead obtained as follows~\cite{NOTALIM} \begin{eqnarray} r_q({s}) &=& \lim_{\epsilon \rightarrow 0} \; \limsup_{T\rightarrow \infty} \; \frac{n_s(T)}{T} \frac{q_s({\epsilon},T)}{n_s(T)} \nonumber \\ &\geqslant &\Big[ \lim_{\epsilon \rightarrow 0} \;
\limsup_{T\rightarrow \infty} \frac{q_s({\epsilon},T)}{n_s(T)}\Big]\; \liminf_{T^\prime \rightarrow \infty} \; \frac{n_s(T^\prime)}{T^\prime}\nonumber \\ &=& Q_s/ {\tau}^{\prime \prime}_s \;, \label{lower1} \end{eqnarray} where ${\tau}^{\prime\prime}_s\geqslant {\tau}^\prime_s$ is the {\em maximum
first-neighbors average distance} among the carriers of ${s}$ defined by \begin{eqnarray} 1/{\tau}^{\prime\prime}_s = \liminf_{T^\prime\rightarrow \infty} \; \frac{n_s(T^\prime)}{T^\prime}= \lim_{T^\prime \rightarrow \infty} \inf_{t\geqslant T^\prime} \frac{n_s(t)}{t}\;. \label{unosutausecondo} \end{eqnarray} If the sequences $s$ is such that $\lim_{T\rightarrow \infty}{n_s(T)}/{T} = 1/\tau_s$ exists, one has $\tau^\prime_s=\tau^{\prime\prime}_s=\tau_s$ with $\tau_s$ being the average first-neighbors distance among the carriers. These are the {\em regular} sequences of the model: for them Eqs.~(\ref{upper1}) and (\ref{lower1}) coincide and the transmission rate is proportional to the quantum capacity of the channel, i.e. \begin{eqnarray} r_q({s}) &=& Q_s/ {\tau}_s \;. \label{ratedis} \end{eqnarray} The same analysis can be repeated also for the classical rate $r_c(s)$ of Eq.~(\ref{formalrate10}). In particular, in this case,
Eqs.~(\ref{upper1}), (\ref{lower1}) and (\ref{ratedis}) still apply by replacing $Q_s$ with the classical capacity $C_s$ of the maps $\{\Phi_s^{(n)}\}_n$ defined by \begin{eqnarray} C_s = \lim_{\epsilon\rightarrow 0} \limsup_{n \rightarrow \infty} \frac{c_s({\epsilon},n)}{n}\;. \label{cdis} \end{eqnarray}
\subsection{Some solvable configurations} \label{s:solvable}
The maximizations implicit in Eqs.~(\ref{qdis}) and (\ref{cdis}) are in general difficult to solve. However, following the analysis of Refs.~\cite{KRETS,BARNUM} one can bound the capacities $Q_s$ and $C_s$ by means of the coherent information~\cite{COHEINFO} and of the Holevo information~\cite{HOLEVO} of $\Phi_s^{(n)}$, respectively. In particular we have \begin{eqnarray} Q_s &\leqslant& \limsup_{N \rightarrow \infty} \max_{R} \; \frac{J (\Phi_{s}^{(N)}, R)}{N} \;,
\label{quantum1111} \end{eqnarray} where the maximization is performed over all density matrices $R$ of $N$ carriers and \begin{eqnarray}
J(\Phi_{s}^{(N)}, R) \equiv S(\Phi_{s}^{(N)}(R)) -S((\Phi_{s}^{(N)}\otimes {\cal I}_A)(\Psi_R)), \label{COHERENT} \end{eqnarray} is the coherent information~\cite{COHEINFO} of $\Phi_{s}^{(N)}(R)$. In the above expression $S(R) = -\mbox{Tr}[R \log_2 R]$ is the von Neumann entropy, $\Psi_R$ is a generic purification of $R$ constructed by adding an ancillary Hilbert space ${\cal H}_A$, and ${\cal I}_A$ is the identical map on ${\cal H}_A$. Analogously one has \begin{eqnarray} C_s &\leqslant& \limsup_{N \rightarrow \infty} \; \max_{\cal P}\; \frac{\chi(\Phi_s^{(N)}, {\cal P})}{N} \label{ccCC} \end{eqnarray} where the maximization is performed over all ensemble ${\cal P} =\{p_k; R_k\}_k$ of $N$ carriers and where \begin{eqnarray} \chi(\Phi_s^{(N)}, {\cal P})&\equiv& S(\Phi_s^{(N)}(\sum_{k} p_k R_{k})) \label{c} \\ &&\qquad -\sum_k p_k S(\Phi_s^{(N)}(R_k))\;, \nonumber \end{eqnarray} is the Holevo information~\cite{HOLEVO} associated with $\Phi_s^{(N)}$. Kretschmann and Werner have identified a class of maps $\{ \Phi_s^{(n)}\}_n$ (the {\em forgetful} channels~\cite{KRETS}) for which the right-hand side term of~(\ref{quantum1111}) and~(\ref{ccCC}) indeed provide the exact value for $Q_s$ and $C_s$. Here we will focus only on the limiting cases discussed in Sec.~\ref{s:memory} for which an expression for $Q_s$ and $C_s$ can be derived without the elegant arguments of Ref.~\cite{KRETS}.
\begin{itemize}
\item[{\bf \em a)}] The simplest configuration is when the sequence $s$ is such that $\tau_j \ll \tau_E$ for all $j$. When this happens the maps $\{\Phi_s^{(n)}\}_n$ describe a perfect memory channel~(\ref{mappaENNE10}) which allows optimal transfer, ensuring $Q_s = C_s =\log_2 D$. Therefore, according to~(\ref{ratedis}) using regular sequences $s$ with $\tau_j\ll \tau_E$, Alice and Bob can achieve transmission rates equal to \begin{eqnarray} r_q({s}) = r_c({s})=\frac{\log_2 D}{ {\tau}_s} \;. \label{ratesperfect} \end{eqnarray}
\item[{\bf \em b)}] For memoryless configurations~(\ref{mappaENNE1}), $Q_s$ and $C_s$ coincide, respectively, with the quantum $Q({\cal N})$ and classical $C({\cal N})$ capacity of the memoryless map $\cal N$ of Eq.~(\ref{mappaENNE2}). On one hand one has~\cite{SETH}, \begin{eqnarray} {Q({\cal N})} = \lim_{N \rightarrow \infty} \max_{R} \; \frac{J ({\cal N}^{\otimes N}, R)}{N} \;, \label{quantum11} \end{eqnarray} where, as in Eq.~(\ref{quantum1111})
the maximization is performed over all density matrices $R$ of $N$ carriers and where $J({\cal N}^{\otimes N}, R)$ is the coherent information~(\ref{COHERENT}) of ${\cal N}^{\otimes N}$. On the other hand one has~\cite{HSW}, \begin{eqnarray} C({\cal N}) = \lim_{N \rightarrow \infty}
\max_{\cal P}\; \frac{\chi({\cal N}^{\otimes N}, {\cal P})}{N} \label{cc} \end{eqnarray} where the maximization is performed over all ensemble ${\cal P} =\{p_k; R_k\}_k$ of $N$ carriers and where $\chi({\cal N}^{\otimes N}, {\cal P})$ is the Holevo information~(\ref{c}) associated with ${\cal N}^{\otimes N}$. Therefore for regular sequences $s$ with $\tau_j \geqslant \tau_E$ we get \begin{eqnarray} r_q({s}) =Q({\cal N}) / {\tau}_s \;, \qquad r_c({s}) = C({\cal N})/ {\tau}_s \;. \label{ratesnomemory} \end{eqnarray}
\item[{\bf \em c)}] The generalized memoryless configurations~(\ref{mappaENNE30}) can be treated in the same way by replacing the quantities $\tau_s^{\prime}, \tau_s^{\prime \prime}$ of Eqs.~(\ref{unosutauprimo}) and (\ref{unosutausecondo}) with the corresponding average first-neighboring {\em group} distances and the map $\cal N$ with the $m$ carriers memoryless map ${\cal M}_s$ of Eq.~(\ref{mappaENNE40}). In particular, for a generalized memoryless sequences $s$ having constant group lengths $T_g=T_s$ and constant group separations $\Delta T_g=\Delta T_s$ for all $g$ one easily verifies the following identities \begin{eqnarray} r_q({s}) &=& {Q({\cal M}_s)}/(T_s + \Delta T_s) \;, \label{ratedisblocchi1} \\ r_c({s}) &=&{C({\cal M}_s)}/(T_s + \Delta T_s) \;.
\label{ratedisblocchi2} \end{eqnarray} \item[{\bf \em d)}] Finally consider the noise attenuation protocols of Sec.~\ref{s:noiseatt}. For the sake of simplicity we will focus on the specific example of Fig.~\ref{f:figu5} where the results for memoryless configuration applies. In this case the rate is given by \begin{eqnarray} r_q({s}) &=& {Q(\overline{\cal N})}/{(n \tau +\tau_E)} \;, \nonumber \\ r_c({s}) &=& {C(\overline{\cal N})}/{(n \tau + \tau_E)} \;,
\label{ratedisblocchi} \end{eqnarray} with $\overline{\cal N}$ being the map~(\ref{MinputoutputNEW1}) and with $n \tau + \tau_E$ being the time intervals which separates two consecutive $A$-carriers. \end{itemize}
\section{Transmission rate for multiple choice of the sequence}\label{s:ratechannel}
In this section we analyze the optimal quantum and classical communication rates $R_{q,c}$ achievable in our model when Alice is not restricted to a single given sequence $s$ but instead she has some freedom in selecting the sequence she will use for the signaling.
For the sake of simplicity we will assume the set $\cal S$ of the allowed sequences to be fully characterized by a single parameter $\tau_{min}$ which bounds the minimum value for the intervals $\tau_j$ of a sequence $s$ of the set. That is ${\cal S}= {\cal S}(\tau_{min})$ will be the set of all sequences $s$ which satisfy $\tau_j \geqslant \tau_{min}$ for all $j$. The need of constraining the minimum value of the $\tau_j$ is fundamental if we want our model to have a non trivial structure (see for instance Sec.~\ref{s:solvable} and Eq.~(\ref{perfect1}) below). From a more practical point of view the introduction of $\tau_{min}$ follows from the physical and technological difficulties in producing sequence of ordered signals that might arise in realistic communication scenarios (for instance, too close packed carriers tend to overlap during their propagation, compromising the time ordering of the sequence).
A natural candidate for $R_{q,c}$ is the maximum of the rates $r_{q,c}(s)$ computed over the sequence $s$ of $\cal S$, i.e. \begin{eqnarray} R_{q,c}^{(1)}(\tau_{min}) = \max_{s\in{\cal S}} r_{q,c}({s}) \label{lowerbound}\;. \end{eqnarray} A detailed analysis of $R_{q,c}^{(1)}$ is presented in Appendix~\ref{s:simplification} where it is shown how Eq.~(\ref{lowerbound}) simplifies in the case in which $\cal S$ contains only regular sequences for which Eq.~(\ref{ratedis}) applies. We will see in a moment that for $\tau_{min} \ll \tau_E$ and $\tau_{min} \geqslant \tau_E$ the function $R_{q,c}^{(1)}(\tau_{min})$ provides indeed the correct values of the achievable rates. For generic $\tau_{min}$ however we claim that the function $R_{q,c}^{(1)}(\tau_{min})$ does not necessarily tell the whole story about $R_{q,c}$. On the contrary we propose to compute $R_{q,c}$ as follows \begin{eqnarray} R_{q}(\tau_{min}) &=& \lim_{\epsilon \rightarrow 0} \; \limsup_{T\rightarrow \infty} \; \max_{s\in{\cal S}}\;\frac{ q_{s}(\epsilon,T)}{T}\;, \label{Formalrate1}\\ R_c(\tau_{min}) &=& \lim_{\epsilon \rightarrow 0} \; \limsup_{T\rightarrow \infty} \; \max_{s\in{\cal S}} \; \frac{c_s({\epsilon},T)}{T}\;, \label{Formalrate10} \end{eqnarray} with $q_s(\epsilon,T)$ and $c_s(\epsilon,T)$ given in Eqs.~(\ref{formularate2}) and (\ref{formularate3}). Equations~(\ref{Formalrate1}) and (\ref{Formalrate10}) define proper rates of the communication line of Sec.~\ref{SEC1} in the sense that, given $\delta >0$ and $\epsilon$ arbitrarily small there is allowed sequence ${s}\in {\cal S}$ which, in the limit of infinite $T$ permit Alice to transfer to Bob at least $(R_q-\delta)T$ qubits with fidelity $>1-\epsilon$.
Since Eq.~(\ref{lowerbound}) is obtained from Eqs.~(\ref{Formalrate1}) and (\ref{Formalrate10})
by inverting the order of the maximization over ${s}$ with the limits in $\epsilon$ and $T$ it follows immediately that $R_{q,c}^{(1)}(\tau_{min})$ is a lower bound for $R_{q,c}(\tau_{min})$ of ${\cal S}$, i.e. \begin{eqnarray} R_{q,c}(\tau_{min}) \geqslant R_{q,c}^{(1)}(\tau_{min}) \label{lowerbound1}\;. \end{eqnarray} An interesting problem is to understand whether or not the inequality in Eq.~(\ref{lowerbound1}) can always be replaced with an identity. Alternatively one may ask under which conditions on the model parameters (i.e. $U_j$, $\cal F$) the transmission rate of ${\cal S}$ can be computed as the maximum of the rates achievable within a specific choice of~$s$. In the next section we provide a partial answer to these questions by showing that for $\tau_{min} \ll \tau_E$ and $\tau_{min}\geqslant \tau_E$ the functions $R_{q,c}(\tau_{min})$ and $R_{q,c}^{(1)}(\tau_{min})$ coincide.
\subsection{Bounds and asymptotic behavior}\label{s:asymp}
Even without solving the maximizations of~(\ref{lowerbound}),~(\ref{Formalrate1}) and~(\ref{Formalrate10}) one expects the resulting expressions $R_{q,c}^{(1)}$, $R_{q,c}$ will depend upon the interplay between the relaxation time $\tau_E$ of LE and the characteristic time $\tau_{min}$ of $\cal S$.
A trivial but useful
upper bound for $R_{q,c}$ follows by observing that the maximum number $n_s(T)$ of carriers that can fit in $[0,T[$ cannot be greater than $T/\tau_{min}$ and that $q_s(\epsilon,T), c_s(\epsilon,T)$ cannot be greater than the $\log_2$ of the dimension of ${\cal H}(T)$, i.e. \begin{eqnarray} q_s(\epsilon,T), c_s(\epsilon,T)\leqslant n_s(T) \log_2 D\;, \label{eqnarray} \end{eqnarray} with $D$ being the dimension of the Hilbert space of a single carrier. Replacing the above relations in Eqs.~(\ref{formalrate1}) and (\ref{formalrate10}) gives \begin{eqnarray} R_{q,c}(\tau_{min}) \leqslant \frac{\log_2 D}{\tau_{min}}\label{upper} \;, \end{eqnarray} for all $\tau_{min}$. From Sec.~\ref{s:solvable} it follows that this bound is achievable at least if $\cal S$ is such that $\tau_{min}\ll \tau_E$. In this case in fact the sequence ${s}_0 $ with $\tau_j=\tau_{min}$ for all $j$ allows for carriers that reliably transfer $\log_2 D$ qubits of information each. Therefore from~(\ref{lowerbound}) and (\ref{lowerbound1}) we get \begin{eqnarray}
R_{q,c}(\tau_{min}) = R_{q,c}^{(1)}(\tau_{min})\Big|_{\tau_{min}\ll \tau_E} \simeq \frac{\log_2 D}{\tau_{min}}\;, \label{perfect1} \end{eqnarray} which shows that the rates
diverge for $\tau_{min}\rightarrow 0$. An explicit expression can also be determined for $\tau_{min}$ greater than $\tau_E$. In fact, according to Sec.~\ref{SEX}, in this case all the allowed sequences ${s}$ yields the same memoryless mapping ${\cal N}^{\otimes n(T)}$. Thus the maximization with respect to ${s}$ becomes a simple optimization with respect to the average time intervals $\tau_s$ and one gets, \begin{eqnarray}
R_q(\tau_{min})=R_q^{(1)}(\tau_{min})\Big|_{\tau_{min}\geqslant \tau_E} &=& \;{ Q({\cal N})}/{\tau_{min}}, \label{formalrate1001}\\
R_c(\tau_{min})=R_ c^{(1)}(\tau_{min})\Big|_{\tau_{min}\geqslant \tau_E} &=& \;{ C({\cal N})}/{\tau_{min}}, \label{formalrate1002}\end{eqnarray} with $Q({\cal N})$ and $C({\cal N})$ the capacities of Eqs.~(\ref{quantum11}) and~(\ref{cc}), respectively.
For intermediate value of $\tau_{min}$ a lower bound for $R^{(1)}_{q,c}$, and thus for $R_{q,c}$, can be obtained for instance by focusing on the generalized memoryless configuration (see Eq.~(\ref{lowerboundmin})) or by considering the noise attenuation strategies. In this last case it is simpler to consider only the configurations described in Fig.~\ref{f:figu5} and maximizing the rates~(\ref{ratedisblocchi}) with respect to the free parameters $\tau\geqslant \tau_{min}$ and $n\geqslant 1$, e.g. \begin{eqnarray} R_{q}^{(1)}(\tau_{min})\geqslant \; \sup_{\begin{subarray}{c} \tau\geqslant \tau_{min}\\ {n\geqslant 1} \end{subarray}} \; \frac{Q(\overline{\cal N})}{n\tau + \tau_E} \;, \nonumber \\ R_{c}^{(1)}(\tau_{min})\geqslant
\; \sup_{\begin{subarray}{c} \tau\geqslant \tau_{min}\\ {n\geqslant 1} \end{subarray}} \;
\frac{C(\overline{\cal N})}{n\tau + \tau_E} \label{RATEMOD} \;. \end{eqnarray}
\begin{figure}
\caption{Plot of the ratio $\Gamma$ of Eq.~(\ref{RATIO}) as a function of the dimensionaless parameter $\tau/\tau_E$, for different values of the $n$ and for different values of the environment-carriers coupling constant $\lambda$. In the strong coupling regime $\lambda \sim 0$, the attenuation noise protocol provides a significative improvement of the transmission rate. For instance for $\lambda = 0.01$, $r$ reaches the maximum value of $\sim 1.3$ for $n=1$ and $\tau\sim \tau_E/2$. }
\label{f:fig3}
\end{figure}
\section{An example with qubits}\label{SEC4}
In this section we analyze an example of dynamical model for memory channels where both the information carriers C$_j$ and the local environment LE are qubits. In this context we will make a comparison between the noise attenuation protocol of Sec.~\ref{s:noiseatt} and the memoryless configuration.
We will assume the carrier-LE interaction $U_j$ of Eq.~(\ref{seconda3}) to be to a control-unitary such that
when the carrier is in $|0\rangle_{C_j}$
nothing happens to LE, while when C$_j$ is in $|1\rangle_{C_j}$ the environment undergoes to the transformation \begin{eqnarray} \Theta(\lambda) \equiv \left( \begin{array}{cc} \sqrt{\lambda} & \sqrt{1-\lambda} \\ \sqrt{1-\lambda} & - \sqrt{\lambda} \end{array} \right) \;, \label{hadamard} \end{eqnarray} with $\lambda\in [0, 1]$ being a parameter which measures the ``intensity'' of the coupling (with low coupling corresponding to $\lambda\sim 1$ and high coupling corresponding to $\lambda \sim 0$). Moreover we will assume the relaxation process ${\cal F}=\{ {\cal E}_\tau\}_\tau$ acting on LE to be
described by amplitude damping maps~\cite{CHUANG} which takes the state $|1\rangle_{LE}$ to $|0\rangle_{LE}$ with probability $1-\eta(\tau)$ where $\eta(\tau) \in [0,1]$ is a non increasing function of $\tau$ with characteristic time $\tau_E$, i.e. \begin{eqnarray}
{\cal E}_\tau(|0\rangle_{LE} \langle 0| ) &=& |0\rangle_{LE} \langle 0| \nonumber \\
{\cal E}_\tau(|1\rangle_{LE} \langle 1| ) &=& \eta(\tau) \;|1\rangle_{LE} \langle 1|
+(1- \eta(\tau) )\;|0\rangle_{LE} \langle 0| \nonumber \\
{\cal E}_\tau(|0\rangle_{LE} \langle 1| ) &=& \sqrt{\eta(\tau)} \;|0\rangle_{LE} \langle 1|\;. \label{DEFNOISE} \end{eqnarray} In this example the stationary state $\sigma_0$ of LE is hence
$|0\rangle_{LE}$. The parameterization of the memory effect is given by $\eta(\tau)$, with $\eta=0$ corresponding to the memoryless case (fast environment relaxation) and $\eta=1$ corresponding to perfect memory case (no environment relaxation). In order to have a well defined threshold between memoryless and memory configuration, in the following we will assume \begin{eqnarray} \eta(\tau) =\left\{ \begin{array}{lll} 1-\tau/\tau_E & &\mbox{for $\tau< \tau_E$} \\ 0 & &\mbox{for $\tau\geqslant \tau_E$.} \end{array} \right. \end{eqnarray} Under the above conditions, it is possible to show that both the map $\cal N$ of the memoryless case and the map $\overline{\cal N}$~(\ref{MinputoutputNEW1}) of the noise attenuation protocol correspond to a phase damping channel ${\cal P}_g$ where the coherence terms of the input qubit $\rho$ are degraded by a positive factor $g\leqslant 1$, i.e.~\cite{CHUANG} \begin{eqnarray}
{\cal P}_g(|\kappa \rangle_{C} \langle \kappa | ) &=& |\kappa \rangle_{C} \langle \kappa
| \qquad \quad \mbox{for $\kappa =0,1$} \nonumber \\
{\cal P}_g(|0\rangle_{C} \langle 1| ) &=& g \;|0\rangle_{C} \langle 1|\;. \label{DEFENNE} \end{eqnarray} In particular Eq.~(\ref{mappaENNE2}) gives ${\cal N}={\cal P}_{g_0}$ with $g_0=\sqrt{\lambda}$. On the other hand, Eq.~(\ref{MinputoutputNEW1}) gives $\overline{\cal N}={\cal P}_{\overline{g}}$ where $\overline{g}$ is a complicated expression~(\ref{gmod}) of $\lambda$ and of the parameters $\rho_0$, $n$ and $\tau$ (see Appendix~\ref{a:example} for details). By appropriately selecting the values of the above quantities one can make the make $\overline{\cal N}$ less noisy than $\cal N$ by having $\overline{g}>g_0$. To see if this corresponds to an increase in the transmission rates $r_{q,c}(s)$ we can use the results of Sec.~\ref{s:solvable}. In the case of the phase damping channels ${\cal P}_g$ the capacities $Q({\cal P}_g)$ and $C({\cal P}_g)$ of Eqs.~(\ref{quantum11}) and (\ref{cc}) can be explicitly computed. For instance since here the noise
does not affect the populations associated with the computational basis, the classical capacity of the phase damping channel~(\ref{DEFENNE}) is optimal for all values of $g$, i.e. $C({\cal P}_g) = 1$. Hence from Eqs.~(\ref{ratesnomemory}) and (\ref{ratedisblocchi}) we get \begin{eqnarray} r_c(s_0) &=& 1 /{\tau_E} \geqslant {1}/{(n \tau + \tau_E)}= \overline{r}_c \;,
\label{confronto} \end{eqnarray} where $s_0$ is the memoryless sequence with uniform interval $\tau_j=\tau_E$ and $\overline{r}_c$ is the classical rate of the noise attenuation protocol of Fig.~\ref{f:figu5}. Equation~(\ref{confronto}) shows that, in the specific example considered here, the noise attenuation protocol does not improve the classical rate of the communication line with respect to the memoryless case. On the other hand the quantum capacity of a phase damping channel~(\ref{DEFENNE}) is equal to~\cite{DEVETAK} \begin{eqnarray} Q({\cal P}_g) = 1-H_2(1/2+g/2)\;, \label{QUANTUMCAPA} \end{eqnarray} where $H_2(x) = -x \log_2 x -(1-x) \log_2 (1-x)$ is the binary entropy function. In this case, higher values of $g$ corresponds to higher $Q({\cal P}_g)$ and the rate $\overline{r}_q$ of the noise attenuation protocol can be higher than the rate $r_q(s_0)$ of the memoryless case. To see this we studied the ratio \begin{eqnarray} \Gamma = \frac{\overline{r}_q}{r_q(s_0)} = \frac{\tau_E}{n\tau + \tau_E} \frac{ 1-H_2(1/2+\overline{g}/2)}{1-H_2(1/2+g_{0}/2)} \label{RATIO}\;,\end{eqnarray} as a function of the variable $\tau/\tau_E$ and for for different values of $n$ and $\lambda$. [Here $\overline{g}$ has been optimized with respect to the no-carrying signal $\rho_0$]. The results have been plotted in Fig.~\ref{f:fig3} which shows that
in the strong coupling limit $\lambda\sim 0$ one can have an appreciable increase of $\Gamma$ for $\tau\sim \tau_E/2 $ and with $n$ of the order of 5.
\section{Conclusion}\label{s:conclusioni}
We have introduced a communication model where memory effects arise from the interaction between the information carriers with the channel environment. Different memory effects can be simulated by varying the time intervals at which the carriers are produced by the sender of the message. The information rates of the model have been defined and computed in some extremal cases.
\appendix \section{}\label{appendixA}
In this appendix we show how a Markovian correlated noise~\cite{BOWEN,KRETS,MEMO} can be derived from the mapping~(\ref{APmappaENNE}) by properly choosing the transformation ${\cal E}_{\tau_j}$.
Consider the case in which for sufficiently big $\tau$ the map ${\cal E}_{\tau}$ describes a decoherent
process of LE where, given $\{|\ell\rangle_{LE}\}$ an orthonormal basis of ${\cal H}_{LE}$, one has \begin{eqnarray}
{\cal E}_{\tau} (|\ell\rangle_{LE}\langle \ell^\prime| ) = \delta_{\ell, \ell^\prime} \;
|\psi_\ell(\tau)\rangle_{LE}\langle \psi_\ell(\tau)| \label{DECO}\;, \end{eqnarray}
with the vectors $\{|\psi_\ell (\tau)\rangle_{LE}\}_\ell$ being not necessarily orthogonal, and $\delta_{\ell,\ell^\prime}$ being the Kronecker delta. The condition~(\ref{relax}) can then be satisfied by identifying $\sigma_0$ with one element of the selected basis (say $|\ell_0\rangle_{LE}$), and imposing
$|\psi_\ell(\tau\geqslant \tau_E)\rangle_{LE} = |\ell_0\rangle_{LE}$ for all $\ell$. In this case the mapping~(\ref{APmappaENNE}) can be expressed in terms of the operators \begin{eqnarray}
A_{\ell_{1}} &\equiv& {_{LE}\langle} \ell_{1} | U_{1}
|{\ell_{0}} \rangle_{LE} \\
A_{\ell_{j+1},\ell_{j}} &\equiv& {_{LE}\langle} \ell_{j+1} | U_{j+1}
|\psi_{\ell_{j}}(\tau_{j}) \rangle_{LE}\;, \end{eqnarray} which act, respectively, on the Hilbert space ${\cal H}_{C_{1}}$ and ${\cal H}_{C_{j+1}}$ for $j=1, \cdots, n-1$. They allow us to define the probability distribution \begin{eqnarray} p^{(1)}_{\ell_1} &\equiv & \mbox{Tr}_{C_{1}} \left\{ A^\dag_{\ell_{1}} A_{\ell_{1}} \right\} \label{PROB1}\end{eqnarray} and the conditional probabilities \begin{eqnarray}
p^{(j+1)}_{\ell_{j+1}|\ell_{j}} &\equiv & \mbox{Tr}_{C_{j+1}} \left\{ A^\dag_{\ell_{j+1},\ell_{j}} A_{\ell_{j+1},\ell_{j}} \right\} \label{PROB2} \;. \end{eqnarray} which satisfies the normalization conditions
$\sum_{\ell_{j+1}} p^{(j+1)}_{\ell_{j+1}|\ell_{j}} =1$
and $\sum_{\ell_{j}} p^{(j+1)}_{\ell_{j+1}|\ell_{j}} <1$. Using these quantities Eq.~(\ref{APmappaENNE}) can be finally expressed in compact Markovian form, \begin{eqnarray} &\Phi^{(n)}_s(R) = \label{redux1} \sum_{\ell_1,\cdots, \ell_n}
p^{(1)}_{\ell_1} \; p^{(2)}_{\ell_2|\ell_1}\;
\cdots \; \; p^{(n)}_{\ell_{n}|\ell_{n-1}} &\label{MARKOV} \\ &\times M_{\ell_n,\ell_{n-1}} \nonumber \cdots M_{\ell_{2}, \ell_{1}} \;M_{\ell_1}\; R \; M^\dag_{\ell_1}\; M^\dag_{\ell_{2}, \ell_{1}} \cdots M^\dag_{\ell_n-1,\ell_n} & \end{eqnarray} with $M_{\ell_1} \equiv A_{\ell_1} /\sqrt{p^{(1)}_{\ell_1}}$ and $$M_{\ell_{j+1},\ell_{j}} \equiv A_{\ell_{j+1},\ell_{j}} /
\sqrt{p^{(j+1)}_{\ell_{j+1}|\ell_j}}\;.$$
\section{} \label{s:simplification} In this section we analyze $R_{q,c}^{(1)}$ showing that, if the set $\cal S$ contains only regular sequences, then the maximization of Eq.~(\ref{lowerbound})
can be solved by focusing on the generalized memoryless configurations.
Consider the subset ${\cal S}_0$ of
the sequence $s \in {\cal S}$ which correspond to the uniform generalized memoryless configurations of Sec.~\ref{SEX1} characterized by constant group distance $\Delta T_s = \max\{\tau_{min}, \tau_E\}$. Since ${\cal S}_0$ is a proper subset of $\cal S$ we have \begin{eqnarray} R_{q}^{(1)}(\tau_{min}) &\geqslant& \max_{s\in{\cal S}_0} r_{q}({s}) \nonumber \\ &=& \max_{s\in{\cal S}_0} \frac{Q({\cal M}_s)}{T_s + \max\{\tau_{min},\tau_E\}} \label{lowerboundmin}\;, \end{eqnarray} where we used Eq.~(\ref{ratedisblocchi1}) to express $r_q(s)$. Now, given $s \in {\cal S}$ from Eqs.~(\ref{upper1}) and (\ref{quantum1111}) one gets \begin{eqnarray} r_q({s}) &\leqslant& (1/ {\tau}^\prime_s ) \limsup_{N \rightarrow \infty} \max_{R} \left\{ J (\Phi_{s}^{(N)}, R)/N \right\} \nonumber \\ &\leqslant& (1/ {\tau}^\prime_s ) \limsup_{N \rightarrow \infty} \left\{ \sup_{k\geqslant 1} \max_{R^\prime} \frac{ J ([\Phi_{s}^{(N)}]^{\otimes k} , R^\prime )}{ k N} \right\} \nonumber \\ &=& (1/ {\tau}^\prime_s ) \limsup_{N \rightarrow \infty} \left\{ \lim_{k\rightarrow \infty} \max_{R^\prime} \frac{ J ([\Phi_{s}^{(N)}]^{\otimes k} , R^\prime )}{ k N} \right\} \nonumber \\ &=& (1/ {\tau}^\prime_s ) \limsup_{N \rightarrow \infty} \left\{ \frac{ Q (\Phi_{s}^{(N)})}{N} \right\} \label{NQ3} \;, \end{eqnarray} where in the second and in the third line the maximization is performed over the density matrix $R^\prime$ of $k\times N$ carriers, $[\Phi_{s}^{(N)}]^{\otimes k}$ are $k$ copies of the map $\Phi_{s}^{(N)}$, and $Q (\Phi_{s}^{(N)})$ is the memoryless quantum capacity~(\ref{quantum11}) of the map $\Phi_{s}^{(N)}$. The second inequality is trivial: it follows from the fact that $ \max_R J (\Phi_{s}^{(N)}, R)/N $ can be obtained from $\max_{R^\prime} J ([\Phi_{s}^{(N)}]^{\otimes k} , R^\prime )/ (k N) $ for $k=1$. The identity on the third line instead is a consequence of the fact that
$\max_{R^\prime} J ([\Phi_{s}^{(N)}]^{\otimes k} , R^\prime )/ (k N) $ achieves its maximum for $k\rightarrow \infty$. We can further simplify the above expression by introducing the time interval $T_s(N-1) =\sum_{j=1}^{N-1} \tau_j$ associated with the first $N-1$ carriers of the sequence $s$ and noticing that \begin{eqnarray} \limsup_{N\rightarrow \infty} \frac{T_s(N-1)}{N} = \tau_s^{\prime\prime} \label{NQ1}\;, \end{eqnarray} with $\tau_s^{\prime\prime}$ defined as in Eq.~(\ref{unosutausecondo}). Using this result, from Eq.~(\ref{NQ3}) we get \begin{eqnarray} r_q(s)&\leqslant & \limsup_{N \rightarrow \infty}
\frac{T_s(N-1)+\max\{\tau_{min},\tau_E\}}{N \tau_s^\prime} \nonumber \\ &&\times \limsup_{N \rightarrow \infty} \frac{ Q (\Phi_{s}^{(N)})}{T_s(N-1)+\max\{\tau_{min},\tau_E\}} \nonumber \\ &\leqslant& \frac{\tau_s^{\prime\prime}}{\tau_s^\prime} \; \sup_{N} \frac{ Q (\Phi_{s}^{(N)})}{T_s(N-1)+\max\{\tau_{min},\tau_E\}} \nonumber \\ &\leqslant& \frac{\tau_s^{\prime\prime}}{\tau_s^\prime} \; \sup_{s \in {\cal S}_0} \frac{ Q ({\cal M}_s)}{T_s+\max\{\tau_{min},\tau_E\}} \;. \label{NQ5} \end{eqnarray} The ratio $\tau_s^{\prime\prime}/\tau_s^\prime$ is always greater than or equal to one. However, if the set $\cal S$ includes only sequences which are regular, than for all $s$ we have $\tau_s^\prime = \tau_s^{\prime\prime}$. In this case the bounds of Eqs.~(\ref{lowerboundmin}) and (\ref{NQ5}) coincides yielding \begin{eqnarray} R_{q}^{(1)}(\tau_{min}) &=&
\max_{s\in{\cal S}_0} \; \frac{Q({\cal M}_s)}{T_s + \max\{ \tau_{min}, \tau_E\}} \label{NLB}\;. \end{eqnarray} The same derivation applies also for the classical rate $R_{c}^{(1)}$. In this case one can show that if $\cal S$ contains only regular sequence then, \begin{eqnarray} R_{c}^{(1)}(\tau_{min}) &=&
\max_{s\in{\cal S}_0} \; \frac{C({\cal M}_s)}{T_s + \max\{ \tau_{min}, \tau_E\}}\label{NLB1}\;. \end{eqnarray}
\subsubsection{Asymptotic limit} It is interesting to note that the above expressions give the correct asymptotic values of Sec.~\ref{s:asymp}. For instance for $\tau_{min}\geqslant \tau_E$ we have ${\cal M}_s = {\cal N}^{\otimes m}$ where $m$ is the number of carriers contained in each group of the sequence and ${\cal N}$ is the memoryless map~(\ref{mappaENNE1}). Given $s\in{\cal S}_0$ this yields \begin{eqnarray} \frac{Q({\cal M}_s)}{T_s + \max\{ \tau_{min}, \tau_E\}} &=& \frac{m Q({\cal N})}{T_s + \tau_{min}} \leqslant \frac{ Q({\cal N})}{\tau_{min}} \end{eqnarray} where we used the additivity property $Q({\cal N}^{\otimes m}) = m Q({\cal N})$ of memoryless channels and the fact that group length~(\ref{imp}) is always greater or equal to $(m-1)\tau_{min}$. Equation~(\ref{formalrate1001}) finally follows by noticing that the rate ${Q({\cal N})}/{\tau_{min}}$ is achieved by the sequence of ${\cal S}_0$ with $\tau_{g,\ell}=\tau_{min}$ for all $g$ and $\ell$.
The limit~(\ref{perfect1}) instead follows by noticing that the rate ${\log_2 D}/{\tau_{min}}$ can be obtained from the set ${\cal S}_0$ by using $\tau_{g,\ell}=\tau_{min}$ for all $\ell =1, \cdots, m-1$ in the limit of large group, i.e. $m\rightarrow \infty$. In this case in fact ${\cal M}_s$ is a tensor product of perfect memory channels and $T_s = (m-1)\tau_{min}$, so that \begin{eqnarray} \frac{Q({\cal M}_s)}{T_s + \max\{ \tau_{min}, \tau_E\}} &=& \frac{m \log_2 D }{(m-1)\tau_{min} + \tau_{E}}\nonumber \\ &\rightarrow& \frac{\log_2 D}{\tau_{min}}\;. \end{eqnarray}
\section{}\label{a:example}
To characterize the modified map of $\overline{\cal N}$ we first solve the system~(\ref{MinputoutputNEW}) by using the following parameterization for the density matrices element of $\sigma_j$ in the canonical basis
$\{ |0\rangle_{LE}, |1\rangle_{LE}\}$, \begin{eqnarray} \sigma_{j} \equiv \left( \begin{array}{cc} 1-{z_j} &{x_j} + i {y_j} \\ {x_j} -i {y_j} & {z_j} \end{array} \right) \;, \label{sigmadec} \end{eqnarray} with ${z_j}\in [0,1]$ and ${x_j}$, ${y_j}$ real for all $j=0,1, \cdots, n$. The resulting recursive equation can be simplified by introducing the column vectors $$\vec{v}_j \equiv (\eta^{-1/4} {z_j}, {x_j})^{T},$$ $$\vec{w}\equiv (1-p)( \eta^{3/4}(1-\lambda) , \eta^{1/4} \sqrt{\lambda (1-\lambda)})^{T}$$ and the $2\times 2$ Hermitian matrix \begin{eqnarray} &&A\equiv (1-p) \left[ \begin{array}{cc} {\eta}( \frac{p}{1-p} -1+2\lambda ) & - 2 \eta^{3/4} \sqrt{\lambda (1-\lambda)}\\ - 2 \eta^{3/4} \sqrt{\lambda (1-\lambda)} &\sqrt{\eta} (\frac{p}{1-p} + 1-2\lambda ) \end{array} \right], \nonumber \end{eqnarray} where $\eta$ stands for $\eta(\tau)$ and
$p$ is the population associated with the $|0\rangle_C$ component of the no-carrying message state $\rho_0$. In this notation Eq.~(\ref{MinputoutputNEW}) gives the following uncoupled equations \begin{eqnarray} y_{j+1} &=& \sqrt{\eta}\; ( 2 p -1 ) \; {y_j} \label{eq2} \\ \vec{v}_{j+1} &=& A \cdot \vec{v}_{j} + \vec{w} \label{eq1} \;, \end{eqnarray}
which can be solved analytically. In particular, imposing the initial condition $\sigma_0 = |0\rangle_{LE}\langle 0|$ (i.e. $x_0=y_0=z_0=0$) the first one gives ${y_j}=0$ for all $j$. The solution of~(\ref{eq1}) instead can be obtained
in terms of the eigenvalues $\lambda_{\pm}$ of $A$ and their corresponding eigenvectors $(\alpha_\pm , \beta_\pm)^T$. Explicitly the eigenvalues of $A$ are \begin{eqnarray} \lambda_\pm &=& \frac{\sqrt{\eta}}{2} [ (1+ \sqrt{\eta}) p \nonumber \\ &&+ (1-p) (1-\sqrt{\eta}) (1-2 \lambda) \pm \Delta ]\;, \end{eqnarray} with \begin{eqnarray} \Delta &=& \{ 4 \sqrt{\eta} (1-2p) \\ &&+ [ (1+ \sqrt{\eta}) p + (1-p) (1-\sqrt{\eta})(1-2\lambda)]^2 \}^{1/2} \nonumber \;. \end{eqnarray} The corresponding eigenvectors $(\alpha_\pm, \beta_\pm)$ have instead the following components \begin{eqnarray} \alpha_\pm &=& \eta^{1/4} (1-p) \sqrt{\lambda (1-\lambda)}/N_\pm\;, \label{eigenv} \\ \beta_\pm &=& [(\sqrt{\eta}-1) p -(1-p) (1-2\lambda)(1+\sqrt{\eta}) \mp \Delta]/N_\pm \;, \nonumber \end{eqnarray} with the normalization coefficient \begin{eqnarray} &N_\pm = \{ 16 (1-p)^2 \sqrt{\eta} \lambda (1-\lambda) & \label{normalizzazione}\\ &+ [ (\sqrt{\eta}-1)p -(1-p) (1-2\lambda) (1+\sqrt{\lambda}) \mp \Delta]^2 \}^{1/2}& . \nonumber \end{eqnarray}
In particular, for $|\lambda_\pm|< 1$ one has~\cite{NOTA11} \begin{eqnarray} \vec{v}_{j} &=& A^j \cdot \vec{v}_{0} + \sum_{k=0}^{j-1} A^k \cdot \vec{w}
= \frac{\openone- A^j}{\openone-A} \cdot \vec{w} \label{AEQ1} \end{eqnarray} and thus \begin{eqnarray} {z_j} &=& \eta^{3/4} (1-p) \;\left[ \eta^{1/4} (1-\lambda) u^{(j)} + \sqrt{\lambda(1-\lambda)} t^{(j)}\right] \nonumber \\ x_j &=& \eta^{1/2} (1-p) \;\left[ \eta^{1/4} (1-\lambda) t^{(j)} + \sqrt{\lambda (1-\lambda) }v^{(j)}\right] \nonumber \;, \end{eqnarray} where $u^{(j)} = \xi^{(j)}_+ \alpha_+^2 + \xi^{(j)}_- \alpha_-^2$, $v^{(j)} = \xi^{(j)}_+ \beta_+^2 + \xi^{(j)}_- \beta_-^2$, and $t^{(j)} = \xi^{(j)}_+ \alpha_+\beta_+ + \xi^{(j)}_- \alpha_-\beta_-$ with \begin{eqnarray} \xi^{(j)}_\pm = \frac{1-(\lambda_\pm)^j}{1-\lambda_\pm} \label{xi}\;. \end{eqnarray} Setting $j=n$ and replacing the above expressions into (\ref{sigmadec}) we obtain the modified state of LE $\sigma_n$ after $n$ successive interactions with $\rho_0$. Using the definition (\ref{MinputoutputNEW1}) one verifies that $\overline{\cal N}$ is a phase damping channel~(\ref{DEFENNE}) characterized by a damping factor \begin{eqnarray} \overline{g}= \sqrt{\lambda} -2 \; ( \sqrt{\lambda} \;z_n - \sqrt{1-\lambda} \; x_n )\;. \label{gmod} \end{eqnarray}
\acknowledgments
I would like to thank Rosario Fazio for remarks and suggestions: without his encouragement this work would never been completed. Moreover I would like to thank Chiara Macchiavello and
Massimo Palma for their comments and discussions. In particular, thank to Massimo for pointing out Refs.~\cite{SCARANI}.
\end{document} | arXiv |
\begin{definition}[Definition:Moving Average Model]
Let $S$ be a stochastic process based on an equispaced time series.
Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t$ be the deviation from a constant mean level $\mu$:
:$\tilde z_t = z_t - \mu$
Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$
Let $M$ be a model where the current value of $\tilde z_t$ is expressed as a finite linear aggregate of the shocks:
:$\tilde z_t = a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$
$M$ is known as a '''moving average (MA) process of order $q$'''.
\end{definition} | ProofWiki |
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Spatiotemporal monitoring of hydrilla [Hydrilla verticillata (L. f.) Royle] to aid management actions
Abhishek Kumar, Christopher Cooper, Caren M. Remillard, Shuvankar Ghosh, Austin Haney, Frank Braun, Zachary Conner, Benjamin Page, Kenneth Boyd, Susan Wilde, Deepak R. Mishra
Journal: Weed Technology / Volume 33 / Issue 3 / June 2019
Hydrilla is an invasive aquatic plant that has rapidly spread through many inland water bodies across the globe by outcompeting native aquatic plants. The negative impacts of hydrilla invasion have become a concern for water resource management authorities, power companies, and environmental scientists. The early detection of hydrilla infestation is very important to reduce the costs associated with control and removal efforts of this invasive species. Therefore, in this study, we aimed to develop a tool for rapid, frequent, and large-scale monitoring and predicting spatial extent of hydrilla habitat. This was achieved by integrating in situ and Landsat 8 Operational Land Imager satellite data for Lake J. Strom Thurmond, the largest US Army Corps of Engineers lake east of the Mississippi River, located on the border of Georgia and South Carolina border. The predictive model for presence of hydrilla incorporated radiometric and physical measurements, including remote-sensing reflectance, Secchi disk depth (SDD), light-attenuation coefficient (Kd), maximum depth of colonization (Zc), and percentage of light available through the water column (PLW). The model-predicted ideal habitat for hydrilla featured high SDD, Zc, and PLW values, low values of Kd. Monthly analyses based on satellite images showed that hydrilla starts growing in April, reaches peak coverage around October, begins retreating in the following months, and disappears in February. Analysis of physical and meteorological factors (i.e., water temperature, surface runoff, net inflow, precipitation) revealed that these parameters are closely associated with hydrilla extent. Management agencies can use these results not only to plan removal efforts but also to evaluate and adapt their current mitigation efforts.
Deformation and failure of the ice bridge on the Wilkins Ice Shelf, Antarctica
A. Humbert, D. Gross, R. Müller, M. Braun, R.S.W. van de Wal, M.R. van den Broeke, D.G. Vaughan, W.J. van de Berg
A narrow bridge of floating ice that connected the Wilkins Ice Shelf, Antarctica, to two confining islands eventually collapsed in early April 2009. In the month preceding the collapse, we observed deformation of the ice bridge by means of satellite imagery and from an in situ GPS station. TerraSAR-X images (acquired in stripmap mode) were used to compile a time series. The ice bridge bent most strongly in its narrowest part (westerly), while the northern end (near Charcot Island) shifted in a northeasterly direction. In the south, the ice bridge experienced compressive strain parallel to its long axis. GPS position data were acquired a little south of the narrowest part of the ice bridge from 19 January 2009. Analysis of these data showed both cyclic and monotonic components of motion. Meteorological data and re-analysis of the output of weather-prediction models indicated that easterly winds were responsible for the cyclic motion component. In particular, wind stress on the rough ice melange that occupied the area to the east exerted significant pressure on the ice bridge. The collapse of the ice bridge began with crack formation in the southern section parallel to the long axis of the ice bridge and led to shattering of the southern part. Ultimately, the narrowest part, only 900 m wide, ruptured. The formation of many small icebergs released energy of >125 ×106 J.
Recent recession of a small plateau ice cap, Ellesmere Island, Canada
Carsten Braun, Douglas R. Hardy, Raymond S. Bradley
Published online by Cambridge University Press: 08 September 2017, p. 154
Using surface velocities to calculate ice thickness and bed topography: a case study at Columbia Glacier, Alaska, USA
R.W. Mcnabb, R. Hock, S. O'Neel, L.A. Rasmussen, Y. Ahn, M. Braun, H. Conway, S. Herreid, I. Joughin, W.T. Pfeffer, B.E. Smith, M. Truffer
Published online by Cambridge University Press: 08 September 2017, pp. 1151-1164
Information about glacier volume and ice thickness distribution is essential for many glaciological applications, but direct measurements of ice thickness can be difficult and costly. We present a new method that calculates ice thickness via an estimate of ice flux. We solve the familiar continuity equation between adjacent flowlines, which decreases the computational time required compared to a solution on the whole grid. We test the method on Columbia Glacier, a large tidewater glacier in Alaska, USA, and compare calculated and measured ice thicknesses, with favorable results. This shows the potential of this method for estimating ice thickness distribution of glaciers for which only surface data are available. We find that both the mean thickness and volume of Columbia Glacier were approximately halved over the period 1957–2007, from 281 m to 143 m, and from 294 km3 to 134 km3, respectively. Using bedrock slope and considering how waves of thickness change propagate through the glacier, we conduct a brief analysis of the instability of Columbia Glacier, which leads us to conclude that the rapid portion of the retreat may be nearing an end.
Radio Synthesis Maps of Large Supernova Remnants
R. Braun, H. van der Laan, R. G. Strom
Journal: Symposium - International Astronomical Union / Volume 101 / 1983
Published online by Cambridge University Press: 04 August 2017, pp. 373-376
Several large (at least 0.°5 diameter) supernova remnants (SNR) located at 2. °5 or more from the galactic plane have been mapped with the Westerbork Synthesis Radio Telescope (WSRT) at 49 cm. The sample, which includes IC443, DA530, VR042.05.01, CTA1 and OA184, is particularly suitable for complementary studies in other spectral regimes. By choosing objects at relatively high galactic latitudes we have consciously selected SNR which are likely to suffer less than average extinction and are probably nearer to the sun than most. This makes them particularly attractive for optical and X-ray studies which, along with IR and further radio observations, are either in progress or being planned. These are summarized in Table 1.
The Kinematics of The SNR G292.0+1.8
R. Braun, W. M. Goss, I. J. Danziger, A. Boksenberg
Optical velocity field mapping of G292.0+1.8 in the [0III] λ5007 å line has been carried out using the IPCS with the 3.6 m ESO telescope at La Silla. Our data are not consistent with the suggestion that the [0III] emitting material in the western portion of this remnant is concentrated in an expanding ring. The existing data on G292.0+1.8 suggests that only the brightest portion of a thick shell of ejecta with high velocity spurs is observed. The expansion centroid, size, velocity and age of this SNR are derived.
The Gem OB1/IC443/S249 Complex: A Case History of Stellar Evolution
R. Braun, R. G. Strom
Published online by Cambridge University Press: 04 August 2017, p. 442
The extended cloud complex containing members of the Gem OB1 association, the supernova remnant IC443, and the H II region S249 has been studied with IRAS observations at 12,25,60 and 100 microns and WSRT observations at 327 and 1400 MHz and in the 21-cm H I line. A skeleton-like framework of cool dust delineates the boundaries of the region, and physical parameters have been derived for the entire complex, individual H II regions and the shocked and recombined gas within IC443 using the radio and infrared data. IC443 is shown to consist of three interconnected, roughly spherical subshells of vastly different radii and centroids. The geometry is fully constrained by the structural and kinematic data. Two of the subshells together define the usually assumed boundaries of IC443, while the third includes the optical filaments which extend beyond the northeastern rim and which are shown to have well-correlated nonthermal radio components. The available evidence implies that the SNR shock has encountered a pre-existing high density shell. It is shown that the system of subshells is fully consistent with formation by stellar wind driven bubbles generated by association members within the inhomogeneous environment of the complex.
Comparison of Radio and X-ray Observations of SNR G109.1–1.0
P. C. Gregory, R. Braun, G. G. Fahlman, S. F. Gull
In this paper, we present a comparison of the radio and X-ray morphology of the supernova remnant G109.1–1.0, based on recent radio observations at 6 and 20 cm and investigate the relationship of the SNR to a neighbouring molecular cloud.
Surface velocity and ice discharge of the ice cap on King George Island, Antarctica
B. Osmanoğlu, M. Braun, R. Hock, F.J. Navarro
Glaciers on King George Island, Antarctica, have shown retreat and surface lowering in recent decades, concurrent with increasing air temperatures. A large portion of the glacier perimeter is ocean-terminating, suggesting possible large mass losses due to calving and submarine melting. Here we estimate the ice discharge into the ocean for the King George Island ice cap. L-band synthetic aperture radar images covering the time-span January 2008 to January 2011 over King George Island are processed using an intensity-tracking algorithm to obtain surface velocity measurements. Pixel offsets from 40 pairs of radar images are analysed and inverted to estimate a weighted average surface velocity field. Ice thicknesses are derived from simple principles of ice flow mechanics using the computed surface velocity fields and in situ thickness data. The maximum ice surface speeds reach >225 m a-1, and the total ice discharge for the analysed flux gates of King George Island is estimated to be 0.720 ± 0.428 Gt a−1, corresponding to a specific mass loss of 0.64 ± 0.38 m w.e. a-1 over the area of the entire ice cap (1127 km2).
Historically unprecedented global glacier decline in the early 21st century
Michael Zemp, Holger Frey, Isabelle Gärtner-Roer, Samuel U. Nussbaumer, Martin Hoelzle, Frank Paul, Wilfried Haeberli, Florian Denzinger, Andreas P. Ahlstrøm, Brian Anderson, Samjwal Bajracharya, Carlo Baroni, Ludwig N. Braun, Bolívar E. Cáceres, Gino Casassa, Guillermo Cobos, Luzmila R. Dávila, Hugo Delgado Granados, Michael N. Demuth, Lydia Espizua, Andrea Fischer, Koji Fujita, Bogdan Gadek, Ali Ghazanfar, Jon Ove Hagen, Per Holmlund, Neamat Karimi, Zhongqin Li, Mauri Pelto, Pierre Pitte, Victor V. Popovnin, Cesar A. Portocarrero, Rainer Prinz, Chandrashekhar V. Sangewar, Igor Severskiy, Oddur Sigurđsson, Alvaro Soruco, Ryskul Usubaliev, Christian Vincent
Observations show that glaciers around the world are in retreat and losing mass. Internationally coordinated for over a century, glacier monitoring activities provide an unprecedented dataset of glacier observations from ground, air and space. Glacier studies generally select specific parts of these datasets to obtain optimal assessments of the mass-balance data relating to the impact that glaciers exercise on global sea-level fluctuations or on regional runoff. In this study we provide an overview and analysis of the main observational datasets compiled by the World Glacier Monitoring Service (WGMS). The dataset on glacier front variations (∼42 000 since 1600) delivers clear evidence that centennial glacier retreat is a global phenomenon. Intermittent readvance periods at regional and decadal scale are normally restricted to a subsample of glaciers and have not come close to achieving the maximum positions of the Little Ice Age (or Holocene). Glaciological and geodetic observations (∼5200 since 1850) show that the rates of early 21st-century mass loss are without precedent on a global scale, at least for the time period observed and probably also for recorded history, as indicated also in reconstructions from written and illustrated documents. This strong imbalance implies that glaciers in many regions will very likely suffer further ice loss, even if climate remains stable.
Thermoelectronic energy conversion: Concepts and materials
R. Wanke, W. Voesch, I. Rastegar, A. Kyriazis, W. Braun, J. Mannhart
Journal: MRS Bulletin / Volume 42 / Issue 7 / July 2017
Thermoelectronic energy conversion can potentially provide an exceptionally efficient way to convert heat into electric power. Key components of such converters are materials with designed, small work functions. We present the principles of thermoelectronic energy conversion and discuss the advantages and challenges of the conversion process, as well the state of the art of the respective research.
Developing one-dimensional implosions for inertial confinement fusion science
HEDP and HPL 2016
J. L. Kline, S. A. Yi, A. N. Simakov, R. E. Olson, D. C. Wilson, G. A. Kyrala, T. S. Perry, S. H. Batha, E. L. Dewald, J. E. Ralph, D. J. Strozzi, A. G. MacPhee, D. A. Callahan, D. Hinkel, O. A. Hurricane, R. J. Leeper, A. B. Zylstra, R. R. Peterson, B. M. Haines, L. Yin, P. A. Bradley, R. C. Shah, T. Braun, J. Biener, B. J. Kozioziemski, J. D. Sater, M. M. Biener, A. V. Hamza, A. Nikroo, L. F. Berzak Hopkins, D. Ho, S. LePape, N. B. Meezan, D. S. Montgomery, W. S. Daughton, E. C. Merritt, T. Cardenas, E. S. Dodd
Journal: High Power Laser Science and Engineering / Volume 4 / 2016
Published online by Cambridge University Press: 12 December 2016, e44
Experiments on the National Ignition Facility show that multi-dimensional effects currently dominate the implosion performance. Low mode implosion symmetry and hydrodynamic instabilities seeded by capsule mounting features appear to be two key limiting factors for implosion performance. One reason these factors have a large impact on the performance of inertial confinement fusion implosions is the high convergence required to achieve high fusion gains. To tackle these problems, a predictable implosion platform is needed meaning experiments must trade-off high gain for performance. LANL has adopted three main approaches to develop a one-dimensional (1D) implosion platform where 1D means measured yield over the 1D clean calculation. A high adiabat, low convergence platform is being developed using beryllium capsules enabling larger case-to-capsule ratios to improve symmetry. The second approach is liquid fuel layers using wetted foam targets. With liquid fuel layers, the implosion convergence can be controlled via the initial vapor pressure set by the target fielding temperature. The last method is double shell targets. For double shells, the smaller inner shell houses the DT fuel and the convergence of this cavity is relatively small compared to hot spot ignition. However, double shell targets have a different set of trade-off versus advantages. Details for each of these approaches are described.
EXISTENCE OF $q$ -ANALOGS OF STEINER SYSTEMS
MSC 2010: Finite geometry and special incidence structures
MSC 2010: Algebraic combinatorics
MICHAEL BRAUN, TUVI ETZION, PATRIC R. J. ÖSTERGÅRD, ALEXANDER VARDY, ALFRED WASSERMANN
Journal: Forum of Mathematics, Pi / Volume 4 / 2016
Published online by Cambridge University Press: 30 August 2016, e7
Let $\mathbb{F}_{q}^{n}$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_{q}$ . A $q$ -analog of a Steiner system (also known as a $q$ -Steiner system), denoted ${\mathcal{S}}_{q}(t,\!k,\!n)$ , is a set ${\mathcal{S}}$ of $k$ -dimensional subspaces of $\mathbb{F}_{q}^{n}$ such that each $t$ -dimensional subspace of $\mathbb{F}_{q}^{n}$ is contained in exactly one element of ${\mathcal{S}}$ . Presently, $q$ -Steiner systems are known only for $t\,=\,1\!$ , and in the trivial cases $t\,=\,k$ and $k\,=\,n$ . In this paper, the first nontrivial $q$ -Steiner systems with $t\,\geqslant \,2$ are constructed. Specifically, several nonisomorphic $q$ -Steiner systems ${\mathcal{S}}_{2}(2,3,13)$ are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of $\text{GL}(13,2)$ . This approach leads to an instance of the exact cover problem, which turns out to have many solutions.
Morphological Characteristics of Compact High-Velocity Clouds Revealed by High-Resolution WSRT Imaging
W. B. Burton, R. Braun
Journal: International Astronomical Union Colloquium / Volume 174 / 2000
A class of compact, isolated high–velocity clouds which plausibly represents a homogeneous subsample of the HVC phenomenon in a single physical state was objectively identified by Braun and Burton (1999). Six examples of the CHVCs, unresolved in single–dish data, have been imaged with the Westerbork Synthesis Radio Telescope. The high–resolution imaging reveals the morphology of these objects, including a core/halo distribution of fluxes, signatures of rotation indicating dark matter, and narrow linewidths constraining the kinetic temperature of several opaque cores. In these regards, as well as in their kinematic and spatial deployment on the sky, the CHVC objects are evidently a dynamically cold ensemble of dark–matter–dominated H ɪ clouds accreting onto the Local Group in a continuing process of galactic evolution.
Prehospital Indicators for Disaster Preparedness and Response: New York City Emergency Medical Services in Hurricane Sandy
Silas W. Smith, James Braun, Ian Portelli, Sidrah Malik, Glenn Asaeda, Elizabeth Lancet, Binhuan Wang, Ming Hu, David C. Lee, David J. Prezant, Lewis R. Goldfrank
Journal: Disaster Medicine and Public Health Preparedness / Volume 10 / Issue 3 / June 2016
We aimed to evaluate emergency medical services (EMS) data as disaster metrics and to assess stress in surrounding hospitals and a municipal network after the closure of Bellevue Hospital during Hurricane Sandy in 2012.
We retrospectively reviewed EMS activity and call types within New York City's 911 computer-assisted dispatch database from January 1, 2011, to December 31, 2013. We evaluated EMS ambulance transports to individual hospitals during Bellevue's closure and incremental recovery from urgent care capacity, to freestanding emergency department (ED) capability, freestanding ED with 911-receiving designation, and return of inpatient services.
A total of 2,877,087 patient transports were available for analysis; a total of 707,593 involved Manhattan hospitals. The 911 ambulance transports disproportionately increased at the 3 closest hospitals by 63.6%, 60.7%, and 37.2%. When Bellevue closed, transports to specific hospitals increased by 45% or more for the following call types: blunt traumatic injury, drugs and alcohol, cardiac conditions, difficulty breathing, "pedestrian struck," unconsciousness, altered mental status, and emotionally disturbed persons.
EMS data identified hospitals with disproportionately increased patient loads after Hurricane Sandy. Loss of Bellevue, a public, safety net medical center, produced statistically significant increases in specific types of medical and trauma transports at surrounding hospitals. Focused redeployment of human, economic, and social capital across hospital systems may be required to expedite regional health care systems recovery. (Disaster Med Public Health Preparedness. 2016;10:333–343)
Multi-Spectral Studies of the Nearby Dwarf Galaxies UGCA 86 and LMC/SMC
G.M. Richter, M. Braun, R. Assendorp
Journal: Highlights of Astronomy / Volume 11 / Issue 1 / 1998
UGCA 86 is an irregular dwarf galaxy in the 1C 342 / Maffei I group, just next to the Local Group. It was first mentioned by Zwicky (1968) as VII Zw 009, but not contained in his "Catalogue of Selected Compact Galaxies and of Post-eruptive Galaxies" (1971). It was independently rediscovered by Nilson (1974) and Rots (1979) as UGCA 86 and A 0355 resp. Rots found it by HI observations, and from peculiarities in the HI morphology and kinematics he suspected that it was interacting with 1C 342. Thus, the tentatively interesting items: a starforming, low surface brightness dwarf galaxy in an interacting system (one of the nearest), triggered us to engage in more detailed studies.
In a first step, we made detailed surface photometry in U, B and V (Richter et al. 1991). UGCA 86 proved to be one galaxy (which was not trivial; Saha & Hoessel 1991 discussed if it could be a chance superposition or a collision of two independent galaxies, due to the very different appearence of the southern and the central starburst; Miller & Hodge 1992 and the distance measurements of Karachentsev & Tikhonov 1993 support our result) with the typical exponential brightness profile of a spheroidal dwarf galaxy, and contains at least two starburst regions of very different color: a central red one and a blue one in the southern outskirts. There is an infrared source in the IRAS Point Source Catalogue coincident with UGCA 86. The amount of dust indicated by this source is in very good quantitative agreement with what would be required to redden the central starburst by the observed color difference compared to the southern burst. Nevertheless, the straightforward hypothesis, that the color of the central burst is due to dust extinction, is contradicted by the improved, higher resolution data.
7 - High-content screening in infectious diseases: new drugs against bugs
By André P. Mäurer, Max Planck-Institüt für Infektionsbiologie, Peter R. Braun, Center for Systems Biomedicine, Kate Holden-Dye, Center for Systems Biomedicine, Thomas F. Meyer, Max Planck-Institüt für Infektionsbiologie
Edited by Florian Markowetz, Michael Boutros
Book: Systems Genetics
Print publication: 02 July 2015, pp 108-138
Despite immense achievements in the past century in hygiene control, and the development of vaccines and antibiotics, infectious diseases continue to pose a tremendous threat to public health globally. There are still devastating infections for which there are no effective vaccines or antimicrobial therapies. Moreover, the problem of drug resistance in bacteria and viral populations and the increasing appreciation that pathologies resulting from infections are responsible for a number of chronic conditions, are creating an ever-growing need for novel preventive and therapeutic approaches. In line with this, a new host-targeted approach has been suggested for antimicrobial drug research that exploits the central role of the host cell during infection. Decades of research have taught us that infections are supported by host cell functions, and that infection pathology is frequently host dependent. Accordingly, the pharmacological targeting of host cell factors promises novel opportunities to prevent and treat infectious disease. Such an approach may be anticipated to expand the number of druggable targets, produce broad-spectrum compounds and impede the generation of resistance. The discovery of RNA interference (RNAi) has created opportunities to explore gene functions in cellular systems in a targeted manner. RNAi loss-of-function approaches have proved invaluable for the identification of host proteins important for pathogen viability. These approaches can be applied on a high-throughput scale, which demands sophisticated liquid handling and high-content image analysis. Here, we provide an overview of the current status of high-content screening (HCS) in loss-of-function analyses in infectious disease research and discuss how these powerful techniques can be applied to identify host factors with previously unknown roles in infection and its pathology.
The challenge of fighting infectious diseases
Infections by pathogenic species of bacteria, viruses, fungi and protozoa have had considerable impact on mankind throughout history. Advances in our understanding of the importance of hygiene control, and later, improvements in diagnostics and the development and successful employment of vaccines and antimicrobial drugs, have substantially benefited human health, and provided social and economic benefits.
Lessons Learned From Hospital Ebola Preparation
Daniel J. Morgan, Barbara Braun, Aaron M. Milstone, Deverick Anderson, Ebbing Lautenbach, Nasia Safdar, Marci Drees, Jennifer Meddings, Darren R. Linkin, Lindsay D. Croft, Lisa Pineles, Daniel J. Diekema, Anthony D. Harris
Hospital Ebola preparation is underway in the United States and other countries; however, the best approach and resources involved are unknown.
To examine costs and challenges associated with hospital Ebola preparation by means of a survey of Society for Healthcare Epidemiology of America (SHEA) members.
Electronic survey of infection prevention experts.
A total of 257 members completed the survey (221 US, 36 international) representing institutions in 41 US states, the District of Columbia, and 18 countries. The 221 US respondents represented 158 (43.1%) of 367 major medical centers that have SHEA members and included 21 (60%) of 35 institutions recently defined by the US Centers for Disease Control and Prevention as Ebola virus disease treatment centers. From October 13 through October 19, 2014, Ebola consumed 80% of hospital epidemiology time and only 30% of routine infection prevention activities were completed. Routine care was delayed in 27% of hospitals evaluating patients for Ebola.
Convenience sample of SHEA members with a moderate response rate.
Hospital Ebola preparations required extraordinary resources, which were diverted from routine infection prevention activities. Patients being evaluated for Ebola faced delays and potential limitations in management of other diseases that are more common in travelers returning from West Africa.
Infect Control Hosp Epidemiol 2015;00(0): 1–5
The Australian Square Kilometre Array Pathfinder: System Architecture and Specifications of the Boolardy Engineering Test Array
A. W. Hotan, J. D. Bunton, L. Harvey-Smith, B. Humphreys, B. D. Jeffs, T. Shimwell, J. Tuthill, M. Voronkov, G. Allen, S. Amy, K. Ardern, P. Axtens, L. Ball, K. Bannister, S. Barker, T. Bateman, R. Beresford, D. Bock, R. Bolton, M. Bowen, B. Boyle, R. Braun, S. Broadhurst, D. Brodrick, K. Brooks, M. Brothers, A. Brown, C. Cantrall, G. Carrad, J. Chapman, W. Cheng, A. Chippendale, Y. Chung, F. Cooray, T. Cornwell, E. Davis, L. de Souza, D. DeBoer, P. Diamond, P. Edwards, R. Ekers, I. Feain, D. Ferris, R. Forsyth, R. Gough, A. Grancea, N. Gupta, J. C. Guzman, G. Hampson, C. Haskins, S. Hay, D. Hayman, S. Hoyle, C. Jacka, C. Jackson, S. Jackson, K. Jeganathan, S. Johnston, J. Joseph, R. Kendall, M. Kesteven, D. Kiraly, B. Koribalski, M. Leach, E. Lenc, E. Lensson, L. Li, S. Mackay, A. Macleod, T. Maher, M. Marquarding, N. McClure-Griffiths, D. McConnell, S. Mickle, P. Mirtschin, R. Norris, S. Neuhold, A. Ng, J. O'Sullivan, J. Pathikulangara, S. Pearce, C. Phillips, R. Y. Qiao, J. E. Reynolds, A. Rispler, P. Roberts, D. Roxby, A. Schinckel, R. Shaw, M. Shields, M. Storey, T. Sweetnam, E. Troup, B. Turner, A. Tzioumis, T. Westmeier, M. Whiting, C. Wilson, T. Wilson, K. Wormnes, X. Wu
Journal: Publications of the Astronomical Society of Australia / Volume 31 / 2014
Published online by Cambridge University Press: 13 November 2014, e041
This paper describes the system architecture of a newly constructed radio telescope – the Boolardy engineering test array, which is a prototype of the Australian square kilometre array pathfinder telescope. Phased array feed technology is used to form multiple simultaneous beams per antenna, providing astronomers with unprecedented survey speed. The test array described here is a six-antenna interferometer, fitted with prototype signal processing hardware capable of forming at least nine dual-polarisation beams simultaneously, allowing several square degrees to be imaged in a single pointed observation. The main purpose of the test array is to develop beamforming and wide-field calibration methods for use with the full telescope, but it will also be capable of limited early science demonstrations.
Of hunters and handles: insights from palaeoanthropology - Lawrence Barham. From hand to handle: the first industrial revolution. xi+357 pages, 53 b&w illustrations, 5 tables. 2013. Oxford: Oxford University Press; 978-0-19-960471-5 hardback £75. - Travis Rayne Pickering. Rough and tumble: aggression, hunting, and human evolution. xiii+208 pages, 12 b&w illustrations. 2013. Berkeley: University of California Press; 978-0-520-27400-6 hardback £19.95.
David R. Braun
Journal: Antiquity / Volume 88 / Issue 341 / 1 September 2014
Print publication: 1 September 2014 | CommonCrawl |
Busemann G-space
In mathematics, a Busemann G-space is a type of metric space first described by Herbert Busemann in 1942.
If $(X,d)$ is a metric space such that
1. for every two distinct $x,y\in X$ there exists $z\in X-\{x,y\}$ such that $d(x,z)+d(y,z)=d(x,z)$ (Menger convexity)
2. every $d$-bounded set of infinite cardinality possesses accumulation points
3. for every $w\in X$ there exists $\rho _{w}$ such that for any distinct points $x,y\in B(w,\rho _{w})$ there exists $z\in (b(w,\rho _{w})-\{x,y\})^{\circ }$ such that $d(x,z)+d(y,z)=d(x,z)$ (geodesics are locally extendable)
4. for any distinct points $x,y\in X$, if $u,v\in X$ such that $d(x,u)+d(y,u)=d(x,u)$, $d(x,v)+d(y,v)=d(x,v)$ and $d(y,u)=d(y,v)$ (geodesic extensions are unique).
then X is said to be a Busemann G-space. Every Busemann G-space is a homogenous space.
The Busemann conjecture states that every Busemann G-space is a topological manifold. It is a special case of the Bing–Borsuk conjecture. The Busemann conjecture is known to be true for dimensions 1 to 4.[1][2]
References
1. M., Halverson, Denise; Dušan, Repovš (23 December 2008). "The Bing–Borsuk and the Busemann conjectures". Mathematical Communications. 13 (2). arXiv:0811.0886. ISSN 1331-0623.{{cite journal}}: CS1 maint: multiple names: authors list (link)
2. Papadopoulos, Athanase (2005). Metric Spaces, Convexity and Nonpositive Curvature. European Mathematical Society. p. 77. ISBN 9783037190104.
| Wikipedia |
Hypergraph
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.
Formally, a directed hypergraph is a pair $(X,E)$, where $X$ is a set of elements called nodes, vertices, points, or elements and $E$ is a set of pairs of subsets of $X$. Each of these pairs $(D,C)\in E$ is called an edge or hyperedge; the vertex subset $D$ is known as its tail or domain, and $C$ as its head or codomain.
The order of a hypergraph $(X,E)$ is the number of vertices in $X$. The size of the hypergraph is the number of edges in $E$. The order of an edge $e=(D,C)$ in a directed hypergraph is $|e|=(|D|,|C|)$: that is, the number of vertices in its tail followed by the number of vertices in its head.
The definition above generalizes from a directed graph to a directed hypergraph by defining the head or tail of each edge as a set of vertices ($C\subseteq X$ or $D\subseteq X$) rather than as a single vertex. A graph is then the special case where each of these sets contains only one element. Hence any standard graph theoretic concept that is independent of the edge orders $|e|$ will generalize to hypergraph theory.
Under one definition, an undirected hypergraph $(X,E)$ is a directed hypergraph which has a symmetric edge set: If $(D,C)\in E$ then $(C,D)\in E$. For notational simplicity one can remove the "duplicate" hyperedges since the modifier "undirected" is precisely informing us that they exist: If $(D,C)\in E$ then $(C,D){\vec {\in }}E$ where ${\vec {\in }}$ means implicitly in.
While graph edges connect only 2 nodes, hyperedges connect an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a k-uniform hypergraph is a hypergraph such that all its hyperedges have size k. (In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. An undirected hypergraph is also called a set system or a family of sets drawn from the universal set.
Hypergraphs can be viewed as incidence structures. In particular, there is a bipartite "incidence graph" or "Levi graph" corresponding to every hypergraph, and conversely, every bipartite graph can be regarded as the incidence graph of a hypergraph when it is 2-colored and it is indicated which color class corresponds to hypergraph vertices and which to hypergraph edges.
Hypergraphs have many other names. In computational geometry, an undirected hypergraph may sometimes be called a range space and then the hyperedges are called ranges.[2] In cooperative game theory, hypergraphs are called simple games (voting games); this notion is applied to solve problems in social choice theory. In some literature edges are referred to as hyperlinks or connectors.[3]
The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms.
Applications
Undirected hypergraphs are useful in modelling such things as satisfiability problems,[4] databases,[5] machine learning,[6] and Steiner tree problems.[7] They have been extensively used in machine learning tasks as the data model and classifier regularization (mathematics).[8] The applications include recommender system (communities as hyperedges),[9] [10] image retrieval (correlations as hyperedges),[11] and bioinformatics (biochemical interactions as hyperedges).[12] Representative hypergraph learning techniques include hypergraph spectral clustering that extends the spectral graph theory with hypergraph Laplacian,[13] and hypergraph semi-supervised learning that introduces extra hypergraph structural cost to restrict the learning results.[14] For large scale hypergraphs, a distributed framework[6] built using Apache Spark is also available.
Directed hypergraphs can be used to model things including telephony applications,[15] detecting money laundering,[16] operations research,[17] and transportation planning. They can also be used to model Horn-satisfiability.[18]
Generalizations of concepts from graphs
Many theorems and concepts involving graphs also hold for hypergraphs, in particular:
• Matching in hypergraphs;
• Vertex cover in hypergraphs (also known as: transversal);
• Line graph of a hypergraph;
• Hypergraph grammar - created by augmenting a class of hypergraphs with a set of replacement rules;
• Ramsey's theorem;
• Erdős–Ko–Rado theorem;
• Kruskal–Katona theorem on uniform hypergraphs;
• Hall-type theorems for hypergraphs.
In directed hypergraphs: transitive closure, and shortest path problems.[17]
Hypergraph drawing
Although hypergraphs are more difficult to draw on paper than graphs, several researchers have studied methods for the visualization of hypergraphs.
In one possible visual representation for hypergraphs, similar to the standard graph drawing style in which curves in the plane are used to depict graph edges, a hypergraph's vertices are depicted as points, disks, or boxes, and its hyperedges are depicted as trees that have the vertices as their leaves.[19][20] If the vertices are represented as points, the hyperedges may also be shown as smooth curves that connect sets of points, or as simple closed curves that enclose sets of points.[21][22][23]
In another style of hypergraph visualization, the subdivision model of hypergraph drawing,[24] the plane is subdivided into regions, each of which represents a single vertex of the hypergraph. The hyperedges of the hypergraph are represented by contiguous subsets of these regions, which may be indicated by coloring, by drawing outlines around them, or both. An order-n Venn diagram, for instance, may be viewed as a subdivision drawing of a hypergraph with n hyperedges (the curves defining the diagram) and 2n − 1 vertices (represented by the regions into which these curves subdivide the plane). In contrast with the polynomial-time recognition of planar graphs, it is NP-complete to determine whether a hypergraph has a planar subdivision drawing,[25] but the existence of a drawing of this type may be tested efficiently when the adjacency pattern of the regions is constrained to be a path, cycle, or tree.[26]
An alternative representation of the hypergraph called PAOH[1] is shown in the figure on top of this article. Edges are vertical lines connecting vertices. Vertices are aligned on the left. The legend on the right shows the names of the edges. It has been designed for dynamic hypergraphs but can be used for simple hypergraphs as well.
Hypergraph coloring
Classic hypergraph coloring is assigning one of the colors from set $\{1,2,3,...,\lambda \}$ to every vertex of a hypergraph in such a way that each hyperedge contains at least two vertices of distinct colors. In other words, there must be no monochromatic hyperedge with cardinality at least 2. In this sense it is a direct generalization of graph coloring. Minimum number of used distinct colors over all colorings is called the chromatic number of a hypergraph.
Hypergraphs for which there exists a coloring using up to k colors are referred to as k-colorable. The 2-colorable hypergraphs are exactly the bipartite ones.
There are many generalizations of classic hypergraph coloring. One of them is the so-called mixed hypergraph coloring, when monochromatic edges are allowed. Some mixed hypergraphs are uncolorable for any number of colors. A general criterion for uncolorability is unknown. When a mixed hypergraph is colorable, then the minimum and maximum number of used colors are called the lower and upper chromatic numbers respectively.[27]
Properties of hypergraphs
A hypergraph can have various properties, such as:
• Empty - has no edges.
• Non-simple (or multiple) - has loops (hyperedges with a single vertex) or repeated edges, which means there can be two or more edges containing the same set of vertices.
• Simple - has no loops and no repeated edges.
• $d$-regular - every vertex has degree $d$, i.e., contained in exactly $d$ hyperedges.
• 2-colorable - its vertices can be partitioned into two classes U and V in such a way that each hyperedge with cardinality at least 2 contains at least one vertex from both classes. An alternative term is Property B.
• Two stronger properties are bipartite and balanced.
• $k$-uniform - each hyperedge contains precisely $k$ vertices.
• $k$-partite - the vertices are partitioned into $k$ parts, and each hyperedge contains precisely one vertex of each type.
• Every $k$-partite hypergraph (for $k\geq 2$) is both $k$-uniform and bipartite (and 2-colorable).
• Downward-closed - every subset of an undirected hypergraph's edges is a hyperedge too. A downward-closed hypergraph is usually called an abstract simplicial complex.
• An abstract simplicial complex with the augmentation property is called a matroid.
Related hypergraphs
Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called subhypergraphs, partial hypergraphs and section hypergraphs.
Let $H=(X,E)$ be the hypergraph consisting of vertices
$X=\lbrace x_{i}\mid i\in I_{v}\rbrace ,$
and having edge set
$E=\lbrace e_{i}\mid i\in I_{e},e_{i}\subseteq X,e_{i}\neq \emptyset \rbrace ,$
where $I_{v}$ and $I_{e}$ are the index sets of the vertices and edges respectively.
A subhypergraph is a hypergraph with some vertices removed. Formally, the subhypergraph $H_{A}$ induced by $A\subseteq X$ is defined as
$H_{A}=\left(A,\lbrace e\cap A\mid e\in E,e\cap A\neq \emptyset \rbrace \right).$
An alternative term is the restriction of H to A.[28]: 468
An extension of a subhypergraph is a hypergraph where each hyperedge of $H$ which is partially contained in the subhypergraph $H_{A}$ is fully contained in the extension $Ex(H_{A})$. Formally
$Ex(H_{A})=(A\cup A',E')$ with $A'=\bigcup _{e\in E}e\setminus A$ and $E'=\lbrace e\in E\mid e\subseteq (A\cup A')\rbrace $.
The partial hypergraph is a hypergraph with some edges removed.[28]: 468 Given a subset $J\subset I_{e}$ of the edge index set, the partial hypergraph generated by $J$ is the hypergraph
$\left(X,\lbrace e_{i}\mid i\in J\rbrace \right).$
Given a subset $A\subseteq X$, the section hypergraph is the partial hypergraph
$H\times A=\left(A,\lbrace e_{i}\mid i\in I_{e},e_{i}\subseteq A\rbrace \right).$
The dual $H^{*}$ of $H$ is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by $\lbrace e_{i}\rbrace $ and whose edges are given by $\lbrace X_{m}\rbrace $ where
$X_{m}=\lbrace e_{i}\mid x_{m}\in e_{i}\rbrace .$
When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an involution, i.e.,
$\left(H^{*}\right)^{*}=H.$
A connected graph G with the same vertex set as a connected hypergraph H is a host graph for H if every hyperedge of H induces a connected subgraph in G. For a disconnected hypergraph H, G is a host graph if there is a bijection between the connected components of G and of H, such that each connected component G' of G is a host of the corresponding H'.
The 2-section (or clique graph, representing graph, primal graph, Gaifman graph) of a hypergraph is the graph with the same vertices of the hypergraph, and edges between all pairs of vertices contained in the same hyperedge.
Incidence matrix
Let $V=\{v_{1},v_{2},~\ldots ,~v_{n}\}$ and $E=\{e_{1},e_{2},~\ldots ~e_{m}\}$. Every hypergraph has an $n\times m$ incidence matrix.
For an undirected hypergraph, $I=(b_{ij})$ where
$b_{ij}=\left\{{\begin{matrix}1&\mathrm {if} ~v_{i}\in e_{j}\\0&\mathrm {otherwise} .\end{matrix}}\right.$
The transpose $I^{t}$ of the incidence matrix defines a hypergraph $H^{*}=(V^{*},\ E^{*})$ called the dual of $H$, where $V^{*}$ is an m-element set and $E^{*}$ is an n-element set of subsets of $V^{*}$. For $v_{j}^{*}\in V^{*}$ and $e_{i}^{*}\in E^{*},~v_{j}^{*}\in e_{i}^{*}$ if and only if $b_{ij}=1$.
For a directed hypergraph, the heads and tails of each hyperedge $e_{j}$ are denoted by $H(e_{j})$ and $T(e_{j})$ respectively.[18] $I=(b_{ij})$ where
$b_{ij}=\left\{{\begin{matrix}-1&\mathrm {if} ~v_{i}\in T(e_{j})\\1&\mathrm {if} ~v_{i}\in H(e_{j})\\0&\mathrm {otherwise} .\end{matrix}}\right.$
Incidence graph
A hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the parts of BG, and (x1, e1) are connected with an edge if and only if vertex x1 is contained in edge e1 in H.
Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above. This bipartite graph is also called incidence graph.
Adjacency matrix
A parallel for the adjacency matrix of a hypergraph can be drawn from the adjacency matrix of a graph. In the case of a graph, the adjacency matrix is a square matrix which indicates whether pairs of vertices are adjacent. Likewise, we can define the adjacency matrix $A=(a_{ij})$ for a hypergraph in general where the hyperedges $e_{k\leq m}$have real weights $w_{e_{k}}\in \mathbb {R} $ with
$a_{ij}=\left\{{\begin{matrix}w_{e_{k}}&\mathrm {if} ~(v_{i},v_{j})\in E\\0&\mathrm {otherwise} .\end{matrix}}\right.$
Cycles
In contrast with ordinary undirected graphs for which there is a single natural notion of cycles and acyclic graphs, there are multiple natural non-equivalent definitions of acyclicity for hypergraphs which collapse to ordinary graph acyclicity for the special case of ordinary graphs.
A first definition of acyclicity for hypergraphs was given by Claude Berge:[29] a hypergraph is Berge-acyclic if its incidence graph (the bipartite graph defined above) is acyclic. This definition is very restrictive: for instance, if a hypergraph has some pair $v\neq v'$ of vertices and some pair $f\neq f'$ of hyperedges such that $v,v'\in f$ and $v,v'\in f'$, then it is Berge-cyclic. Berge-cyclicity can obviously be tested in linear time by an exploration of the incidence graph.
We can define a weaker notion of hypergraph acyclicity,[5] later termed α-acyclicity. This notion of acyclicity is equivalent to the hypergraph being conformal (every clique of the primal graph is covered by some hyperedge) and its primal graph being chordal; it is also equivalent to reducibility to the empty graph through the GYO algorithm[30][31] (also known as Graham's algorithm), a confluent iterative process which removes hyperedges using a generalized definition of ears. In the domain of database theory, it is known that a database schema enjoys certain desirable properties if its underlying hypergraph is α-acyclic.[32] Besides, α-acyclicity is also related to the expressiveness of the guarded fragment of first-order logic.
We can test in linear time if a hypergraph is α-acyclic.[33]
Note that α-acyclicity has the counter-intuitive property that adding hyperedges to an α-cyclic hypergraph may make it α-acyclic (for instance, adding a hyperedge containing all vertices of the hypergraph will always make it α-acyclic). Motivated in part by this perceived shortcoming, Ronald Fagin[34] defined the stronger notions of β-acyclicity and γ-acyclicity. We can state β-acyclicity as the requirement that all subhypergraphs of the hypergraph are α-acyclic, which is equivalent[34] to an earlier definition by Graham.[31] The notion of γ-acyclicity is a more restrictive condition which is equivalent to several desirable properties of database schemas and is related to Bachman diagrams. Both β-acyclicity and γ-acyclicity can be tested in polynomial time.
Those four notions of acyclicity are comparable: Berge-acyclicity implies γ-acyclicity which implies β-acyclicity which implies α-acyclicity. However, none of the reverse implications hold, so those four notions are different.[34]
Isomorphism, symmetry, and equality
A hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge.
A hypergraph $H=(X,E)$ is isomorphic to a hypergraph $G=(Y,F)$, written as $H\simeq G$ if there exists a bijection
$\phi :X\to Y$
and a permutation $\pi $ of $I$ such that
$\phi (e_{i})=f_{\pi (i)}$
The bijection $\phi $ is then called the isomorphism of the graphs. Note that
$H\simeq G$ if and only if $H^{*}\simeq G^{*}$.
When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism. One says that $H$ is strongly isomorphic to $G$ if the permutation is the identity. One then writes $H\cong G$. Note that all strongly isomorphic graphs are isomorphic, but not vice versa.
When the vertices of a hypergraph are explicitly labeled, one has the notions of equivalence, and also of equality. One says that $H$ is equivalent to $G$, and writes $H\equiv G$ if the isomorphism $\phi $ has
$\phi (x_{n})=y_{n}$
and
$\phi (e_{i})=f_{\pi (i)}$
Note that
$H\equiv G$ if and only if $H^{*}\cong G^{*}$
If, in addition, the permutation $\pi $ is the identity, one says that $H$ equals $G$, and writes $H=G$. Note that, with this definition of equality, graphs are self-dual:
$\left(H^{*}\right)^{*}=H$
A hypergraph automorphism is an isomorphism from a vertex set into itself, that is a relabeling of vertices. The set of automorphisms of a hypergraph H (= (X, E)) is a group under composition, called the automorphism group of the hypergraph and written Aut(H).
Examples
Consider the hypergraph $H$ with edges
$H=\lbrace e_{1}=\lbrace a,b\rbrace ,e_{2}=\lbrace b,c\rbrace ,e_{3}=\lbrace c,d\rbrace ,e_{4}=\lbrace d,a\rbrace ,e_{5}=\lbrace b,d\rbrace ,e_{6}=\lbrace a,c\rbrace \rbrace $
and
$G=\lbrace f_{1}=\lbrace \alpha ,\beta \rbrace ,f_{2}=\lbrace \beta ,\gamma \rbrace ,f_{3}=\lbrace \gamma ,\delta \rbrace ,f_{4}=\lbrace \delta ,\alpha \rbrace ,f_{5}=\lbrace \alpha ,\gamma \rbrace ,f_{6}=\lbrace \beta ,\delta \rbrace \rbrace $
Then clearly $H$ and $G$ are isomorphic (with $\phi (a)=\alpha $, etc.), but they are not strongly isomorphic. So, for example, in $H$, vertex $a$ meets edges 1, 4 and 6, so that,
$e_{1}\cap e_{4}\cap e_{6}=\lbrace a\rbrace $
In graph $G$, there does not exist any vertex that meets edges 1, 4 and 6:
$f_{1}\cap f_{4}\cap f_{6}=\varnothing $
In this example, $H$ and $G$ are equivalent, $H\equiv G$, and the duals are strongly isomorphic: $H^{*}\cong G^{*}$.
Symmetry
The rank $r(H)$ of a hypergraph $H$ is the maximum cardinality of any of the edges in the hypergraph. If all edges have the same cardinality k, the hypergraph is said to be uniform or k-uniform, or is called a k-hypergraph. A graph is just a 2-uniform hypergraph.
The degree d(v) of a vertex v is the number of edges that contain it. H is k-regular if every vertex has degree k.
The dual of a uniform hypergraph is regular and vice versa.
Two vertices x and y of H are called symmetric if there exists an automorphism such that $\phi (x)=y$. Two edges $e_{i}$ and $e_{j}$ are said to be symmetric if there exists an automorphism such that $\phi (e_{i})=e_{j}$.
A hypergraph is said to be vertex-transitive (or vertex-symmetric) if all of its vertices are symmetric. Similarly, a hypergraph is edge-transitive if all edges are symmetric. If a hypergraph is both edge- and vertex-symmetric, then the hypergraph is simply transitive.
Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity.
Partitions
A partition theorem due to E. Dauber[35] states that, for an edge-transitive hypergraph $H=(X,E)$, there exists a partition
$(X_{1},X_{2},\cdots ,X_{K})$
of the vertex set $X$ such that the subhypergraph $H_{X_{k}}$ generated by $X_{k}$ is transitive for each $1\leq k\leq K$, and such that
$\sum _{k=1}^{K}r\left(H_{X_{k}}\right)=r(H)$
where $r(H)$ is the rank of H.
As a corollary, an edge-transitive hypergraph that is not vertex-transitive is bicolorable.
Graph partitioning (and in particular, hypergraph partitioning) has many applications to IC design[36] and parallel computing.[37][38][39] Efficient and scalable hypergraph partitioning algorithms are also important for processing large scale hypergraphs in machine learning tasks.[6]
Further generalizations
One possible generalization of a hypergraph is to allow edges to point at other edges. There are two variations of this generalization. In one, the edges consist not only of a set of vertices, but may also contain subsets of vertices, subsets of subsets of vertices and so on ad infinitum. In essence, every edge is just an internal node of a tree or directed acyclic graph, and vertices are the leaf nodes. A hypergraph is then just a collection of trees with common, shared nodes (that is, a given internal node or leaf may occur in several different trees). Conversely, every collection of trees can be understood as this generalized hypergraph. Since trees are widely used throughout computer science and many other branches of mathematics, one could say that hypergraphs appear naturally as well. So, for example, this generalization arises naturally as a model of term algebra; edges correspond to terms and vertices correspond to constants or variables.
For such a hypergraph, set membership then provides an ordering, but the ordering is neither a partial order nor a preorder, since it is not transitive. The graph corresponding to the Levi graph of this generalization is a directed acyclic graph. Consider, for example, the generalized hypergraph whose vertex set is $V=\{a,b\}$ and whose edges are $e_{1}=\{a,b\}$ and $e_{2}=\{a,e_{1}\}$. Then, although $b\in e_{1}$ and $e_{1}\in e_{2}$, it is not true that $b\in e_{2}$. However, the transitive closure of set membership for such hypergraphs does induce a partial order, and "flattens" the hypergraph into a partially ordered set.
Alternately, edges can be allowed to point at other edges, irrespective of the requirement that the edges be ordered as directed, acyclic graphs. This allows graphs with edge-loops, which need not contain vertices at all. For example, consider the generalized hypergraph consisting of two edges $e_{1}$ and $e_{2}$, and zero vertices, so that $e_{1}=\{e_{2}\}$ and $e_{2}=\{e_{1}\}$. As this loop is infinitely recursive, sets that are the edges violate the axiom of foundation. In particular, there is no transitive closure of set membership for such hypergraphs. Although such structures may seem strange at first, they can be readily understood by noting that the equivalent generalization of their Levi graph is no longer bipartite, but is rather just some general directed graph.
The generalized incidence matrix for such hypergraphs is, by definition, a square matrix, of a rank equal to the total number of vertices plus edges. Thus, for the above example, the incidence matrix is simply
$\left[{\begin{matrix}0&1\\1&0\end{matrix}}\right].$
See also
Wikimedia Commons has media related to Hypergraphs.
• BF-graph
• Combinatorial design
• Factor graph
• Greedoid
• Incidence structure
• Multigraph
• P system
• Sparse matrix–vector multiplication
• Petri Net
Notes
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4. Feige, Uriel; Kim, Jeong Han; Ofek, Eran (2006). Witnesses for non-satisfiability of dense random 3CNF formulas. 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06). IEEE. pp. 497–508. doi:10.1109/FOCS.2006.78. Archived from the original on 2021-01-27. Retrieved 2021-01-20.
5. Beeri, C.; Fagin, R.; Maier, D.; Yannakakis, M. (1983). "On the Desirability of Acyclic Database Schemes" (PDF). Journal of the ACM. 30 (3): 479–513. doi:10.1145/2402.322389. S2CID 2418740. Archived (PDF) from the original on 2021-04-21. Retrieved 2021-01-03.
6. Huang, Jin; Zhang, Rui; Yu, Jeffrey Xu (2015). "Scalable Hypergraph Learning and Processing". 2015 IEEE International Conference on Data Mining (PDF). pp. 775–780. doi:10.1109/ICDM.2015.33. ISBN 978-1-4673-9504-5. S2CID 5130573. Archived (PDF) from the original on 2021-01-26. Retrieved 2021-01-08.
7. Brazil, M; Zachariasen, M (2015). "Steiner Trees in Graphs and Hypergraphs". Optimal Interconnection Trees in the Plane. Algorithms and Combinatorics. Vol. 29. Springer. pp. 301–317. doi:10.1007/978-3-319-13915-9_5. ISBN 978-3-319-13915-9. Archived from the original on 2021-01-29. Retrieved 2021-01-20.
8. Zhou, Dengyong; Huang, Jiayuan; Scholkopf, Bernhard (2006), "Learning with hypergraphs: clustering, classification, and embedding", Advances in Neural Information Processing Systems, MIT Press, pp. 1601–8, ISBN 9780262256919, archived from the original on 2021-10-22, retrieved 2021-07-24
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10. Tan, Shulong; Bu, Jiajun; Chen, Chun; Xu, Bin; Wang, Can; He, Xiaofei (October 2011), "Using rich social media information for music recommendation via hypergraph model", ACM Transactions on Multimedia Computing, Communications, and Applications, 7S (1), Article 22, Bibcode:2011smma.book..213T, doi:10.1145/2037676.2037679, S2CID 432036
11. Liu, Qingshan; Huang, Yuchi; Metaxas, Dimitris N. (2013), "Hypergraph with sampling for image retrieval", Pattern Recognition, 44 (10–11): 2255–2262, doi:10.1016/j.patcog.2010.07.014
12. Patro, Rob; Kingsoford, Carl (2013), "Predicting protein interactions via parsimonious network history inference", Bioinformatics, 29 (10–11): 237–246, doi:10.1093/bioinformatics/btt224, PMC 3694678, PMID 23812989
13. Gao, Tue; Wang, Meng; Zha, Zheng-Jun; Shen, Jialie; Li, Xuelong; Wu, Xindong (2013), "Visual-textual joint relevance learning for tag-based social image search", IEEE Transactions on Image Processing, 22 (1): 363–376, Bibcode:2013ITIP...22..363Y, doi:10.1109/tip.2012.2202676, PMID 22692911, S2CID 7432373, archived from the original on 2017-09-23, retrieved 2017-09-22
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15. Goldstein, A. (1982). "A Directed Hypergraph Database: A Model for the Local Loop Telephone Plant". Bell System Technical Journal. 61 (9): 2529–54. doi:10.1002/j.1538-7305.1982.tb03439.x. S2CID 11290643.{{cite journal}}: CS1 maint: date and year (link)
16. Ranshous, Stephen; Joslyn, Cliff; Kreyling, Sean; Nowak, Kathleen; Samatova, Nagiza; West, Curtis; Winters, Samuel (2017). Exchange Pattern Mining in the Bitcoin Transaction Directed Hypergraph (PDF). Financial Cryptography and Data Security. Springer. doi:10.1007/978-3-319-70278-0_16. Archived (PDF) from the original on 2021-07-15. Retrieved 2021-01-20.
17. Ausiello, Giorgio; Laura, Luigi (2017). "Directed hypergraphs: Introduction and fundamental algorithms - A survey". Theoretical Computer Science. 658: 293–306. doi:10.1016/j.tcs.2016.03.016.
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23. Naheed Anjum, Arafat; Bressan, Stéphane (2017), "Hypergraph Drawing by Force-Directed Placement", 28th International Conference on Database and Expert Systems Applications (DEXA 2017), Lecture Notes in Computer Science, vol. 10439, Springer International Publishing, pp. 387–394, doi:10.1007/978-3-319-64471-4_31, ISBN 978-3-319-64470-7.
24. Kaufmann, Michael; van Kreveld, Marc; Speckmann, Bettina (2009), "Subdivision drawings of hypergraphs", Proc. 16th International Symposium on Graph Drawing (GD 2008), Lecture Notes in Computer Science, vol. 5417, Springer-Verlag, pp. 396–407, doi:10.1007/978-3-642-00219-9_39, ISBN 978-3-642-00218-2.
25. Johnson, David S.; Pollak, H. O. (2006), "Hypergraph planarity and the complexity of drawing Venn diagrams", Journal of Graph Theory, 11 (3): 309–325, doi:10.1002/jgt.3190110306.
26. Buchin, Kevin; van Kreveld, Marc; Meijer, Henk; Speckmann, Bettina; Verbeek, Kevin (2010), "On planar supports for hypergraphs", Proc. 17th International Symposium on Graph Drawing (GD 2009), Lecture Notes in Computer Science, vol. 5849, Springer-Verlag, pp. 345–356, doi:10.1007/978-3-642-11805-0_33, ISBN 978-3-642-11804-3.
27. "Vitaly Voloshin: Mixed Hypergraph Coloring Website". spectrum.troy.edu. Archived from the original on 2022-01-20. Retrieved 2022-04-27.
28. Lovász, László; Plummer, M. D. (1986), Matching Theory, Annals of Discrete Mathematics, vol. 29, North-Holland, ISBN 0-444-87916-1, MR 0859549
29. Berge, Claude (1973). Graphs and Hypergraphs. Amsterdam: North-Holland. ISBN 0-7204-2450-X.
30. Yu, C. T.; Özsoyoğlu, M. Z. (1979). "An algorithm for tree-query membership of a distributed query" (PDF). Proc. IEEE COMPSAC: 306–312. doi:10.1109/CMPSAC.1979.762509. Archived (PDF) from the original on 2018-09-02. Retrieved 2018-09-02.
31. Graham, M. H. (1979). "On the universal relation". Technical Report. Toronto, Ontario, Canada: University of Toronto.
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36. Karypis, G., Aggarwal, R., Kumar, V., and Shekhar, S. (March 1999), "Multilevel hypergraph partitioning: applications in VLSI domain", IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 7 (1): 69–79, CiteSeerX 10.1.1.553.2367, doi:10.1109/92.748202.{{citation}}: CS1 maint: multiple names: authors list (link)
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References
• Berge, Claude (1984). Hypergraphs: Combinatorics of Finite Sets. Elsevier. ISBN 978-0-08-088023-5.
• Berge, C.; Ray-Chaudhuri, D. (2006). Hypergraph Seminar: Ohio State University, 1972. Lecture Notes in Mathematics. Vol. 411. Springer. ISBN 978-3-540-37803-7.
• "Hypergraph", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Bretto, Alain (2013). Hypergraph Theory: An Introduction. Springer. ISBN 978-3-319-00080-0.
• Voloshin, Vitaly I. (2002). Coloring Mixed Hypergraphs: Theory, Algorithms and Applications: Theory, Algorithms, and Applications. Fields Institute Monographs. Vol. 17. American Mathematical Society. ISBN 978-0-8218-2812-0.
• Voloshin, Vitaly I. (2009). Introduction to Graph and Hypergraph Theory. Nova Science. ISBN 978-1-61470-112-5.
• This article incorporates material from hypergraph on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
External links
• PAOHVis: open-source PAOHVis system for visualizing dynamic hypergraphs.
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| Wikipedia |
\begin{document}
\title{The Gross-Zagier-Zhang formula over function fields} \author{Congling Qiu} \subjclass[2010]{Primary 11F52; Secondary 11F67, 11G09, 11G40} \maketitle
\begin{abstract}
We prove the Gross-Zagier-Zhang formula over global function fields of arbitrary characteristics. It is an explicit formula which relates the \Neron-Tate heights of CM points on abelian varieties and
central derivatives of associated quadratic base change $L$-functions.
Our proof is based on an arithmetic variant of a relative trace identity of Jacquet.
This approach is proposed by W. Zhang. We apply our results to the Birch and Swinnerton-Dyer conjecture for abelian varieties of $\GL_2$-type. In particular, we prove the conjecture for elliptic curves of analytic rank 1. \end{abstract} \tableofcontents
\section{Introduction}\label{Introduction}
\subsection{Motivation}\label{Mot} For an elliptic curve $A$ defined over a global field $F$, there are two associated objects of fundamental importance. First, its set of rational points $ A(F)$, which form a finitely generated abelian group. Second, its $L$-function $L(s,A)$. The conjecture of Birch and Swinnerton-Dyer asserts that $$\rank_\BZ A(F)=\ord _{s=1}L(s,A).$$
Then it is natural to ask the following question: if $ \ord _{s=1}L(s,A)=1 $, how to find a non-torsion point?
In the case that $F=\BQ$, Gross and Zagier \cite{GZ} established a formula which relates the \Neron-Tate height of a Heegner point on $A$ associated to an imaginary quadratic field $E$, and the derivative
$L'(1,A_E)$.
In particular, if $\ord _{s=1}L(1,A_E)=1$, then the Heegner point is non-torsion.
Over global function fields of odd characteristics, an analog of the Gross-Zagier formula was established by R\"{u}ck and Tipp \cite{RT}.
The work of Gross and Zagier was first generalized by S. Zhang \cite{Zha1} \cite{Zha01} to Shimura curves over totally real number field.
Later the general form of the Gross-Zagier-Zhang formula was obtained by Yuan, W. Zhang and S. Zhang \cite{YZZ} in the automorphic framework. This general formula applies to arbitrary cuspidal automorphic representations of $\GL_2$ over totally real fields holomorphic of weight 2.
In this paper, over global function fields of arbitrary characteristics, we fully generalize the analog of the Gross-Zagier formula in the format of \cite{YZZ}. Our result applies to arbitrary cuspidal automorphic representations of $\GL_{2}$. Besides, we prove the Waldspurger formula \cite{Wal}.
As an application of our Gross-Zagier-Zhang formula, we prove the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank 1 in arbitrary positive characteristics. Indeed, this is a special case of one of our results for abelian varieties of $\GL_2$-type. The Birch and Swinnerton-Dyer conjecture in the analytic rank 1 case in characteristic $>3$ was considered by Ulmer \cite{Ulm} \cite{Ulm1}.
To prove our Gross-Zagier-Zhang formula and Waldspurger formula, we employ the relative trace formulas of Jacquet \cite{Jac86}\ \cite{Jac87}\cite{JN} and
their arithmetic analogs following W. Zhang \cite{Zha12}. The main advantage of the relative trace formula approach is its applicability in arbitrary characteristics. Over global function fields of odd characteristics, it is also possible to approach our results using the theta lifting machinery. For example, Chuang and Wei \cite{CW} proved the Waldspurger formula in odd characteristics following Waldspurger's original approach closely. A key ingredient is the Siegel-Weil formula \cite{WFT}. Similarly, it is also possible to prove the Gross-Zagier-Zhang formula in odd characteristics
following \cite{YZZ}.
Below in this introduction, we first state our main results. Then we describe the relative trace formula strategy and
the structure of this paper.
Finally, we discuss higher dimensional generalizations of the Gross-Zagier formula via the relative trace formula approach.
\subsection{Main results}
\subsubsection{Modular curves}
Let $X$ be a smooth proper connected curve over a finite field $\BF_q$ of characteristic $p>0$. Let $F$ be the function field of $X$. Let $|X|$ be the set of closed points of $X$, equivalently the set of places of $F$.
Let $\BA_F$ be the ring of adeles of $F$.
Let $\BB$ be an incoherent quaternion algebra over $\BA_F$, i.e.
the set ${\mathrm{Ram}}\subset |X|$ of ramified places of $\BB$ is finite and of odd cardinality. Distinguish a place $\infty\in {\mathrm{Ram}}$, which plays the role of the infinite place as in the number field case.
Fix a ``level structure" at $\infty$, which is given by an arbitrary open normal subgroup $U_\infty\subset \BB_\infty^\times$ containing a uniformizer of $F_\infty$. In \ref{Moduli spaces of}, we define a smooth proper modular curve
$M_I$ over $F$ \cite{DriEll1} \cite{DriEll2} \cite{LLS} for every finite and nonempty closed subscheme $I\subset |X|-\{\infty\}$.
Let $M$ be the procurve $\vpl_I M_I$ where the transition maps are natural projections. Then $M$ is endowed with the right action of $\BB ^\times$. Let $J_I$ be the Jacobian of $M_I$, and let $J(F^\sep)_\BQ=\vpl _I J_I(F^\sep)_\BQ$, where $F^\sep$ is a separable closure of $F$. Using the normalized Hodge divisor class (see \ref{Hodge classes}) on each $ M_I$, we have a map \begin{equation}M(F^\sep)\to J(F^\sep)_\BQ .\label{MJ}\end{equation}
\subsubsection{Abelian varieties and $L$-functions}\label{Abelian varieties parametrized by modular curvesint}
Let $A$ be a simple abelian variety over $F$. Assume that $A$ is modular (w.r.t. $\BB/U_\infty$) in the sense that $$\pi=\pi_A:=\varinjlim \Hom(J_I,A)_\BQ$$ is nontrivial. Then $\pi$ is an irreducible $\BQ$-coefficient representation of $\BB^\times$. Let $K:=\End(A)_\BQ$ which is a number field and acts on $\pi$. Then $L(s,\pi,\ad)$ a polynomial on $q^{-s}$ with coefficients in $K$.
Let $E$ be a quadratic field extension of $F$. Let $\Omega$ be a continuous character of $\Gal(E^\ab/E)$ valued in a finite field extension $K'$ of $K$. Also regard $\Omega$ as a Hecke character of $E^\times$ via the reciprocity map
$\BA_E^\times/E^\times\to \Gal(E^\ab/E).$ Then the $L$-function $$L(1/2,\pi,\Omega):=L(1/2,\pi_{E}\otimes \Omega).$$
is a polynomial on $q^{-s}$ with coefficients in $K'$.
The twisted $L$-function $L(s,A_E,\Omega)$ of $A_E$ satisfies
$$L(s,A_E,\Omega)=L(s-1/2,\pi , \Omega ).$$
\subsubsection{Global and local periods}
Let $E/F$ be nonsplit at $\infty$.
Fix an embedding \begin{equation}e_0:E \hookrightarrow \BB \label{e0}\end{equation} of $F$-algebras. Let $P_0\in M^{E^\times}( F^\sep)$ be a CM point.
Then $P_0$ is defined over $E^{\ab}$, the maximal abelian extension of $E$ in $F^\sep$ (see \ref{CM theory}). Regard $P_0$ as a point in $J(E^\ab)_\BQ$ via \eqref{MJ}. For $\phi\in \pi_A$, we have a CM point $\phi(P_0)\in A(E^\ab)_\BQ.$ Define $$P_\Omega(\phi):=\int_{\Gal(E^\ab/E)}\phi(P_0)^\tau\otimes_K\Omega(\tau)d\tau\in A(E^\ab)_\BQ\otimes_KK'$$
where the Haar measure on $\Gal(E^\ab/E)$ is of total volume 1. This integral is essentially a finite sum. For $\varphi\in \pi_{A^\vee}$, our global height period of $\phi$ and $\varphi$ is
$$ \pair{P_\Omega(\phi),P_{\Omega^{-1}}(\varphi)}_\NT^{K'}.$$
Here $\pair{\cdot,\cdot}_\NT^{K'}$ is the natural extension by scalar of
$K$-bilinear \Neron-Tate height pairing
$$\pair{\cdot,\cdot}_\NT^{K}:A(F^\sep)_\BQ\otimes_K A^\vee(F^\sep)_\BQ \to K\otimes \BC$$ such that
$\tr_{K\otimes_\BQ \BC/\BC}\pair{\cdot,\cdot}_\NT^K$ is the usual \Neron-Tate height pairing.
Now we define local periods.
Regard $E^\times/F^\times$ as an algebraic group over $F$. We always use the Tamagawa measure on $(E^\times/F^\times) (\BA_F)$, and fix a local-global decomposition of the measure.
Let $(\cdot,\cdot)_v$ be the natural pairing on $\pi_v\otimes \tilde \pi_v$ which takes values in $K$. For $ \phi_v\in \pi_v$ and $ \varphi_v\in \tilde \pi_v$,
define the local period to be $$\alpha_{\pi_v} (\phi_v,\varphi_v) := \int_{E_v^\times/F_v^\times} (\pi_v(t)\phi_v,\varphi_v)_v\Omega_v(t)d t\in K' .$$
This integral is essentially a finite sum.
Let $\eta$ be the quadratic Hecke character of $F^\times$ associated to the quadratic extension $E/F$.
Define the normalized local period to be
\begin{equation*} \alpha_{\pi_v}^\sharp:=\frac{L(1,\eta_v)L(1,\pi_v,\ad)}{L(2,1_{F_v})L(1/2,\pi_v,\Omega_v)} \alpha_{\pi_v}.
\end{equation*}
Then $\alpha_{\pi_v}^\sharp$ takes values in $K'$. Let \begin{equation}\alpha_{\pi_A}:=\bigotimes_{v\in |X|}\alpha_{\pi_v}^\sharp.\label{aglobal} \end{equation}
\begin{thm}[The Gross-Zagier-Zhang formula]\label{GZ}
Let $\phi \in \pi_A,\varphi\in \pi_{A^\vee}$, then
\begin{equation}\pair{P_\Omega(\phi),P_{\Omega^{-1}}(\varphi)}_\NT^{L'}=\frac{L(2,1_F)L'(1/2,\pi_{A },\Omega)}{4L(1,\eta)^2 L(1,\pi_A,\ad)} \alpha_{\pi_A} (\phi ,\varphi ) \label{GZeq}
\end{equation} as an identity in $K'\otimes_\BQ\BC$. \end{thm}
Here, we use the following pairing \eqref{ duality pairing} to identify $\pi_{A^\vee}$ with the contragradient $\tilde\pi$ of $\pi $. For $\phi\in \Hom(J_I,A^\vee)$, let $\phi^\vee \in \Hom(A,J_I^\vee)$ be the dual morphism. Regard $\phi^\vee$ as in $\Hom(A,J_I)$ by canonically identifying $J_I$ and $ J_I^\vee$. We have a perfect $\BB^\times$-invariant pairing \begin{equation}(\cdot,\cdot) : \pi_A
\times\pi_{A^\vee}\to \End(A)_\BQ = K\label{ duality pairing}
\end{equation} defined by $$(\phi_1,\phi_2):=\Vol(M_I)^{-1} \phi_{1,I}\circ \phi_{2,I}^\vee.$$
Here $\Vol(M_I) $ is the degree of the Hodge class on $M_I$.
\subsubsection{Generality of Theorem \ref{GZ}}\label{choices}
Let $A$ be an abelian variety over $F$ of strict $\GL_2$-type which does not have potential good reduction \footnote{ Potential good reduction includes good reduction.} at $\infty$. The latter condition ensures that the corresponding automorphic representation is a discrete series at $\infty$ (so comes from $\BB^\times_\infty$ via the Jacquet-Langlands correspondence). Since $U_\infty $ contains a uniformizer of $F_\infty$,
for $A$ to be modular (w.r.t. $\BB/U_\infty$), the value of the corresponding automorphic representation on this uniformizer should be identity. We twist $A$ by a suitable character so that it is modular. Then Theorem \ref{GZ} can be applied to $A$ thanks to the freedom on $\Omega$ in Theorem \ref{GZ}.
From the representation-theoretical perspective, our result applies to arbitrary cuspidal automorphic representations of $\GL_{2}$, see \ref{choices'}.
\subsubsection{The Birch and Swinnerton-Dyer Conjecture}
A direct consequence of Theorem \ref{GZ} is a twisted Birch and Swinnerton-Dyer Conjecture for $A_E$ in the analytic rank 1 case, see Theorem \ref{BSDTE}.
With some more effort, we can remove the base change. \begin{thm}
The Birch and Swinnerton-Dyer conjecture holds for elliptic curves over $F$ in the analytic rank 1 case. \end{thm} Indeed, this theorem is a special case of Theorem \ref{BSDT}, which applies to $A$ as in \eqref{choices}.
The condition that the automorphic representation corresponding to $A$ is a discrete series at $\infty$ is also necessary for
the existence of a quadratic base change $A_E$ which has analytic rank 1 (see Remark \ref{notgod}), so that we can apply Theorem \ref{GZ}.
\subsubsection{Waldspurger formula over function fields} \label{The global relative trace formulaintro} Let $B$ be a quaternion algebra over $F$, and let $E$ be a quadratic extension of $F$ embedded in $B$ (not necessary nonsplit over $\infty$). Let $\pi$ be a cuspidal automorphic representation of $B^\times $ and $\Omega$ a Hecke character of $E^\times$ (both are $\BC$-valued).
Let $\alpha_\pi$ be defined as in \eqref {aglobal}. For $\phi\in \pi$, define the toric period of $\phi$ by \begin{equation}\label{toric period}P_\Omega(\phi):=\int_{ E^\times\backslash \BA_E^\times /\BA_F^\times}\phi(t)\Omega(t)dt. \end{equation}
\begin{thm}[The Waldspurger formula]\label{The Waldspurger formula over function fieldsintro} Let $\phi \in \pi ,\varphi\in \tilde \pi $, then
\begin{equation} P_\Omega(\phi) P_{\Omega^{-1}}(\varphi) =\frac{L(2,1_F)L(1/2,\pi,\Omega)}{ 2 L(1,\pi ,\ad)} \alpha_{\pi } (\phi ,\varphi ). \label{Waldeq}\end{equation}
\end{thm}
\subsection{Relative trace formula strategy}
Instead of studying periods directly, relative trace formulas treat distributions on adelic groups related to the periods. We will compare
relative trace formulas for unit groups of quaternion algebras and $\GL_{2,E}$. \subsubsection{Distribution version of Theorem \ref{The Waldspurger formula over function fieldsintro}}
The Hecke action of $f\in C_c^\infty(B^\times(\BA_F))$ on $\pi$ is \begin{equation*}\pi(f)\phi:=\int_{ B ^\times(\BA_F)} f (g)\pi(g)\phi dg. \end{equation*} Define a distribution on $ B^\times(\BA_F)$ by assigning to $f\in C_c^\infty(B^\times(\BA_F))$ the value
\begin{equation*}\CO_\pi(f):=\sum_{\phi }P_\Omega(\pi(f)\phi)P_{\Omega^{-1}}( \tilde \phi),
\end{equation*} where the sum is over an orthonormal basis $\{\phi\}$ of $\pi$, and $\{\tilde \phi \}$ is the dual basis of $\tilde \pi$. For $f_v\in C_c^\infty(B_v^\times)$, define $$\rho_{\pi_v}(f_v):=\sum_{\phi }\pi_v(f_v)\phi \otimes \tilde \phi,$$ where
the sum is over an orthonormal basis $\{\phi\}$ of $\pi_v$, $\{\tilde \phi \}$ is the dual basis of $\tilde \pi_v$,
and
$$\pi_v(f_v)\phi :=\int_{ B ^\times(F_v) } f_v (g)\pi_v(g)\phi dg.$$
Abusing notation, we use $\alpha^\sharp_{\pi_v}$ to denote the distribution on $ B_v^\times$ which assigns to $f_v $ the value \begin{equation}\alpha^\sharp_{\pi_v}( f_v) :=\alpha^\sharp _{\pi_v}(\rho_{\pi_v}(f_v)).\label{alpha}\end{equation}
Let $\omega$ be the restriction of $\Omega$ to $\BA_F^\times/F^\times$.
Assume that the central character of $\pi$ is $\omega^{-1}$, otherwise both sides of \eqref{Waldeq} are 0. Then for every $v\in |X|$, $ \vep(1/2,\pi_{v,E_v}\otimes \Omega_v)=\pm1$, and it is 1 for all but finitely many places (see \cite{Tun}).
Define \begin{equation}\Sigma(\pi,\Omega):=\{v\in |X|:\vep(1/2,\pi_{v,E_v}\otimes \Omega_v)\neq \Omega_v(-1) \}.\label{sigset} \end{equation}
Let ${\mathrm{Ram}}(B)$ be the ramification set of $B$.
Theorem \ref{The Waldspurger formula over function fieldsintro} implies the following theorem.\begin{thm}\label{The Waldspurger formula over function fieldsintrof}\label{zhl}
Assume that the central character of $\pi$ is $\omega^{-1}$
and ${\mathrm{Ram}}(B)=\Sigma(\pi,\Omega)$.
There exists $f=\bigotimes_{v\in |X| }f_v\in C_c^\infty(B^\times(\BA_F))$ such that
\begin{equation*}
2\CO_\pi (f ) =\frac{L(2,1_F)L(1/2,\pi, \Omega)}{ L(1,\pi,\ad)} \prod_{v\in |X| } \alpha_{\pi_v}^\sharp (f_v)
\end{equation*} and $ \alpha_{\pi_v}^\sharp (f_v)\neq 0$
for every $v\in |X|$.
\end{thm}
Let $\CP_{\Omega }(\pi ):=\Hom_{ \BA_E^\times }(\pi\otimes \Omega ,\BC).$ Then both $P_\Omega \otimes P_{\Omega^{-1}} $ and $ \alpha_{\pi } $ are
elements in $\CP_\Omega(\pi)\otimes \CP_{\Omega^{-1}}(\tilde\pi)$. The theorem of Tunnell-Saito \cite{Tun}\cite{Sai} says that $\CP_{\Omega}(\pi)\neq \{0\}$ if and only if ${\mathrm{Ram}}(B)=\Sigma(\pi,\Omega)$ and in this case, $\CP_{\Omega}(\pi )$ is of dimension 1.
It follows that Theorem \ref{zhl} implies
Theorem \ref{The Waldspurger formula over function fieldsintro}.
\subsubsection{Distribution version of Theorem \ref{GZ}}\label{1.3.2}
Let $\CH_{\BC} $ be the Hecke algebra of locally constant $\BC$-valued bi-$U_\infty$-invariant functions on $\BB ^\times$ with compact support modulo $U_\infty\times U_\infty$. Define a distribution on $ \CH_{\BC}$ by assigning to $f\in \CH_{\BC}$ the height pairing $H (f)$ of the images of the CM cycle given by $e_0:E \hookrightarrow \BB $ and its Hecke translation by $f$ in $J$, twisted by $\Omega$ and $\Omega^{-1}$ respectively (see Definition \ref{CM height}).
Let $\pi$ be an irreducible admissible representation of $\BB^\times/U_\infty$ whose Jacquet-Langlands correspondence to $\GL_{2,F} $ is cuspidal. Let $ H_\pi^\sharp (f )$ be the normalized $\pi$-component of $H (f)$ (see \eqref{HHpi0}). Let $\alpha^\sharp_{\pi_v}( f_v)$ be defined as in \eqref{alpha}, suitably modified when $v=\infty$ (see \ref{The CM height distribution}).
\begin{thm} [Theorem \ref{GZdis'}]\label{GZdis} Assume that the central character of $\pi$ is $\omega^{-1}$
and ${\mathrm{Ram}}=\Sigma(\pi,\Omega).$
There exists $f=\bigotimes_{v\in |X| }f_v\in \CH_\BC$ such that \begin{equation*} H_\pi^\sharp(f ) =\frac{L(2,1_F)L'(1/2,\pi,\Omega)}{ L(1,\pi,\ad)} \prod_{v\in |X|} \alpha_{\pi_v}^\sharp (f_v),\end{equation*} and $ \alpha_{\pi_v}^\sharp (f_v)\neq 0$
for every $v\in |X|$.
\end{thm}
\subsubsection{Distributions on $\GL_{2}(\BA_E)$}
Let $A$ be the diagonal torus of $G=\GL_{2,E}$, and let $Z$ be the center of $G$.
Let $H\subset G$ be the similitude unitary group with respect to the Hermitian matrix $\begin{bmatrix}0&1\\
1&0\end{bmatrix}$
and $\kappa$ the associated similitude character of $H$. Let $\omega_E$ be the base change of $\Omega$ to $E$.
Let $f'\in C_c^\infty(G(\BA_E))$. Consider the Hecke action of $f'$ on the space of automorphic forms on $ G(\BA_E)$ which transform
by $\omega_E^{-1}$ under the action of the center. The action of $f'$ is given by a kernel function $K_{\omega_E,f'}$ on $$G(E)\backslash G(\BA_E)\times G(E)\backslash G(\BA_E).$$
Let $\sigma$ be an automorphic representation of $G$.
Define $\CO (s, f')$ (resp. $\CO_\sigma(s, f')$) to be the integral of $K_{\omega_E,f'}$ (resp. the $\sigma$-component of $K_{\omega_E,f'}$)
on $$Z(\BA_E)A(E)\backslash A(\BA_E)\times Z(\BA_E)H(F)\backslash H(\BA_F)$$
against the character $\Omega|\cdot |_E^s \boxtimes(\eta\cdot(\omega^{-1}\circ\kappa)) $. Let $\CO (f')=\CO (0,f')$ (resp. $\CO_\sigma( f')=\CO_\sigma(0, f')$)
\subsubsection{Theorem \ref{The Waldspurger formula over function fieldsintrof}}\label{skw}
Let $\sigma=\pi_E$. For functions $f\in C_c^\infty(B^\times(\BA_F))$ and $f'\in C_c^\infty(G(\BA_E))$ with matching local orbital integrals, $\CO(f') $ equals its counterpart $\CO(f)$ for $B^\times$.
(see \eqref{RTF}). The spectral decomposition gives an equation relating $\CO_\pi(f)$ and $\CO_\sigma(f') $.
In the number field case \cite{JN}, by the smooth matching for $f'$, the function $f'$ can be arbitrary (but not $f$). If $ L(1/2,\pi,\Omega)\neq 0$, one can choose $f'$ such that $ \CO_\sigma(f') $ is nonzero. Moreover, under the condition $ L(1/2,\pi,\Omega)\neq 0$, the local factors of $\CO_\sigma$ and $ \alpha_{\pi_v}^\sharp$'s can be compared, which is called the spherical character identity.
Thus Theorem \ref{The Waldspurger formula over function fieldsintrof} follows in this case.
In our proof of Theorem \ref {The Waldspurger formula over function fieldsintrof}, we take explicit pairs of matching functions $f$ and $f'$ such that $ \alpha_{\pi_v}^\sharp (f_v)\neq 0$, and prove the spherical character identity for $f_v$ and $f_v'$ (see \ref{the proof is more important for us}, \ref{Explicit computations for smooth matching}). Thus Theorem \ref{The Waldspurger formula over function fieldsintrof} holds without the condition $ L(1/2,\pi,\Omega)\neq 0$.
\subsubsection{Theorem \ref{GZdis}}
Let $|X|_s$ be the set of places of $F$ split in $E$, $\Xi_\infty:=F_\infty^\times\cap U_\infty$ and $S$ a finite set of ``bad places". We have an arithmetic relative trace identity
(see Theorem \ref{jacrtf'}): \begin{equation}2 H(f) =\CO' (0,f' )+\sum_{v\in S-|X|_s } \text{distributions on }B(v)^\times(\BA_F)
\label{bbb}\end{equation} for good matching functions $f$ and $f'$.
Here $B(v)$ is a quaternion algebra over $F$.
An easy but essential observation is that under the condition ${\mathrm{Ram}}=\Sigma(\pi,\Omega),$ the $\pi$-component of second term on the right hand side is 0 by the theorem of Tunnell-Saito.
Then Theorem \ref{GZdis} follows from \eqref{bbb} by taking explicit pairs of matching functions $f$ and $f'$, such that $ \alpha_{\pi_v}^\sharp (f_v)\neq 0$
and the spherical character identity holds for $f$ and $f'$.
Now we sketch the proof of \eqref{bbb}. To compute $H(f)$, we apply the theory of admissible pairing \cite{Zha01} to an integral model $\CN$ of a modular curve over a certain field extension of $E$. A vanishing condition on the average of a local component of $f$ makes the contribution in $H(f)$ from the Hodge class vanish. Then $H(f)$ equals the intersections of admissible extensions of CM points. Decompose $H(f)$ into a sum of local intersection numbers $H(f)_v$, $v\in |X|$. We call the intersection number of horizontal divisors in $H(f)_v$ the $i$-part, and the rest the $j$-part.
For $v\in |X|_s$, let $f^v$ be regularly supported. Then the $i$-part in $H(f)_v $ vanishes. The $j$-part in $H(f)_v $ comes from intersections on $\CN$ of horizontal divisors with components in the special fiber with moduli interpretations. The integral Hecke actions on these components are easy to understand. Let the average of $f^v$ vanish. Then the $j$-part in $H(f)_v $ vanishes.
For $v\in |X|-|X|_s$, let $f^v$ be regularly supported. We compute local intersections on the Lubin-Tate or Drinfeld uniformization spaces on which $B(v)^\times$ acts.
Then $ H(f)_v$ is decomposed into a sum over regular $E^\times\times E^\times $-orbits in $B(v)^\times$.
For each regular orbit, we compare the local component at $v$ of the corresponding summand with the local orbital integral of $f_v'$ at a matching orbit of $G$.
Outside a large enough finite set $S\subset |X|$, we prove the arithmetic fundamental lemma for the full
spherical Hecke algebra.
For $v\in S-|X|_s$, we prove the arithmetic smooth matching for both the $i$-part and the $j$-part. Then \eqref{bbb} follows.
\subsection{Structure of the paper}
In Part 1, we define global objects which will be studied locally in Part 2.
We first define the modular curves and CM points in Section \ref{MCM}. In Section \ref{Height distributions, Rational Representations, and Abelian varieties}, we define the height distribution on $\BB^\times$, study its spectral decomposition, and reduce the Gross-Zagier-Zhang formula to the distribution version. Finally in Section \ref{The automorphic distributions}, we define automorphic distributions on $\GL_{2}$ and unit groups of quaternion algebras. (The automorphic distributions on the quaternion groups will be used to prove the Waldspurger formula.) We study their orbital and spectral decompositions. The orbital terms are decomposed into local orbital integrals.
The most important section of Part 2 is Section \ref{Local intersection multiplicity}, where we study the orbital decomposition of the height distribution into local intersection numbers. Besides, Section \ref{review} and Section \ref{amafl} also concern the orbital side. In Section \ref{review}, we give
explicit functions on $\GL_2$ and the quaternion groups over local fields
with matching orbital integrals. In Section \ref{amafl}, we compare the derivatives of local orbital integrals and the local intersection numbers associated to the matching functions. For the spectral sides,
we introduce local distributions for $\GL_2$ and the quaternion groups in Section \ref{local relative trace formula},
compare their values at the matching functions in Section \ref{review},
and relate the local and global distributions (as well as $L$-functions) for $\GL_2$ in Section \ref{Global and local periods} (then the Waldspurger formula follows).
Finally in
Section \ref{Proof of Theorem}, we use the above ingredients to prove the Gross-Zagier-Zhang formula.
In part 3, we apply the Gross-Zagier-Zhang formula to the Birch and Swinnerton-Dyer conjecture.
\subsection{Related works in higher dimensions}\label{Remarks on related works}
\subsubsection{} Using the geometrized relative trace formula, Yun and W. Zhang \cite{YZ} \cite{YZ2} proved an amazing formula relating higher
derivatives of $L$-functions and intersection numbers on the moduli stacks of shtukas in odd characteristics. Their formula applies to representations with trivial central character and Iwahori level-structures. Our modular curve is closely related to
a special case of the moduli stack of shtukas when $\BB$ is only ramified at one place.
We hope to generalize this higher derivative formula to more general representations, using the arithmetic relative trace formula as in this work. Indeed, one obstacle in Yun and W. Zhang's geometric approach is that they need to construct precise matching functions. However, our choices of matching functions are more flexible, by allowing an extra term (i.e., the last term in \eqref{bbb}).
\subsubsection{}
For groups over number fields other than $\GL_2$, a generalization of the Gross-Zagier formula was conjectured by Gan-Gross-Prasad \cite{GGP} and refined by S. Zhang \cite{Zha10}.
W. Zhang \cite{Zha12} proposed the arithmetic relative trace formula approach to attack this conjecture. During the revision of this paper, W. Zhang \cite{Zha19} proved
the arithmetic fundamental lemma over $\BQ$ for large enough primes. Cases of the arithmetic smooth matching were proved by Rapoport, Smithling, Terstiege and W. Zhang
\cite{RSZ15} \cite{RSZ16} \cite{RSZ17}.
The spherical character identity over $p$-adic fields, conjectured in \cite{Zha14b}, was proved by
Beuzart-Plessis \cite{BP}\cite{BP1}.
\subsection{Notations}\label{measures}
The Haar measures on $\BA_F$, $\BA_E$, $\BA_F^\times$, $\BA_E^\times $, $\GL_2(\BA_F)$, $G(\BA_E)$, $H(\BA_F)$ and $\BB^\times$ takes values in $\BQ$ on open compact subgroups, and will be specified in Section \ref{notations and measures }.
For $S\subset |X|$ and a decomposable adelic object $Z$ over $\BA_F$, we use $Z_S$ (resp. $Z^S$) to denote the $S$-component (resp. component away from $S$) of $Z$.
Let $\BA_{F,\mathrm{f}} =\BA_F^{{\infty}}$ and $\BB_{\mathrm{f}}=\BB^{{\infty}}$.
We have defined $\Xi_\infty =U_\infty \cap F_\infty^\times$ where $ F_\infty^\times$ is regarded as the center of $\BB_\infty^\times$. For an open compact subgroup $U$ of $\BB_{\mathrm{f}}^\times$, let $\Xi_U:=U \cap \BA_{F,\mathrm{f}}^\times$ where $ \BA_{F,\mathrm{f}}^\times$ is regarded as the center of $\BB_{\mathrm{f}}^\times$. Let $\Xi=\Xi_U\Xi_\infty\subset\BA_F^\times.$
Let $\tilde U:=UU_\infty\subset \BB^\times .$
Let $l\neq p$ be a prime number. Let $\CA_{U_\infty} (\BB^\times,\bar \BQ_l)$ (resp. $\CA_{U_\infty} (\BB^\times,\BC)$) be the set of isomorphism classes of $\bar \BQ_l$ (resp. $\BC$)-coefficient irreducible admissible representations of $\BB^\times/U_\infty$, whose Jacquet-Langlands correspondence to $\GL_{2,F}$ is cuspidal.
For a field extension $K/\BQ$, let $\CH_K$ be the Hecke algebra of $K$-valued locally constant bi-$U_\infty$-invariant functions on $\BB^\times $ with compact support modulo $U_\infty\times U_\infty$. For an open compact subgroup $U$ of $\BB^\times_{\mathrm{f}}$, let $\CH_{U ,K} \subset \CH_K$ be the subalgebra of bi-$\tilde U $-invariant functions.
\section{Modular curves and CM points}\label{MCM} We at define modular curves and review their properties, following Drinfeld \cite{DriEll1} \cite{DriEll2} \cite{DriCar}, Laumon, Rapoport, and Stuhler \cite{LLS}, et al.. Then we define CM points, and show their algebraicity.
\subsection{Modular curves}\label{Moduli spaces of}
For a sheaf $\CF$ of $\CO_X$-modules and $x\in |X|$, let $\CF_x$ be the completion of the stalk of $\CF$ at $x$.
An order of $D$ is a locally free coherent sheaf $\cD$ of $\CO_X$-algebras on $X$ whose stalk at the generic point is isomorphic to $D$. The set of local orders $\{\cD_x\subset D_x \}_{x\in |X|}$ satisfies the following property: there exists an $F$-basis $R$ of $D$ such that $\cD_x=\CO_{F_x}R$ for almost all $x$. Let $Ord$ be the set of sets of local orders satisfying this property.
\begin{lem}[{\cite[Section 1] {LLS}}]\label{order}
The map $\cD\mapsto\{\cD_x\}_{x\in |X|} $
is a bijection between the set orders of $D$ and the set $Ord$.
\end{lem}
An order $\cD$ of $D$ is called a maximal order of $D$ if
$\cD_x$ is a maximal order of $D_x$ for every $x\in |X|$. Let $\cD$ be a maximal order of $D$.
\subsubsection{Moduli spaces without level structures at $\infty$}
Let $S$ be an $\BF_q$-scheme. For a sheaf $\CE$ on $X\times S$, let ${}^\tau\CE:=(1\times {\mathrm{Frob}}_S)^*\CE$.
\begin{defn}[{\cite[(2.2)]{LLS}}] \label{DES}
A $\cD$-elliptic sheaf, on $X$ with respect to $\infty$, over $S$ is the following data: a morphism ${\mathfrak{zero}}_\BE:S\to X$ with image away from ${\mathrm{Ram}}$ and
a sequence of commutative diagrams $$
\xymatrix{
{}^\tau \CE_{i-1} \ar[r]^{{}^\tau j_{i-1}}\ar[d]_{ t_{i-1}} &{}^\tau \CE_{i} \ar[d]^{t_{i} } \\
\CE_i \ar [r]_{j_i} & \CE_{i+1} } $$ indexed by $i\in\BZ$,
where each $\CE_i$ is a locally free $\CO_{X\times S}$-module of rank 4 equipped with a \textit{right} action of $\cD$ compatible with the $\CO_{X}$ action such that $$\CE_{i+2\cdot\deg(\infty)}=\CE_i(\infty),$$ and $j_i,t_i$ are injections compatible with $\cD$-actions and satisfy the following conditions : \begin{itemize} \item[(1)] the composition $j_{i+2\cdot\deg(\infty)-1}\circ\cdot\cdot\cdot\circ j_{i+1}\circ j_i$ is the canonical inclusion $\CE_i\hookrightarrow\CE_i(\infty)$; \item[(2)] let $\pr_S:X\times S\to S$ be the projection, then $\pr_{S,*}(\coker j_i)$ is a locally free $\CO_S$-module of rank $2$; \item[(3)] $\coker t_i$ is the direct image of a locally free $\CO_S$-module of rank 2 by the graph morphism $({\mathfrak{zero}}_\BE,\id): S\to X\times S$.
\end{itemize} A morphism between two $\cD$-elliptic sheaves $\BE,\BF$ is a number $n\in \BZ$ and a sequence of morphisms $\phi_i:\CE_{i }\to\CF_{i+n}$ of right $\cD$-modules satisfying the obvious compatibility with the other data.
\end{defn}
For a sheaf $\CF$ of $\CO_X$-modules and a finite closed subscheme $I$ of $X-\{\infty\}$,
let $\CF|_{I}$ be the restriction of $\CF$ to $I$.
Let $\hat\CO_F :=\prod_ {v\in |X|-\{\infty\}}\CO_{F_v}$. Let $$\CF \otimes\hat\CO_F:=\vpl_I \CF|_{I}=\prod_{v\in |X|-\{\infty\}}\CF_v$$ where the inverse limit is over all finite closed subschemes of $ X-\{\infty\} $.
For a $\cD$-elliptic sheaf $\BE$ such ${\mathfrak{zero}}_\BE(S)\cap I=\emptyset$, the restrictions of $\CE_i$ and $t_i$ to $I\times S$ are independent of $i$. Let $\BE|_{I}$ and $t|_{I}$ be these restrictions. Define a level-$I$-structure on $\BE$
to be an isomorphism $$\kappa:\cD|_{I}\boxtimes\CO_S\simeq \BE|_{I}$$ of right $\cD|_{I}\boxtimes\CO_S$-modules such that the following diagram is commutative: $$ \xymatrix{
& \cD|_{I}\boxtimes\CO_S \ar[dr]^{\kappa} \ar[dl]_{{}^\tau\kappa}\\
{}^\tau \BE|_{I} \ar[rr]^{t|_{I}} & & \BE|_{I}.}$$
Let $\Ell_I$ be the set-valued functor $$S\mapsto \{\cD\mbox{-elliptic sheaves over }S \mbox{ with level-}I \mbox{ structures}\}/\simeq$$ on the category of $\BF_q$-schemes.
Note that there is a morphism of functors $$ \Ell_I\to X-{\mathrm{Ram}}-I $$ by mapping a $\cD$-elliptic sheaf over $S$ to ${\mathfrak{zero}}_\BE$ (which is an $S$-point of $X-{\mathrm{Ram}}-I $).
\begin{thmdefn}[{\cite{DriEll1}\cite[(4.1,5.1,6.1)]{LLS}}]\label{LLSsmooth}
Assume that $I$ is nonempty.
(1) The functor $\Ell_I$ is represented by a smooth $\BF_q$-scheme, which we denote by $\CM_I$.
(2) The morphism $ \Ell_I\to X-{\mathrm{Ram}}-I $ is represented by a smooth morphism $ \CM_I\to X-{\mathrm{Ram}}$ of relative dimension 1 which factors through $X-{\mathrm{Ram}}-I$. Moreover, if $D$ is a division algebra, the morphism $ \CM_I\to X-{\mathrm{Ram}}-I$ is proper.
\end{thmdefn}
Define the modular curve $M_I$ to be the smooth compactification of the generic fiber of $\CM_I$.
The smooth compactification is only needed when $D$ is a matrix algebra. The points in $M_I$ added by the smooth compactification are called cusps of $M_I$.
There is a right action of $(\cD\otimes \hat\CO_F)^\times $ on $M_I$ by acting on level structures (extended to the compactification).
For $J\supset I$, let $\pi_{J,I}:M_J\to M_I$ be the natural finite morphism which is \etale outside cusps.
Define an $F$-procurve \begin{equation*}M:=\vpl_I M_I\end{equation*} where the transition maps are $\pi_{J,I}$'s.
If $D$ is a matrix algebra, points in $M$ whose images in $M_I$ are cusps are called cusps of $M$.
From now on, we only consider $\BE$ such that ${\mathfrak{zero}}_\BE$ factors through the generic point of $X$. Let
$\BE \otimes\hat\CO_F:=\vpl_I\CE_i |_{I},$ where the inverse limit is over all finite closed subschemes $I\subset X-\{\infty\} $. This definition is independent of $i$. We have the induced morphism $$t \otimes\hat\CO_F:=t _i \otimes\hat\CO_F:{}^\tau\BE \otimes\hat\CO_F\to \BE \otimes\hat\CO_F$$ which is independent of $i$. Define an infinite level structure on $\BE$ to be an isomorphism $$\kappa:(\cD \otimes \hat\CO_F)\boxtimes\CO_S\simeq \BE \otimes\hat\CO_F $$ of right $(\cD \otimes \hat\CO_F)\boxtimes\CO_S$-modules such that the following diagram is commutative: $$ \xymatrix{ & (\cD \otimes \hat\CO_F)\boxtimes\CO_S\ar[dr]^{\kappa} \ar[dl]_{{}^\tau\kappa}\\
{}^\tau \BE \otimes\hat\CO_F\ar[rr]^{t \otimes\hat\CO_F} & & \BE \otimes\hat\CO_F.}$$ Distinguish the notion``infinite level structures" here and ``level structures at $\infty$" in \ref{LSI}.
The following lemma is easy to prove. \begin{lem}\label{infmod} The procurve $M$, excluding cusps if $D$ is a matrix algebra, is the moduli space of $\cD$-elliptic sheaves over $F$-schemes with infinite level structures. \end{lem} In particular, there is a natural action of $(\cD\otimes \hat\CO_F)^\times $ on $M$.
We summarize \cite[Section 5, D)]{DriEll1} \cite[Proposition 9.3]{DriEll1} and \cite[(7.1)-(7.4)]{LLS} as follows.
\begin{prop}\label{BBaction}
The action of $(\cD\otimes \hat\CO_F)^\times $ on $M$ extends to a right action of $D^\times(\BA_{\mathrm{f}})$.
\end{prop} Let us describe this construction since it will be referred below.
\begin{proof}
Let $\BE=\{\CE_i : i\in \BZ\}$ be a $\cD$-elliptic sheaf with infinite level structure $\kappa$. The construction is divided into two parts. First, let $g\in \BA_{\mathrm{f}} ^\times$, which corresponds to a line bundle $\CL$ on $X$ with ``infinite level structure".
The collection $\{\CE_i\otimes \CL: i\in \BZ\}$ is naturally a $\cD$-elliptic sheaf with infinite level structure. This gives the action of $g$. Second, let $g\in \BB_{\mathrm{f}}^\times \cap \cD \otimes \hat\CO_F$. Combined with $\kappa$, $g$ gives an endomorphism $[g]$ on $ \BE \otimes\hat\CO_F $. This endomorphism $[g]$ produces
another $\cD$-elliptic sheaf $\BE'$ as follows. Define $\CE_i'$ by the following cartesian diagram:
$$
\xymatrix{ \CE_i' \ar[r]^{ }\ar[d]_{ } &\CE_i \otimes\hat\CO_F \ar[d]^{ [g] } \\
\CE_i \ar [r]_{ } & \CE_i \otimes\hat\CO_F }; $$ the definitions of $t_i',j_i'$ are obvious.
The top morphism induces an isomorphism $$\alpha :\BE' \otimes\hat\CO_F\simeq\BE \otimes\hat\CO_F .$$
The level structure $\kappa'$ on $\BE'$ is defined to be the composition of $\kappa$ and $\alpha^{-1}.$ This gives the action of $g$. \end{proof}
\subsubsection{Level structures at $\infty$}\label{LSI}
When $D=\RM_2$, Drinfeld \cite{DriEll2} \cite{DriCar} introduced level structures of elliptic sheaves at $\infty$. When $D$ is a division algebra, the definition of level structures of $\cD$-elliptic sheaves at $\infty$ is given in \cite[Section 8]{LLS}, and depends on the choice of a uniformizer $\varpi_\infty$ of $F_\infty$, which we fix from now on.
We do not recall the definitions here, but only note that the level structures at $\infty$ there should be considered as ``infinite level structures at $\infty$".
Let $\tilde\Ell_I$ be the set-valued functor on the category of $\BF_q$-schemes: $$S\mapsto \{\cD\mbox{-elliptic sheaves over }S \mbox{ with level-}I \mbox{ structures and level structures at } \infty\}/\simeq.$$
\begin{thmdefn}[{\cite{DriEll2}\cite{DriCar}\cite[(8.10)]{LLS}}]\label{LLSsmooth'} (1) The functor $\tilde\Ell_{I}$ is represented by an $\BF_q$-scheme which we denote by $\tilde\CM_I$.
(2) The natural morphism $ \tilde\CM_I\to \CM_I $ is pro-finite pro-Galois with Galois group $\BB_\infty^\times/\varpi_\infty^\BZ$.
\end{thmdefn}
Let $U(I)$ be the principal congruence subgroup of level $I$ in $(\cD\otimes \hat\CO_F)^\times$.
For $U_\infty\subset \BB^\times_\infty$ open subgroup containing $\varpi_\infty^\BZ$, let $ \CM_{U(I)U_\infty}$ be the quotient of
$\tilde \CM_I$ by $U_{\infty}$.
Then the morphism $\CM_{U(I)U_\infty}\to \CM_I$ is finite \'etale.
In particular, the generic fiber of $\CM_{U(I)U_\infty}$ is smooth. Define the modular curve $ M_{U(I)U_\infty}$ to be the smooth compactification of the generic fiber of $\CM_{U(I)U_\infty}$, and call the points added by the smooth compactification cusps. Define an $F$-procurve $$ M_{U_\infty}:=\vpl_{I} M_{U(I)U_\infty}.$$ Fix an isomorphism $ D^\times(\BA_{\mathrm{f}})\simeq \BB_{\mathrm{f}}^\times $. Then $M_{U_\infty }$ is endowed with a right action of $ \BB^\times /U_\infty$, lifting the one in Proposition \ref{BBaction} (see \cite[(8.7)]{LLS}).
Define $$T_g: M_{U_\infty}\to M_{ U_\infty}$$
to be the right action of $g\in \BB ^\times$.
For an open compact subgroup $U$ of $ (\cD\otimes \hat\CO_F)^\times\simeq \BB_{\mathrm{f}}^\times $, we always assume that $U$ is contained in the conjugation of a certain $U(I)$ with $I$ nonempty. Let $$ M_{UU_\infty}= M_{U_\infty}/U,$$ which is a smooth projective curve over $F$. Points in $M_{UU_\infty}$ from cusps of $M_{U_\infty}$ are called cusps of $M_{UU_\infty}$. Let $\cusp$ be the closed subscheme of cusps of $M_{UU_\infty}$
\subsubsection{Equivalence} \label{Equivalence} Let $\cD'$ be another maximal order of $D$. Then $\cD$ and $\cD'$ are locally isomorphic.
By \cite[Proposition 3.1, Remarks 4.3 (g), Proposition 5.10]{Spi}, the moduli stacks of $\cD$-elliptic sheaves and $\cD'$-elliptic sheaves are isomorphic. It is easy to verify that the isomorphism extends to moduli spaces with level structures (as level structures are local properties), and are compatible with $\BB^\times$-actions.
In particular,
if $D=\RM_{2,F}$ is the matrix algebra, we take $\cD=\RM_{2}(\CO_X)$. Then
the moduli spaces are isomorphic to the ones considered by Drinfeld \cite{DriEll1} \cite{DriEll2} \cite{DriCar}.
\subsubsection{Decomposition of cohomology}\label{LLSmain}
Let
$$H^1(M_{U_\infty,F^\sep},\bar\BQ_l)=\varinjlim_U H^1(M_{UU_\infty, F^\sep},\bar\BQ_l) $$ where the limit is over open compact subgroups of $\BB_{\mathrm{f}}^\times$.
For $g\in \BB ^\times$, let $T_g^*
$ be the pullback by $T_g$.
For $f\in \CH_{\bar\BQ_l}$, let $f$ act on $H^1(M_{U_\infty, F^\sep},\bar\BQ_l)$ by
$$T(f):=\int_{\BB^\times/\Xi_\infty } f(g) T_g^*.$$
If $f\in \CH_{U ,\bar\BQ_l}$, then
$$T(f)(H^1(M_{ U_\infty,F^\sep},\bar\BQ_l))\subset H^1(M_{ UU_\infty, F^\sep},\bar\BQ_l) .$$
For $\pi\in \CA _{U_\infty}(\BB^\times,\bar\BQ_l)$, let $\LC(\pi)$ be the
unique irreducible representation of $\Gal(F^\sep/F)$ over $\bar\BQ_l$ of dimension 2 such that $L(s,\pi)=L(s+1/2,\LC(\pi)).$
It is the (suitably normalized)
Langlands correspondence of the Jacquet-Langlands correspondence of $\pi$ to $\GL_{2,F}$.
Recall that the notion ``Langlands correspondence" only means the local compatibility of automorphic and Galois representations
for almost all places and the existence for $\GL_{2,F}$ was established by Drinfeld \cite{DriEll2}\cite{Dri3}. The compatibility over all places under ``Langlands correspondence" is a theorem of L.Larfforgue \cite[Corollaire VII.5]{LL}.
\begin{thm}[Drinfeld, Laumon-Rapoport-Stuhler]\label{semisimple} Let
$U_\infty$ be an open normal subgroup of $\BB_\infty^\times$ containing $\varpi^\BZ$.
There is an isomorphism of $\BB^\times\times \Gal(F ^\sep/F )$-representations:
$$H^1(M_{U_\infty, F^\sep},\bar\BQ_l) \simeq
\bigoplus_{\pi\in \CA_{U_\infty}(\BB^\times,\bar\BQ_l)} \pi \boxtimes \LC(\pi).$$
\end{thm}
\begin{proof}If $D$ is the matrix algebra, the theorem follows from \cite{DriEll2} by \ref{Equivalence}. If $D$
is a division algebra, we use \cite[(13.8)]{LLS}.
The functions satisfying the second condition in \cite[(13.8)]{LLS} are already used in \cite[p. 166]{DriEll2}.
The first condition in \cite[(13.8)]{LLS} follows from the second by the Weyl integration formula \cite[(7.2.2)]{JL} (see \cite[Lemma 2]{Fli} for the argument).
\end{proof}
\subsubsection{Rigid analytic uniformization at $\infty$}\label{Rigid analytic uniformization}
Let $ \Omega_\infty $ be Drinfeld's rigid analytic upper half plane over $F_\infty$.
For an integer $n\geq 0$, let $\Sigma_n $ be Drinfeld's $n$-th covering of the base change of $\Omega_\infty$ to
the (separable) unramified quadratic extension $F'_\infty$ of $F_\infty$
(see \cite{Gen}).
Suitably choose the deformation and level structure data defining $\Omega_\infty$ and $\Sigma_n $ such that
they are equipped with (necessary compatible) left actions of $\GL_2(F_\infty) $ and right actions of $\BB_\infty^\times. $
Moreover, we require that the left action of $\GL_2(F_\infty) $ on $\Omega_\infty$ is the
action by fractional linear transformations.
Let $D^\times$ act on $\Omega_\infty$ via an isomorphism $D_\infty \simeq \RM_2(F_\infty)$ and act on $\BB_{\mathrm{f}}^\times$ via the isomorphism $D^\times(\BA_{\mathrm{f}})\simeq \BB_{\mathrm{f}} $
\begin{prop}
\label{riguni}
Let $U_\infty\subset \BB^\times_\infty$ be generated by $\varpi^\BZ_\infty$ and the principal congruence subgroup of level $n\geq 0$. For every open compact subgroup $U\subset \BB_{\mathrm{f}}^\times$, there are isomorphisms of rigid analytic space over $F_\infty$:
$$M_{U\BB^\times_\infty}^\an-\{\cusp\}\simeq D^\times \backslash\Omega_\infty \times \BB_{\mathrm{f}} ^\times /U,
$$
and
$$M_{UU_\infty}^\an-\{\cusp\}\simeq D^\times \backslash\Sigma_n \times \BB_{\mathrm{f}} ^\times /U
$$
such that
the actions of $ \BB^\times$ on the inverse systems $( M_{U\BB^\times_\infty}^\an)_U$ and $( M_{UU_\infty}^\an)_U$ are compatible with
the natural actions of $ \BB^\times$ on the inverse systems $(D^\times \backslash\Omega_\infty \times \BB_{\mathrm{f}} ^\times /U)_U$ and $(D^\times \backslash\Sigma_n \times \BB_{\mathrm{f}} ^\times /U)_U.$
\end{prop}
\begin{proof} When ${\mathrm{Ram}}=\{\infty\}$, the proposition
is proved in \cite {DriEll1}\cite{DriCar}. Let ${\mathrm{Ram}}\neq \{\infty\}$. For $M_{U\BB_\infty^\times}$, the isomorphism is given in \cite[Theorem 4.4.11]{BS} and the compatibility holds. Let $U_\infty\subset \BB^\times_\infty$ be generated by $\varpi^\BZ_\infty$ and the principal congruence subgroup of level $n\geq 0$. The isomorphism for $M_{UU_\infty }$ is obtained as follows.
If there exists $v\in {\mathrm{Ram}}-\{\infty\}$ such that $U_v$ is maximal, apply \cite[Proposition 4.28]{Spi} to \cite[Theorem 8.3]{Hau} to get the isomorphism for $M_{UU_\infty }$. In general, let $v\in {\mathrm{Ram}}-\{\infty\}$, and let $U'=U^v\cD_v^\times$. Then $$M_{UU_\infty}= M_{U'U_\infty}\times_{M_{U'\BB_\infty^\times}}M_{U\BB_\infty^\times}.$$ Then the isomorphism for $M_{UU_\infty}$ is obtained from isomorphisms for all three modular curves on the right hand side.
\end{proof}
\subsubsection{Redefine notations}\label{useU}
If not specified, let $U_\infty\subset \BB^\times_\infty$ be $ \BB^\times_\infty$ or be generated by $\varpi^\BZ_\infty$ and the principal congruence subgroup of $\BB_\infty^\times$ of level $n>0$.
We use the symbol $M$ for $M_{U_\infty}$ and $M_U$
for $M_{UU_\infty}$. Let $\BC_\infty$ be the completion of the algebraic closure of $F_\infty$. We use the symbol $[z,h]$ (resp. $[z,h]_U $), where $z\in \Omega_\infty $ or $\Sigma_n $ and $h\in \BB_{ \mathrm{f}}^\times $, to represent a point in $M $ (resp. $M_U $) via Proposition \ref{riguni}.
\subsubsection{Jacobians and Height pairings} \label{Jacobians and Height pairings}
We define two Jacobians
$J$ and $J^\vee$ as in \cite[3.1.6]{YZZ}.
For an open compact subgroup ${U }$ of $\BB ^\times$.
Let $J_U$ be the Jacobian variety of $M_U$.
By \cite[Proposition 6.9]{MFK}, there is a canonical isomorphism
\begin{equation}J_U\simeq J_U^\vee.\label{GIT}\end{equation}
\begin{defn}
Let $J $ be the inverse system $(J_U)_U$ where the transition morphisms are induced by pushforwards of divisors, $J^\vee $ be the direct system $(J_U)_U$ where the transition morphisms are induced by pullbacks of line bundles. For a field extension $F'/F$ and a $\BZ$-algebra $R$,
let $$J(F')_R:=\vpl_U J_U(F') \otimes_\BZ R=\vpl_U\Cl^0(M_{U,F'}) \otimes_\BZ R,$$
$$ J^\vee(F')_R:=\varinjlim _UJ_U^\vee(F')\otimes_\BZ R=\varinjlim_U\Pic^0(M_{U,F'}) \otimes_\BZ R .$$
If $R=\BZ$, the subscript $R$ is omitted.
Let $$\Pic(M\times M)_R:=\varinjlim _U\Pic(M_U\times M_U)\otimes _\BZ R,$$
where the transition morphisms are pullbacks of line bundles.
Define \begin{equation}\Hom(J,J^\vee)_R:=\varinjlim_U \Hom(J_U,J_U^\vee )_R,\label{HJJ}\end{equation}
where
the transition map for $U'\subset U$ is
$\phi\mapsto \pi_{U',U}^*\circ \phi\circ\pi_{U',U,*}.$
\end{defn}
By \cite[Lemma 3.2]{YZZ}, the pushforward by a correspondence defines a map \begin{equation}\Pic(M\times M)_R\to \Hom(J,J^\vee)_R.\label{PICJ}\end{equation}
Define the \Neron-Tate height pairing $$\pair{\cdot,\cdot}_\NT :J_U(F^\sep)_\BQ \times J_U^\vee(F^\sep)_\BQ\to \BR$$ as in \cite[7.1]{YZZ}. By the projection formula, this pairing extends to the \Neron-Tate height pairing $$\pair{\cdot,\cdot}_\NT:J (F^\sep)_\BQ \times J^\vee (F^\sep)_\BQ\to \BR$$ which further induces $$ \pair{\cdot,\cdot}_\NT:J (F^\sep)_\BC \times J ^\vee(F^\sep)_\BC\to \BC.$$ We swap $J (F^\sep)_\BQ $ and $ J^\vee (F^\sep)_\BQ $ in the definition of $\pair{\cdot,\cdot}_\NT$ when necessary. Let $Z \in \Pic(M\times M)_\BQ $, and
$x,y\in J (F^\sep)_\BQ$, then
$\pair{Z_* x,y}_\NT$ is well-defined.
\subsubsection{Hodge classes}\label{Hodge classes}
For our purpose, we only need to consider the case when $U$ is small enough so that there is no ``elliptic points" issue.
Let $\omega_{M_U/F}$ be the canonical bundle of $M_U$ over $F$.
If $U_\infty=\BB_\infty^\times$, define the Hodge class of $M_U$ to be
\begin{equation*}L_U:=\omega_{M_U/F}(2 \cdot \cusp). \end{equation*}
In general, define the Hodge class $L_U$ to be the pullback of the Hodge class of $M_{U\BB_\infty^\times}$. \begin{lem} \label{nmb}For every $U'\subset U$, $L_{U'}=\pi_{U',U}^*L_U$. \end{lem} \begin{proof} If $D$ is a division algebra, the lemma follows from the fact that $\pi_{U',U}$ is \'{e}tale. (In fact, we always have $L_U=\omega_{M_U/F}$.) If $D$ is the matrix algebra, the lemma follows from an explicit computation of the ramifications at cusps \cite[VII, Theorem 5.11]{Gek}.
\end{proof} \begin{defn}Define $ \Vol (M_U):=\deg L_U$. \end{defn}
By Lemma \ref{nmb} and the projection formula, we have $\Vol(M_{U'})/\Vol(M_U) =\deg\pi_{U',U} $
for $U'\subset U$. Then a direct computation gives the following corollary. \begin{cor}\label{the constant}
The number
$$\frac{\Vol (\tilde U/\Xi )\Vol(M_U)}{ |F^\times\backslash \BA_F^\times/ \Xi|} $$ is independent of $ U$. (The notations are as in \ref{measures}.) \end{cor} Indeed, after normalization of measures, this number is 4 (see Lemma \ref{the constant'}).
For $\alpha\in \pi_0(M_{U,F^\sep})$, let $M_{U,\alpha}$ be the corresponding geometrically connected component and $L_{U,\alpha}:=L_U|_{M_{U,\alpha}}$. Define normalizations $\xi_{U,\alpha}:=\frac{1 }{\deg L_{U,\alpha}} L_{U,\alpha}$. For $ \alpha=(\alpha_U)\in \pi_0(M_{ F^\sep})$, the sequence $(\xi_{U,\alpha_U})_U$, indexed by $U$, defines an element $$ \xi_\alpha\in\vpl \Cl(M_{U,\alpha_U})_\BQ. $$ For $x\in M(F^\sep)$ in the connected component of $\alpha$. Then $x-\xi_\alpha\in J(F^\sep)_\BQ .$ This defines a map \begin{equation}M(F^\sep)\hookrightarrow J(F^\sep)_\BQ .\label{xia}\end{equation}
\subsection{CM points}\label{CM}
We define CM points, and prove the algebraicity of CM points.
\subsubsection{Endomorphisms of $\cD$-elliptic sheaves} Let $\BE$ be a $\cD$-elliptic sheaf.
For $f\in \CO_F:=H^0(X-\{\infty\}, \CO_X) $, multiplication by $f$ on each $\CE_i$ gives an endomorphism of $\BE$.
In particular, $\End(\BE)$ is a $\CO_F$-algebra.
Let $\BC_\infty$ be the completion of the algebraic closure of $F_\infty$.
Suppose that $\BE$ is defined over $\BC_\infty$ and ${\mathfrak{zero}}_\BE:\Spec \BC_\infty\to X$ of $\BE$ factors through the generic point of $X$.
\begin{lem}There is an embedding $\End(\BE) \hookrightarrow \cD\otimes{\CO_F}$, and $\End(\BE)\otimes_{\CO_F}F$ is a field extension of $F$ which is not split over $\infty$, and of degree at most 2.
\end{lem}
\begin{proof} By the analytic uniformization \cite[2.13, 3.6]{Tae2}, there is a rank one $\cD\otimes{\CO_F}$-lattice $\Lambda$ in a $D$-representation on $\BC_\infty^2$ such that $$\End(\BE)\simeq \{ \lambda\in \BC_\infty:\lambda\Lambda\subset \Lambda \}\subset \End_{\cD\otimes{\CO_F}}(\Lambda)\simeq\cD\otimes{\CO_F}.$$
Thus $\End(\BE)\otimes_{\CO_F}F$ is a commutative subalgebra of $D$. Then the lemma follows.
\end{proof}
\subsubsection{CM $\cD$-elliptic sheave and CM points}
Let $E/F$ be a quadratic field extension nonsplit over $\infty$, which is fixed from now in this and next section.
\begin{defn}A $\cD$-elliptic sheaf $\BE$ has CM by $E$ if $\End(\BE)\otimes_{\CO_F}F\simeq E$.
\end{defn} A point in $M(\BC_\infty)$ or $M_U(\BC_\infty)$ is called a CM point if it
corresponds to a $\cD$-elliptic sheaf with CM by $E$.
Let $CM$ (resp. $CM_U$) be the set of all CM points in $M$ (resp. $M_U$).
We will prove that all CM points are defined over $E^\ab$, the maximal abelian extension of $E$. Then regard $CM$ (resp. $CM_U$) as a subset of $M(E^\ab)$ (resp. $M_U(E^\ab)$).
Let $x \in M(\BC_\infty)$, and let $\BE$ be the corresponding elliptic sheaf. Associated to $x$ is an infinite level structure $\kappa$ on $\BE$ and a level structure $\kappa_\infty$ at $\infty$.
The actions of endomorphisms of $\BE$ on $\kappa$ and $\kappa_\infty$ defines a group morphism \begin{equation}(\End(\BE)\otimes_{\CO_F}F)^\times \to \BB^\times/U_\infty \label{eq241}.\end{equation} The following lemmas are easy to be verified.
\begin{lem} The image of $(\End(\BE)\otimes_{\CO_F}F)^\times$ under \eqref{eq241} fixes $x$.
\end{lem}
In particular, if $x\in CM$, $x\in M(\BC_\infty)^{e(E^\times)}
$ for some embedding $e:E \hookrightarrow \BB $ of $F$-algebras.
\begin{lem} For an embedding $e:E \hookrightarrow \BB $ of $F$-algebras, a point in $M(\BC_\infty)$ fixed by $e(E^\times)$ corresponds to a CM $\cD$-elliptic sheaf $\BE$ with a infinite level structure over all finite places and a level structure at $\infty$ and such that image of \eqref{eq241} is $e(E^\times)$.
\end{lem}
To sum up, we have a decomposition of the set CM of points\begin{equation}CM=\bigcup_{e:E \hookrightarrow \BB } M(\BC_\infty)^{e(E^\times)}.\label{allcm}\end{equation}
\subsubsection{CM points under the rigid analytic uniformization}\label{CMuni0}
We describe the CM points in $M(\BC_\infty)$ under the rigid analytic unformization at $\infty$ in \ref{Rigid analytic uniformization}. We have two isomorphisms $i_\infty:D_\infty \simeq \RM_2(F_\infty)$ and $i_{\mathrm{f}}:D (\BA_{\mathrm{f}})\simeq \BB_{\mathrm{f}}$ in the definition of unformization.
Let $e_\infty:E\hookrightarrow \BB_\infty $ and $d:E\hookrightarrow D$ be embeddings of $F$-algebras. Let $e$ be the product of $e_\infty$ and the composition $i_{\mathrm{f}}\circ d$.
Let $z_0\in \Sigma_n(\BC_\infty)$ be a fixed point of $E^\times $ via $$\left((i_{\infty}\circ d)^{-1}, e_\infty \right): E^\times \hookrightarrow \GL_2(F_\infty)\times \BB_\infty^\times.$$
By the Noether-Skolem theorem, there exists $j \in \BB^\times $ such that $jgj^{-1}=\bar g$ for every $g\in e(E)$ where $\bar g$ is the Galois conjugate of $g$. Then the normalizer $H$ of $e(E^\times)$ in $\BB^\times$ is isomorphic to $ \BA_E^\times \cup \BA_E^\times j$. It is not hard to prove that \begin{equation*}M (\BC_\infty)^{e(E^\times)}=\{[z_0h_\infty,h_{\mathrm{f}}]:h\in H\}. \end{equation*}
For a general embedding $E\hookrightarrow \BB$, by the Noether-Skolem theorem, there exists $g\in \BB^\times$ such that the embedding is $ g^{-1}e g$. Then \begin{equation*}M (\BC_\infty)^{e(E^\times)}=\{[z_0h_\infty g_\infty,h_{\mathrm{f}}g_{\mathrm{f}}]:h\in H\}. \end{equation*} In particular, from \eqref{allcm} we have \begin{equation*}CM=\{[z_0g_\infty,g_{\mathrm{f}}]:g\in \BB ^\times\}. \end{equation*}
\subsubsection{Construction of CM $\cD$-elliptic sheaves}\label{CM theory}
Let $\pi:X'\to X$ be a double cover which is a smooth projective model of $E/F$. Let $\infty'$ be the unique preimage of $\infty$. Let $\CO_E =H^0(X'-\{\infty'\},\CO_{X'})$ the ring of integers of $E$ away from $\infty'$.
Let $S$ be an $\BF_q$-scheme.
\begin{defn}
An elliptic sheaf $\BL$, on $X'$ with respect to $\infty'$, of rank 1 over $S$ is the following data: a morphism ${\mathfrak{zero}}_\BL:S\to X'$ and
a sequence of commutative diagrams $$
\xymatrix{
{}^\tau \CL_{i-1} \ar[r]^{{}^\tau j_{i-1}}\ar[d]_{ t_{i-1}} &{}^\tau \CL_{i} \ar[d]^{t_{i} } \\
\CL_i \ar [r]_{j_i} & \CL_{i+1} } $$ indexed by $i\in\BZ$,
where each $\CL_i$ is a line bundle on $X'\times S$ such that $$\CL_{i+\deg(\infty')}=\CL_i(\infty'),$$
and $j_i,t_i$ are injections compatible with $\cD$-actions and satisfy the following conditions: \begin{itemize} \item[(1)] the composition $j_{i+\deg(\infty')-1}\circ\cdot\cdot\cdot\circ j_{i+1}\circ j_i$ is the canonical inclusion $\CL_i\hookrightarrow\CL_i(\infty')$; \item[(2)] let $\pr_S:X\times S\to S$ be the projection, then $\pr_{S,*}(\coker j_i)$ is a locally free $\CO_S$-module of rank $1$; \item[(3)] $\coker t_i$ is the direct image of a locally free $\CO_S$-module of rank 1 by the graph morphism $({\mathfrak{zero}}_\BL,\id): S\to X'\times S$.
\end{itemize}
\end{defn}
\begin{lem}\label{cdl}There exists a maximal order $\cD$ which admits an embedding of $\CO_X$-algebras: \begin{equation}\cD\hookrightarrow \pi_*( \End(\CO_X\oplus \CL)).\label{ebd}\end{equation} where $\CL$ is a line bundle on $X'$.
\end{lem} \begin{proof}Embed $D$ in $ \RM_2(E)$ as the subalgebra of matrices $ \begin{bmatrix}a&b\ep\\
\bar b&\bar a\end{bmatrix} $ where $\ep\in F^\times$ and $a,b\in E$ (see \eqref{(5.1)}). For $x\in |X|$, let
$\fp_{E_x}$ be the maximal ideal of $\CO_{E_x}$. For an integer $n$, we have an order $\CO_n=\End_{\CO_{E_x}}(\CO_{E_x}\oplus \fp_{E_x}^{n_x})$ of $ \RM_2(E_x)$. There exists an integer $n_x$ such that
$D_x\bigcap \CO_{n_x}$ is a maximal order of $D_x$ and $n_x=0$ outside a finite set $I\subset |X|$.
Let $\cD$ correspond to $\{D_x\bigcap \CO_{n_x}\}_{x\in |X|}$ via Lemma \ref{order} and
$\CL=\otimes_{x\in I} \CO(-n_x x)$.
\end{proof}
By \ref {Equivalence}, we can let $\cD$ be as in Lemma \ref{cdl}. \begin{lem} \label{pil}
The $\cD$-action on $\pi_*(\CL_i \oplus (\CL_i\otimes \CL))$ by the embedding \eqref{ebd} makes
$$ ( \pi\circ {\mathfrak{zero}}_\BL, \pi_*(\CL_i \oplus (\CL_i\otimes \CL)),\pi_*(j_i \oplus (j_i\otimes \id_{\CL} )),\pi_*(t_i \oplus( t_i\otimes \id_{\CL} )): i\in \BZ)$$
a $\cD$-elliptic sheaf on $X$ of rank 2 over $S$.
\end{lem}
\begin{proof}
The condition at $\infty$ follows from the projection formula and that $\pi^*\CO_{X}(\infty)= \CO_{X'}(\infty')$.
The remaining verification is trivial. \end{proof}
Let $M^1 $ be the moduli space over $E$ of rank 1 elliptic sheaves over $E$-schemes with all level structures (defined similarly as in \ref{Moduli spaces of}).
From Lemma \ref{pil}, we have an $E$-scheme morphism $\Pi:M^1\to M_E,$ where $M_E$ is the base change to $M$ to $E$. We describe the map when $U_\infty=\BB_\infty^\times.$ Let $\hat\CL=\CL \otimes\hat\CO_E$.
By \cite[(18.7) Theorem]{Rei}, we can fix an isomorphism of right $\cD \otimes \hat \CO_F$-modules:
\begin{equation} \hat\CO_E \oplus \hat\CL\simeq \cD \otimes \hat \CO_F\label{CMisom},\end{equation}
where $\cD \otimes \hat \CO_F$ acts on the left hand side by \eqref{ebd}. For an $E$-scheme $S$ and a point $(\BL,\kappa)$ in $M^1(S)$ forgetting the level structure at $\infty$,
where $\kappa$ is an infinite level structure over all finite places,
let $\BE $ be the $\cD$-elliptic sheaf on $X$ of rank 2 defined in Lemma \ref{pil}.
Then tautologically \begin{equation}\BE\otimes \hat \CO_F \simeq \hat\CO_E \oplus \hat\CL \label{CMisom0}\end{equation} as right $\cD\boxtimes \CO_S$-modules, where the $\cD \otimes \hat \CO_F$-module structure on the left hand side is give by \eqref{ebd} and $\kappa$.
Thus \eqref{CMisom} and \eqref {CMisom0} give an isomorphism $$ (\cD \otimes \hat\CO_F)\boxtimes\CO_S\simeq \BE \otimes\hat\CO_F , $$ i.e. an infinite level structure on $\BE$. This gives the morphism $\Pi:M^1\to M_E.$
There is an $\BA_E^\times/E^\times$-action on $M^1$ by acting on level structures. This is related to the $\BB^\times$-action on $M_E$ as follows. Define an embedding $\BA_{E}\hookrightarrow \BB $ as follows.
By \eqref{CMisom}, we have an isomorphism $$\End_{\cD \otimes \hat \CO_F}( \hat\CO_E \oplus \hat\CL)\simeq \End_{\cD \otimes \hat \CO_F}( \cD \otimes \hat \CO_F)\simeq \cD \otimes \hat \CO_F.$$
The diagonal left action of
$\hat\CO_E$ on $\hat\CO_E \oplus \hat\CL$ gives an embedding
$\hat\CO_E\hookrightarrow \cD \otimes \hat \CO_F.$ Then we have an embedding $\BA_{E,\mathrm{f}}\hookrightarrow \BB_{\mathrm{f}} $. The construction at $\infty $ is defined in the same way. The embedding gives a group morphism \begin{equation}\BA _E^\times\to \BB^\times\to \BB^\times/U_\infty\label{CMebd'}.\end{equation} Let $\BA _E^\times$ act on $ M$ via \eqref{CMebd'}. By the description of the action of $\BB^\times$ on $M$ \cite[(8.7)]{LLS} (see the proof of Proposition \ref{BBaction} when $U_\infty=\BB_\infty^\times$),
we have the follow lemma. \begin{lem}\label{CMcpt} The morphism
$ \Pi: M^1\to M_E$ is compatible with the actions of $\BA _E^\times$.
In particular, $\Pi(M^1)\subset ( M_E)^{E^\times}$.
\end{lem}
\subsubsection{Algebraicity of CM points} Let $E^{\ab,\varpi_\infty}\subset E^\ab$ be the maximal subfield fixed by the image of $\varpi_
\infty$ under the reciprocity map $\BA_E^\times/E^\times\to \Gal(E^\ab/E).$
\begin{prop}[{\cite[Corollary of Proposition 2.2]{DriEll2}}]\label{CFT}
We have $ M^1 \simeq \Spec E^{\ab,\varpi_\infty}$. Moreover, the $\BA_E^\times /E^\times$-action on $ M^1$ coincides with the $ \Gal(E^{\ab }/E)$-action via the reciprocity map.
\end{prop}
Combining \ref{CMuni0} with Lemma \ref{CMcpt} and Proposition \ref{CFT}, we have the following corollary.
\begin{cor} \label{TSCM}
(1) The scheme $( M_E)^{E^\times}$
consists of two $\BA_E^\times /E^\times$ orbits and the image of $\Pi $ is one of them.
(2) All CM points, in particular all points in $( M_E)^{E^\times}(\BC_\infty)$, are defined over $E^\ab$. Moreover, the $\BA_E^\times /E^\times$-action on $( M_E)^{E^\times}$ coincides with the $ \Gal(E^{\ab }/E)$-action via the reciprocity map.
\end{cor}
\section{Height distribution}
\label{Height distributions, Rational Representations, and Abelian varieties}
In this section, we first introduce the height distribution on $\BB^\times$ and study its spectral decomposition. Then we reduce Theorem \ref{GZ} to its distribution version, namely Theorem \ref{GZdis'}.
\subsection{Height distribution} \subsubsection{Hecke correspondences}\label{Hecke correspondences}
Continue to use the notations in the last section. For $g\in \BB ^\times,$ let $T_g:M\to M$ be the right action by $g $, $\pi_U:M\to M_U$ be the natural projection.
Let $Z(g)_U'$ be the image of the graph of $T_g$ in $M_U\times M_U$ under $\pi_U\times \pi_U$, as a reduced closed subscheme of $M_U\times M_U$. Define a $\BQ$-coefficient divisor of $M_U\times M_U$:
\begin{equation*}Z(g)_U :=\frac{|\tilde U g \tilde U/ \tilde U|} {|F^\times\backslash F^\times \tilde U g \tilde U/ \tilde U|}Z(g)_U'. \end{equation*} Then the pushforward by $Z(g)_U$ on divisors on $M_{\BC_\infty}$ is the usual Hecke correspondence: for example, under the rigid analytic uniformization at $\infty$, the pushforward by $Z(g)_U$ is \begin{equation*} [z,x]_U\mapsto \sum_{y\in \tilde U g \tilde U/ \tilde U}[zy_\infty,xy_{\mathrm{f}}]_U .\end{equation*}
Let $K/\BQ$ be a field extension and $\CH_{U,K}$ the Hecke algebra of $\BB^\times$ (see \ref {measures}). For $f\in \CH_{U,K}$, define $$Z(f)_U:=\sum_{g\in \tilde U\backslash \BB^\times/\tilde U}f(g)Z(g)_U,$$
and define a normalization $$\tilde Z(f)_U:=\Vol(\Xi_U)|F^\times\backslash \BA_F^\times/\Xi|Z(f)_U.$$
The effect of the normalization factor $\Vol(\Xi_U)|F^\times\backslash \BA_F^\times/\Xi|$ is given by the following lemma.
\begin{lem} [{\cite[ Lemma 3.18]{YZZ}}] \label{normhecke} The line bundles defined by $ \tilde Z(f)_U $'s are compatible under pull back. In particular, the sequence
$(\tilde Z(f)_U )_U $ defines an element in $ \Pic(M\times M)_K.
$
\end{lem}
Denote this element by $\tilde Z(f)\in \Pic(M\times M)_K.
$
Define $\tilde Z(f)_*\in \Hom(J ,J^\vee )_{K }$ by \eqref{PICJ}.
\subsubsection{Cohomological projectors}
We follow \cite[3.3.1]{YZZ}.
Identifying $H^1(M_{U,F^\sep},\bar\BQ_l)$ with $H^1(J_{U,F^\sep},\bar\BQ_l)$, then we have an injection \begin{equation}\Hom(J_U ,J_U )_\BQ\hookrightarrow \Hom(H^1(M_{U,F^\sep},\bar\BQ_l),H^1(M_{U,F^\sep},\bar\BQ_l)) \label{3121} \end{equation} which maps a morphism $\phi\in \Hom(J_U,J_U)$ to its action $\phi^*$ on $H^1$ via pullback. Let $(\phi_U)_U$ be a sequence of elements in $\Hom(J_U ,J_U )_\BQ$ indexed by small enough $U$ such that
\begin{equation} \phi_U(\pi_{U',U,*}(x_{U'}))=\pi_{U',U,*}( \phi_{U'}(x_{U'})).\label{31215} \end{equation}
Then the image of $(\phi_U)_U$ via \eqref{3121} gives an element in $\Hom(H^1(M_{ F^\sep},\bar\BQ_l),H^1(M_{ F^\sep},\bar\BQ_l) )$.
\begin{lem}\label{cu}
Let $ (c_U )_U$ be a sequence of rational numbers such that $c_{U'}/c_U =\deg\pi_{U',U} .$ Let $\psi=(\psi_U)_U\in\Hom(J,J^\vee)_\BQ$. Then the sequence $ (c_{U}^{-1}\psi_U)_U $ satisfies \eqref{31215}.
\end{lem}
\begin{proof} By the definition of $\Hom(J,J^\vee)_\BQ$ (see \eqref{HJJ}), we have $$\pi_{U',U}^*\psi_U(\pi_{U',U,*}(x_{U'}))=\psi_{U'}(x_{U'})$$ for $x_{U'}\in J_{U'}$. Apply $\pi_{U',U,*}$ to both sides, we have $$ \deg \pi_{U',U}\cdot \psi_U(\pi_{U',U,*}(x_U')))=\pi_{U',U,*}\psi_{U'}(x_{U'}).$$ This gives \eqref{31215}.
\end{proof}
The sequence $C:=(c_U)_U$
where \begin{equation}\label{cu1}c_U:= |F^\times\backslash \BA_F^\times/\Xi|/\Vol(\tilde U/\Xi)\end{equation}
satisfies the property in Lemma \ref{cu}. So we have a morphism
\begin{equation} \Hom(J ,J^\vee )_{\bar \BQ_l}\hookrightarrow \Hom(H^1(M_{ F^\sep},\bar\BQ_l),H^1(M_{ F^\sep},\bar\BQ_l) )\label{inclJ} \end{equation} by $$\psi=(\psi_U)_U\mapsto (C^{-1}\psi)^*:=(c_U^{-1}\psi_U^*)_U. $$
\begin{lem}\label{compareHecke}
Let
$ T(f) \in \Hom(H^1(M_{ F^\sep},\bar\BQ_l),H^1(M_{ F^\sep},\bar\BQ_l) )$ be defined as in \ref{LLSmain}, then $$ (C^{-1} \tilde Z (f)_* ) ^*= T(f).
$$
\end{lem}
\begin{proof}
For an open compact subgroup $U$ of $ (\cD\otimes \hat\CO_F)^\times $ such that $f$ is bi-$U$-invariant, we have the restriction of $T(f)$ to ${H^1(M_{U,F^\sep},\bar\BQ_l)}$: $$T(f)|_{H^1(M_{U,F^\sep},\bar\BQ_l)} : H^1(M_{U,F^\sep},\bar\BQ_l)\to H^1(M_{U,F^\sep},\bar\BQ_l) .$$
We show that the pullback action of $\Vol (\tilde U/\Xi_\infty)Z(f)_{U,*}\in \Hom (J_U,J_U)$ on $H^1(J_{U,F^\sep},\bar\BQ_l) $, which is identified with $H^1(M_{U,F^\sep},\bar\BQ_l)$, equals $T(f)|_{H^1(M_{U,F^\sep},\bar\BQ_l)}$. We may assume $f=1_{\tilde Ug\tilde U}$ for some $g\in \BB^\times$.
Consider the following diagram $$
\xymatrix{ H^1(M_{U,F^\sep},\bar\BQ_l) \ar@{.>}[r] \ar[d]_{\Vol(\tilde U/\Xi_\infty) \bigoplus_{h\in \tilde Ug\tilde U/\tilde U} T_h^*} & \ar[d] H^1(M_{U,F^\sep},\bar\BQ_l) \\
\bigoplus_{h\in \tilde Ug\tilde U/\tilde U} H^1(M_{hU h^{-1}, F^\sep},\bar\BQ_l) \ar [r]_{ } & H^1(M_{ F^\sep},\bar\BQ_l) }, $$
then $T(f)|_{H^1(M_{U,F^\sep},\bar\BQ_l)}$ is the dotted arrow making the diagram above commute. So we need to check that $\Vol(\tilde U/\Xi_\infty)Z(f)_{U,*}$ is the dotted arrow making the diagram below commute. $$
\xymatrix{ J_U & \ar@{.>}[l] J_U \\
\ar[u]_{ \bigoplus_{h\in \tilde Ug\tilde U/\tilde U} T_{h,*}} \bigoplus_{h\in \tilde Ug\tilde U/\tilde U} J_{hU h^{-1}} & \ar [l]_{ } \ar[u] J }. $$
This follows from the definition of $Z(g)_{U,*}$. \end{proof}
Let $\pi\in \CA_{U_\infty} (\BB ^\times,\bar\BQ_l)$. The Hecke action of $f\in \CH _{\bar\BQ_l}$ on $\phi\in\pi$ is defined by \begin{equation} \pi(f)\phi(x):=\int_{ \BB ^\times/\Xi_\infty} f(g)\phi(xg)dg\label{pixi}.\end{equation} Let $\tilde \pi$ be the contragradient representation of $\pi$. Let $ \rho_{\pi }:\CH _{\bar\BQ_l}\to \pi\otimes \tilde\pi$ be given by $$\rho_\pi(f)=\sum_{\phi }\pi(f)\phi \otimes \tilde \phi$$ where the sum is over an orthonormal basis $\{\phi\}$ of $\pi$, and $\{\tilde \phi \}$ is the dual basis of $\tilde \pi$. Note that the sum has only finitely many nonzero terms. It is well known that
$$\rho = \bigoplus_{\pi\in \CA_{U_\infty}(\BB^\times,\bar \BQ_l)} \rho_{\pi }: \CH _{\bar\BQ_l}\to \bigoplus_{\pi\in \CA_{U_\infty}(\BB^\times,\bar \BQ_l)} \pi\otimes \tilde\pi $$ is surjective.
\begin{cor}\label{ebdj} The natural embedding
$ \pi \otimes\tilde \pi \hookrightarrow \Hom(H^1(M_{ F^\sep},\bar\BQ_l),H^1(M_{ F^\sep},\bar\BQ_l))$ given by Theorem \ref{semisimple} factors through $\Hom(J,J^\vee)_{\bar\BQ_l}$ via \eqref{inclJ}.
\end{cor}
\begin{proof}By the discussion above, the image of any $$x\in \pi \otimes\tilde \pi \hookrightarrow
\bigoplus_{\pi'\in \CA_{U_\infty}(\BB^\times,\bar \BQ_l)} \pi' \otimes\tilde \pi' ,$$ equals $\rho (f)$ for a certain $f\in \CH _{\bar\BQ_l}$. Since Theorem \ref{semisimple} identifies the usual Hecke action $\rho (f)$ and the Hecke action $T (f) $ on $H^1(M_{F^\sep},\bar \BQ_l)$, the image of $x$ in $\Hom(H^1(M_{ F^\sep},\bar\BQ_l),H^1(M_{ F^\sep},\bar\BQ_l))$ equals $T (f)$. Then the Corollary follows from Lemma \ref{compareHecke} by letting $\tilde Z(f)_*$ be the image of $x$ in $\Hom(J,J^\vee)_{\bar\BQ_l}$. \end{proof}
Fix an isomorphism $c:\BC\simeq \bar\BQ_l$. For $\pi\in \CA_{U_\infty}(\BB ^\times,\BC)$, let $\pi^c:=\pi\otimes_{\BC,c}\bar\BQ_l \in \CA_{U_\infty}(\BB ^\times,\bar\BQ_l)$. Corollary \ref{ebdj}, applied to $\pi^c$,
defines an injective morphism
$$ T_{\pi}':\pi\otimes \tilde\pi \to \Hom(J ,J^\vee )_\BC.$$
Then for $f\in \CH_{\BC}$, we have an equation in $\Hom(J ,J^\vee )_\BC$: \begin{equation}\label{specdecomht} \tilde Z(f)_{ *}=
\bigoplus_{\pi\in \CA_{U_\infty}(\BB^\times,\BC)} T'_{\pi } \circ \rho_{\pi }(f).
\end{equation}
Let $c_U$ be as in \eqref{cu1}, then $ \Vol(M_U)/c_U$ is the number in
Corollary \ref{the constant}, and independent of $U$.
\begin{defn}\label{cohproj}Define the cohomological projector associated to $\pi$ to be \begin{equation*}T_\pi:=\frac{ \Vol(M_U)}{c_U}T_{\pi}':\pi\otimes \tilde\pi \to \Hom(J ,J^\vee )_\BC.\end{equation*} \end{defn}
\subsubsection{The Gross-Zagier-Zhang formula, projector version} \label{Spectral decomposition of cohomology'}
\begin{defn}[{\cite[1.6.7]{YZZ} }]
\label{regint} Let $f$ be a locally constant function on $E^\times\backslash \BA_E^\times $ invariant by $\Xi_\infty$. Define $$ \fint _{\BA_F^\times} f(z)dz= \frac{1}{ |F^\times\backslash \BA_F^\times/ \Xi_\infty V|}\sum_{z\in F^\times\backslash \BA_F^\times/ \Xi_\infty V}f(z) $$ where $V$ is any open compact subgroup of $\BA_{F,\mathrm{f}}^\times$ such that $f$ is invariant by $V$. Define the regularized integral
$$\int_{E^\times\backslash \BA_E^\times }^* f(t) dt=\int_{E^\times\backslash \BA_E^\times / \BA_F^\times} \fint _{\BA_F^\times} f(z)dz .$$
\end{defn}
Fix a CM point $P_0\in M^{e_0(E^\times)}(F^\sep)$.
For $h \in \BB^\times$, let $h^\circ$ be the image of $T_{h}P_0$ in $J(F^\sep)_\BQ$ via \eqref{xia}. Let $\Omega$ be a Hecke character of $E^\times$ valued in $\BC$. Define \begin{equation} H_\pi^{\sharp,\proj}(\phi\otimes\varphi):=\int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int^*_{E^\times\backslash \BA_E^\times } \pair{ T_\pi( \phi\otimes \varphi)(t_1^\circ), t_2^\circ}_{\NT} \Omega^{-1}(t_2)\Omega(t_1) dt_1dt_2.\label{Hproj}\end{equation}
Let $\omega$ be the restriction of $\Omega$ to $\BA_F^\times/F^\times$.
Assume that the central character of $\pi$ is $\omega^{-1}$. Otherwise both sides of \eqref{Waldeq} are 0.
Regard $E^\times/F^\times$ as an algebraic group over $F$.
Fix the Tamagawa measure on $(E^\times/F^\times) (\BA_F)$.
Then $\Vol(E^\times\backslash \BA_E^\times / \BA_F^\times)=2L(1,\eta)$.
\begin{thm}[The Gross-Zagier-Zhang formula, projector version]
\label{projversion}
For $\phi \in \pi ,\varphi\in\tilde \pi $,
\begin{equation} H_\pi^{\sharp,\proj}(\phi\otimes\varphi) =\frac{L(2,1_F)L'(1/2,\pi,\Omega)}{ L(1,\pi,\ad)} \alpha_{\pi}(\phi ,\varphi ) . \label{eqprojversion} \end{equation}
\end{thm}
Now we reduce Theorem \ref{projversion} to its distribution version (see Theorem \ref{GZdis'} below). \subsubsection{Height distribution } \label{The CM height distribution}
\begin{defn} \label{CM height}
Define the height distribution on $\BB^\times$ by assigning to $f\in \CH_{\BC}$ the complex number \begin{align} H(f) :&= \int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{E^\times\backslash \BA_E^\times }^* \pair{\tilde Z(f) _*t_1^\circ,t_2^\circ}_\NT\Omega^{-1}(t_2)\Omega(t_1) dt_2dt_1 .\label{3.8} \end{align}
Here
the \Neron-Tate height pairing is the one on $J^\vee\times J $.
\end{defn}
Let $\pi\in \CA_{U_\infty}(\BB ^\times,\BC)$. Define a linear functional $H_\pi ^\sharp$ on $\CH_\BC $ by \begin{equation}H_\pi^\sharp (f) = \int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{E^\times\backslash \BA_E^\times }^* \pair{T_\pi\circ \rho_\pi(f)t_1^\circ,t_2^\circ}_\NT \Omega^{-1}(t_2)\Omega(t_1) dt_2dt_1 . \label{HHpi0}
\end{equation} Define \begin{equation}H_\pi (f) = \frac{ [F^\times\backslash \BA_F^\times/\Xi]}{\Vol(\tilde U/\Xi )\Vol(M_U)} H_\pi^\sharp (f). \label {specdecomht0} \end{equation} Here the coefficient is independent of $U$ (see Corollary \ref{the constant}). By \eqref {specdecomht}, we have \begin{equation}H(f )= \sum_{\pi\in \CA_{U_\infty}(\BB^\times,\bar \BQ_l)} H_\pi(f ).\label {specdecomht'} \end{equation}
Immediately from the definition of $H_\pi^{\sharp,\proj}$ in \eqref{Hproj}, we have \begin{equation*}H_\pi^\sharp(f)=H_\pi^{\sharp,\proj}(\rho_\pi(f)). \end{equation*} According to the global Hecke action \eqref{pixi}, define the action of $f_\infty$ on $\phi_\infty\in \pi_\infty$ to be
\begin{equation}\pi_\infty(f_\infty)\phi_\infty(x):=\int_{ \BB_\infty ^\times/\Xi_\infty} f_\infty (g)\phi_\infty(xg)dg.\label{HHprojpi}\end{equation}
Let $\alpha^\sharp_{\pi_v}( f_v)$ be defined as in \eqref{alpha} (for $v=\infty$, with the action of $f_\infty$ as in \eqref{HHprojpi}).
Let $\omega$ be the restriction of $\Omega$ to $\BA_F^\times/F^\times$. Let $\Sigma(\pi,\Omega)$ be defined by the formula in \eqref{sigset}. By the theorem of Tunnell-Satio (see Theorem \ref{TSlocal}) and the same reasoning as in \ref{The global relative trace formulaintro},
Theorem \ref{projversion} is implied by the following theorem.
\begin{thm}[The Gross-Zagier-Zhang formula, distribution version]\label{GZdis'}
Assume that the central character of $\pi$ is $\omega^{-1}$ and ${\mathrm{Ram}}=\Sigma(\pi,\Omega).$
There exists $f=\bigotimes_{v\in |X| }f_v\in \CH_\BC$, such that \begin{equation} H_\pi^\sharp(f ) =\frac{L(2,1_F)L'(1/2,\pi_{E_v})}{ L(1,\pi,\ad)} \prod_{v\in |X| } \alpha_{\pi_v}^\sharp (f_v),\label{GZdiseq}\end{equation} and $ \alpha_{\pi_v}^\sharp (f_v)\neq 0$
for every $v\in |X|$.
\end{thm}
We will prove Theorem \ref{GZdis'} in Section \ref{Proof of Theorem}.
\subsection{Reduction of Theorem \ref{GZ}}
\label{Abelian varieties parametrized by modular curves}
We reduce Theorem \ref{GZ} to Theorem \ref{GZdis'}.
\subsubsection{Modular abelian varieties}\label{3.3.1Abelian varieties parametrized by modular curves}
\begin{defn}An abelian variety $A$ over $F$ is modular if $\Hom(J,A )_\BQ$ is nontrivial. \end{defn} For $
\pi\in \CA_{U_\infty}(\BB^\times,\bar \BQ_l)$, we construct a simple modular abelian variety parametrized by $M$. For simplicity, assume that $\dim \pi_\infty =1$. (Otherwise, replace $U_\infty$ by an open, not necessary normal, subgroup of $\BB^\times/U_\infty$ such that $\dim \pi_\infty^{U_\infty'} =1$.) Choose $U$ such that $\dim \pi^{U }=1$. (For the existence of such $U$ and $U_\infty'$, see \cite{Cas}.)
Let $S $ the subset of $v\in |X| $ such that $U_v$ is not maximal or $\BB_v$ is a division algebra.
Let $\BS^S$ be the subalgebra of $\End(J_{U}) $ generated by the image of $Z(x)_{U,*}$ such that $x_v=1$ at all places $v\in S$. Let $K_\pi\subset \bar \BQ_l$
be the image of the spherical Hecke character associated to $\pi$ on the
$\BQ$-valued spherical Hecke algebra of $(\BB^S)^\times$. Below, we understand $K_\pi$ as an abstract field and ignore the embedding into $\bar\BQ_l$.
Lemma \ref{compareHecke}
gives a morphism $\BS^S \to K_\pi $. Let $\ker_\pi $ be the kernel. Let $A_\pi=J_{U }/\ker_\pi J_{U }$.
Define $\pi^\BQ:=\Hom(J,A_\pi)_\BQ,$
which is a $\BQ$-coefficient representation of $\BB^\times.$ Then we have inclusions of $\BQ$-algebras: \begin{equation}K_\pi\hookrightarrow \End(A_\pi)_\BQ\hookrightarrow \End_{\BB^\times}(\pi^\BQ)\label{lpiact}.\end{equation}
In particular, $K_\pi $ is a finite field extension of $\BQ$.
(The finiteness can also be obtained from \cite[Proposition 10.5]{JL}, see \cite[p 73]{YZZ}.)
By the strong multiplicity one theorem, the $\Aut(\bar\BQ_l)$-orbit $O_\pi$ of $\pi$ has $[K_\pi:\BQ]$ elements, indexed by
embeddings of $K_\pi$ into $\bar\BQ_l$ which are given by their spherical Hecke characters.
\begin{lem}\label{Jdecom}
(1) The $\Gal(F^\sep/F)$-representation associated to $H^1(A_{\pi,F^\sep},\bar\BQ_l) $ is the direct sum of $\LC( \pi')$ for $\pi'\in O_\pi$. Here $\LC$ is as in \ref{LLSmain}.
(2) The inclusions \eqref{lpiact} are isomorphisms. In particular, the abelian variety $A_\pi$ is simple of dimension $[K_\pi:\BQ]$, and the $\BB^\times$-representation $\pi^\BQ $ is irreducible.
\end{lem} \begin{proof} By Theorem \ref{semisimple} and that $\dim \pi^U=1$, the subspace of $H^1(J_{U ,F^\sep},\bar\BQ_l) $
where $\BS^S $ acts via $K_\pi$ is a direct sum of $\LC( \pi')$ for $\pi'\in O_\pi$. By the definition of $A_\pi$, this subspace is also $H^1(A_{\pi,F^\sep},\bar\BQ_l) $.
Then (1) follows. (1) implies that the $\BB^\times$-representation $$\Hom_{\Gal(F^\sep/F)}(H^1(A_{F^\sep} ,\bar\BQ_l),H^1(J_{\pi,\sep},\bar\BQ_l))$$ is the direct sum of representations in $ O_\pi$.
By the natural inclusion $$ \Hom(J,A_\pi)_{\bar\BQ_l}\hookrightarrow \Hom_{\Gal(F^\sep/F)}(H^1(A_{F^\sep} ,\bar\BQ_l),H^1(J_{\pi,\sep},\bar\BQ_l)),$$ we have $\dim_{\bar \BQ_l} \End_{\BB^\times}(\pi^\BQ)\otimes_{\BQ}{\bar \BQ_l}\leq [K_\pi:\BQ].$ Then (2) follows. \end{proof}
Conversely, let $A$ be a simple modular abelian variety over $F$, and $\pi_A:=\Hom(J,A) _ \BQ$. By Zarhin's theorem on the Tate conjecture and Theorem \ref{semisimple}, $\pi_A\otimes_\BQ\bar\BQ_l$ is the direct sum of $ \pi\in \CA_{U_\infty}(\BB^\times,\bar \BQ_l)$ such that $\LC(\pi)$ is a direct summand of the $\Gal(F^\sep/F)$-representation $H^1(A_{\pi,\sep},\bar\BQ_l)$. Let $\pi$ be an direct summand of $\pi_A\otimes_\BQ\bar\BQ_l$. By Lemma \ref{Jdecom} (2) and the simplicity of $A$, $A$ is isogeny to $A_\pi$ and $\pi_A\otimes_\BQ\bar\BQ_l$ is
the direct sum of $\pi'\in O_\pi$.
\subsubsection{$L$-functions}\label{Lf}
Denote $\pi_A$ by $\pi$ in this paragraph. Let $K= \End_{\BB^\times}(\pi ).$ Use the duality pairing \eqref{ duality pairing} to identify $\pi_{A^\vee}$ with the contragradient $\tilde\pi$ of $\pi_{A }$.
Then for every $\iota: K\hookrightarrow \BC$, $\tilde\pi \otimes _{K,\iota }\BC$ is irreducible, and $\tilde\pi \otimes _{K,\iota }\BC$ is the contragradient of $\pi\otimes _{K,\iota }\BC$.
Then for every $\iota: K\hookrightarrow \BC$, $\tilde\pi \otimes _{K,\iota }\BC$ is irreducible. Let $\Omega$ be a Hecke character of $E^\times$ with coefficients in a finite field extension $K'$ of $K$.
For $v\in |X|$, define the local $L$-factor
$L(s,\pi_v,\Omega_v)$ to be the unique rational function of $q_v^{-s}$ with coefficients in $K'$ such that for every embedding $\iota:K'\hookrightarrow \BC$, we have
$$ \iota(L(s,\pi_v,\Omega_v))=L(s ,(\pi^\iota)_{E_v}\otimes \Omega_v^\iota),$$ where $ \iota$ acts on the coefficients of $L(s,\pi_v ,\Omega_v)$, see \cite[3.2.2]{YZZ}.
Define
$$L(1/2,\pi,\Omega):=\prod_{v\in |X|} L(s,\pi_v,\Omega_v).$$ which is a polynomial on $q^{-s}$ with coefficients in $K'$. It
satisfies a functional equation $$L(s,\pi,\Omega)=\vep(s,\pi,\Omega)L(1-s,\tilde \pi,\Omega^{-1})$$ where
$\vep(s,\pi,\Omega)$ is an exponential $\pm c^{s-1/2}$ where $c$ is a positive number.
Assume that $\Omega$ comes from a continuous character $\Omega$ of $ \Gal(E^\ab/E)$
via the reciprocity map
$\BA_E^\times/E^\times\to \Gal(E^\ab/E).$
For a place
$v\in |X|$ nonsplit in $E$, regard $v$ as a place of $E$. Let
$I_v$ be the inertia group of $E$ at $v$, $\Fr_v $ the geometric Frobenius at $v$.
Let
\begin{equation}\label{ldef0}P_v(T):=\det_{K'\otimes_\BQ \BQ_l}(1-\Fr_v\cdot T|(H^1(A_{F^\sep}, \BQ_l) \otimes _K\Omega)^{I_v}) \in K'\otimes_\BQ \BQ_l[T].\end{equation}
The definition for $v $ split in $E$ is similar.
By Deligne's work on the Weil conjecture \cite{Del} (or by Lemma \ref{Jdecom} (1)),
\begin{equation}\label{ldef}L(s,A_E,\Omega):=\prod_{v\in |X|} P_v(q_v^{-s})^{-1}\end{equation}
is well-defined and valued in $K'\otimes_\BQ\BC$. Moreover, Lemma \ref{Jdecom} (1) implies that $$L(s,A_E,\Omega)=L(s-1/2,\pi , \Omega ).$$
\subsubsection{Height pairing} \label{Height pairing} We follow \cite[1.2.4]{YZZ}. By \cite[Proposition 7.3]{YZZ}, the usual \Neron-Tate height pairing
on $A(F^\sep)_\BQ \otimes A^\vee(F^\sep)_\BQ$ descends to a pairing
$$\pair{\cdot,\cdot}_\NT :A(F^\sep)_\BQ \otimes_K A^\vee(F^\sep)_\BQ \to \BC.$$
For $x\in A(F^\sep)_\BQ, y\in A^\vee(F^\sep)_\BQ$,
$$(a\mapsto\pair{ax,y}_\NT)\in \Hom_\BQ(K,\BC)\simeq K\otimes_\BQ \BC, $$ where the last isomorphism is from the trace map $K\otimes K\to \BQ$.
Thus we have a pairing $$\pair{\cdot,\cdot}_\NT^K :A(F^\sep)_\BQ \otimes_K A^\vee(F^\sep)_\BQ \to K\otimes_\BQ \BC.$$
Let $K'/K$ be a finite field extension. We have the extension of $\pair{\cdot,\cdot}_\NT^K$ by scalar
$$\pair{\cdot,\cdot}_\NT^{K'} :(A(F^\sep)_\BQ\otimes_K K')\otimes_{K'} (A^\vee(F^\sep)_\BQ\otimes_K K') \to K' \otimes_\BQ \BC.$$
For an embedding $\iota:K\hookrightarrow \BC$, there is a canonical isomorphim
$$(A(F^\sep)_\BQ \otimes_K A^\vee(F^\sep)_\BQ )\otimes_{K,\iota} \BC\simeq (A(F^\sep)_\BQ\otimes_{K,\iota}\BC) \otimes_\BC (A^\vee(F^\sep)_\BQ\otimes_{K,\iota}\BC).$$
Let $\BC \otimes_\BQ \BC\to \BC$ be the multiplication map. Then we define the $\iota$-component of $ \pair{\cdot,\cdot}_\NT^K$ by \begin{equation}\pair{\cdot,\cdot}_\NT^\iota:(A(F^\sep)_\BQ\otimes_{K,\iota}\BC) \otimes_\BC (A^\vee(F^\sep)_\BQ\otimes_{K,\iota}\BC)\to \BC \otimes_\BQ \BC\to \BC.\label{pairiota}\end{equation}
\subsubsection{Height identity} \label{Duality pairing, algebraic projectors and the height identity}
Now we deduce Theorem \ref{GZ} from Theorem \ref{projversion}, following \cite[3.3.3, 3.3.4]{YZZ}.
Let $\iota: K'\hookrightarrow \BC$ be an embedding and we also use $\iota$ to deduce its restriction to $K$.
Let $\phi=\sum_{i}\phi_i\otimes_{K,\iota } c_i\in \pi_A^\iota.$
Each $\phi_i$ gives a morphism $\phi_i:M(F^\sep)\hookrightarrow J(F^\sep)\to A(F^\sep)$.
Then we have an induced map $$\phi:M(F^\sep)\to A(F^\sep)_\BQ\otimes_{K,\iota}\BC.$$ Similarly for $\varphi\in \pi_{A^\vee}^\iota.$
On the targets of $\phi$ and $\varphi$ we have the $\iota$-component of the \Neron-Tate height pairing $\pair{\cdot,\cdot}_\NT^\iota $ (see \eqref{pairiota}). Define \begin{equation} \pair{P_\Omega(\phi),P_{\Omega^{-1}} (\varphi)}_{\NT} ^\iota:=\int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int^*_{E^\times\backslash \BA_E^\times }
\pair{ \phi(t_1),\varphi(t_2)}_{\NT} ^\iota\Omega^{\iota,-1}(t_2)\Omega^\iota(t_1) dt_1dt_2.\label{GZcomplex}\end{equation}
The complex version of Theorem \ref{GZ} is as follows (compare with \cite[Theorem 3.13]{YZZ}).
\begin{thm}\label{GZiota}
For $\phi \in \pi _A^\iota$ and $\varphi\in\tilde \pi _{A^\vee}^\iota$, we have \begin{equation} \pair{P_\Omega(\phi),P_{\Omega^{-1}} (\varphi)}_{\NT} ^\iota=\frac{L(2,1_F)L'(1/2,\pi_A^\iota,\Omega^\iota)}{ L(1,\pi_A^\iota,\ad)} \alpha_{\pi_A^\iota } (\phi ,\varphi ) . \label{eqcomversion} \end{equation}
\end{thm}
Applying Theorem \ref{GZiota} to all embeddings $\iota: K'\hookrightarrow \BC$, Theorem \ref{GZ} follows from Theorem \ref{GZiota} and Corollary \ref{TSCM}.
Now we deduce Theorem \ref{GZiota} from Theorem \ref{projversion}.
Abusing notations, we use $\iota$ to denote the natural inclusions $$\iota:\pi_A \hookrightarrow \pi_A\otimes _{K,\iota}\BC,\ \iota: \pi_{A^\vee}\hookrightarrow \pi_{A^\vee}\otimes _{K,\iota}\BC.$$ The embedding $\iota:K\to \BC$ also induces natural inclusions
$$\iota:A(F^\sep)_\BQ \hookrightarrow A(F^\sep)_\BQ\otimes_{K,\iota}\BC,\ \iota:A^\vee(F^\sep)_\BQ \hookrightarrow A^\vee(F^\sep)_\BQ\otimes_{K,\iota}\BC.$$
By the definition of the cohomological projector $T_{\pi_A^\iota}$ (see Definition \ref{cohproj}),
Lemma \ref{compareHecke} and the same discussion as in \cite[3.3.3, 3.3.4]{YZZ}, we have the following lemma.
\begin{lem} \label{lheightid}
Let $\phi\in\pi_A,\ \varphi\in \pi_{A^\vee}$, $x,y\in J (F^\sep) $,
then $$\pair{T_{\pi_A^\iota}(\iota(\phi)\otimes\iota(\varphi))x,y}_\NT=\pair{ \iota(\phi(x)),\iota(\varphi(y))}_\NT^\iota$$
where the height pairing on the left hand side is the one on $J (F^\sep)_\BC \times J ^\vee(F^\sep)_\BC$ (see \ref{Jacobians and Height pairings}).
\end{lem}
Then Theorem \ref{GZiota} follows from Theorem \ref{projversion}, and we have the following Corollary.
\begin{cor} \label{GZfromproj} Theorem \ref{GZ} follows from Theorem \ref{projversion}.
\end{cor}
\subsubsection{Choices of $\infty$ and $\varpi_\infty$}\label{choices'}
Let $\pi$ be an irreducible $\BC$-coefficient admissible representation of $\BB^\times$. \begin{lem}\label{chi} For a place $\infty$ of $F$ and a uniformizer $\varpi_\infty$ of $F_\infty$, there exists a Hecke character $\tau$ of $F^\times$ such that $(\pi\otimes \tau)(\varpi_\infty)=1$. Moreover, if the central character of $\pi_\infty$ has finite order, such $\tau$ can be chosen to have finite order. \end{lem} \begin{proof} Let $a$ a square root of the inverse of $\pi(\varpi_\infty)$. It is enough to find $\tau$ such that $\tau_\infty(\varpi_\infty)=a$. Let $V$ be an open compact subgroup of $\BA_F^\times$. The Pontryagin dual of the exact sequence $$1\to \varpi_\infty^{\BZ}\to F^\times\backslash\BA_F^\times/V\to F^\times\backslash\BA_F^\times/V\varpi_\infty^{\BZ}\to 1$$ is the exact sequence $$1\to (F^\times\backslash\BA_F^\times/V\varpi_\infty^ \BZ )\hat{} \to (F^\times\backslash\BA_F^\times/V)\hat{}\to (\varpi_\infty^\BZ )\hat{}\to 1.$$ Choose $\tau_\infty\in (\varpi_\infty^\BZ )\hat{}$ such that $\tau_\infty(\varpi_\infty)=a$. Let $\tau$ be in the preimage of $\tau_\infty$ in $(F^\times\backslash\BA_F^\times/V)\hat{}$, then $\tau_\infty(\varpi_\infty)=a$. This proves the first statement. Since $F^\times\backslash\BA_F^\times/V\varpi_\infty^ \BZ $ is a finite group, if $a$ is a root of unity, $\tau$ is of finite order. This proves the second statement. \end{proof}
Let $\Omega$ a Hecke character of $E^\times$.
Let $\pi'=\pi\otimes \chi$, and $\Omega'=\Omega \chi_E^{-1}$.
Then $$L(s,\pi,\Omega)=L(s,\pi',\Omega'),\ L(1,\pi ,\ad)=L(1,\pi',\ad),$$ and similar relations hold for local periods.
Assume that the Jacquet-Langlands correspondence of $\pi$ to $\GL_{2,F}$ is cuspidal.
Choose the place $\infty$ in Lemma \ref{chi} inside the ramification set ${\mathrm{Ram}}$ of $\BB$.
Then
Theorem \ref{GZiota} can be applied to get a complex Gross-Zagier-Zhang formulas for $\pi$.
If $\pi$ is an irreducible $\BQ$-coefficient representation of $\BB^\times$, then choose $\chi$ to be of finite order and define
$\pi'=\pi\otimes \chi$ by regarding $\chi$ as a finite dimensional
$\BQ$-coefficient representation. Then
Theorem \ref{GZ} can be applied to get a Gross-Zagier-Zhang formulas for $\pi$.
\section{Automorphic distributions}\label{The automorphic distributions} We review the relative trace formulas in \cite{Jac87}\cite{JN}.
In \ref{matchorb}, \ref{local orbital integrals0}, \ref{Split case}, we classify orbits and define local orbital integrals. In \ref{Automorphic distributions},
\ref{Decomposition under the pure matching condition}, we define the automorphic distributions and study their orbital decompositions.
In \ref{Spectral decomposition of the automorphic distributions}, we study the spectral decompositions.
\subsection{Orbits}\label{matchorb} Let $F$ be a field, $E$ a separable quadratic field extension of $F$, and ${\mathrm{Nm}}:E^\times\to F^\times$ the norm map. Let $B$ be a quaternion algebra over $F$ containing $E$. By the Noether-Skolem theorem, there exists $j\in B^\times$ such that $B=E\oplus Ej$, $j^2=\ep\in F^\times$ and $jz=\bar zj$ for $z\in E$, where $\bar z$ is the Galois conjugate of $z$. Then $B$ is determined by $\ep$, and we denote $B$ by $\left(\frac{E,\ep}{F}\right)$.
Embed $B$ in $ \RM_2(E)$ as an $F$-subalgebra via \begin{equation}a+bj\mapsto \begin{bmatrix}a&b\ep\\
\bar b&\bar a\end{bmatrix}\label{(5.1)}\end{equation} where $a,b\in E$.
We use the symbol $\det$ to denote the reduced norm map on $B^\times$. Then for $a,b\in E$, $\det (a+bj)={\mathrm{Nm}} (a)-{\mathrm{Nm}} (b)\cdot\ep$.
Under the embedding \eqref{(5.1)},
the reduced norm map
is just the determinant map.
Note that if $c\in E^\times$, then $cjcj=c\bar cjj=c\bar c\ep$, and $cjz=\bar z cj$. So $B\simeq \left(\frac{E,{\mathrm{Nm}} (c) \cdot \ep}{F}\right)$. Thus we have a bijection $\ep\mapsto \left(\frac{E,\ep}{F}\right)$
between $F^\times/ {\mathrm{Nm}} (E^\times)$ and the set of isomorphism classes of quaternion algebras over $F$ containing $E$.
For $\ep\in F^\times$, $G_\ep=B^\times$ where $B = \left(\frac{E,\ep}{F}\right)$.
Let $T_\ep\subset G_\ep$ be the subgroup induced by the canonical embeddings of $E^\times$, and let $Z_\ep\subset G_\ep$ be the center.
Let $T_\ep\times T_\ep$ act on $G_{\ep }$ by $$(h_1,h_2)\cdot \gamma=h_1^{-1}\gamma h_2.$$ Let $ \inv_{T_\ep}(a+bj)= \ep {\mathrm{Nm}} (b)/{\mathrm{Nm}} (a).$
This defines a bijection $$\inv_{T_\ep} : T_\ep \backslash G_{\ep } /T_\ep \simeq \ep {\mathrm{Nm}} E^\times \cup\{0,\infty\} -\{1\}.
$$ Regard $G_\ep$ as a subgroup of $\GL_2(E)$ via \eqref{(5.1)}. Then $$\inv_{T_\ep}\left(\begin{bmatrix}a&b\ep\\
\bar b&\bar a\end{bmatrix}\right)=\ep\frac{ b \bar b }{a\bar a} .$$
Let $\delta=\begin{bmatrix}a&b\ep\\
\bar b&\bar a\end{bmatrix}\in G_\ep $. We say that $\delta$ is $T_\ep$-regular (regular for short) if $\inv_{T_\ep}(\delta)\in \ep {\mathrm{Nm}} (E^\times)-\{1\}$, equivalently if
$a b \neq 0$. The stabilizer of the $T_\ep\times T_\ep$-action on a regular element is the diagonal embedding of $Z_\ep$.
Let $G_{\ep,\reg}\subset G_\ep$ be the subset of regular elements.
For $x \in \ep {\mathrm{Nm}} (E^\times) -\{1\}$, choose $b\in E^\times$ such that $x=\ep {\mathrm{Nm}}(b)$. Let $$\delta(x) :=\begin{bmatrix}1&b\ep\\
\bar b& 1\end{bmatrix} =1+bj\in G_{\ep,\reg}$$ which is
a representative of the corresponding $T_\ep\times T_\ep$-orbit, i.e. $\inv_{T_\ep}(\delta(x))=x$.
\begin{rmk}This definition depends on the choice between $b$ and $\bar b$.
However, the orbital integrals of $\delta(x)$ (which will be defined in the next subsection) do not depend on this choice.
\end{rmk}
Define $\inv_{T_\ep}' : G_\ep \to F$ by $\inv_{T_\ep}':= \inv_{T_\ep}/(1-\inv_{T_\ep})$, i.e. \begin{equation}\label{inv'}\inv_{T_\ep}'(a+bj)=\ep\frac{ {\mathrm{Nm}} (b)}{{\mathrm{Nm}} (a+bj)}. \end{equation}
Now we turn to the $\GL_2$-side. Let $G=\GL_{2,E}$. Let $\CS\subset G $ be the subset of invertible Hermitian matrices over $F$ with respect to the separable quadratic extension $E$. We also regard $\CS$ as a subvariety of $G$ if necessary. Let $E^\times\times F^\times$ act on $\CS$ via $$(a,z)\cdot s= \begin{bmatrix}a&0\\ 0&1\end{bmatrix}s\begin{bmatrix}\bar a&0\\ 0& 1\end{bmatrix}z.$$
There is a bijection $$\inv_{\CS} : E^\times\backslash \CS/ F^\times \simeq F^\times \cup\{0,\infty\} -\{1\},
$$$$\inv_{\CS}\left(\begin{bmatrix}a&b \\
\bar b&d\end{bmatrix} \right )= \frac{ad}{b\bar b}.$$
Define $\gamma\in S$ to be regular if $\inv_{\CS}(\gamma)\in F^\times-\{1\}$. The stabilizer of the $E^\times\times F^\times$-action on a regular element is trivial.
Let $\CS_{ \reg}\subset \CS$ be the subset of regular elements.
For $x\in F^\times -\{1\}$, let $$\gamma(x) = \begin{bmatrix}x&1\\ 1&1\end{bmatrix}\in \CS_\reg $$ which is
a representative of the corresponding regular $E^\times\times F^\times$-orbit.
Define \begin{equation}\inv_{\CS}'=\frac{\inv_{\CS}}{1-\inv_{\CS}}:\CS\to F \label{xi'} .\end{equation}
Let $g\in G$ act on $\CS$ by $g\cdot s:=gs \bar g^t$ where $ \bar g^t$ is the Galois conjugate of the transpose of $g$. Let $H_0\subset G$ be the unitary group associated to $$w=\begin{bmatrix}0&1\\
1&0\end{bmatrix},$$ i.e. the stabilizer of $w$ in $G$. Let $H\subset G$ be the similitude unitary group associated to $w$, and
let $\kappa$ be the similitude character.
If $F$ is a local field, for $f\in C_c^\infty(G )$, let $ \Phi_{f}\in C_c^\infty(\CS )$ supported on $Gw$ such that
\begin{equation}\label{Phi}\Phi _f(g\cdot w )=\int_{H_0 } f(gh) dh.\end{equation} If $F$ is a global field, apply this definition to $f\in C_c^\infty(G(\BA_F) )$.
\subsection{Local orbital integrals: nonsplit case}\label{local orbital integrals0}
Let $F$ be a non-archimedean local field.
Let $E$ be a separable quadratic field extension, $\eta$ the associated quadratic character of $F^\times$.
\begin{lem}\label{goodx} Let $\gamma\in \CS$, then $\gamma\in Gw$ if and only if $-\det(\gamma)\in {\mathrm{Nm}}(E^\times)$. For $x\in F^\times -\{1\}$,
$\gamma(x) \in Gw$ if and only if $ 1-x\in {\mathrm{Nm}}(E^\times)$. \end{lem}
Let $\Omega$ be a continuous character of $E^\times$ and $\omega$ be its restriction to $F^\times$.
Endow $\CS\subset G$ with the subspace topology. For $\Phi\in C_c^\infty(\CS)$, $\gamma\in \CS_\reg$, and $s\in \BC$, define the orbital integral \begin{equation}\CO(s, \gamma,\Phi):=\int_{E^\times}\int_{F^\times} \Phi\left(z\begin{bmatrix}a&0\\ 0&1\end{bmatrix}\gamma\begin{bmatrix}\bar a&0\\
0&1 \end{bmatrix} \right)\eta\omega^{-1}(z)\Omega^{-1}(a) |a|_E^sd zd a. \label{int0} \end{equation} For $x\in F^\times-\{1\}$, define $\CO(s, x,\Phi):=\CO(s, \gamma(x),\Phi).$ Let $\CO(x,\Phi):=\CO(0, x,\Phi).$
\begin{lem}\label{intwell} The integral \eqref{int0} converges absolutely and defines a holomorphic function on $s$. Its derivative at $s=0$ is the following
convergent integral: \begin{equation*}\CO'(0, x,\Phi)=\int_{E^\times}\int_{F^\times} \Phi\left( \begin{bmatrix}za\bar a x&za\\
z\bar a&z\end{bmatrix}\right) \eta\omega^{-1}(z)\Omega^{-1}(a)\log |a|_E d zd a. \end{equation*} \end{lem} \begin{proof}
The map $$E^\times \times F^\times\to G,\ (a,z)\mapsto z\begin{bmatrix}a&0\\ 0&1\end{bmatrix}\gamma(x)\begin{bmatrix}\bar a&0\\ 0&1 \end{bmatrix} $$ is a closed embedding since $\gamma(x)$ is regular. Therefore $$\supp (\Phi)\cap\left \{z\begin{bmatrix}a&0\\ 0&1\end{bmatrix}\gamma\begin{bmatrix}\bar a&0\\ 0&1 \end{bmatrix} :(a,z)\in E^\times \times F^\times\right \}$$ is compact, and so is its preimage in $E^\times \times F^\times.$ The lemma follows. \end{proof}
As an example, we have the following lemma which is easily proved by a direct computation. Let $K=\GL_2(\CO_E)$ be the standard maximal compact subgroup of $G$.
\begin{lem}\label{sint=1} Let $\Phi=1_{K\cap \CS}$, and let $\eta$ and $\Omega$ be unramified. Suppose $v(x)=v(1-x)=0$, then $$\CO(s,\gamma(x),\Phi ) =\Vol(\CO_F^\times)\Vol(\CO_E^\times) .$$\end{lem}
Now we turn to $ G_\ep $. We do not need the parameter $s$.
For $f\in C _c^\infty (G_\ep)$ and $\delta\in {G_{\ep, \reg}} $, define \begin{align*}\CO( \delta,f): =\int_{T_\ep/Z_\ep}\int_{T_\ep } f(h_1^{-1}\delta h_2) \Omega(h_1) \Omega^{-1}(h_2) dh_2dh_1. \end{align*} For an open subgroup $\Xi$ of $Z_\ep$
such that $\Omega$ is $\Xi$-invarian, define \begin{align*}\CO_\Xi( \delta,f):= \int_{T_\ep/Z_\ep}\int_{T_\ep/\Xi } f(h_1^{-1}\delta h_2) \Omega(h_1) \Omega^{-1}(h_2) dh_2dh_1. \end{align*} Finally, for $x \in \ep{\mathrm{Nm}} (E^\times)-\{1\}$, let $\CO(x,f):= \CO(\delta(x),f) $
and
$\CO_\Xi(x,f):= \CO_\Xi(\delta(x),f)$.
It is easy to check that the integrals defining $\CO(\delta ,f) , \CO_\Xi( \delta,f)$ converge absolutely.
\begin{defn} \label{matchingdef}(1) Let $\Phi\in C_c^\infty (\CS)$, and $f_\ep\in C_c^\infty(G_\ep)$. We say that $\Phi$ and $f_\ep $ have matching orbital integrals (match, for short) if for every $x\in \ep{\mathrm{Nm}}(E^\times)-\{1\}$, the following equation holds: $$ \CO( x,\Phi)=\CO(x,f_\ep).$$
(2) We say that $\Phi$ and $(f_\ep)_{\ep\in F^\times/ {\mathrm{Nm}} (E^\times)}$ have matching orbital integrals (match, for short) if $\Phi$ and $f_\ep $ match for each $\ep$.
(3) We say that $\Phi$ and $f_\ep$ purely match if $\Phi$ matches $(f_\ep, 0)$.
(4) For $f\in C_c^\infty(G )$, the definitions in (1) (2) (3) apply if they apply to $\Phi_{f}$.
(5)
The definitions in (1) (2) (3) (4) apply to $f_\ep\in C _c ^\infty (G_\ep/\Xi )$ if corresponding conditions hold when $\CO(x,f_\ep)$ is replaced by $\CO_\Xi(x,f_\ep)$. \end{defn}
\subsection{Local orbital integrals: split case}\label {Split case} Let $F$ be a non-archimedean local field as in \ref{local orbital integrals0}, and let $E=F\times F$.
Let $a\mapsto \bar a$ be the standard involution on $E$ w.r.t. $F$.
Let $G=\GL_2(E)$. Then there is a canonical isomorphism \begin{equation}G \simeq\GL_2(F)\times \GL_2(F).\label{541}\end{equation}
With respect to the involution $a\mapsto \bar a$, the space $\CS $ of invertible hermitian matrices and the unitary group (resp. similitude unitary group) $H_0$ (resp. $H$) associated to $w=\begin{bmatrix}0&1\\
1&0\end{bmatrix}$ as in \ref{matchorb} is well defined.
Under the isomorphism \eqref{541}, $H_0\subset G$ is the subgroup of elements of the form $ (h,w(h^t)^{-1} w).$ Thus we have an isomorphism \begin{equation}H_0\simeq \GL_2(F) \label{543}\end{equation}
given by $ (h,w(h^t)^{-1} w)\mapsto h.$
Similarly, under the isomorphism \eqref{541}, $\CS\subset G$ is the subset of elements of the form $(s,s^t).$ Thus
we have an isomorphism \begin{equation}\CS\simeq \GL_2(F) \label{544}\end{equation}
given by $ (s,s^t) \mapsto s.$
Let $f_1,f_2\in C_c^\infty(\GL_2(F))$, which induce a function $f_1\otimes f_2\in C_c^\infty(G)$ under the isomorphism \eqref{541}. Define $\Phi_{f_1\otimes f_2}\in C_c^\infty(\CS)$ as in \eqref{Phi}. Regarded as a function on $\GL_2(F)$ via \eqref{544}, $\Phi_{f_1\otimes f_2} $ has the following expression: let $s\in \GL_2(F)$, and $(g,g_2)\in G$ such that $$(g,g_2)(w,w)(g_2^t,g^t)=(s,s^t),$$ then
\begin{align*} \Phi_{f_1\otimes f_2}(s)&=\int_{\GL_2(F)} f_1(gh )f_2(g_2w(h ^t)^{-1} w)dh . \end {align*}
Take $g_2=1$ , we have
\begin{align} \Phi_{f_1\otimes f_2}(g w)=\int_{\GL_2(F)} f_1(gh)f_2(w(h^t)^{-1}w)dh=(f_1\ast \tilde f_2)(g)\label{phi12}, \end{align}
where $\tilde f_2$ is defined by
$$\tilde f_2(g):=f_2(wg^tw).$$
Let $\Omega=\Omega_1\boxtimes \Omega_2$ be a continuous unitary character of $E^\times=F^\times\times F^\times$. Then $\omega:=\Omega_1\otimes \Omega_2$ is the restriction of $\Omega$ to the diagonal embedding $F^\times\hookrightarrow E^\times$. Let $\Phi\in C_c^\infty(\CS)$. For $x\in F^\times-\{1\}$, define $\CO(x,\Phi) $ as in \eqref{int0} (with $s=0$). Regarded $\Phi$ as a function on $\GL_2(F)$ via \eqref{544}. Using the isomorphism \eqref{541}, we have \begin{equation}\CO(x,\Phi)=\int _{F^\times}\int _{F^\times}\int _{F^\times}
\Phi\left(\begin{bmatrix}z&0\\ 0&z\end{bmatrix}\begin{bmatrix}a&0\\ 0&1\end{bmatrix}\begin{bmatrix}x&1\\ 1&1\end{bmatrix}\begin{bmatrix}b&0\\ 0&1\end{bmatrix}\right)\Omega_1^{-1}(a)\Omega_2^{-1}(b)\omega^{-1}(z) d a d bd z.\label{545}\end{equation}
Similar to Lemma \ref{sint=1}, we have the following lemma.
\begin{lem}\label{sint=1'}Let $\Phi=1_{K\cap \CS}$, $\eta$ and $\Omega$ be unramified. Suppose $v(x)=v(1-x)=0$, then $$\CO(s,\gamma(x),f ) =\Vol(\CO_F^\times)\Vol(\CO_E^\times) .$$\end{lem}
Let $G_\ep=\GL_2(F)$. Here $\ep$ is just an abstract subscript.
Let $T_\ep$ be the diagonal torus of $\GL_2(F)$. For $f_\ep\in C_c^\infty(G_\ep)$, define $\CO(x,f_\ep) $ as in \ref{local orbital integrals0}.
Then \begin{equation*}\CO(x,f_\ep)=\int _{F^\times}\int _{F^\times}\int _{F^\times}
f_\ep\left(\begin{bmatrix}c^{-1}&0\\ 0&1\end{bmatrix}\begin{bmatrix}x&1\\ 1&1\end{bmatrix}\begin{bmatrix}a&0\\ 0&b\end{bmatrix}\right)\Omega_1(c)\Omega_1^{-1}(a)\Omega_2^{-1}(b)d cd a d b.\label{546}\end{equation*}
Change variable: $b=at $,
we have \begin{equation}\CO(x,f_\ep)=\int _{F^\times}\int _{F^\times}\int _{F^\times}
f_\ep\left(\begin{bmatrix}a&0\\ 0&a\end{bmatrix} \begin{bmatrix}c&0\\ 0&1\end{bmatrix} \begin{bmatrix}1&x\\ 1&1\end{bmatrix} \begin{bmatrix}t&0\\ 0&1\end{bmatrix} w\right)\Omega_1^{-1}(c)\Omega_2^{-1}(t)\omega^{-1}(a)d c d t d a.\label{547}\end{equation}
Define $\Phi\in C_c^\infty (\CS)$ and $f_\ep\in C_c^\infty(G_\ep)$ to have matching orbital integrals (match, for short) if for every $x\in F^\times-\{1\}$, $ \CO( x,\Phi)=\CO(x,f_\ep).$
By \eqref{545} and \eqref{547}, we have the following lemma.
\begin{lem} \label{afneq000}
Let $f_\ep(g)=\Phi(gw),$ then
$f_\ep$ matches $\Phi$.
\end{lem}
\begin{lem} \label{afneq00}
(1) Let $f_\ep \in C_c^\infty(G_\ep) $. Then there
exists $ f_1,f_2\in C_c^\infty(\GL_2(F))$ such that \begin{equation}f_\ep=f_1\ast \tilde f_2 \label{splitplaces}.\end{equation}
(2) For every $f_1,f_2$ as in (1), $f_\ep(g)=\Phi_{f_1\otimes f_2}(gw)$.
(3) For every $f_1,f_2$ as in (1), $f_\ep$ matches $\Phi_{f_1\otimes f_2}$. \end{lem}
\begin{proof} (1) Suppose $f_\ep$ is bi-$U$-invariant for a open compact subgroup $U\subset G_\ep $, choose $f_1=f_\ep$, $f_2= {1_U}/\Vol( U)$. (2) is a restatement of \eqref{phi12}. (3) follows from (2) and Lemma \ref{afneq000}.\end{proof}
\subsection{Automorphic distributions}\label{Automorphic distributions}
We come back to the global situation. Let $F$ be a global function field, and let $E$ be a separable quadratic field extension.
Let $\Omega$ be a Hecke character of $E^\times$, and let $\omega$ be its restriction to $\BA_{F}^\times$, and $\omega_E:=\omega\circ{\mathrm{Nm}}$.
\subsubsection{} Let $G=\GL_{2,E}$, let $A$ be the diagonal torus and let $Z$ be the center of $G$. Let $\CS\subset G$ be the subvariety of invertible Hermitian matrices. Let $H,H_0,\kappa $ be defined as in the end of \ref{matchorb}. For $f'\in C_c^\infty(G(\BA_E))$, define a kernel function on $G(\BA_E)\times G(\BA_E)$: \begin{equation}\label{autker}K(x,y)=K_{\omega_E,f'}(x,y):=\int_{ Z(E)\backslash Z(\BA_E) }\left(\sum_{g\in G(E)} f'(x^{-1}gzy) \right)\omega_E^{-1}(z)dz.\end{equation}
\begin{lem} \label{finitesum3} The inner sum of \eqref{autker} has only finitely many nonzero terms. \end{lem} \begin{proof}
If $f'(x^{-1}gy)\neq 0$, $g\in x\supp (f') y^{-1}$ which is compact. Since $G(E)\subset \GL_2(\BA_E)$ is closed and discrete with the subspace topology, $x\supp (f') y^{-1}\cap G(E)$ is finite. \end{proof}
For $a= \begin{bmatrix}a_1&0\\ 0&a_2\end{bmatrix}$, let $\Omega(a)=\Omega(a_1/a_2)$. For $s\in \BC$, formally define the distribution $\CO(s,\cdot)$ on $G(\BA_E)$ by assigning to $f' \in C_c^\infty( G(\BA_E))$ the integral \begin{align}\CO( s, f')= \int_{Z(\BA_E)A(E)\backslash A(\BA_E)} \int_{Z(\BA_E)H(F)\backslash H(\BA_F)} K_{\omega_E,f'}(a,h)\Omega(a) \eta\omega^{-1}(\kappa(h) ) |a|_E^sdh da\label{45} .\end{align}
\begin{lem} \label{5.5.4} Assume $\Phi_{f'}(g)=0$ for $g\in \BA_E^\times (\CS(F)-\CS(F)_\reg) \BA_F^\times .$ Then
the integral $\CO( s, f')$ in \eqref{45} converges absolutely.
\end{lem}
For $\Phi\in C_c^\infty(\CS(\BA_F))$ and $x\in F^\times-\{1\}$, define $$\CO(s,x,\Phi):=\int _{\BA_E^\times}\int_{\BA_F^\times} \Phi\left( \begin{bmatrix}za\bar a x&za\\
z\bar a&z\end{bmatrix}\right) \eta\omega^{-1}(z)\Omega^{-1}(a)|a|_E^s d zd a.$$
By the same reasoning as in the proof Lemma \ref{intwell}, we have the following lemma.
\begin{lem}\label{AV} Let $\Phi\in C_c^\infty(\CS(\BA_F))$. The integral $\CO(s,x,\Phi)$ converges absolutely.
\end{lem}
\begin{proof} [Proof of Lemma \ref{5.5.4}] Formally, we have \begin{align}
\CO( s, f')=\sum_{x\in F^\times-\{1\}}\CO(s,x,\Phi_{f'}).\label{hohoho} \end{align}
Extend $\inv_\CS'$ defined in \eqref{xi'} to $\CS(\BA_F)\to \BA_F$. Since $\inv_\CS'(\supp \Phi_{f'})$ is compact, it has finite intersection with the closed and discrete subset $F=\inv_\CS'(\CS(F))$ of $\BA_F$. In particular, the right hand side is a finite sum. The by Lemma \ref{AV}, and Fubini's theorem, the integral $\CO( s, f')$ converges absolutely and \eqref{hohoho} holds.
Thus the lemma follow.\end{proof}
\begin{asmp} We only use pure tensors $f'\in C_c^\infty(G(\BA_E))$
and $\Phi\in C_c^\infty(\CS(\BA_F))$: $$\Phi=\bigotimes_{v\in |X|}\Phi_v,\ f'=\bigotimes_{v\in |X|}f_v',$$ where $\Phi_v\in C_c^\infty(\CS(F_v))$
and
$f'_v\in C_c^\infty(G(E_v))$.
\end{asmp}
Lemma \ref{sint=1}, \ref {sint=1'} show that $\CO(s,x,\Phi_v )=1$ for almost all $v$.
Thus $$\CO(s,x,\Phi) = \prod_{v\in |X|}\CO(s,x,\Phi_v) ,$$ and the infinite product converges absolutely.
And the following sum is a finite sum: \begin{align} \CO'(0,x,\Phi)=\sum_{v\in |X|}\CO'(0,x,\Phi_v)\CO(x ,\Phi^v)\label{513} .\end{align}
\begin{asmp}\label{freg} Assume that $\Phi_{f'}(g)=0$ for $g\in \BA_E^\times (\CS(F)-\CS(F)_\reg) \BA_F^\times .$ \end{asmp}
By \eqref{hohoho} and \eqref{513}, we have a decomposition
\begin{align}\CO'( 0, f')&=\sum_{x\in F^\times-\{1\}} \sum_{v\in |X|}\CO'(0,x,\Phi_{f',v})\CO(x ,\Phi_{f'}^v) , \label{decder} \end{align}
By the same reasoning as in the proof of Lemma \ref{5.5.4}, there
are only finitely many $x$ such that $\CO'(0,x,\Phi_{f',v})\CO(x ,\Phi_{f'}^v)$ is nonzero for some $v$. In particular, the sum is a finite sum.
\subsubsection{} Let $B$ be a quaternion algebra over $F$. For $f\in C_c^\infty (B^\times(\BA_F))$, define a kernel function on $B^\times(\BA_F)\times B^\times(\BA_F)$: \begin{equation} k (x,y):=\sum_{g\in B^\times} f(x^{-1}gy).\label{kxy}\end{equation}
Define a distribution $\CO(\cdot)$ on $B^\times(\BA_F)$ by assigning to $f\in C_c^\infty( B^\times(\BA_F))$ the integral $$\CO( f):=\int_{E^\times \backslash \BA_E^\times / \BA_F^\times} \int_{E^\times \backslash \BA_E^\times } k(h_1, h_2) \Omega(h_1)\Omega^{-1}(h_2) dh_2dh_1.$$
\begin{asmp}\label{freg'} Assume that $f$ vanishes on $\BA_E^\times (B^\times-B^\times_\reg) \BA_E^\times $. \end{asmp}
\begin{lem} \label{decquat} (1) The integral defining $\CO( f)$ converges absolutely.
(2) We have a decomposition \begin{align*} \CO( f) =\sum_{x\in\ep {\mathrm{Nm}} ( E^\times)-\{1\}}\CO(x,f ), \end{align*} where $$\CO( x,f)=\int_{ \BA_E^\times / \BA_F^\times}\int_{ \BA_E^\times } f(h_1^{-1}\delta(x) h_2)\Omega(h_1)\Omega^{-1}(h_2) dh_2dh_1 .$$
\end{lem}
We also need the following modification of $\CO(\cdot)$. For $\infty\in |X|$ and $\Xi_\infty \subset F_\infty^\times$, the image of $B^\times\hookrightarrow B^\times(\BA_F)/\Xi_\infty$ is closed and discrete.
Thus, for $f \in C_c^\infty (B^\times(\BA_{F })/\Xi_\infty)$, the formula \eqref{kxy} gives a well-defined function $ k (x,y) $
on $B^\times(\BA_F)\times B^\times(\BA_F)$.
\begin{defn}\label{COXi} The distribution $\CO_{\Xi_\infty}$ on $B^\times(\BA_F)/\Xi_\infty$ assigns to $f\in C_c^\infty( B^\times(\BA_F)/\Xi_\infty)$ the integral $$\CO_{\Xi_\infty}( f):=\int_{E^\times \backslash \BA_E^\times / \BA_F^\times}\int_{ E^\times\backslash \BA_E^\times/\Xi_\infty } k(h_1, h_2) \Omega(h_1)\Omega^{-1}(h_2) dh_2dh_1 .$$
\end{defn}
\subsection{Pure matching conditions}
\label{Decomposition under the pure matching condition}
\begin{asmp} \label{fpure} Assume that $f\in \CH_\BC$
is a pure tensor: $f=\bigotimes_{v\in |X|}f_v.$
\end{asmp}
Recall that $|X|_s\subset |X|$ is the subset of places split in $E$. \begin{defn} \label{globalpurematch} Let $f\in \CH_\BC$ or $ C_c^\infty ( B^\times(\BA_F) ) $ and $\Phi\in C_c^\infty (\CS (\BA_F))$. We say that $f$ and $\Phi$ purely match if they purely match
at $v\in |X|-|X|_s$ and match at $v\in |X|_s$.
\end{defn}
Let $f\in \CH_\BC$ and $\Phi\in C_c^\infty (\CS (\BA_F))$ purely match. Suppose that there exists $f'\in C_c^\infty(G(\BA_E)$ satisfying Assumption \ref{freg} such that $\Phi=\Phi_{f'}$. We rearrange the decomposition of $\CO'( 0, f')$ in \eqref{decder} according to the decomposition $$F^\times-\{1\}=\coprod
\inv_{E^\times}( B ^\times_\reg) ,$$
where the union is over all quaternion algebras over $F$ containing $E$ as an $F$-subalgebra. Let $B$ be such a quaternion algebra, and $x\in \inv_{E^\times}( B ^\times_\reg) $. Consider $$\CO'(0,x,\Phi)=\sum_{v\in |X|}\CO'(0,x,\Phi_v)\CO(x ,\Phi^v) .$$
If $B $ and $ \BB$ are not isomorphic at more than one place (which must be in $|X|-|X|_s$), then by the pure matching condition, for every place $v$, the infinite product $$\CO(x ,\Phi^v)=\prod_{u\neq v} \CO(x ,\Phi_u) $$ contains at least one local component
with value 0. So $\CO(x ,\Phi^v)=0$, thus $\CO'(0,x ,\Phi)=0.$
Suppose $B=B(v)$ is a $v$-nearby quaternion algebra of $\BB$
for some place $v\in |X|-|X|_s$. For $x\in \inv_{E^\times}( B ^\times_\reg) $, let $\CO(x,f^v) $ be the orbital integral defined by regarding $f^v$ as a function on $B^\times(\BA_F^v)$. By the pure matching condition, $\CO(x,\Phi^u)\neq 0 $ only if $u=v$. In this case $ \CO(x,\Phi^v)=\CO(x,f^v)$.
To sum up, we have the following lemma. \begin{lem}\label{decpure} There is a decomposition
$$\CO'( 0,f')=\sum_{v\in |X|-|X|_s}\sum_{x\in\inv_{E^\times}( B(v)^\times_\reg) } \CO'(0,x,\Phi_v) \CO(x,f^v).$$
\end{lem}
Let $f\in C_c^\infty ( B^\times(\BA_F) )$ and $\Phi\in C_c^\infty (\CS (\BA_F))$ purely match as in Definition \ref{globalpurematch}.
\begin{lem}\label{decpure'} There is a decomposition $$\CO( f')= \sum_{x\in\inv_{E^\times}( B ^\times_\reg) } \CO(x,f).$$
\end{lem}
\subsection{Spectral decomposition}\label{Spectral decomposition of the automorphic distributions}
Let $\CA_c(G,\omega_E^{-1})$ be the set of all $\BC$-coefficient cuspidal representations of $\GL_2(\BA_E)$ with central character $\omega_E^{-1}$.
From now on, we always identify the complex conjugate of
$\sigma\in \CA_c(G,\omega_E^{-1})$
with the contragradient representation $\tilde\sigma$ of $\sigma$ via the Petersson pairing.
Let $\CF_c(G,\omega_E^{-1})$ be the space of $\BC$-valued cusp forms which transform by the Hecke character $\omega_E^{-1}$ under the action of $Z(\BA_E)$. For $f'\in C_c^\infty(\GL_2(\BA_E))$,
let $K (x,y) $ be the associated kernel function (see \eqref{autker}).
Let $$K_{c}(x,y):=\sum_{\phi}\rho(f')\phi(x)\bar\phi(y)$$ where the sum is over an orthonormal basis of $\CF_c(G,\omega_E^{-1})$ w.r.t the Petersson pairing. By \cite[Proposition 10.5]{JL},
this sum contains only finitely many nonzero terms. Then $K_{c}(x,y)$ is the kernel function of the Hecke action of $f'$ on $\CF_c(G,\omega_E^{-1})$.
For $\sigma\in \CA_c(G,\omega_E^{-1})$, let $$K_{\sigma}(x,y)=\sum_{\phi}\sigma(f')\phi(x)\bar \phi(y),$$
where the sum is over an orthonormal basis of $\sigma$. By the finiteness of above sums, for $*=c,\sigma $, $K_*$ has compact support in $(Z(\BA_E)\GL_2(E)\backslash\GL_2(\BA_E))^2$. In particular,
\begin{equation} \CO_* (s, f') := \int_{Z(\BA_E)A(E)\backslash A(\BA_E)} \int_{Z(\BA_E)H(F)\backslash H(\BA_F)} K_*(a,h)\Omega(a) \eta\omega^{-1}(\kappa(h) ) |a|_E^sdh da . \label{co} \end{equation} converges absolutely. By the multiplicity one theorem, we have
$$K_{c}(x,y)=\sum_{\sigma\in \CA_c(G,\omega_E^{-1}) } K_{\sigma}(x,y).$$ Similarly, define the kernel functions $K_\Sp$ and $ K_\Eis$ of the Hecke actions of $f'$ on the residual spectrum and continuous
spectrum of representations with central character $\omega_E^{-1}$ (see \cite[5.5]{Jac87}, \cite[8.2]{JN}).
Then $$K (x,y)=K_c(x,y)+K_\Eis(x,y)+K_\Sp(x,y).$$
For $\sigma\in\CA_c(G,\omega_E^{-1})$, $\phi\in \sigma$, $\chi:\BA_E^\times/E^\times\to \BC^\times$, define the toric period
$$\lambda (s,\phi)=\int_{Z(\BA_E)A(E)\backslash A(\BA_E)}\phi(a)\Omega(a) |a|_E^sda$$ which absolutely converges for all $s\in \BC $.
Also define the base change period $$\CP( \phi)= \int_{Z(\BA_E)H(F)\backslash H(\BA_F)} \phi(h) \eta\omega (\kappa (h) ) dh .$$
Then \begin{equation} \CO_\sigma (s,f') =\sum_{\phi} \lambda(s,\sigma( f')\phi)\overline {\CP ( \phi)} \label{COSigma}\end{equation} where the sum is over an orthonormal basis of $\sigma$.
However, for $*= \Eis, \Sp$, the same integral as in \eqref{co} needs to be regularized. Apply the truncation operators $\Lambda_d^{T}$ and $ \Lambda_m^{T}$ (\cite[Section 8]{JN}) on the kernel
$K_*$ and take \begin{equation*} \CO_{*,T_1,T_2} (s, f') := \int_{Z(\BA_E)H(F)\backslash H(\BA_F)} \int_{Z(\BA_E)A(E)\backslash A(\BA_E)} \Lambda_d^{T_1} \Lambda_m^{T_2} K_*(a,h)\Omega(a) \eta\omega^{-1}(\kappa(h) ) |a|_E^sda dh. \end{equation*} Define \begin{equation} \CO_{* } (s, f'):=\lim_{T_1\to 0}\lim_{T_2 \to 0} \CO_{*,T_1,T_2}' (0, f') \label{co*} . \end{equation} Define a regularized integral $\CO_{\reg } (s, f')$ similar to $\CO_{* } (s, f')$ with $K_* $ replaced by $K .$ \begin{lem}\label{modi} Assume that $f'$ satisfies Assumption \ref{freg}, then $\CO_{\reg} (s, f')=\CO (s, f')$. \end{lem} \begin{proof} In the number field setting, this is implied by \cite[Lemma 10]{JN} and \cite{Jac87} (see also the end of \cite[Section 3]{JN}). We modify the proof in our setting. There is another truncation $\Lambda_c^{T}$ in \cite{Jac87}.
By Lemma \ref{5.5.4} and \cite[p. 49]{JN}, the regularized integral of $K$ using $\Lambda_c^{T}$ is the same as $\CO (s, f')$. Then we use discussion in \cite[8.1]{JN} with the following modifications: \begin{itemize}\item[(1)] "rapid decreasing" in \cite[p. 71]{JN} is replaced by ``of compact support" (see \cite[I.2.9]{MW}); \item[(2)] to prove \cite[Lemma 9]{JN}, we note that the function $m(g)$ on $Z(\BA_E)\GL_2(E)\backslash\GL_2(\BA_E) $ defined below
\cite[Lemma 9]{JN} has compact support (which follows from (1) and definition). \end{itemize} \end{proof} Below, when citing a result from \cite{Jac87}\cite{JN}, we mean the same result hold in our setting with a modification of the proof, as we do in the proof of Lemma \ref{modi}. By \cite[(5.7)]{Jac87}\cite[p. 81, p. 83]{JN}, $\CO_\Sp (s, f')=0$. Thus \begin{equation}\CO(s, f')=\sum_{\sigma\in \CA_c(G,\omega_E^{-1})} \CO_\sigma (s, f')+ \CO_\Eis(s, f') . \label{COSigma'}\end{equation}
Let $N$ be the upper unipotent subgroup and let $K$ be the standard maximal compact compact subgroup of $G(\BA_E)$.
We use the standard notation: for $t\in \BC$ and $g\in G(\BA_E)$, let
$$e^{\pair{t+\rho,H(g)}}:=|a/d|_E^{1/2+t} $$ if $\begin{bmatrix}a&0\\ 0&d\end{bmatrix}^{-1}g \in NK.$
Fix $\alpha\in \BA_E^\times$ with $|\alpha|=q$. Let $\Lambda_E$ be the set of all Heche characters $\lambda$ of $E^\times$ such that $\lambda(\alpha)=1$ (which corresponds to the condition ``normalized" in \cite{JN}) modulo the equivalence relation $\lambda\simeq \lambda^{-1}\omega_E^{-1}$. For $\lambda\in \Lambda_E$, $t\in \BC$, the admissible representation of $G(\BA_E)$ associated to the data $(t,\lambda,\lambda^{-1}\omega_E^{-1})$ is realized
on the space of smooth functions $\phi: G( \BA_E )\to \BC$ such that
$$\phi\left(\begin{bmatrix}a&b\\ 0&d\end{bmatrix} x\right)=\lambda(a)\lambda^{-1}\omega_E^{-1}(d)\phi(x) $$ and the action of $g\in G(\BA_E)$ is given by $$g\cdot \phi(x)=e^{\pair{t+\rho,H(g)}}\phi(xg).$$
Let $S$ be a finite subset of $|X|$.
Let $\CT'^S $ be the spherical Hecke algebra of $G$ away from the set of places of $E$ over $S$, i.e. the algebra of bi-$K^S$-invariant functions in $C_c^\infty(G(\BA_E^S)$).
\begin{thm}[{\cite[(5.6)]{Jac87}\cite[Theorem 3]{JN}}] \label{Eisterm} Suppose that $S$ contains all ramified places of $E/F$ and all places below ramified places of $\Omega$. Given $ f'_{S }\in C_c^\infty(G(\BA_{E,S }))$, there exists \begin{itemize}
\item[(1)] for each $\chi\in\Lambda_E$ which is a lift of a Hecke character of $F^\times$,
a continuous function $\Phi_\chi(s,t)$ on $\BC\times [0,2\pi i/\log q]$, entire on the first variable with $\partial_s\Phi_\chi(s,t)$ continuous. Moreover, only finitely many $\Phi_\chi\neq 0$; \item[(2)] for each $ {\xi} \in {\Lambda}_E$ which is not a lift of a Hecke character of $F^\times$ and has restriction $\eta\omega^{-1}$ to $\BA_F^\times$,
an entire function $\Phi_ {\xi}(s )$.
Moreover, only finitely many $\Phi_ {\xi}\neq 0$; \item[(3)] an entire function $\Phi_\Omega(s )$;
\end{itemize}
such that as a linear functional on $\CT'^S$, we have
\begin{equation}\begin{split} \CO_\Eis (s, f'_Sf'^S)&=\sum_{\chi}\int_{0}^{2\pi i/\log q}\Phi_\chi(s,t) \widehat{f'^S}(t,\chi,\chi^{-1}\omega_E^{-1}) dt\\ &+\sum_{ {\xi}}\Phi_ {\xi}(s) \widehat{f'^S}(0, {\xi}, {\xi}^{-1}\omega_E^{-1}) \\ &+\Phi_\Omega(s) \widehat{f'^S}(1/2, \Omega, \Omega) . \label{COEis}
\end{split}
\end{equation} Here $\widehat{f'^S}$ is the Satake transform of $f'^S$, $\widehat{f'^S}(t,\chi,\chi^{-1}\omega_E^{-1})$ is nonzero only when representation of $G(\BA_E^S)$ associated to the data $(t,\chi^S,\chi^{S,-1}\omega_E^{S,-1})$ is unramified, and in this case, $\widehat{f'^S}(t,\chi,\chi^{-1}\omega_E^{-1})$ is the
value of $\widehat{f'^S}$ on the Satake parameter of the representation of $G(\BA_E^S)$ associated to the data $(t,\chi^S,\chi^{S,-1}\omega_E^{S,-1})$, etc..
\end{thm}
\begin{cor}\label{Eisterm'} As a linear functional on $\CT'^S$, we have \begin{align*}(\CO_\Eis)' (0, f'_Sf'^S)&=\sum_{\chi}\int_{0}^{2\pi i/\log q}\Phi_\chi'(0,t) \widehat{f'^S}(t,\chi,\chi^{-1}\omega_E^{-1}) dt\\ &+\sum_{ {\xi}}\Phi_ {\xi}'(0) \widehat{f'^S}(0, {\xi}, {\xi}^{-1}\omega_E^{-1}) \\ &+\Phi_\Omega'(0) \widehat{f'^S}(-1/2, \Omega, \Omega) . \end{align*}
\end{cor}
Let $\sigma=\sigma_{\xi}$ be the representation of $G(\BA_E)$ associated to the data $(0, {\xi}, {\xi}^{-1}\omega_E^{-1})$. Define $\CO_\sigma(s,f')$ by a trucantion process as in \eqref{co*}. If $f'=f_S'f'^S$, then $\CO_\sigma(s,f')$ equals the term $\Phi_ {\xi}(s) \widehat{f'^S}(0, {\xi}, {\xi}^{-1}\omega_E^{-1})$ in \eqref{COEis}.
\part{Local theory}
\section{Notations and measures}\label{notations and measures }
\subsection{Local setting}\label{localnotations and measures }
Let $F$ be a non-archimedean local field of residue characteristic $p>0$. Let $\varpi $ be a uniformizer, and $k =\CO_F/\varpi$ the residue field with $q $ elements.
Let $ v $ be the discrete valuation on $F$ such that $v(\varpi)=1$, and $|\cdot| $ the absolute value on $F$ such that $|\varpi|=q^{-1}$. Let $E $ be a separable quadratic extension of $F$,
and $\eta$ the associated quadratic character field of $F^\times$. We use subscript to distinguish notations for $E$ and $F$ when necessary,
such as $|\cdot|_F $ and $|\cdot| _E$.
Let $E^1\subset E^\times$ be the subgroup of norm 1.
Make the following assumption from now on.
\begin{asmp}\label{asmpep} We fix a representative $\ep$ for each coset in $F^\times/{\mathrm{Nm}}(E^\times)$. If $E/F$ is unramified, choose $\ep=1$ or $\varpi$.
If $E/F$ is ramified,
choose $\ep=1$ or $\ep\in \CO_{F}^\times$ and $ \ep\neq 1 (\mod {\mathrm{Nm}} ( E^\times))$.
\end{asmp}
Let $G,H,H_0,\CS$ and $G_\ep $ be defined as in \ref{matchorb}. Let $K=\GL_2(\CO_E)$ be the standard maximal compact subgroup of $G=\GL_2(E)$, and let $K_{H_0}=K\cap H_0$.
Note that $G_\ep\simeq G_{\ep{\mathrm{Nm}} (c)}$ for every $c\in E^\times$, and
$F^\times/ {\mathrm{Nm}} (E^\times)$ consists of two elements. Embed $G_\ep$ into $G $ via \eqref{(5.1)}. Let \begin{equation*}K_\ep:=G_\ep\cap \GL_2(\CO_E). \end{equation*} Then if $E/F$ is unramified, then $K_\ep$ is a maximal compact subgroup of $G_\ep$.
Let the tori $T_\ep,Z_\ep$ of $G_\ep$ be as in \ref{matchorb}.
We fix measures as in \cite[2.1, 2.2]{JN}. Let $\psi$ be a nontrivial additive character of $F$. Let $ d_Fx$ be the self-dual Haar measure on $F$ w.r.t. $\psi$, then $ L(1,1_F)|x|_F^{-1}d_Fx$ is a Haar measure on $F^\times$. If not confusing, we just use $dx$ to indicate these Haar measures on $F$ or $F^\times$. Let $c(\psi)$ be the conductor of $\psi$.
Then we have $\Vol (\CO_F)=\Vol (\CO_F^\times)=q^{c(\psi)/2} $ where the volumes are computed w.r.t. the additive measure on $F$ and the multiplicative measure on $F^\times$.
Let $\tr=\tr_{E/F}$ be the trace map from $E$ to $F$, and let $\psi_E:=\psi\circ \tr $ which is an additive character on $E $.
Define measures on $E$ and $E^\times$ in the same way using $\psi_E$.
Endow $E^\times/F^\times$ with the quotient measure.
Let the measures on $T_\ep $ and $Z_\ep $ be induced from the Haar measures on $E^\times$ and $F^\times$.
Define the measure on $G$ by $$dg=L(1,1_E)\frac{\prod\limits_{i,j=1}^2d_Eg_{i,j}}{|\det g|_E^2} .$$
Then $\Vol(K)=L(2,1_E)^{-1}\Vol(\CO_E)^4. $ The same formula also gives the measure on $\GL_2(F)$.
Define the measure on $G_\ep$ by $$dg=L(1,1_F)|\ep|_F\frac{d_Ead_Eb}{|\det g|_F^2} $$
if $g=\begin{bmatrix}a&b\ep\\
\bar b&\bar a\end{bmatrix}$.
In particular, if $\ep=1$, this recovers the measure on $\GL_{2}(F)$.
Define the measure on $H$ as follows. Let $H'\subset G$ be the image $\GL_{2,F}$ under the natural embedding.
If $p>2$, let $\xi\in E-F$ be a trace free element. Then \begin{equation}\begin{bmatrix}\xi&0\\ 0&a\end{bmatrix}H\begin{bmatrix}\xi^{-1}&0\\ 0&1\end{bmatrix}=ZH'.\label{HH'}\end{equation}
Define the measure on $H'Z$ by
$$dg=L(1,1_E) \frac{d_Fad_Ez}{|a|_F|z|_E}d_Fxd_Fy $$ if $$g=\begin{bmatrix}z&0\\ 0&z\end{bmatrix}\begin{bmatrix}a&0\\ 0&1\end{bmatrix}\begin{bmatrix}1&0\\ y&1\end{bmatrix}\begin{bmatrix}1&x\\ 0&1\end{bmatrix}.$$ Define the measure on $H$ by \eqref{HH'} and the measure on $HZ'$. If $p=2$, then $ H =ZH'$. The measure on $H$ is defined to be the measure on $HZ'$.
\begin{lem}\label{Kmeasure}If $E/F$ and $\psi$ are unramified, then
$$
\Vol(K_1)=\Vol(K_{H_0})=L(2,1_F).$$ \end{lem}
The following lemma plays an important role in \cite{JN} and \cite[(4.3)]{Jac86}.
\begin{lem}[{\cite[(15)]{JN}}]\label{JC15} Let $f$ be an integrable function on $G_\ep$, then
$$\int_{G_\ep}f(g)d g =\frac{1}{L(1,\eta)^2}\int_{ {x=\ep a\bar a\in F^\times} } \int_{T_\ep/Z_\ep}\int_{T_\ep}f\left(h_1^{-1} \begin{bmatrix}1&a\ep\\ \bar a &1\end{bmatrix} h_2\right) dh_1dh_2 \frac{d_F x}{|1-x|_F^2} .$$
\end{lem}
\subsection{Global setting}\label{Global measures}
For $v\in |X|$, let $\varpi_v$ be a uniformizer of $\CO_{F_v}$, $k(v)$ the residue field with cardinality $q_v $. The absolute value on $\BA_F$ is the product of the local ones. Similar notations apply to finite extensions of $F$.
For $v\in |X|$, $\BB_v^\times$ is denoted by $G_\ep$ (for the corresponding $\ep$) in the local setting \ref{Split case} $\ep$ is an abstract subscript and \ref{localnotations and measures } ($\ep$ is as in Assumption \ref{asmpep}).
Fix an isomorphism $ D^\times(\BA_{\mathrm{f}})\simeq \BB_{\mathrm{f}}^\times $ such that the image of $\cD_v^\times$ is $K_\ep\subset G_\ep$ for every $v\in |X|-\{\infty\}$.
Fix a nontrivial additive character $\psi$ on $F\backslash \BA_F$, and a decomposition $\psi=\prod_ {v\in |X|}\psi_v$. Choose local measures on $F_v^\times,$ $E_v^\times,$ $\BB_v^\times$, $G(E_v)$, and $\GL_2( F_v)$ as in \ref{localnotations and measures }.
Take the product measures on adelic objects.
\section{Local distributions}\label{local relative trace formula}
Let $F$ be a local field,
$E/F$ a separable quadratic field extension. Let $\Omega$ be a unitary character of $E^\times$, $\omega$ the restriction of $\Omega$ to $F^\times$ and $\omega_E:=\omega\circ{\mathrm{Nm}}.$
We define distributions on $G$ and $G_\ep$ associated to their representations. (These distributions are called spherical characters in \cite{Zha14b}). The values of these distributions at certain matching functions will satisfy a ``spherical character identity" (following the terminology of \cite{Zha14b}). The split case will be dealt in \ref{secSplit places}. \subsection{A distribution on $G$ }\label{the local distribution I(f)}
Let $\sigma $ be an infinite dimensional irreducible unitary representation of $G=\GL_{2,E}$ with central character $\omega_E^{-1}$. Let $W(\sigma,\psi_E)$ be the $\psi_E$-Whittaker model of $\sigma$.
Define a $G$-invariant inner product on $W(\sigma,\psi_E)$ by \begin{equation}\pair{W_1,W_2}:= \int_{E^\times}W_1\left(\begin{bmatrix}x&0\\ 0&1\end{bmatrix}\right )\overline {W_2\left(\begin{bmatrix}x&0\\ 0&1\end{bmatrix}\right )}d x.\label{inprod}\end{equation}
For $W\in W(\sigma,\psi_E)$, define the local toric period
\begin{equation}\lambda (s,W):=\int_{E^\times}W \left(\begin{bmatrix}x&0\\ 0&1\end{bmatrix}\right )|x|_E^s\Omega(x)d x\label{lambda}\end{equation}
and the local base change period
\begin{equation}\CP (W):=\int_{F^\times}W \left(\begin{bmatrix}x&0\\ 0&1\end{bmatrix}\right ) \eta\omega(x)d x\label{cp}. \end{equation}
Assume that $\sigma$ is tempered, then \eqref {lambda} converges absolutely for $\Re(s)>-1/2$. \begin{defn} For $f\in C_c^\infty(G)$ and $\Re (s)>-1/2$, define $$ I_\sigma(s, f)= \sum_W \lambda (s,\pi(f)W) \overline {\CP( W)}$$ where the sum is over an orthonormal basis of $W(\sigma,\psi_E)$.
Denote $I_\sigma(0,f)$ by $I_\sigma(f)$.
\end{defn}
\subsection{Distributions on $G_\ep$}\label{ local distribution J(f)}
Let
$\pi $ be an irreducible unitary representation of $G_\ep$ with central character $\omega^{-1}$. Let $\vep(1/2,\pi,\Omega):=\vep(1/2,\pi_E\otimes \Omega)$.
Define $$\CP_\Omega(\pi):=\Hom_{T_\ep}(\pi\otimes \Omega,\BC).$$
\begin{thm} [{Tunnell \cite{Tun}, Saito \cite{Sai}}]\label{TSlocal} The space $ \CP_\Omega(\pi )$ is at most one dimensional. Moreover, $\dim\CP_\Omega(\pi )=1$ if and only if the $\vep$-factor satisfies: $$\vep(1/2,\pi,\Omega) =\eta(\ep)\Omega(-1).$$
\end{thm}
\begin{rmk}The proof in \cite{Tun} for non-supercuspidal representations holds for all local fields. The proof in \cite{Sai} for supercuspidal representations holds for all local fields.
\end{rmk}
Assume that $\vep(1/2,\pi,\Omega) =\eta(\ep)\Omega(-1)$. Let $\tilde e_\pi\in\tilde \pi$ (resp. $e_{\pi}\in \pi$) be the unique up to scalar vector of $\tilde \pi$ (resp. $ \pi$) such that the linear form $$v\mapsto (v,\tilde e_\pi)\mbox{ (resp. }\tilde v\mapsto ( e_\pi, \tilde v) \mbox{)}$$ generates $\CP_\Omega(\pi) $ (resp. $\CP_{\Omega^{-1}}(\tilde \pi) $). Here $(\cdot,\cdot)$ is the natural pairing between $\pi$ and $\tilde \pi$.
Assume that $(e_\pi,\tilde e_\pi)=1$. For $f\in C_c^\infty(G_\ep)$, define $$J_\pi(f):=\sum_v( \pi(f)v, \tilde e_\pi) ( e_\pi,\tilde v) $$ where the sum is over an orthonormal basis $\{v\}$ of $\pi$ and $\{\tilde v\}$ is the dual basis of $\tilde \pi$.
Let $w_{\pi}$ be the function on $G_\ep$ defined by $$w_\pi(g)=(\pi(g)e_\pi,\tilde e_\pi).$$ Then $w_{\pi}$ is $T_\ep\times T_\ep$-invariant, locally constant with $w_{\pi}(1)=1$, and
\begin{equation}J_\pi(f)=\int_{G_\ep}f(g)w_\pi(g)dg.\label{7.5}\end{equation}
Let $f=1_{U}$ where $U$ is a small enough open compact subgroup of $G_\ep$, then \begin{equation}J_\pi(f)=J_{\pi\otimes \eta}=\int_{G_\ep}f(g)dg\neq 0.\label{7.4}\end{equation}
Let $\alpha_{\pi}\in \CP_\Omega(\pi)\otimes \CP_{\Omega^{-1}}(\tilde \pi) $ be
$$\alpha_\pi(u,\tilde v)=\int_{T_\ep/Z_\ep} (\pi(t) u,\tilde v) \Omega(t)dt ,$$ which is essentially a finite sum. By Theorem \ref{TSlocal}, $\alpha_{\pi}({\cdot ,\cdot })$ is a multiple of $(\cdot,\tilde e_\pi)( e_\pi,\cdot )$. Taking the first variable to be $e_\pi$, we find that the ratio is $ \Vol(E^\times/F^\times)$.
\begin{defn} \label{api}For $f\in C_c^\infty(G_\ep)$, we
abuse notation and define $$\alpha_{\pi}(f):=\sum_v\alpha_{\pi}( \pi(f)v ,\tilde v)$$ where the sum is over an orthonormal basis $\{v\}$ of $\pi$ and $\{\tilde v\}$ is the dual basis of $\tilde \pi$. \end{defn}Then $\alpha_{\pi}(f)= \Vol(E^\times/F^\times)J_{\pi}(f).$ By \eqref{7.4}, the following lemma holds.
\begin{lem}\label{pipieta} Let $f=1_{U}$ where $U$ is a small enough open compact subgroup of $G_\ep$. Then
$\alpha_\pi(f)=\alpha_{\pi\otimes \eta}(f)\neq 0.$
\end{lem}
\subsection{Spherical character identity} \label{the proof is more important for us}
Assume that $\vep(1/2,\pi,\Omega) =\eta(\ep)\Omega(-1)$. Let
$\sigma$ be the base change of
$\pi$ to $E$. The values of the distributions on $G$ and $G_\ep$ for $\sigma$ and $\pi$ at matching functions are expected to satisfy a ``spherical character identity" (following the terminology of \cite{Zha14b}). Such an identity is proved in \cite[Proposition 5]{JN} conditionally.
In particular, in \cite[Proposition 5]{JN},
$\sigma$ is a local component of a global representation with non-vanishing central $L$-value. Thus the case we need for Theorem \ref{GZ} is not covered.
The main result of this subsection is Corollary \ref{7.9}, which is an explicit example of matching functions satisfying
the spherical character identity. We find this example using a construction from the proof of \cite[Proposition 5]{JN}, which we recall now.
\begin{prop} \label{localRTF}
Suppose there is a constant $c$ such that for every pair of (not necessarily purely) matching functions
$f\in C_c^\infty(G)$ and $f_\ep\in C_c^\infty(G_\ep)$ supported on $\{g\in G_\ep:\det g\in {\mathrm{Nm}} (E^\times)\}$, the following equation holds: $ I_\sigma(f)=c J_{\pi}(f_\ep). $Then
\begin{align*} c=2 \vep(1,\eta,\psi) L(0,\eta) . \end{align*}
\end{prop}
\begin{proof}
We follow the proof of \cite[Proposition 5]{JN}.
Let $f_n \in C_c^\infty(G)$ approximate the Dirac function $\delta_1$ at $1\in G$, then $\Phi_{f_n}$ approximates $\delta_w$. Let $\phi\in C_c^\infty(E)$, $\hat\phi$ be the Fourier transform of $\phi$ w.r.t. $\psi$. Choose $\phi$ such that $\hat\phi(0)=0$ and $\int_{F^\times}\hat \phi (b)\eta(b)d b\neq 0.$ Such $\phi$ exists (see \ref{7.3.1}). Let \begin{equation}f^n(g):=\int_{E}\phi(x)f_n\left(\begin{bmatrix}1&x\\ 0&1\end{bmatrix} g\right)dx.\label{fn}\end{equation} As in \cite[p 57]{JN}, for $n$ large enough w.r.t. the choice of $\phi$, we have $$I_\sigma(f^n)=\int_{F^\times}\hat \phi (b)\eta(b)d b\neq 0.$$
Let $f_\ep^n$ match $f^n$. To find the constant $c$ between $ I_\sigma(f^n)$ and $J_{\pi}(f_\ep^n)$,
the key is to compute $J_{\pi}(f_\ep ^n)$ using $f^n$. We have the following result deduced from Lemma \ref{JC15} and \eqref {7.5}.
\begin{lem}[{\cite[5.3]{JN}}]\label{JC5.3} Let $f$ and $f_\ep$ match, then
$$J_{\pi}(f_\ep) =\frac{1}{L(1,\eta)^2}\int_{x=\ep a\bar a\in F^\times }\omega_{\pi}\left(\begin{bmatrix}1&a\ep\\ \bar a &1\end{bmatrix}\right)\CO(x,\Phi_{f})\frac{d _Fx}{|1-x|_F^2}.$$
\end{lem}
The computation of $J_{\pi}(f_\ep)$ for $p>2$ is similar to \cite[5.2]{JN}.
To integrate on $E$, one write $E=F+F\xi$ where $\xi$ is a trace free element. When $p=2$, the difference is as follows. Write $E=F+F\xi$ where $\xi$ has trace 1.
So if we write $t\in E$ as $a+b\xi$, the roles of $a$ and $b\xi$ in the cases $p>2$ and $p=2$ are exactly opposite.
\end{proof}
The proof of Proposition \ref{localRTF} implies the following proposition.
\begin{prop} \label{localRTF'}
Let $f_n \in C_c^\infty(G)$ approximate $\delta_1$. Let $\phi\in C_c^\infty(E)$ such that $\hat\phi(0)=0$ and $\int_{F^\times}\hat \phi (b)\eta(b)d b\neq 0.$ Let $ f^n$ be defined as in \eqref{fn}. Then for $n$ large enough w.r.t. the choice of $\phi$, the following equation holds
for any $f_\ep\in C_c^\infty(G_\ep)$ matching $f^n$: \begin{align*} I_\sigma(f^n)=\frac{2 \vep(1,\eta,\psi) L(0,\eta)}{\Vol(E^\times/F^\times)} \alpha_{\pi}(f_\ep).\end{align*}
\end{prop}
\subsubsection{An explicit example}\label{7.3.1}
For $f_n\in C_c^\infty(G) $, let $f^{\phi,n}$ denote the function $f^n$ defined in \eqref{fn}, indicating the role of $\phi$.
We construct $f_n$ and $\phi$ satisfying the requirements in Proposition \ref{localRTF'}.
Then we compute $\Phi_{f^{\phi,n}}$ explicitly, which will be useful for smooth matching in \ref{Explicit computations for smooth matching}.
Let $\fp_E$ be the maximal ideal of $\CO_E$. Let
\begin{equation*} K_{n}:=\left\{\begin{bmatrix}a&b\\ c&d\end{bmatrix} \in \GL_2( \CO_E):\begin{bmatrix}a&b\\ c&d\end{bmatrix} \equiv w (\mod \fp_E^n)\right \}. \end{equation*}
For $n$ large enough,we have $$-\det( K_{n}) =1+\fp_E^n\subset {\mathrm{Nm}} (E^\times).$$ By Lemma \ref{goodx}, $K_{n}\cap \CS\subset Gw .$ Endow $Gw=G/H_0$ with the quotient measure.
Then $1_{K_{n}\cap \CS}/\Vol(K_{n}\cap \CS)$ approximates $\delta_w$.
By \cite[V.2, Theorem 11]{HC}, we have the following lemma.
\begin{lem} \label{HCused} There exists $f_n\in C_c^\infty(G) $ approximating $\delta_1$ such that $$\Phi_{f_n}=\frac{1_{K_{n}\cap \CS}}{\Vol(K_{n}\cap \CS) }. $$ \end{lem}
Now we construct the function $\phi$ we want.
For $l\in \BZ$, $\xi\in E$, let
$$\phi_{l,\xi}:=\frac{1_{\xi+\fp_E^l}}{\Vol(\fp_E^l)}.$$ Let $c(\psi_{E})$ be the conductor of $\psi_{E}$.
If $b\in \fp_E^{-(l+c(\psi_{E}))} $, then $$\widehat {1_{\xi+\fp_E^l}}(b)=\psi_E(\xi b) \int_{\fp_E^l}\psi_E(ab)da=\psi_E(\xi b)\Vol(\fp_E^l);$$
otherwise $\widehat {1_{\xi+\fp_E^l}}(b)=0.$ Let $\phi :=\phi_{l',\xi'}-\phi_{l,\xi}$
where $l>l'$, then
$$\widehat {\phi}(b)=\psi_E(\xi'b)-\psi_E({\xi b})$$ on $\fp_E^{-(l'+c(\psi_{E}))} $.
In particular, \begin{equation}\widehat {\phi}(0)=0\label{phi0}\end{equation}
Now we compute the integral $\int_{F^\times}\hat \phi (b)\eta(b)d b .$ Recall the following fact about Gauss sums.
\begin{lem}\label{Gsum} (1) Let $ \chi $ be a unitary character of $ F^\times$ of conductor $c(\chi)$, and
let $$\tau_n(\chi,\psi):=\int_{\CO_F^\times} \chi(\varpi^n x)\psi(\varpi^nx)d x.$$
Then $\tau_n(\chi,\psi)\neq 0$ if and only if $n= -(c(\chi)+c(\psi))$.
(2) For $a\in \CO_F^\times$, let $\psi_a$ be the character $x\mapsto \psi(ax)$. Then
$$\tau_n(\chi,\psi_a)=\chi(a^{-1})\tau_{n }(\chi,\psi).
$$
(3) If $\chi$ is unramified, then $$\tau_{-c(\psi)}(\chi,\psi)= \chi(\varpi^{-c(\psi)} )\Vol(\CO_F^\times).$$
\end{lem}
Note that the restriction of the character $b\mapsto \psi_E(\xi b)$ to $F$ is
$\psi_{ \tr(\xi)} $. Let $c=c(\eta)+c(\psi)$.
We always choose $l,l'$ such that $$l>l'> c+\max\{v_F( \tr(\xi)), v_F( \tr(\xi')) \}.$$
Choose $\xi,\xi' \in E $ with nonzero trace as follows. Note that the trace map from $E$ to $F$ is surjective.
If $E/F $ is ramified, then we choose $\xi$ and $\xi'$ such that $ \tr(\xi) , \tr(\xi')\in \CO_F^\times $ and $ \tr(\xi)\neq \tr(\xi)(\mod {\mathrm{Nm}}(E^\times)).$ By Lemma \ref{Gsum} (2), we have \begin{equation}|\int_{F^\times} \hat\phi (b)\eta(b)d b |=|2\tau_{n}(\eta,\psi)|\neq 0 \label{gausssum0}.
\end{equation}
If $E/F $ is unramified, then we choose $\xi,\xi'$ such that $v_F( \tr(\xi)) $ and $ v_F( \tr(\xi')) $ are nonnegative and have different parities.
By Lemma \ref{Gsum} (3), we have \begin{equation}|\int_{F^\times} \hat\phi (b)\eta(b)d b| =2\Vol(\CO_F^\times)\neq 0 \label{gausssum}.
\end{equation}
Let $n$ be large enough depending on the choices of $l,l'$, $\xi$ and $\xi'$. Since we have \eqref{phi0}, \eqref{gausssum0} and \eqref{gausssum}, Proposition \ref{localRTF'} implies the following corollary. \begin{cor}\label{7.9} For any $f_\ep\in C_c^\infty(G_\ep)$ matching $f^{\phi,n}$, we have
\begin{align*} I_\sigma(f^{\phi,n})=\frac{2 \vep(1,\eta,\psi) L(0,\eta)}{\Vol(E^\times/F^\times)} \alpha_{\pi}(f_\ep).\end{align*}
\end{cor} Now we compute $\Phi_{f^{\phi,n}}$. We use the relation \begin{equation}\Phi_{f^{\phi,n}}(s) =\int_{E}\phi(x)\Phi_{f_n}\left(\begin{bmatrix}1&x\\ 0&1\end{bmatrix} s\begin{bmatrix}1&0\\ \bar x&1\end{bmatrix}\right )dx.\label{79}\end{equation}
Let \begin{equation} K_{l,\xi,n}:=\left\{\begin{bmatrix}a&b\\ c&d\end{bmatrix} \in \GL_2( \CO_E):\begin{bmatrix}0&b\\ c&d\end{bmatrix} \equiv \begin{bmatrix}0&1\\ 1&0\end{bmatrix} (\mod \fp_E^n),\ a\in -\tr(\xi)+\tr(\fp_E^l) \right\}.\label{orbintK1st}\end{equation} Let $X_n:=\tr^{-1}(\CO_F\cap \fp_E^n)\cap \fp_E^l$.
A direct computation using \eqref{79} shows that \begin{equation*}\Phi_{f^{\phi_l,n}}(s)=\frac{ \Vol(X_n)}{\Vol(K_{n}\cap \CS)\Vol(\fp_E^l)} 1_{\CS\cap K_{l,\xi,n}} .\end{equation*} Thus \begin{equation}\Phi_{f^{\phi,n}}(s)= \frac{ \Vol(X_n)}{\Vol(K_{n}\cap \CS) } \left (\frac{1_{\CS\cap K_{l',\xi',n}}}{ \Vol(\fp_E^{l'})} -\frac{1_{\CS\cap K_{l,\xi,n}}}{\Vol(\fp_E^l)}\right).\label{ppn}\end{equation}
\subsection{Split extension}\label{secSplit places}
Recall the setting in \ref{Split case}.
Let $G= \GL_2(F)\times \GL_2(F)$
and $G_\ep= \GL_2(F). $ Here $\ep$ is just an abstract subscript.
Let
$\pi$ be an infinite dimensional irreducible unitary representation of $G_{\ep }$. Let $\sigma=\pi\boxtimes \pi$ which is an irreducible unitary representation of $G $.
Let $\Omega=\Omega_1\boxtimes \Omega_2$ be a unitary character of $ F^\times\times F^\times$. Then $\omega:=\Omega_1\otimes \Omega_2$ is the restriction of $\Omega$ to the diagonal embedding of $F^\times$ in $ F^\times\times F^\times$.
Let $W(\sigma,\psi\boxtimes \psi)=W(\pi,\psi)\boxtimes W(\pi,\psi)$ be the Whittaker model of $\sigma$, on which we have the inner product \eqref{inprod}. Let $W= W_1\otimes W_2\in W(\sigma,\psi\boxtimes \psi)$. Define two periods on $W(\sigma,\psi\boxtimes \psi)$:
\begin{equation}\lambda _\sigma( s, W):=\prod_{i=1,2}\int_{F^\times}W_i \left(\begin{bmatrix}x&0\\ 0&1\end{bmatrix}\right ) |x|^s\Omega_i(x)d x\label{lambda1},\end{equation}
and
\begin{equation}\CP (W):=\int_{F^\times}W_1W_2 \left(\begin{bmatrix}x&0\\ 0&1\end{bmatrix}\right ) \eta\omega(x)d x\label{lambda2}. \end{equation}
Assume that $\sigma$ is tempered, then the integral \eqref {lambda1} converges for $\Re(s)>-1/2$.
\begin{defn} For $f\in C_c^\infty(G)$ and $\Re(s)>-1/2$, define $$ I_\sigma(s, f)= \sum_W \lambda (s,\pi(f)W) \overline {\CP( W)}$$ where the sum is over an orthonormal basis of $W(\pi,\psi)\boxtimes W(\pi,\psi)$. Denote $I_\sigma(0,f)$ by $I_\sigma(f)$.
\end{defn}
Now we turn to the $G_\ep $-side.
For $W_1,W_2\in W(\pi,\psi)$, we have an absolute convergent integral \begin{align*} \alpha_\pi(W_1,W_2):&= \int_{E^\times/F^\times} \pair{\pi(t)W_1,W_2}\Omega(t)dt .\end{align*}
\begin{defn} \label{def58} For $f\in C_c^\infty(G_\ep)$, we
abuse notation and define $$ \alpha_\pi ( f_\ep)= \sum_W\alpha_\pi(\pi(f_\ep)W , W)$$ where the sum is over an orthonormal basis of $W(\pi,\psi) $.
\end{defn}
The following spherical character identity can be directly verified.
\begin{prop}[{\cite[Proposition 6]{JN}}]\label{localRTF2} Let $ f_1,f_2\in C_c^\infty(\GL_2(F))$ and
$f_\ep=f_1\ast \tilde f_2 \in C_c^\infty(G_\ep)$ (see \eqref{splitplaces}), then $ I_\sigma(f_1\otimes f_2)= \alpha_\pi(f_\ep).$
\end{prop}
\begin{lem}[{\cite[Theorem A.2]{Zha14}}] \label{nontrisplit} There exists $ f_\ep\in C_c^\infty(G_{\ep,\reg})$ such that $\alpha_{\pi }(f_\ep)\neq 0.$ \end{lem}
\section{Smooth matching and fundamental lemma} \label{review}
We have defined the local orbital integrals $\CO( x,\Phi)$ and $\CO( x,f_\ep)$ for
$\Phi\in C^\infty_c(\CS)$ and $ f_\ep\in C^\infty_c( G_\ep)$ respectively in \ref{local orbital integrals0}. We compute explicit examples of $\CO( x,\Phi)$ to prove special cases of the smooth matching. Using the results in \ref {7.3.1}, we prove the spherical character identity for these functions. Finally we prove the fundamental lemma for the full Hecke algebra.
\subsection{Local orbital integrals on $G_\ep$}
We have the following characterization of $\CO(x,f)$. \begin{prop}[{\cite[p. 332, Proposition]{Jac87}}]\label{Proposition 2.4, Proposition 3.3Jac862}
Let $\phi$ be a function on $\ep {\mathrm{Nm}} (E^\times)-\{1\}$. Then $\phi(x)=\CO( x,f)$ for some $f\in \CC_{ c}^\infty(G _\ep)$ if and only if the following conditions hold: \begin{itemize}
\item[(1)]the function $\phi$ is locally constant on $\ep E^\times-\{1\}$;
\item[(2)] the function $\phi$ vanishes near 1;
\item[(3)]there exists a constant $A$, such that $\phi(x)=A $ for $ x$ near $0$;
\item[(4)]
there exists a constant $B$ such that $\phi(x)=\Omega(a)B $ for $x =\ep a\bar a$ and near $\infty$;
\end{itemize}
\end{prop}
We remind the reader that \cite[p. 332, Proposition]{Jac87} contains a mistake: it swapped the behaviors of $\CO( x,f)$ for $x$ near 0 and near $\infty$.
Indeed, it is easy to detect by taking $f$ to be supported near 1
and $x$ near 0.
\subsection{Explicit computations for smooth matching}\label{Explicit computations for smooth matching}
To avoid confusion, let $\Vol^\times$ indicate the volume of an open subset of $F^\times$ or $E^\times$ w.r.t. the multiplicative measure,
let
$\Vol^+$ indicate the volume of an open subset of $F$ or $E$ w.r.t. the additive measure. Let $\xi\in E$ with $\tr(\xi)\neq 0$, $n,l $ be integers which are large enough. Let $K_{l,\xi,n}$ be as in \eqref{orbintK1st}. Similarly define \begin{equation} K'_{l,\xi,n}:=\left\{\begin{bmatrix}a&b\\ c&d\end{bmatrix} \in \GL_2( \CO_E):\begin{bmatrix}a&b\\ c&0\end{bmatrix} \equiv \begin{bmatrix}0&-1 \\ -1 &0\end{bmatrix} (\mod \fp_E^n),\ d\in -\tr(\xi)+\tr(\fp_E^l)\right \}.\label{orbintK'}\end{equation}
\begin{eg} \label{fepm1} Let $l, n$ be large enough such that $\Omega(1+\fp_E^n)=1$ and $\eta(-\tr(\xi)+\tr(\fp_E^l))=\eta(-\tr(\xi))$.
Then for $x\in F^\times-\{1\}$, $\CO(x,1_{ K_{l,\xi,n}\cap \CS})=0$ unless $v_E(x)\geq n+v_E(\tr(\xi))$.
In this case, we have $$\CO(x, 1_{ K_{l,\xi,n}\cap \CS})=\eta(-x\tr(\xi))\Vol^{\times}(1+\fp_E^{n })\Vol^{\times}(-\tr(\xi)+\tr(\fp_E^l)) .$$
\end{eg}
\begin{eg} \label{fepm2} Let $l, n$ be large enough such that $\Omega(1+\fp_E^n)=1$ and $\eta(-\tr(\xi)+\tr(\fp_E^l))=\eta(-\tr(\xi))$.
Then for $x\in F^\times-\{1\}$, $\CO( x,1_{ K_{l,\xi,n}'\cap \CS})=0$ unless $v_E(x)\geq n+v_E(\tr(\xi))$.
In this case, we have $$\CO(x, 1_{ K_{l,\xi,n}'\cap \CS})=\eta(-\tr(\xi)) \Vol^{\times}(1+\fp_E^{n })\Vol^{\times}(-\tr(\xi)+\tr(\fp_E^l))\cdot \Omega(-1).$$
\end{eg}
We also have some explicit computations on $G_\ep$ following
\cite[2.3, 2.4]{Guo}.
Recall that $G_\ep$ is embedded in $G$ via \eqref{(5.1)} and Assumption \ref{asmpep}.
For an integer $m\geq 1$, let \begin{equation} K_{\ep,m}:=\left\{\begin{bmatrix}a&b\\ c&d\end{bmatrix} \in K_\ep=G_\ep \cap \GL_2( \CO_E):\begin{bmatrix}a&b\\ c&d\end{bmatrix} \equiv 1 (\mod \fp_E^m) \right\}.\label{orbint1K}\end{equation}
This is a congruence subgroup of the maximal compact subgroup $K_\ep$ in the usual sense.
\begin{eg} \label{fepm} Let $x\in \ep {\mathrm{Nm}}(E^\times)-\{1\}$.
If $v_E(x)\geq 2 m+v_F(\ep)$,
then $$\CO(x, 1_{ K_{\ep,m}})=\Vol^{\times}(1+\fp_E^{m })\Vol (E^\times/F^\times).$$ Otherwise, $\CO(x,1_{ K_{\ep,m}})=0$.
\end{eg}
Let $\pi $ be an irreducible unitary representation of $G_\ep$ such that $\vep(1/2,\pi,\Omega) =\eta(\ep)\Omega(-1).$
Let $\sigma$ be the base change of
$\pi$ to $E$.
Let $I_\sigma $ (resp. $ \alpha_{\pi} $) be the local distribution on $G$ (resp. $G_\ep$) defined in Section \ref{local relative trace formula}.
\begin{prop}\label{expmat}Let $m$ be large enough. There exists $f\in C_c^\infty(G)$ such that \begin{itemize}\item[(1)] $ f$ purely matches $1_{ K_{\ep,m}}$; \item[(2)] the following equation holds: \begin{align*} I_\sigma(f)=\frac{2 \vep(1,\eta,\psi) L(0,\eta)}{\Vol(E^\times/F^\times)} \alpha_{\pi}(1_{ K_{\ep,m}}).\end{align*} \end{itemize}
\end{prop}
\begin{lem} \label{822}Let $m$ be large enough. There exists $f_1\in C_c^\infty(G)$ such that \begin{itemize}\item[(1)] for $x\in \ep {\mathrm{Nm}}(E^\times)-\{1\}$, if $v_E(x)\geq 2 m+v_F(\ep)$,
then $$\CO(x, \Phi_{f_1})=\eta(\ep x) \Vol^\times(1+\fp_E^{m })\Vol(E^\times/F^\times),$$ otherwise $\CO(x,\Phi_{f_1})=0$;
\item[(2)] the following equation holds: \begin{align*} I_\sigma(f_1)=\frac{2 \vep(1,\eta,\psi) L(0,\eta)}{\Vol(E^\times/F^\times)} \alpha_{\pi}(1_{ K_{\ep,m}}).\end{align*} \end{itemize}
\end{lem}
\begin{proof}We use the results in \ref{7.3.1}.
Choose $\xi,\xi' \in E^\times$ and positive integers $l,l'$ as under Lemma \ref{Gsum}. In particular, $\eta(\tr(\xi'))/\eta(\tr(\xi))=-1$.
If $E/F$ is unramified, we further require that $v_E(\tr(\xi)=1,v_E(\tr(\xi')=0$. Let $m$ be large enough w.r.t. $l,l'$, $\xi$ and $\xi'$. Let $n=2m+v_F(\ep)-v_E(\tr(\xi)) $. Let $x_{l}, x_{l'}\in \BR$ such that \begin{equation}\begin{split}&x_{l'} \eta(-\tr(\xi')) \Vol^\times(1+\fp_E^{n' })\Vol^\times(-\tr(\xi')+\tr(\fp_E^{l'})) \\-&x_l \eta(-\tr(\xi)) \Vol^\times(1+\fp_E^{n })\Vol^\times(-\tr(\xi)+\tr(\fp_E^l)) =\eta(\ep) \Vol^\times(1+\fp_E^{m })\Vol^\times(E^\times/F^\times) \label{8242'}\end{split}\end{equation} and \begin{equation}x_{l'} \Vol^+(\fp_E^{l'})=x_l \Vol^+(\fp_E^{l}).\label{8242}\end{equation}
Indeed, since \begin{align*}&\frac{\eta(\tr(\xi'))\Vol^\times(1+\fp_E^{n' }) \Vol^\times(-\tr(\xi')+\tr(\fp_E^{l'}))}{\eta(\tr(\xi)) \Vol^\times(1+\fp_E^{n })\Vol^\times(-\tr(\xi)+\tr(\fp_E^l))}\\%=&-\frac{\Vol^\times(\tr(\xi')+\tr(\fp_E^{l'}))}{\Vol^\times(\tr(\xi)+\tr(\fp_E^l))}\\
\neq & \frac{\Vol^+(\fp_E^{l'})}{ \Vol^+(\fp_E^{l})},\end{align*}
such $x_{l}, x_{l'}$ exists.
Let $$\Phi :=x_{l'}1_{ K_{l',\xi',n}\cap \CS}-x_l 1_{ K_{l,\xi,n}\cap \CS}.$$
By \eqref{ppn} and condition \eqref{8242}, there exists $f_1\in C_c^\infty(G)$ such that $\Phi=\Phi_{f_1}$.
By our choice of $n$, Example \ref{fepm1} and condition \eqref{8242'}, (1) holds.
By Example \ref{fepm} and (1), $f_1$ and $1_{ K_{\ep,m}}$ match.
By
Corollary \ref{7.9}, (2) holds. \end{proof}
\begin{lem}\label{823} Let $m$ be large enough. There exists $f_2\in C_c^\infty(G)$ such that \begin{itemize}\item[(1)] for $x\in F^\times-\{1\}$, $\CO(x, \Phi_{f_2})= \eta(\ep x) \CO(x, \Phi_{f_1}),$
\item[(2)] the following equation holds: \begin{align*} I_\sigma(f_2)=\frac{2 \vep(1,\eta,\psi) L(0,\eta)}{\Vol(E^\times/F^\times)} \alpha_{\pi}(1_{ K_{\ep,m}}).\end{align*} \end{itemize}
\end{lem}
\begin{proof} Let $w'=\begin{bmatrix}0&1 \\- 1 &0\end{bmatrix}$.
Let $f_1$ be as in Lemma \ref{822}, and $f_2(g):=f_1(w'g)\cdot \eta(\ep)\Omega(-1).$ Then $$\Phi_{f_2}(s)=\Phi_{f_1}(w'sw'^t)\eta(\ep)\Omega(-1).$$ By Example \ref{fepm1} and \ref{fepm2}, (1) holds. By the condition $\vep(1/2,\pi,\Omega) =\eta(\ep)\Omega(-1)$, the local functional equation of $ L(s,\pi,\Omega)$ and Lemma \ref{822} (2), (2) holds. \end{proof}
\begin{proof}[Proof of Proposition \ref{expmat}] Let $f_1$ and $f_2$ be as in Lemma \ref{822}, \ref{823}. Let $f=\frac{1}{2} (f _1+f_2).$
By
Example \ref{fepm}, Lemma \ref{822} (1) and \ref{823} (1), (1) holds.
By Lemma \ref{822} (2) and \ref{823} (2), (2) holds. \end{proof}
We have a mild modification when $1_{ K_{\ep,m}}$ is replaced by $ 1_{ K_{\ep,m}\varpi^\BZ }$.
Let $\Xi=K_{\ep,m}\varpi^\BZ\cap Z_\ep$.
Redefine the Hecke action $\pi(f_\ep)$ as in \eqref{HHprojpi}, $\alpha_{\pi}(f_\ep)$ be as in Definition \ref{api} w.r.t. the new Hecke action $\pi(f_\ep)$. Proposition \ref{expmat} has the following variant.
\begin{prop}\label{expmat1}Let $m$ be large enough. There exists $f\in C_c^\infty(G)$ such that \begin{itemize}\item[(1)] $ f$ purely matches $f_\ep$ (see Definition \ref{matchingdef}); \item[(2)] the following equation holds: \begin{align*} I_\sigma(f)=\frac{2 \vep(1,\eta,\psi) L(0,\eta)}{\Vol(E^\times/F^\times)} \alpha_{\pi}(1_{ K_{\ep,m}\varpi^\BZ }).\end{align*} \end{itemize}
\end{prop}
\subsection{Fundamental lemma}
Assume $E/F$, $\psi$ and $\Omega$ (so $\omega$) are
unramified through this subsection. Recall that $K$ is the standard maximal compact subgroup $\GL_2(\CO_E)$ of $G$,
and $K_{H_0}=H_0\cap K$ (see \ref{localnotations and measures }).
Let $v=v_F$.
Let ${\mathrm{bc}}$ be the base change homomorphism from the spherical Hecke algebra of $G$ to the spherical Hecke algebra of $G_1$ (see \cite[Section 1]{Lan}).
\begin{prop} [Fundamental lemma]\label{FLgeneral}
Let $f\in C_c^\infty(G)$ be bi-$K$-invariant, then $${\mathrm{bc}}\left(\frac{\Vol(K_{H_0}) \Vol(K_1)}{\Vol(K)}f\right)$$ purely matches $\Phi_{f}$. In particular, $\frac{1_{K_1}}{ \Vol(K_1)}$ purely matches $\Phi_{\frac{1_{K}}{\Vol(K_{H_0})\Vol(K_1)}}.$
\end{prop}
In odd characteristic, the fundamental lemma is proved in \cite[Section 4]{Jac87}, and another proof is given in
\cite[Proposition 3]{JN} using
the reduction method in \cite[Section 3]{JLR}.
This the reduction method works in arbitrary characteristic, and we prove only the following cases of Proposition \ref {FLgeneral} incharacteristic $2$.
\begin{prop} \label{FLgeneral1} Let $m\geq 0$ and be even.
Let $f$ be the characteristic function of the set of matrices $g\in G_1$ with integral entries such that $v(\det g)=m$,
$\Phi$ be the characteristic function of the set of matrices $g\in \CS$ with integral entries such that $v(\det g)=m$.
Then for $x\in {\mathrm{Nm}} (E^\times)-\{1\}$, we have $$ \CO(x,\Phi)= \CO( x, f ) ;$$
for $x\in F^\times-{\mathrm{Nm}} (E^\times)$, we have $$ \CO(x,\Phi)=0.$$
\end{prop}
We compute $\CO(x,\Phi)$ explicitly. Let $\xi:=\Omega^{-1}(\varpi)$. \begin{eg}\label{m=0} Let $\Phi=1_{K\cap \CS} $.
Suppose $v(x)>0$. If $v(x)$ is odd, then $\CO(x,\Phi)=0.$ If $v(x)$ is even, then $\CO(x,\Phi)=1.$
Suppose $v(x)<0$. If $v(x)$ is odd, then $\CO(x,\Phi)=0.$ If $v(x)$ is even, then $\CO(x,\Phi)=\xi^{-v(x)/2}.$
Suppose $v(x)=0$. If $v(1-x)> 0$, then $\CO(x,\Phi)=0.$ If $v(1-x)=0$, then $\CO(x,\Phi)=1 .$
\end{eg}
\begin{eg}\label{935}
Let $m>0
$ and be even.
Let $\Phi$ be the characteristic function of the set of matrices $g\in \CS$ with integral entries such that $v (\det g)=m$.
Suppose $v(x)> 0$. If $v(x)$ is odd, then $\CO(x,\Phi) = 0.$ If $v(x)$ is even, then $\CO(x,f)= \xi^{m/2}. $
Suppose $v(x)<0$. If $v(x)$ is odd, then $\CO(x,\Phi) = 0.$ If $v(x)$ is even, then $\CO(x,\Phi)
= \xi^{(m-v(x))/2}. $
Suppose $v(x)= 0$, then $\CO(x,\Phi) = \xi^{(m-v(1-x))/2}.$
\end{eg}
Now we compute the orbital integrals for the $G_1$-side in characteristic 2. We follow \cite[5.4, 5.5]{Jac86}. Let $p=2$.
Let $E=F[\xi]$ be the Artin-Schreier unramified quadratic field extension of $F$, where $\xi\in E$ satisfies $\xi^2+\xi+\tau=0$ for some $\tau\in k^\times$. Let $E^\times\hookrightarrow G_1\simeq \GL_2(F_v)$ be given by \begin{equation}\label{echar2} a+b\xi\mapsto \begin{bmatrix}a&b \\ b\tau&a+b\end{bmatrix} \end{equation} where $a,b\in F$. Let $j= \begin{bmatrix}1&0 \\ 1 &1\end{bmatrix}.$ The $j$ satisfies the conditions in \ref{matchorb}.
The embedding $E^\times\hookrightarrow G_1\simeq \GL_2(F) $ is as \eqref{echar2}. Let $x=a^2+ab+b^2\tau ={\mathrm{Nm}} (a+b\xi)$. Take \begin{equation}\delta(x)=1+ (a+b\xi)j= 1_2+ \begin{bmatrix}a&b \\ b\tau&a+b\end{bmatrix}j=\begin{bmatrix}1+a+b&b \\ a+b+b\tau&1+a+b\end{bmatrix}\label{dx2}.\end{equation} Then $\det \delta(x)=1+x.$ \begin{lem}\label{v(x)} We have $ v(x)=\min\{v(a^2),v(b^2)\}.$
In particular $v(x)$ is even.
\end{lem}
\begin{proof} If $v(a)\neq v(b)$, $v(ab)$ is between $v(a^2)$ and $v(b^2)$. Then
$ v(x)=\min\{v(a^2),v(b^2)\}$. If $v(a)=v(b)$, we may assume $b=1$. Then $v(a^2+ab+b^2\tau)=0$.
\end{proof}
Let $$ C_m:=K_1\begin{bmatrix}\varpi^m&0 \\ 0&1\end{bmatrix}K_1.$$ \begin{prop}\label{5.4,5.5'}
(1) Suppose $m>0$, then $\CO (x,1_{C_m})=0$ unless $v(x)=0$ and $v(1+x)=m$, in which case $\CO (x,1_{C_m})=1 $.
(2) Suppose $m=0$.
If $v(x)>0$, then $\CO(x,1_{C_m})= 1.$
If $v(x)<0$, then $\CO(x,1_{C_m})=\xi ^{-v(x)/2}.$
If $v(x)=0$, and $v(1-x)=0$, then $\CO(x,1_{C_m})=1 .$
If $v(x)=0$, and $v(1-x)> 0$, then $\CO(x,1_{C_m})=0.$
\end{prop} \begin{proof}
Now we begin to compute $\CO (x,1_{C_m})$.
Let $\xi=\Omega^{-1}(\varpi)$.
Since $T_1 \subset K_1Z_1$, we have
$$ \CO(x,1_{C_m} )= \int_{Z}1_{C_m}(z\delta(x))dz= \sum_{k\in\BZ} 1_{C_m}(\varpi^k \delta(x))\Omega^{-1}(\varpi^k).$$
The conditions \cite[(5.4.7),(5.4.8),(5.4.9)]{Jac86} on $1_{C_m}(\varpi^k \delta(x))\neq0$ (so $=1$), i.e. $\varpi^k \delta(x)\in K_1$ are exactly the same.
In particular, \cite[(5.4.7)]{Jac86} says that if $ \CO_E(\delta(x),1_{C_m} )\neq 0$, then $ n:=m-v(1+x)$ is even. In this case, only the term for $k=n/2$ possibly contributes to the sum.
The proof of (1). The cases $v(x)<0,v(x)>0$ are exactly the same as in \cite[(5.4)]{Jac86}. Let us prove the case $v(x)=0$. Suppose $\CO (\delta(x),1_{C_m})\neq0$.
The entries of $\varpi^{n/2} \delta(x)$, where $ n:=m-v(1+x)$ is even, are $$ \varpi^{n/2}(1+a+b),\ \ \ \varpi^{n/2}b,\ \ \ \varpi^{n/2}(a+b+b\tau),\ \ \ \varpi^{n/2}(1+a+b).$$ Since $v(x)= 0$. By Lemma \ref{v(x)}, at least one of $v(a),v(b)$ is 0, and the other is non-negative. So at least one of $$ b,\ \ \ a+b+b\tau $$
is a unit in $\CO_F$.
By \cite[(5.4.8),(5.4.9)]{Jac86}, $n=0$. Thus, $\varpi^{n/2} \delta(x)\in K_1$ and
$\CO_E(\delta,1_{C_m})=1$.
The proof of (2). We only compute the case $v(x)=0$, $v(1+x)>0$. Suppose $\CO_E(\delta(x),1_{C_m})\neq0$. The entries of $\varpi^{n/2} \delta(x)$, where $ n:= -v(1+x)<0$ and even, are $$ \varpi^{n/2}(1+a+b),\ \ \ \varpi^{n/2}b,\ \ \ \varpi^{n/2}(a+b+b\tau),\ \ \ \varpi^{n/2}(1+a+b).$$ Since $v(x)= 0$, by Lemma \ref{v(x)}, at least one of $v(a),v(b)$ is 0. So at least one of $$\varpi^{n/2}b,\ \ \ \varpi^{n/2}(a+b+b\tau)$$ is not contained in $\CO_F$. By \cite[ (5.4.8) ] {Jac86},
$\CO_E(\delta(x),1_{C_m})=0$, a contradiction.
\end{proof}
Comparing Example \ref{m=0} and Example \ref{935} with Proposition \ref{5.4,5.5'}, Proposition \ref{FLgeneral1} for $p=2$ follows.
\section{Global and local periods, proof of Theorem \ref{The Waldspurger formula over function fieldsintro} }
\label{Global and local periods}
We come back to the global situation. We decompose the global automorphic periods into products of local distributions and $L$-functions. Then we prove the Waldspurger formula. \subsection{Cuspidal representations}
Let $\sigma\in\CA_c(G,\omega_E^{-1})$ (see \ref{Spectral decomposition of the automorphic distributions}) and $\Omega$ a Hecke character of $E^\times$. We have the global distribution $ \CO_\sigma (s, \cdot) $ (see \eqref{co}) and the local distribution $I_{\sigma_v} ( s, \cdot )$ (see \ref{the local distribution I(f)}).
Let $S\subset |X|$ be a finite set containing all ramified places of $B$, $E/F$, $\psi $, $\pi$, and all places below ramified places of $\Omega$. Let $K_{H_0}^S$ (resp. $K^S$) be the product of local maximal compact groups of $H_0$ (resp. $G$) defined in \ref{localnotations and measures } outside $S$ (resp. places of $E$ over $S$). \begin{prop}\label{CO1} Assume that $\sigma $ is the base change of a cuspidal representation of $ \GL_{2,F}$. Let $f'^S=\frac{1_{K^S}}{\Vol(K_{H_0}^S)\Vol(K'^S)}$, then
\begin{equation*} \CO_\sigma (s, f'_Sf'^S) =\left(\prod_{v\in S}I_{\sigma_v} ( s, f'_v)\right) L_S(1,\eta) \frac{L^S(1/2+s,\pi,\Omega)L^S(2,1_F)}{L^S(1,\pi,\ad)}. \end{equation*} \end{prop} This proposition is the analog of \cite[Proposition 4]{JN} for function fields. We follow the proof in loc. cit..
We have two global periods $\lambda (s,\phi)$ and $\CP( \phi)$ for $\phi\in \sigma$ (see \ref{Spectral decomposition of the automorphic distributions}). Let
$W $ be the $\psi_E$-th coefficient of $\phi$. Fix a decomposition $W=\prod_{v\in |X|} W_v$ where $W_v$ is the Whittaker newform of $\sigma_v$ for $v\not\in S$ (in the sense of \cite[2.3]{Zha01}).
Then we have a decomposition \begin{equation}\lambda (s,\phi)=\prod_{v\in S}\lambda_v (s,W_v) L^S(s+1/2,\sigma_E\otimes\Omega)\label{decomlambda}\end{equation} for $\Re(s)>0$,
where $\lambda_v$ is the local period defined in \eqref{lambda} and \eqref{lambda1}.
Now we decompose $\CP$. Let $H'=\GL_{2,F} $ which is naturally a subgroup of $G$. Let $ B'\subset H'$ be the subgroup of upper triangular matrix,
$N'
$ be the upper unipotent subgroup, and $Z'\subset H'$ be the center.
If $p>2$, let $\xi\in E-F$ be a trace free element. Then $$\begin{bmatrix}\xi&0\\ 0&1\end{bmatrix} H\begin{bmatrix}\xi^{-1}&0\\ 0&1\end{bmatrix} =ZH'.$$
The pullback of the similitude character $\kappa $ on $H$ to $H'=\GL_{2,F}$ is the determinant.
Thus $$\CP( \phi)= \int_{H'(F)\backslash H'(\BA_F)} \phi\left(h\begin{bmatrix}\xi&0\\ 0&1\end{bmatrix}\right ) \eta\omega (\det(h) ) dh .$$
If $p=2$, let $ H =ZH'$, and $$\CP( \phi)= \int_{H'(F)\backslash H'(\BA_F)} \phi(h ) \eta\omega (\det(h) ) dh .$$
Let $\Phi=\otimes'\Phi_v\in C_c^\infty(\BA_F^2)$ be a pure tensor. Suppose that $\Phi^S$ is the characteristic function of $(\hat\CO_F^S)^2$.
Let $e_2=(0,1)\in \BA_F^2 $, and $H'(\BA_F)$ act on $\BA_F^2$ from right naturally.
Let $E(g,\Phi,s)$ be the Eisenstein series associated to $\Phi$ (see \cite[Section 4]{JN}).
Consider the integral
$$\Psi(s, \phi,\Phi):=\int_{Z'(\BA_F)H'(F)\backslash H' (\BA_F)}E(g,\Phi,s)\phi(g)\eta\omega (\det(g) ) |\det g|_E^sdg. $$
\begin{lem}[{\cite[p. 50]{JN}}]\label{tangent}If $p>2$ ,let $\xi\in E-F$ be a trace free element. Then
$$\Psi(s,\phi,\Phi)=\int_{N' (\BA_F)\backslash H' (\BA_F)}W\left(\begin{bmatrix}\xi&0\\ 0&1\end{bmatrix} g\right)\Phi(e_2g)\eta\omega (\det(g) )|\det g|_E^sdg.$$
If $p=2$, then
$$\Psi(s,\phi,\Phi)=\int_{N' (\BA_F)\backslash H' (\BA_F)}W( g)\Phi(e_2g)\eta\omega (\det(g) )|\det g|_E^sdg.$$ \end{lem}
The expressions of $\Psi(s,\phi,\Phi)$ in the above lemma have obvious local-global decompositions. Define $\Psi^S(s,W,\Phi) $ to be the away from $S$ part. Computing the residue of $\Psi$ at $s=1$, we have a decomposition (see \cite[(23)]{JN}) \begin{align}\CP(\phi)=\prod_{v\in S}\CP_v(W_v) \frac{2\Res_{s=1}\Psi^S(s,W,\Phi) }{\Res_{s=1} L(s,1_F)}\label{101}\end{align} where $\CP_v$ is the local period associated to $\sigma_v$ defined in \eqref{cp} and \eqref{lambda2}. We remark that the proof of \cite[(23)]{JN} uses a certain invariance property of the local period $\CP_v$ which can be established for all local fields. The same proof of \cite[(23)]{JN} works for \eqref{101}. If $p>2$, let $\xi$ be as in Lemma \ref{tangent}, and enlarge $S$ such that $\xi$ is a unit in $\CO_{E_v}$ for $v\not\in S$. Applying the formula in \cite{Sch} for the unramified Whittaker newforms, we have the following lemma (for all $p$). \begin{lem} \label{cp'} Assume that $\sigma $ is the base change of $\pi\in \CA_c(H')$. Then $$ \frac{ \Psi^S(s,W,\Phi) }{L^S(s,\sigma\times \tilde \sigma)}=\frac{\Vol(K'^S)}{L^S(s,\eta)L^S(s,\pi,\ad)}.$$
Here $K'^S$ is the standard maximal compact subgroup of $H'(\BA_F^S)$, and $L^S(s,\sigma\times \tilde \sigma)$ is the product of local $L$-factors outside the places of $E$ over $S$. \end{lem}
Recall that the Petersson pairing on $\sigma$ satisfies the following local-global decomposition \cite[Proposition 2.1]{CST}: let $\phi_i\in \sigma$, where $i=1,2$, and $W_i $ be its $\psi_E$-th Whittaker coefficient, then \begin{equation}\int_{Z(\BA_E)G(E)\backslash G(\BA_E)}\phi_1\bar \phi_2 (g)dg=2\prod_{v }\frac{1}{L(1,1_{E_v})}\pair{W_1,W_2}_v ,\label{peterdecom}\end{equation} where the product is over all places of $E$.
By the explicit formula of $W_v$ for $v\not\in S$ (see \cite{Sch}), we have \begin{equation}\frac{1}{L(1,1_{E_v})}\pair{W_v,W_v}_v= \frac{L(1,\sigma_v,\ad) }{L(2,1_{E_v})}.\label{decompet}\end{equation} By \eqref{COSigma}, Lemma \ref{Kmeasure}, \eqref{decomlambda}, \eqref{101}, Lemma \ref{cp'}, \eqref{peterdecom} and \eqref{decompet}, Proposition \ref{CO1} follows.
\subsection{Non-cuspidal representations}\label{Local-global decomposition of periods 2}
Let $\sigma=\sigma_{\xi}$ be the admissible representation of $G(\BA_E)$ associated to the data $( 0, {\xi}, {\xi}^{-1}\omega_E^{-1})$ (see \ref{Spectral decomposition of the automorphic distributions}). The non-cuspidal version of Proposition \ref{CO1} is Proposition \ref {CO1d} below, which is the analog of \cite[Appendiex, Corollary 1]{MW} for function fields. The proof is the same with \cite[Appendiex, Lemma 2]{MW} with mild modifications when $p=2$.
\begin{prop} \label{CO1d} Let $f'_Sf'^S$ be as in Proposition \ref{CO1}. Then
\begin{equation*}4 \CO_\sigma ( s, f'_Sf'^S) =\left(\prod_{v\in S}I_{\sigma_v} ( s,f'_v)\right) L_S(1,\eta) \frac{L^S(1/2+s,\pi,\Omega)L^S(2,1_F)}{L^S(1,\pi,\ad)}. \end{equation*} \end{prop}
\subsection{The Waldspurger formula over function fields}\label{The Waldspurger formula over function field}
In this subsection, let notations be as in \ref{The global relative trace formulaintro}. We prove Theorem \ref{The Waldspurger formula over function fieldsintrof}. We choose $ f_v$'s in Theorem \ref{The Waldspurger formula over function fieldsintrof} as follows.
Let $S\subset |X| $ be a finite set. Suppose $S$ contains all ramified places of $B$, $E/F$, $\psi $, $\pi$, all
places below ramified places of $\Omega$, and $S \bigcap |X|_s$ is nonempty. Here $|X|_s\subset |X|$ is the subset of places split in $E$.
Let the functions $f^S\in C_c^\infty( B^\times(\BA_F^S))$ and $f'^S\in C_c^\infty (G(\BA_E^S))$ be as follows:
\begin{itemize}
\item[(1)] for $v\in |X|-S-|X|_s $, let $ f'_v $ be spherical,
and $f_v$ be the matching function of $f_v'$ on $B_v ^\times$ given by Proposition \ref{FLgeneral};
\item[(2)] for $v\in |X|_s-S $, let $ f'_v=(f_{1,v},f_{2,v}) $ be spherical,
and $f_v$ be the matching function of $f_v'$ on $B_v ^\times$ as in \eqref{splitplaces}.
\end{itemize}
For $v\in |X|_s\cap S$, choose $ f_v$ as in Lemma \ref{nontrisplit}. Then $\alpha^\sharp_{\pi_v}(f_v)\neq 0$ and $f$ satisfies Assumption \ref{freg'}.
Choose $f'_v $ as in Lemma \ref{afneq00} (1). Then $f'_v$ purely matches $f_v$.
By Lemma \ref{afneq00} (2),
$f'$ satisfies Assumption \ref{freg}.
For $v\in S-|X|_s$, let $f_v=1_{K_{\ep,m}}$ and $f_v'$ its pure matching function in Proposition \ref{expmat}.
Let $m$ be large enough, such that $\det (K_{\ep,m})\subset {\mathrm{Nm}}(E_v^\times)$ and
$\alpha^\sharp_{\pi_v}(f_v)\neq 0$
(see Lemma \ref{pipieta}).
By Lemma \ref{decquat} and \ref{decpure'}, we have the relative trace formula identity:
\begin{align}\CO(f)=\CO(f')\label{RTF}\end{align}
Suppose that the Jacquet-Langlands correspondence of $\pi$ to $\GL_{2,F}$ is not of the form $\pi_\xi$ as in the end of \ref{Local-global decomposition of periods 2}. Then $\pi\not \simeq \pi\otimes\eta$,
and $\sigma:=\pi_E$ is cuspidal. By \eqref{COSigma}, \eqref{COSigma'}, Theorem \ref{Eisterm} (for $s=0$), and similar results for $B^\times$ (see \cite[(7.6)]{Jac87} and \cite[Section 9]{JN} in the number field setting, and the same proof applies here),
we have the following spectral decomposition of \eqref{RTF}:
\begin{align*}\CO_\pi(f) +\CO_{\pi\otimes\eta}(f)= \CO_\sigma(f') . \end{align*} Now choose $$f'^S=\frac{1_{K^S}}{\Vol(K_{H_0}^S)\Vol(K'^S)},\ f^S= \frac{1_{K_1^S}}{ \Vol(K'^S)}= \frac{1_{K_1^S}}{ \Vol(K_1^S)} .$$
Then $f$ is supported on $\{g\in B^\times(\BA_F):\det (g)\in {\mathrm{Nm}} (\BA_E^\times )\}$. In particular, we have $\CO_\pi(f) =\CO_{\pi\otimes\eta}(f).$
Thus $2 \CO_\pi(f) = \CO_\sigma(f').$
From Proposition \ref{CO1}, we have
\begin{align*}2 \CO_\pi(f) = \CO_\sigma(f')=\left(\prod_{v\in S}L(1,\eta_v)I_{\sigma_v} ( f'_v)\right) \frac{L^S(1/2,\pi,\Omega)L^S(2,1_F)}{L^S(1,\pi,\ad)}. \end{align*}
By Proposition \ref{localRTF2}, \ref{expmat}, the computation of $\ep$ (or $\gamma$)-factor
\cite[2.5]{Tat}, and that the product of the local root numbers of $\eta$ is 1, we have
\begin{align*}2 \CO_\pi(f)&= \CO_\sigma(f')= \frac{L (1/2,\pi,\Omega)L (2,1_F)}{L (1,\pi,\ad)} \prod_{v\in |X| } \alpha_{\pi_v}^\sharp (f_v) . \end{align*}
Suppose that the Jacquet-Langlands correspondence of $\pi$ to $\GL_{2,F}$ is the representation $\pi_\xi$ as in the end of \ref{Local-global decomposition of periods 2}. Then $\pi\simeq \pi\otimes \eta $.
Let $\sigma=\sigma_\xi $ be the base change of $\pi_\xi$ as in the last subsection.
We have the following spectral decomposition of the relative trace formula
\begin{align*} \CO_\pi(f)= \CO_{\sigma_\xi}(f')+ \CO_{\sigma_{{\xi}^{-1}\omega_E^{-1}}}(f')= 2 \CO_\sigma(f') . \end{align*}
From Proposition \ref{localRTF2}, \ref{expmat} and \ref{CO1d}, we have
\begin{align*} 2\CO_\pi(f) = 4\CO_\sigma(f')&=\left(\prod_{v\in S}I_{\sigma_v} ( f'_v)\right) L_S(1,\eta) \frac{L^S(1/2,\pi,\Omega)L^S(2,1_F)}{L^S(1,\pi,\ad)}\\
&=\frac{L (1/2,\pi,\Omega)L (2,1_F)}{L (1,\pi,\ad)} \prod_{v\in |X| } \alpha_{\pi_v}^\sharp (f_v) . \end{align*}
\section{Decomposition of the height distribution}\label{Local intersection multiplicity} Let $I$ be a nonempty finite closed subscheme of $X-\{\infty\}$, $U=U(I)\subset \cD\otimes\hat \CO_F $ the corresponding principal congruence subgroup. Identify $U$ as a subgroup of $\BB_{\mathrm{f}}^\times$ via the isomorphism $ D^\times(\BA_{\mathrm{f}})\simeq \BB_{\mathrm{f}}^\times $ fixed in \ref{Global measures}. Let $\CH_{U,\BC}$ the Hecke algebra of $\BB^\times$ (see \ref {measures}). For $f \in \CH_{U,\BC}$, we compute the height distribution $H(f)$ (see Definition \ref{CM height}) under the following assumptions.
\begin{asmp}\label{asmpe0}
For every $v\in |X|-|X|_s$, the inclusion $ E _v^\times \hookrightarrow \BB_v^\times=G_\ep $ (for the corresponding $\ep$ in Assumption \ref{asmpep}) induced by $ e_0:E \hookrightarrow \BB $ (see \eqref{e0}) is $T_\ep\hookrightarrow G_\ep$.
\end{asmp}
By the discussion in \ref{CMuni0} and invariance of the \Neron-Tate height pairing $\pair{\cdot,\cdot}_\NT$ on $J$ under the diagonal $\BB^\times$-action on the two variables, the truth of Theorem \ref{GZ} for one embedding $ e_0:E \hookrightarrow \BB $ is equivalent to the truth of Theorem \ref{GZ} for every such embedding by the Noether-Skolem theorem.
Thus we can choose $e_0$ such that Assumption \ref{asmpe0} holds.
Let $ \BB_v^n:=\{g\in \BB_v^\times: v(\det (g))=n\}$.
\begin{asmp} \label{fvan} Assume that $f$ is a pure tensor with $f_\infty=1_{U_\infty}$ and there exists two disjoint finite subsets $S_{s,\reg} $ and $S_{s,{\mathrm{ave}}}$ of $|X|_s$, both of cardinality $\geq 2$, such that
\item[(1)] for $v\in S_{s,\reg}$, $f_v$ is supported on the regular locus of $ \BB_v^\times$ for the $E _v^\times$-action; \item[(2)] for $v\in S_{s,{\mathrm{ave}}}$, $U_{v}$ is maximal and for every $n\in \BZ$, the following equation holds: $$ \sum_{g_{v}\in \BB_{v}^n /U_{v}} f_{v}(g_{v}) =0.$$ \end{asmp}
The main results of this section are summarized in the following theorem.
\begin{thm}\label{summary}Assume Assumption \ref{asmpe0} and Assumption \ref{fvan}. Then we have \begin{equation*}H(f)= \sum_{v\in |X|} -( i(f)_v+j(f)_v))\log q_{v}, \end{equation*}
where $i(f)_v$ and $j(f)_v$ are given as follows:
\begin{itemize}
\item[(1)] For $v\in |X|-|X|_s-\{\infty\}$, we have $$i(f)_v=\sum_{\delta\in E^\times\backslash B(v)^\times_\reg/E^\times} \CO _{\Xi_\infty}(\delta,f^v) i (\delta,f_v),$$ where $ i (\delta,f_v) $ is given by an orbital integrals of an intersection multiplicity function $m_v $ on $B(v)_v^\times\times \BB_v^\times $ (see Definition \ref{mf1}, \ref{mf2}) weighted by $f_v$ as in Definition \ref{iterm1}. For nonvanishing conditions on $m_v$, see Lemma \ref{vdetun1}, \ref{vdetun"}, \ref{vdetun'}. For computations on $m_v(g_1,1)$, see Lemma \ref{U'}, \ref{U''}.
\item[(2)] For $v\in |X|-|X|_s-\{\infty\}$, there exists $\overline {f_v}\in C_c^\infty(B_v^\times)$ such that
$$j(f)_v= \sum_{\delta\in E^\times\backslash B_\reg^\times/E^\times}\CO _{\Xi_\infty}(\delta,f^v) \CO (\delta ,\overline {f_v}) .$$
\item[(3)] For $\infty$, similar properties for $ i(f)_\infty$ and $j(f)_\infty$ hold (see \ref{ijinfty}).
\item[(4)] For $v$ split in $E$, $ i(f)_v=j(f)_v=0$.
\end{itemize}
\end{thm}
To prove Theorem \ref{summary},
we first define integral models of the modular curve $M_U$, and use intersection theory on the integral models to decompose $H(f)$ over places of $F$. The intersection number of horizontal divisors is the $i$-part in Theorem \ref{summary}, and the rest the $j$-part.
Then we decompose the $i$-part and the $j$-part at different places case by case.
\subsection{CM points and models}\label{Points and models} \subsubsection{CM points}\label{CM points} Let $P_0 \in M^{E^\times}(F^\sep )$.
For $h\in \BB ^\times$, the image of $T_{h}P_0$ under the rigid analytic uniformization of $M$ at $\infty$ is $[z_0h_\infty,h_{\mathrm{f}}]$.
Let $(T_{h}P_0)_U $ be the image of $T_{h}P_0$ in $M_U$.
The map $ h \mapsto (T_{h}P_0)_U $
induces a bijection \begin{equation} \label{111} E^\times\backslash \BB^\times/ \tilde U\simeq CM_U. \end{equation}
Thus we identify $CM_U$ with $ E^\times\backslash \BB^\times/ \tilde U$, and denote $ (T_{h}P_0)_U $ by $h$. Let $\tilde U_ E=\tilde U\cap \BA_E^\times$. Regard $E^\times\backslash \BA_E^\times /\tilde U_E$ as a subset of $CM_U$. Let $H$ be a finite abelian extension of $E$ such that all geometrically connected components of $M_U$
and all points in $E^\times\backslash \BA_E^\times /\tilde U_E$ are defined over $H$.
For $v\in |X|$, let $\overline v$ be an extension of $v$ to $ F^\sep$, and let $w$ be the restriction of $\overline v$ to $H$, Let $\CO_{\overline v}$ be the ring of integers of the completion of $F^\sep $ at $\overline v$.
Let $ \hat F_v^\ur $ be the completion of the maximal unramified extension of $ {F_v}$ w.r.t. the restriction of $\overline v$,
$ \hat \CO_{F_v}^\ur $ be its ring of integer.
Define $ \hat H_w^\ur $ and $ \hat \CO_{H_w}^\ur $ simiarly.
In particular, we have embeddings $\hat \CO_{F_v}^\ur \hookrightarrow \hat \CO_{H_w}^\ur\hookrightarrow \CO_{\overline v}.$ \subsubsection{Models}
Let $I'=I- {\mathrm{Ram}} $ which we assume to be nonempty by enlarging $I$. We first define a regular projective model $\CM_{U(I')\BB_\infty^\times}$ of $M_{U(I')\BB_\infty^\times}$ over $X$ as follows.
If ${\mathrm{Ram}}\neq \{\infty\}$,
the notion of $\cD$-elliptic sheaves with level-$I'$ structures,
is generalized in \cite{BS}, \cite{Boy} and \cite{Hau}.
The image of the morphism ${\mathfrak{zero}}_\BE$ of a $\cD$-elliptic sheaf $\BE$ (see Definition \ref{DES}) is allowed to be in $X-I'-({\mathrm{Ram}}-\{\infty\})$, $X-{\mathrm{Ram}}$, and $X-I'-\{\infty\}$
respectively.
Moreover, the corresponding moduli spaces are obtained over respective open subschemes of $X$ such that every two of these moduli spaces are isomorphic over the open subschemes of $X$ where both are defined. The define
$\CM_{U(I')\BB_\infty^\times}$ by gluing these moduli spaces.
If ${\mathrm{Ram}}= \{\infty\}$, define $\CM_{U(I')\BB_\infty^\times}$ using the moduli spaces defined in \cite{DriEll1} and \cite{BS}.
\begin{defn} (1) Let $\CM_{U} $ be the minimal desingularization of the normalization of $\CM_{U(I')\BB_\infty^\times} $ in the function field of $M_{U} $. Let $\cusp$ be the Zariski closure in $\CM_{U} $ of the cusps in $M_U$.
(2) Let $\CN_{U} $ be the minimal desingularization of the normalization of $\CM_{U(I')\BB_\infty^\times} \times_X X'$ in the ring of rational functions of $M_{U}\otimes_F H $. Here $X'$ is the smooth projective curve corresponding to $H$.
\end{defn}
Then there is a natural morphism from $\CN_{U} $ to $\CM_U$.
\subsubsection{Moduli interpretations of the $\CM_U$ outside ${\mathrm{Ram}}$} \label{mio}
For $v\in |X|-{\mathrm{Ram}}$, let $X_{(v)}$ be the spectrum of the localization of $\CO_X$
at $v$. In this paragraph, our schemes over $X$ are restricted to $X_{(v)}$.
For $v\not \in I\bigcup {\mathrm{Ram}}$, then $\CM_U$ equals the moduli space $\CM_{U(I)U_\infty}$ (over $X_{(v)}$) defined in \ref{LSI}.
For $v\in I $, if ${\mathrm{Ram}}=\{\infty\}$, then $\CM_{U } $ equals the quotient by $U_\infty$ of the smooth compactification of the moduli space of $\cD$-elliptic sheaves with Drinfeld level-$I$ structures and level structures at $\infty$ (see \ref{Equivalence} and \cite{DriCar}).
Now assume that ${\mathrm{Ram}}\neq \{\infty\}$.
Let $\CM_{U\BB_\infty^\times} $ be the moduli space of $\cD$-elliptic sheaves with Drinfeld level-$I$ structures in \cite{Boy}, which is a regular projective model of $M_{U\BB_\infty^\times}$ over $X_{(v)}$.
Note that the natural morphism from $\CM_{U(I^v)U_\infty } $ to $\CM_{U(I^v)\BB_\infty^\times}$ is finite \etale. Then it is easy to check that $\CM_{U } =\CM_{U(I^v)U_\infty } \times_{\CM_{U(I^v)\BB_\infty^\times}} \CM_{U\BB_\infty^\times} $.
Thus $\CM_{U } $ is the quotient by $U_\infty$ of the moduli space of $\cD$-elliptic sheaves with Drinfeld level-$I$ structures and level structures at $\infty$.
\subsection{Local-global decomposition of the height distribution}
\subsubsection{Admissible pairing} If $U_\infty=\BB_\infty^\times$, define the arithmetic Hodge class $\CL_U$ be the sum of the divisor classes of $2\cdot\cusp$ and of the relative dualizing sheaf of $\omega_{\CM_U/X}$. In general, let $\CL_U$ be the pullback of the arithmetic Hodge class of $\CM_{U\BB_\infty^\times}/X$. (If ${\mathrm{Ram}}\neq \{\infty\}$, then $\CL_U=\omega_{\CM_U/X}$.) Then the generic fiber of $\CL_U$ is the hodge class defined in \ref{Hodge classes}.
\begin{defn} \label{ZNU}
Let $\CZ $ be the pullback of $\CL_U$ to $\CN_{U} $, divided by the degree of the restriction of $L_U$ to any connected component of $M_{U,H}$. \end{defn}
We interpret the \Neron-Tate height pairing on $J_U$ as the $\CZ$-admissible pairing in the sense of \cite[7.1.6]{YZZ}.
For a divisor (class) of $M_{U,H}$ or $ \CN_U$ and $\alpha\in \pi_0(M_{U,H})$, use the subscript $\alpha$ to denote the restriction of this divisor (class) to the connected component of $M_{U,H}$ indexed by $\alpha$.
Let $\xi_\alpha$ be the generic fiber of $\CZ_\alpha$.
\begin{defn}\label{admext} For $D \in \Div(M_{U,H})$, let $\overline D$ be the Zariski closure. A divisor $\widehat D=\overline D+V_D$ of $\CN_U$, where $V_D$ is a vertical divisor is called the admissible extension of $ D$ if \begin{itemize} \item[(1)] the intersection of $\widehat D-\sum_{\alpha\in \pi_0(M_{U,H})} \deg D_\alpha \cdot \CZ_\alpha$ with every vertical divisor is 0; \item[(2)] $V_D\cdot \CZ_\alpha=0$ for every $\alpha\in \pi_0(M_{U,H})$. \end{itemize}
\end{defn} The admissible extension of $D$ exists and is unique. Using a regular model of $M_U$ over a finite extension $L$ of $H$ which dominates $\CN_U$ and the pullback of $\CZ$, extend Definition \ref{admext} to $\Div(M_{U,L})$. For $D_1,D_2\in \Div(M_{U,L})$, define the $\CZ$-admissible pairing $$\pair{D_1,D_2} :=-\frac{1}{[L:F]}\widehat {D_1}\cdot\widehat{ D_2}.$$
Then by the arithmetic Hodge index theorem (see \cite[7.1.4]{YZZ}), we have
\begin{align} \pair{ \tilde Z(f)_{*}t_1^\circ ,t_2^\circ }_\NT = \pair{ \tilde Z(f)_{U,*}t_1 ,t_2 } - \pair{ \tilde Z(f)_{U,*} \xi_{t_1} ,t_2 }- \pair{ \tilde Z(f)_{U,*}t_1^\circ ,\xi_{t_2} }.\label{Zttdecom}\end{align}
Here the left hand side is (part of) the integrand in the integral \eqref{3.8} defining $H(f)$, and $\xi_{t}=\xi_{\alpha}$ if
$t$ is in the connected component of $M_{U,H}$ indexed by $\alpha$.
\begin{lem}\label{van23} The second and third term in the right hand side of \eqref{Zttdecom} vanish. In particular, the following equation holds \begin{equation}H(f)= \int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{E^\times\backslash \BA_E^\times }^* \pair{ \tilde Z(f)_{U,*}t_1 ,t_2 } \Omega^{-1}(t_2)\Omega(t_1) dt_2dt_1 .\label{nocirc}\end{equation}\end{lem} \begin{proof} For $g\in \BB^\times$ with $g_\infty\in U_\infty$, we have $
Z(g)_{U,*}( \xi_{t_1})=\deg Z(g)_U\cdot \xi _{t_1}$. (See \cite[Lemma 7.6]{YZZ}. Indeed, this is true for all $g\in \BB^\times$ if ${\mathrm{Ram}}\neq \{\infty\}$.)
Then for $v\in |X|$, the term $\pair{ \tilde Z(f)_{U,*} \xi_{t_1} ,t_2 } $ in \eqref{Zttdecom} equals
\begin{equation}\Vol(\Xi_U)|F^\times\backslash \BA_F^\times/\Xi|\sum_{n\in \BZ}( \sum_{g_v\in U_v\backslash \BB_v^n /U_v} f_v(g_v)\deg Z(g_v) \pair{ Z(f^v)_{U,*} \xi_{t_1 g_v} ,t_2 })\label{ccc} .\end{equation} Let $v\in S_{s,\reg}$. Fix $h_n\in \BB_v^n$. Then for every $g_v\in \BB_v^n$, we have $\xi_{t_1 g_v}=\xi_{t_1 h_n}$. Thus the inner sum of \eqref{ccc} equals \begin{align*}& ( \sum_{g_v\in U_v\backslash \BB_v^n /U_v} f_v(g_v)\deg Z(g_v))\cdot \pair{ Z(f^v)_{U,*} \xi_{t_1 h_n} ,t_2 }\\ =&( \sum_{g_v\in \BB_v^n /U_v} f_v(g_v)) \cdot \pair{ Z(f^v)_{U,*} \xi_{t_1 h_n} ,t_2 }=0\end{align*} where the last equation follows from Assumption \ref{fvan} (2). For the third term, similar to \cite[Lemma 7.7]{YZZ}, we have $$\pair{Z(g_v)_{U,*} t_1^\circ ,\xi_{t_2} }=\deg Z(g_v)\pair{Z(f^v)_{U,*} t_1^\circ,\xi_{t_2}}. $$ Note that here we use the fact that $Z(f^v)_{U,*} t_1^\circ$ has degree 0 which implies that the second term in the right hand side of \cite[Lemma 7.7]{YZZ} vanishes.
So the term $ \pair{ \tilde Z(f)_{U,*}t_1^\circ ,\xi_{t_2} }$ in \eqref{Zttdecom} vanishes by the same reasoning for the second term.
\end{proof}
\subsubsection{Decomposition of the height distribution}\label{Decomposition of the height distribution}
Let $\overline v $ be an extension of $v$ to $ F^\sep$. For $D_1\in \Div(M_{U,F^\sep})$ and $D_2\in \Div(M_{U,H})$, let $i_{\overline v }(D_1,D_2)$ and $j_{\overline v }(D_1,D_2)$ be defined as in \cite[7.1.7]{YZZ}. More precisely, let $Y$ be a finite cover of $X'$ such that $D_1$ is defined over the function field of $Y$. Let $\CN_U'$ be the base change of $\CN_U$ to $Y$. Define
$i_{\overline v }(D_1,D_2) $ to be the intersection number (normalized by the ramification index) of the Zariski closures of $D_1,D_2$ in $\CN'$,
and define $j_{\overline v }(D_1,D_2)$ to be the intersection number of the Zariski closure of $D_1$ with the pullback to $\CN_U'$ of the vertical part in the $\CZ$-admissible extension of $D_2$ in $\CN_U$.
If $v$ is not split in $E$, let \begin{align*} i(f)_v= \Vol(\Xi_U)|F^\times\backslash \BA_F^\times/\Xi|\int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{E^\times\backslash \BA_E^\times }^* \sum_{g\in \BB^\times/\tilde U} f(g) i_{\overline v}( t_1g,t_2) \Omega^{-1}(t_2)\Omega(t_1) dt_2dt_1 \end{align*}
and \begin{align*} j(f)_v = \Vol(\Xi_U)|F^\times\backslash \BA_F^\times/\Xi|\int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{E^\times\backslash \BA_E^\times }^* \sum_{g\in \BB^\times/\tilde U} f(g) j_{\overline v}( t_1g,t_2) \Omega^{-1}(t_2)\Omega(t_1) dt_2dt_1 . \end{align*}
Here the regularized integral is as in Definition \ref{regint}.
If $v$ is split in $E$, let $v_1,v_2$ be the two places over $v$. Let $\overline { v_i}$ be an extension of $v_i$ to $ F^\sep$. Let \begin{align*}i(f)_{v}=\frac{1}{2}\sum_{n=1,2} i(f)_{v_n}, \ j(f)_{v}=\frac{1}{2}\sum_{n=1,2} j(f)_{v_n} \end{align*}
where \begin{align*} i(f)_{v_n} = \Vol(\Xi_U)|F^\times\backslash \BA_F^\times/\Xi|\int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{E^\times\backslash \BA_E^\times }^* \sum_{g\in \BB^\times/\tilde U} f(g) i_{\overline {v_n}}( t_1g,t_2) \Omega^{-1}(t_2)\Omega(t_1) dt_2dt_1 , \end{align*}
and
\begin{align*} j(f)_{v_n} = \Vol(\Xi_U)|F^\times\backslash \BA_F^\times/\Xi|\int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{E^\times\backslash \BA_E^\times }^* \sum_{g\in \tilde U\backslash \BB^\times/\tilde U} f(g) j_{\overline {v_n}}(Z(g)_{U,*} t_1,t_2) \Omega^{-1}(t_2)\Omega(t_1) dt_2dt_1 . \end{align*}
The expression of $ j(f)_{v_n}$ is useful for the computation (see \ref{11.4.3}). Then similar to \cite[7.2.2]{YZZ}, by Lemma \ref{van23} and
the CM theory (see Corollary \ref{TSCM}), we have
\begin{equation*}H(f)= \sum_{v\in |X|} -( i(f)_v+j(f)_v))\log q_{v}. \end{equation*}
\subsection{Supersingular case}\label{nonsplit}
Let $v\in |X|-\infty$ be split in $\BB$, and let $B=B(v)$ be the $v$-nearby quaternion algebra of $\BB.$ Then $B_v$ is a division algebra.
Let $n$ be the level of the principal congruence subgroup $U_v\subset \BB_v^\times$. To each $\cD$-elliptic sheaf $\BE$ over $\CO_{F_v}$ (resp. $\overline {k(v)}$) with Drinfeld level structure of level-$nv$, one can associate to
it a divisible $\CO_{F_v}$-module $ M$ over $\CO_{F_v}$ (resp. $\overline {k(v)}$) of height 4 with $\cD_v$-action and Drinfeld level structure of level $n$ (see \cite[6.3]{Boy}). Fix an isomorphism $\cD_v\simeq \RM_2(\CO_{F_v})$. Let \begin{align} M_1:=\begin{bmatrix}1&0 \\0 &0\end{bmatrix}M,
\
M_2:= \begin{bmatrix}0 &0 \\0&1\end{bmatrix}M
\label{special fibers'} \end{align} which are divisible $\CO_{F_v}$-modules of height 2. Then $M=M_1\oplus M_2$ and $\begin{bmatrix}0&1 \\1 &0\end{bmatrix}$ gives an isomorphism $M_1\simeq M_2$ of divisible $\CO_{F_v}$-modules.
Recall that for each positive integer $m$, there is a unique connected formal $\CO_{F_v}$-module of height $m$ over $\overline {k(v)}$. If $M_1 $ is not connected, it is the direct sum of the connected divisible $\CO_{F_v}$-module of height 1 and the constant divisible $\CO_{F_v}$-module $F_v/\CO_{F_v}$.
\begin{defn}(1) A $\cD$-elliptic sheaf $\BE$ over $\overline {k(v)}$ is called supersingular if $M_1$ is connected.
Otherwise $\BE$ is called ordinary.
A point in $ \CM_{U ,\overline {k(v)}}$ is called supersingular if the underlying $\cD$-elliptic sheaf is supersingular.
(2) Let $\CM_{U ,\overline {k(v)}}^{\mathrm{sing}}\subset \CM_{U ,\overline {k(v)}}$ be the subset of supersingular points. Let $\CM_{U,\overline {k(v)}}^{\mathrm{sing}}\subset \CM_{U,\overline {k(v)}}$ be the subset of points whose images in $\CM_{U ,\overline {k(v)}}$ is contained in $\CM_{U ,\overline {k(v)}}^{\mathrm{sing}}$.
Points in $\CM_{U,\overline {k(v)}}^{\mathrm{sing}}$ are called superinsgular points.
Points
in $ \CM_{U,\overline {k(v)}}$ outside $\CM_{U,\overline {k(v)}}^{\mathrm{sing}}$ are called ordinary points.
(3)
An irreducible component in the special fiber of $\CN_{U,\hat\CO_{H_w}^\ur} $ is called a supersingular component if its
image in the special fiber of $\CM_{U,\hat\CO_{F_v} ^\ur}$ is a supersingular point. Otherwise it is called an ordinary component.
\end{defn}
By \cite[(10.6)]{LLS} and \cite[Proposition 10.2.2]{Boy}, we have a bijection
\begin{equation}\label{ss12'}\CM_{U,\overline {k(v)}}^{\mathrm{sing}}\simeq B^\times\backslash \BZ \times \BB^{v,\times}/ \tilde U^v.
\end{equation}
The following lemma is easy to prove.
\begin{lem} \label{lieord}
If $v$ is split in $E$, points in $CM_U$ have reductions outside supersingular components on $\CN_{U,\CO_{H_w}}$.
Otherwise they have reductions in supersingular components.
\end{lem}
\subsubsection{Components} \label{special fibers}
Let $F_U$ be the abelian extension of $F$ corresponding to $F ^\times\backslash \BA_{F }^\times/\det (\tilde U)$ by the class field theory.
Let $\CM_{U,(v)}$ be the restriction of $\CM_U$ to $X_{(v)}$ (see \ref{mio}). By \ref{mio} and the determinant construction of $\cD$-elliptic sheaves (see \cite{LL97}), there is a natural projection $${\mathrm{Det}}_U:\CM_{U,(v)}\to\Spec \CO_{F_U,(v)} $$ of $X_{(v)}$-schemes. Here $\Spec \CO_{F_U,(v)} $ is identified with a moduli space of rank 1 elliptic sheaves with (Drinfeld) level structures associated to $\det (\tilde U)$.
The base change of ${\mathrm{Det}}_U$ via the embedding $F\to \BC_\infty$ gives a morphism $${\mathrm{Det}}_{U,\BC_\infty}: M_{U,\BC_\infty}\to\Spec \BC_\infty\times F ^\times\backslash \BA_{F }^\times/\det (\tilde U) . $$ After rigid analytification, ${\mathrm{Det}}_{U,\BC_\infty}$ coincides with the combination of the rigid analytic uniformization Proposition 6.6\ref{riguni} and the determinant morphism (over $\BC_\infty$) on the Drinfeld's covering space in \cite[IV]{{Gen}}. The fibers of the determinant morphism on Drinfeld's covering space are connected.
In particular, the fibers of ${\mathrm{Det}}_{U,\BC_\infty}$ are the connected components of $M_{U,\BC_\infty}$.
Let $\CV^\ord$ be the set of ordinary components in the special fiber of $\CN_{U,\hat\CO_{H_w}^\ur} $.
Then ${\mathrm{Det}}_{U,\hat\CO_{H_w}^\ur}$ (combined with
$\CN_{U,\hat\CO_{H_w}^\ur}\to \CM_{U,\hat\CO_{H_w}^\ur}$) induces a map $\CV^\ord\to F ^\times\backslash \BA_{F }^\times/\det (\tilde U) $ whose fibers are connected components of $\CN_{U,\hat\CO_{H_w}^\ur} $
by Zariski's connectedness theorem.
Recall that a Drinfeld level structure of level $n$ on the divisible $\CO_{F_v}$-modules $M_1$ is an $\CO_{F_v}$-module morphism
$$((\varpi_v^{-n}\CO_{F_v})/\CO_{F_v})^2\to M_1[\varpi_v^n]$$ satisfying certain conditions (see \cite[Secton 4, B)]{DriEll1}).
The $ \CO_{F_v}$-submodules
of $((\varpi_v^{-n}\CO_{F_v})/\CO_{F_v})^2$ of rank 1 are indexed by $ B_v\backslash \GL_2(F_v)/U_v,$
where $B_v$ is the standard Borel subgroup. By the argument in \cite[9.4]{Car} (see also \cite[10.4]{Boy}), we have the following results. On each irreducible component of $\CM_{U,\overline {k(v)}}$, the Drinfeld level structure vanishes on a corresponding rank-1 $ \CO_{F_v}$-submodule. In particular, we have a map from irreducible components of $\CM_{U,\overline {k(v)}}$ to $B_v\backslash \GL_2(F_v)/U_v$. Combined with
$\CN_{U,\overline {k(w)}}\to \CM_{U,\overline {k(v)}}$, we have a map $\CV^\ord\to B_v\backslash \GL_2(F_v)/U_v$.
\begin{prop}\label{123}
Irreducible components of $\CM_{U,\overline {k(v)}}$ (with reduced scheme structures) are smooth. The ones in the same connected component
intersect at supersingular points. Moreover, we have a bijection $$\CV^\ord\simeq B_v\backslash \GL_2(F_v)/U_v \times F ^\times\backslash \BA_{F }^\times/\det (\tilde U) .$$
\end{prop}
\subsubsection{Uniformizations}\label{Serre-Tate theory, the multiplicity function} Assume that $v$ is not split in $E$.
Let $\Art_{\hat \CO_{F_v}^\ur}$ be the category of complete artinian $\hat \CO_{F_v}^\ur$-algebras with residue fields isomorphic to $\overline {k(v)}\simeq\hat \CO_{F_v} ^\ur/\varpi_v$.
Let $ \CS_{U_v}$ be the level $n$ deformation space of the unique formal $\CO_{F_v} $-module $\BM$ of height 2 over $\overline {k(v)}$.
In other words, an element in $ \CS_{U_v}(R)$, where $R\in \Art_{\hat\CO_{F_v} ^\ur}$, is a(n isomorphism class) of formal $\CO_{F_v} $-module $M$ over $R$ with Drinfeld level structure of level $n$
and a quasi-isogeny \begin{equation}M_{\overline {k(v)}}\to\BM.\label{left}\end{equation} Then there is a natural projection
\begin{equation}\CS_{U_v}\to\BZ\label{CSBZ}\end{equation} which maps an element in $ \CS_{U_v}(R)$ to the (valuation of the) degree of the quasi-isogeny. Let
$g\in B_v^\times$ act by $v(\det(g))$.
There is a natural left action of $$B_v^\times \simeq (\End(\BM)\otimes _{\CO_{F_v}} F_v)^\times$$ on $\CS_{U_v} $ which respect the morphism \eqref{CSBZ} and
the actions of $ B_v^\times$ on both sides.
Let $\hat \CM_{U }$ be the formal completion of $\CM_{U, \hat \CO_{F_v}^\ur}$ along the supersingular locus.
The Serre-Tate theory for $\cD$-elliptic sheaves
(see \cite[Proposition 5.4]{DriEll1}, \cite[Theorem 7.4.4]{Boy}) gives a formal uniformization of $\hat\CM_U$ (see \cite[Proposition 14.1]{Boy}): \begin{equation}\label{ST}
\hat \CM_{U } \simeq B^\times\backslash \CS_{U_v} \times \BB^{v,\times}/\tilde U^v.
\end{equation}
Under the isomorphisms \eqref{ss12'} and \eqref{ST}, the reduction map $\hat \CM_{U } \to \CM_{U,\overline {k(v)}}^{\mathrm{sing}}$ is given by the map $\CS_{U_v} \to \BZ.$
Let $\hat \CN_U$ be the formal completion of $\CN_{U,\hat\CO_{H_w} ^\ur}$ along the union of supersingular components. Let $\tilde \CS_{U_v}$ be the minimal desingularization of the normalization of $\CS_{U_v }$ in the ring of meromorphic functions of $ \CS_{U_v}\hat\otimes_{\hat\CO_{F_v}^\ur}\hat \CO_{H_w}^\ur$. Then $\tilde\CS_{U_v} $ admits an action of $B_v^\times $ by the functoriality of minimal desingularization. Then \eqref{ST} induces an isomorphism of formal schemes:
\begin{equation} \label{CNU}\hat \CN_U \simeq B^\times\backslash \tilde \CS_{U_v} \times \BB^{v,\times}/\tilde U^v.
\end{equation}
Define $\CH_{U_v}$ to be the set of (isomorphism classes of) quasi-canonical liftings of $\BM$ to $\CO_{\overline v}$ with Drinfeld level structures of level $n$. Here a lifting is called quasi-canonical if its endomorphism ring contains $E_v$. Then there is a canonical $ B_v^\times $-action on $\CH_{U_v}$, and a $ B_v^\times $-equivariant map
\begin{equation}\CH_{U_v}\to \CS_{U_v}(\CO_{\overline v}) .\label{HUv'}
\end{equation}
Consider maps \begin{equation}CM_U\to \CM_U(\CO_{\overline v}) \to \hat \CM_U(\CO_{\overline v})\label{113}\end{equation} where the first map sends a CM point to the base change to $\CO_{\overline v}$ of its Zariski closure in $\CM_U$ and the second map is the natural one. We have a ``uniformization" of $CM_U$ by $\CH_{U_v}$ and a ``uniformization" of the composition of \eqref{113} by \eqref{HUv'} as follows.
Let $t\in E_v^\times $ act on $B_v^\times$ via right multiplication by $t^{-1}$, and act on $\BB_v^\times$ via left multiplication by $t$.
By \cite[5.5]{Zha01},
there is a $ B_v^\times $-equivariant bijection \begin{equation*}\CH_{U_v}\simeq B_v^\times\times_{E_v^\times}\BB_v^\times/U_v , \end{equation*}
where
$ B_v^\times $-action on the right hand side is by left multiplication on $B_v^\times$. This bijection depends on the choice of the preimage of $(1,1)$ in $ \CH_{U_v}$. We take $(1,1)$ to be the formal $\CO_{F_v}$-module of height 2 with Drinfeld level structure of level $n$ associated to $\cD$-elliptic sheaf represented by $P_0$ (see \ref{CM points}). We also have an embedding $E_v^\times\hookrightarrow \CH_{U_v}$ by $t\mapsto (t ,1)=(1,t)$.
The bijection \eqref{111} induce a bijection \begin{equation}CM_U\simeq B^\times\backslash \left ( (B ^\times\times_{E ^\times}\BB_v^\times/U_v) \times \BB^{v,\times}/\tilde U^v\right),\label{CMUv}
\end{equation} by sending $\beta\in E^\times\backslash \BB^\times/ \tilde U$ to $((1,\beta_v),\beta^v)$. Under the natural inclusion \begin{equation}B ^\times\times_{E ^\times}\BB_v^\times/U_v\hookrightarrow \CH_{U_v},\label{HUv}
\end{equation}
we regard $\CH_{U_v}$ as an ``uniformizing space" of $CM_U$.
By
\eqref{CMUv} and \eqref{ST}, the composition of \eqref{HUv} and \eqref{HUv'} induces
a map \begin{align} CM_U \to B^\times\backslash \CS_{U_v} (\CO_{\overline v}) \times \BB^{v,\times}/\tilde U^v \simeq \hat \CM_U(\CO_{\overline v}) .\label{118} \end{align} Up to choices of the data away from $v$ defining \eqref{ST}, we have the following result. \begin{lem} \label{11.2.6}The map \eqref{118} is the composition of \eqref{113}.
\end{lem}
A point in $\CS_{U_v}(\CO_{\overline v})$ lifts to a point in $\tilde \CS_{U_v}(\CO_{\overline v})$ ($\tilde \CS_{U_v}$ is regarded as over $\hat\CO_{F_v}^\ur$) by strict transform. Thus the map $\CH_{U_v}\to \CS_{U_v}(\CO_{\overline v}) $ lifts to a $ B_v^\times $-equivariant map \begin{align}\CH_{U_v}\to \tilde\CS_{U_v}(\CO_{\overline v}).\label{119}\end{align} This map induces a map \begin{align}\begin{split}&CM_U\simeq B^\times\backslash \left ( (B ^\times\times_{E ^\times}\BB_v^\times/U_v) \times \BB^{v,\times}/\tilde U^v\right)\\
\to &\hat \CN_U(\CO_{\overline v})\simeq B^\times\backslash \tilde\CS_{U_v} (\CO_{\overline v}) \times \BB^{v,\times}/\tilde U^v .\label{120}\end{split}\end{align} Lemma \ref {11.2.6} implies the following result.
\begin{lem} \label{11211}The map \eqref{120} coincides with the composition of maps $$CM_U\to\CN_U (\CO_{\overline v})\to \hat \CN_U (\CO_{\overline v}) $$ where the first map maps a CM point to the base change to $\CO_{\overline v}$ of its Zariski closure in $\CM_U$ and the second map is the natural one. \end{lem}
\subsubsection{Multiplicity function}
\begin{defn}\label{mf1} Define the multiplicity function $m_v$ on $ \CH_{U_v}-\{(1,1)\}$ as follows: for $(g_1 ,g_2 )\in \CH_{U_v} $
and $(g_1,g_2)\neq (1,1)$, let $m_v(g_1 ,g_2 )$
to be the intersection number of the images of the
points $(g_1,g_2)$ and $(1,1)$ in $\tilde \CS(\CO_{\overline v}) $ under \eqref{119}.
\end{defn} We have the following properties of the multiplicty function $m_v$.
\begin{lem}\label{bequiv1} (1) The following equation holds: $ m_v(g_1^{-1} ,g_2^{-1} )= m_v(g_1 ,g_2 ). $
(2) The number $m_v(g_1,g_2)$ is the intersection number of the images of the
points $(gg_1,g_2)$ and $ (g,1)$ in $\tilde \CS(\CO_{\overline v}) $ for every $g\in B_v^\times .$
\end{lem}
\begin{proof} (1) follows from definition.
(2) follows from the $B_v^\times$-equivariance of the map \eqref{119}.
\end{proof}
Now we consider the nonvanishing of $m_v$.
Let $F_{U_v}$ be the totally ramified abelian extension of $\hat F_v^\ur$ corresponding to $\det(U_v)$ via local class field theory, so that $(\Spec \CO_{F_{U_v}})(\CO_{\overline v})\simeq \CO_{F_v}^\times/\det(U_v)$.
There is a natural morphism $\CS_{U_v} ^0\to \Spec \CO_{F_{U_v}}$ constructed in \cite{Str}. (In fact, this morphism comes from the determinant construction of formal modules, see \cite{Hed}.) Its composition with the natural morphism $\tilde \CS_{U_v} ^0\to \CS_{U_v} ^0$ is denoted by \begin{equation}\label{Detuv}{\mathrm{Det}}_{U_v} : \tilde \CS_{U_v}^0 \to \Spf \CO_{F_{U_v}}.\end{equation} The composition of $B_v^\times\times_{E_v^\times}\BB_v^\times/U_v\simeq \CH_{U_v} $ with \begin{equation} \CH_{U_v} \to \tilde \CS_{U_v}(\CO_{\overline v})\to\CS_{U_v}(\CO_{\overline v}) \to \BZ\label{CCBB}\end{equation}
is given $(g_1,g_2)\mapsto v(\det(g_1)\det(g_2)).$ Let $\CH_{U_v} ^0$ be the preimage of 0. Then the composition of $$\CH_{U_v} ^0\to \tilde \CS^0(\CO_{\overline v}) \to \tilde \CS_{U_v}^0 (\CO_{\overline v}) \xrightarrow{{\mathrm{Det}}_{U_v}}(\Spec \CO_{F_{U_v}})(\CO_{\overline v})\simeq \CO_{F_v}^\times/\det(U_v) $$ is given by $(g_1,g_2)\mapsto \det(g_1)\det(g_2).$
Thus we have the following lemma. \begin{lem}\label{vdetun1}The multiplicty function $m_v(g_1,g_2)\neq 0$ only if $ \det(g_1)\det(g_2) \in \det(U_v)$.
\end{lem}
\subsubsection{Compute $i(f)_v$} Let $i_{\overline v}(t_1g,t_2)$ be as in the definition of $i(f)_v$
(see \ref{Decomposition of the height distribution}).
\begin{lem}\label{heightpullpack} Let $x,y\in CM_U$ be \textit{distinct} CM points. Assume $x=t_1g,y=t_2$ for $t_1,t_2\in \BA_E^\times $ and $g\in \BB^\times$, then $$i_{\overline v}(x,y)=\sum_{\delta\in B^\times} m_v(t_{1,v}^{-1}\delta t_{2,v},g_v ^{-1}) 1_{\tilde U^v}(((t_1 g )^{-1} \delta t_2 )^v).$$
Moreover, this is a finite sum.
\end{lem} \begin{proof} The decomposition is standard, see \cite[Lemma 8.2]{YZZ}. The particular expression here follows from Lemma \ref{bequiv1}. Only need to show that the sum is a finite sum. Since $B_v $ is a division algebra, the preimage of an open compact subset of $F_v^\times$ under the reduced norm $\det$ is compact. By Lemma \ref{vdetun1},
the function $$m_v(t_{1,v}^{-1}\delta t_{2,v},g_v ^{-1}) 1_{\tilde U^v}(((t_1 g )^{-1} \delta t_2 )^v)$$ for $\delta\in B^\times(\BA_F)/\varpi_\infty^\BZ$ is only nonvanishing on a compact subset.
Since the image of the inclusion $B^\times\hookrightarrow B^\times(\BA_F)/\varpi_\infty^\BZ$ is discrete and closed, the sum is a finite sum.
\end{proof}
To compute $i(f)_v$, we first deal with the regularized integral involved in its definition (see \ref{Decomposition of the height distribution}).
By a direct computation and Fubini's theorem, we have the following lemma.
\begin{lem} \label{innersum} Let $V$ be an open compact subgroup of $\BA_{F,\mathrm{f}}^\times$ whose intersection with $F^\times$ is $\{1\}$,
$\delta \in B^\times_\reg$, and let $\phi$ be a function on $ \BA_E^\times\delta \BA_E^\times $ which is $V$-invariant and $\Xi_\infty$-invariant.
Then if either side of the equation \begin{align*}& \Vol(V) \int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{E^\times\backslash \BA_E^\times / \BA_F^\times} \sum_{t\in F^\times\backslash \BA_F^\times/\Xi_\infty V}\sum_{x\in E^\times \delta E^\times} \phi(t_1^{-1}xtt_2)d{t_2}d{t_1} \\
=& \int_{ \BA_E^\times / \BA_F^\times}\int_{ E^\times(\BA_{F,\mathrm{f}}) }\int_{ E^\times(F_\infty)/\Xi_\infty } \phi(t_1^{-1}\delta t_{2,\mathrm{f}} t_{2,\infty})d_{t_{2,\infty}}d{t_{2,\mathrm{f}}}d{t_1} \end{align*}
converges absolutely, the other side also converges absolutely,
and in this case the equation holds.
\end{lem}
Use Assumption \ref{fvan} (1) to get rid of contributions from singular orbits in $i(f)_v$ as follows. \begin{lem}\label{asmp4im}(1) If $t_1g=t_2$ as points in $ M_U$, then $f(g)=0.$
(2) Let ${g\in \BB^\times/\tilde U} $, we have \begin{align*}
& f(g) \sum_{\delta\in B^\times} m_v(t_{1,v}^{-1}\delta t_{2,v},g_v ^{-1}) 1_{\tilde U^v}(((t_1 g )^{-1} \delta t_2 )^v)\\
=& f(g)\sum_{\delta\in B_\reg^\times} m_v(t_{1,v}^{-1}\delta t_{2,v},g_v ^{-1}) 1_{\tilde U^v}(((t_1 g )^{-1} \delta t_2 )^v).\end{align*}
\end{lem} \begin{proof}(1) If $t_1g=t_2$, then $g\in \BA_E^\times \tilde U$. Thus $f(g)=0$ by Assumption \ref{fvan} (1) and the $\tilde U$-invariance of $f$. (2) Let $\delta\in B^\times_{\mathrm{sing}}$. If $1_{\tilde U^v}(((t_1 g )^{-1} \delta t_2 )^v)\neq 0$, then $g^v\in t_1 ^{-1} \delta t_2 ^v \tilde U ^v$. So $f(g)=0$ by Assumption \ref{fvan} (1) and the $\tilde U$-invariance of $f$. The equation follows. \end{proof}
Now we compute $i(f)_v$. Lemma \ref{asmp4im} validates the third ``$=$" in the equation
\begin{align*}
i(f)_v&=\int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{E^\times\backslash \BA_E^\times }^* i_{\overline v}( \tilde Z(f)_{U,*}t_1,t_2)\Omega^{-1}(t_2)\Omega(t_1) dt_2dt_1\\
&=\Vol (\Xi_U) |F^\times\backslash \BA_F^\times/ \Xi|\int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{E^\times\backslash \BA_E^\times }^*\sum_{g\in \BB^\times/\tilde U} f(g) i_{\overline v}( t_1g,t_2)\Omega^{-1}(t_2)\Omega(t_1) dt_2dt_1\\
&= \Vol (\Xi_U) |F^\times\backslash \BA_F^\times/ \Xi|\int_{E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{E^\times\backslash \BA_E^\times / \BA_F^\times} \frac{1}{|F^\times\backslash \BA_F^\times/ \Xi|}
\\
&\sum_{t\in F^\times\backslash \BA_F^\times/\Xi_\infty \Xi_U}\sum_{g\in \BB^\times/\tilde U} f(g) \sum_{\delta\in B_\reg^\times}m_v(t_{1,v}^{-1}\delta tt_{2,v},g_v ^{-1}) 1_{\tilde U^v}(((t_1 g )^{-1} \delta tt_2 )^v) \omega^{-1}(t)\Omega^{-1}(t_2)\Omega(t_1) dt_2dt_1. \end{align*}
Applying Lemma \ref{innersum} and Fubini's theorem, we have
\begin{align} \begin{split}
i(f)_v
& = \sum_{\delta\in E^\times\backslash B_\reg^\times/E^\times} \int_{ \BA_E^{v,\times} / \BA_F^{v,\times}}\int_{ \BA_E^{v,\times} /\Xi_\infty } f^v( (t_{1 } ^{v})^{-1} \delta t_{2}^v ) \Omega^{-1}(t_2^v)\Omega(t_1^v) d {t_{2}^v}d{t_1^v}\\
&\ \ \ \ \ \sum _{g_v\in \BB_v^\times/U_v}f_{v}(g_v) \int_{ E^\times_v/F_v^\times}\int_{ E^\times_v} m_v( t_{1,v} ^{-1} \delta t_{2,v} , g_v^{-1} ) \Omega_v^{-1}( t_{2,v})\Omega_v(t_{1,v}) dt_{2,v}d t_{1,v}\label{1118}
\end{split} \end{align}
provided the right hand side is absolutely convergent. The absolutely convergence is as follows. First, the integral of $m_v$ is absolutely convergent by Lemma \ref{vdetun1}.
Second, the integral over $\BA_E^{v,\times} / \BA_F^{v,\times}\times \BA_E^{v,\times} /\Xi_\infty $
decomposes into a product of local orbital integrals such that for almost all places the values are 1 (see Lemma \ref{sint=1}, \ref {sint=1'}). So we only need to prove that the right hand side is a finite sum. Let the invariant map $\inv_{E^\times}':B^\times(\BA_F)\to\BA_F$ be the product of the local ones $$\inv_{E_v^\times}':B_v^\times\to F_v$$defined in \eqref{inv'}. Since ${\mathrm{Nm}} (E_v^\times)\not\in \inv _{E^\times}(B_v^\times)$, an open neighborhood of 1 is not contained in $\inv _{E^\times}(B_v^\times)$. So $ \inv _{E^\times}'(B_v^\times)$
is contained in a compact subset of $F_v$. In particular, with $g_2$ fixed, the subset $\inv_{E^\times}'(\{g_1\in (B_v^\times)_\reg :m_v(g_1 ,g_2 ) \neq0\}) $ of $ \inv _{E^\times}'(B_v^\times)$
is contained in a compact subset of $F_v$. By Proposition \ref{Proposition 2.4, Proposition 3.3Jac862}, there is a compact subset of $\BA_F$, such that the summand in the right hand side is nonzero only
if $\inv_{E^\times}'(\delta)$ is in this compact subset. Since $\inv_{E^\times}'(B^\times)$ is a discrete closed subset of $F$, the sum on the right hand side is a finite sum.
\begin{defn}\label{iterm1}For $\delta\in B^\times_\reg$, define the arithmetic orbital integral of the multiplicty function $m_v$ weighted by $f_v$ to be
$$ i (\delta,f_v) := \int_{ E^\times_v/F_v^\times}\int_{ E^\times_v} \sum _{g_v\in \BB_v^\times/U_v}f_{v}(g_v) m_v( t_{1,v} ^{-1} \delta t_{2,v} , g_v^{-1} )
\Omega_v^{-1}( t_{2,v})\Omega_v(t_{1,v}) dt_{2,v}d t_{1,v} . $$ \end{defn}
By \eqref{1118}, we have the following proposition.
\begin{prop}\label{regwell}
Under Assumption \ref{fvan}, we have
\begin{equation*} i(f)_v= \sum_{\delta\in E^\times\backslash B_\reg ^\times/E^\times} \CO_{\Xi_\infty} (\delta,f^v)
i(\delta,f_v).
\end{equation*}
\end{prop}
\subsubsection{Compute the multiplicity function $m_v$}
We will use some computations in \cite{Zha01} \cite{YZZ}. They are based on Gross' results \cite{Gro}, which works in arbitrary characteristics.
We first assume that $U_v$ is maximal.
Then $\CM_{U,\CO_{F_v}} $ is a smooth model. So $\CN_{U,\CO_{H_w}} \simeq\CM_{U,\CO_{F_v}} \otimes_{\CO_{F_v}} \CO_{H_w} $ and is a smooth model.
Then $m_v$ can be computed
using Gross's theory of quasi-canonical lifting.
Fix an isomorphism $\GL_2(F_v)\simeq \BB_v^\times$ which maps $\GL_2(\CO_{F_v})$ to $U_v$.
\begin{lem} \label{GL2decom}
Let $h_c=\begin{bmatrix}\varpi_v^c&0 \\0 &1\end{bmatrix}\in \BB_v^\times$. Under Assumption \ref{asmpe0},
there is a decomposition
$$\BB_v^\times =\coprod_{c\in \BZ_{\geq 0}} E_v^\times h_c U_v.$$
\end{lem} \begin{lem} [{\cite[Lemma 5.5.2]{Zha01} \cite[Lemma 8.6]{YZZ}}]\label{unramified multiplicity function} The multiplicity function $m_v$
on $\CH_{U_v} $ is nonzero
only if $\det (g_1)\det(g_2)\in \CO_{F_v}^\times $.
In this case, assume $g_2\in E_v^\times h_c U_v $, then
\begin{itemize}
\item[(a)] if $c=0$ then $m_v(g_1,g_2)=\frac{1}{2}(v(\inv_{E_v^\times}' (g_1))+1)$;
\item[(b)] if $c>0$ and $E_v/F_v$ is unramified, then $m_v(g_1,g_2)=q_v^{1-c}(q_v+1)^{-1};$
\item[(c)] if $c>0$ and $E_v/F_v$ is ramified, then $m_v(g_1,g_2)=\frac{1}{2}q_v^{ -c}.$
\end{itemize}
\end{lem}
Now suppose $U_v $ is principal of level $n>0$. Assume $g_2=1$. Then
by Lemma \ref{vdetun1}, $m(g_1,1)$ is supported on $B_{v,0}:=\{g_1\in B_v^\times: \det (g_1)\in \det U_v\}$ (and not defined on $ E_v^\times$).
\begin{lem}\label{U'}There is an open compact subgroup $U'\subset B_{v,0}$ such that
\begin{itemize}
\item[(1)] as subgroups of $E_v^\times$, $U'\cap E_v^\times=U_v\cap E_v^\times$:
\item[(2)] the function $$m (g_1,1 ) - \frac{v (\inv_{E_v^\times}'(g_1)) }{2} 1_{U'}(g_1) $$
on $ B_{v,0} - E_v^\times$
can be extended to a locally constant function on $B_{v,0} $.
\end{itemize}
\end{lem}
\begin{proof}
For $ (g_1,g_2)\in \CH_{U_v}$, let $\red(g_1,g_2)$ be the reduction of its image in $\tilde \CS_{U_v}$. Let $U'$ be the maximal open compact subgroup through which the restriction of the $B_{v,0}$-action on $\red(1,1)$ factors. Then the inclusion $U'\cap E_v^\times\supset U_v \cap E_v^\times$ and (2) follows from the argument in \cite[Lemma 5.5.3, Lemma 5.5.4]{Zha01}. Now we prove (1). For $t\in B_{v,0} \cap E_v^\times-U_v\cap E_v^\times $, the reductions of the images of $(t^{-1},1)$ and $(1,1)$ in
$ \CS_{U_v}\hat\otimes_{\hat\CO_{F_v}^\ur}\hat \CO_{H_w}^\ur$ are the same. Thus the blowing-up process separates the images of $(t^{-1},1)$ and $(1,1)$ in $\tilde \CS_{U_v}$, i.e. $\red(t^{-1},1)\neq \red(1,1)$. Thus $U'\cap E_v^\times\subset U_v \cap E_v^\times$ and (1) follows.
\end{proof}
\subsubsection{Compute $j(f)_v$} Now we compute the $j(f)_v$.
\begin{prop}\label{nonsplitj} There exists $\overline {f_v}\in C_c^\infty(B_v^\times)$ such that
$$j(f)_v= \sum_{\delta\in E^\times\backslash B_\reg^\times/E^\times}\CO _{\Xi_\infty}(\delta,f^v) \CO (\delta ,\overline {f_v}) .$$
\end{prop}
We will prove Proposition \ref{nonsplitj} in \ref{cadmissible extensions} after some preparations.
Let $\CV^{\mathrm{sing}}$ be the set of supersingular components $\CN_{U,\hat\CO_{H_w}^\ur} \otimes_{\hat \CO_{H_w}^\ur}\overline{k(w)}$. Let $ \CV $ be the set of exceptional (irreducible reduced) curves of $\tilde \CS_{U_v}$ (from the desingulariation), contained in $\tilde \CS_{U_v} \otimes_{\hat \CO_{H_w}^\ur}\overline{k(w)}$, then there is a natural action of $B_v^\times$ on $\CV$. From \eqref{CNU}, we have a bijection
\begin{equation} \CV^{\mathrm{sing}} \simeq B^\times\backslash \CV \times \BB^{v,\times}/\tilde U^v.\label{vsing}\end{equation} For $C\in \CV$, $g\in\BB^{v,\times}$, let $[C,g]$ be the corresponding element in $\CV^{\mathrm{sing}}.$
\begin{defn} \label{66}Let $C\in \CV$, define a function $l_{ C}$ on $ \CH_{U_v}$
as follows. For $(g_1,g_2)\in \CH_{U_v}$, let $l_{ C}(g_1,g_2)$ be the intersection number of $C$ and the image of the
point $(g_1,g_2)$ in $\tilde \CS(\CO_{\overline v})$.
\end{defn}
Since points in $ \CH_{U_v}$ are in fact defined over $\hat \CO_{H_w}^\ur$, the image of the first map
in \eqref{CCBB} is in $ \tilde \CS(\hat \CO_{H_w}^\ur) $. Since $ \tilde\CS $ is regular, the reductions of points in $ \tilde\CS(\hat \CO_{H_w}^\ur) $ the are in the smooth locus of $ \tilde\CS \otimes_{\hat \CO_{H_w}^\ur}\overline{k(w)}$. Thus we have a map $ \CH_{U_v}\to \CV$.
The maps in \eqref{CCBB} also induce a map $ \CV\to \BZ .$
\begin{lem} \label{9123}The function $l_C$ satisfies the following properties:
\begin{itemize}
\item[(1)] for $h \in \CH _{U_v}$, $l_{C}(h)\neq 0$
only if the image of
$ C$ in $\BZ$ under the map $ \CV\to \BZ$ and
the image of $h$ under $\CH_{U_v}\to \BZ$ (the composition of \eqref{CCBB}) are the same;
\item[(2)] for $b\in B_v^\times $, $h \in \CH_{U_v} $, $l_{bC}(bh)=l_C(h)$;
\item[(3)] $l_C$ is locally constant. \end{itemize}
\end{lem}
\begin{proof} (1) follows from Definition \ref{66}. (2) follows from the fact that the action by $b$ is an isomorphism.
(3) follows from (2) and the fact that the $B_v^\times$-action on the special fiber of $\tilde\CS$ factors through the open compact subgroup $U'$ as in Lemma \ref{U'}.
\end{proof}
Similar to Lemma \ref{heightpullpack}, we have the following lemma.
\begin{lem}\label{heightpullpack'} Let $g \in( \BB ^{v})^{\times}$ and $y\in CM_U \simeq B^\times\backslash \left ( (B ^\times\times_{E ^\times}\BB_v^\times/U_v) \times \BB^{v,\times}/\tilde U^v\right)$, the intersection number of $(C,g)\in \CV^{\mathrm{sing}}$ and the Zariski closure of $y $ in $\CN_U \otimes _{\CO_{H_w}}\CO_{\overline v}$ is given by
$$\sum_{\delta\in B^\times} l_C ( \delta^{-1} ,y_v)1_{\tilde U^v}(g^{-1}\delta^{-1} y^v).$$
Moreover, this is a finite sum.
\end{lem}
The finiteness of the sum is implied by Lemma \ref{9123} (1).
\subsubsection{Compute $\CZ$-admissible extensions}
For $t \in \BA_E^\times $, regarded as a point $ CM_U$, let $\overline t $ be the
Zariski closure of $t$ in $\CN_{U,\hat\CO_{H_w}^\ur}$, and $
\hat t $ the $\CZ$-admissible extension $\hat t$ of $t $ in $\CN_U
$. There exists $ C_i\in \CV $
such that $$\hat 1=\bar 1+\sum_{i=1}^n a_i[ C_i,1] -(C_{\ord,1},1).$$ Here $C_{\ord,1} $ is a linear combination of elements in $B_v\backslash \GL_2(F_v)/U_v$ and $(C_{\ord,1},1)$ represents a linear combination ordinary components via the canonical bijection in Proposition \ref{123}:
$$\CV^\ord\simeq B_v\backslash \GL_2(F_v)/U_v \times F ^\times\backslash \BA_{F }^\times/\det (\tilde U) .$$
Let $Z_{t_v} $ be the image of $ (1,t_v)\in \CH$ in $\tilde \CS_{U_v} $, then the image of $\bar t$ in $\hat \CN_{U} $
under the isomorphism
\begin{equation} \hat \CN_U \simeq B^\times\backslash \tilde \CS_{U_v} \times \BB^{v,\times}/\tilde U^v\label{wawa}
\end{equation} is $[Z_{t_v} ,t^v]$.
Let $D\in \CV$, $g^v\in \BB^{v,\times}$. Since $[D,g^v]$ is an exceptional divisor, $ \CZ \cdot [D,g^v]=0$ by the definition of $\CZ$ (see Definition \ref{ZNU}). Then
Definition \ref{admext} (1) for $ \hat 1$ and $[D,g^v]$ implies \begin{align} \begin{split}\sum_{\gamma \in \tilde U^v(g^v)^{-1}\cap B^\times} ( Z_{1} +\sum_{i=1}^n a_i C_i )\cdot \gamma D = (C_{\ord,1},1)\cdot [D,g^v],
\label{ext1} \end{split}\end{align}
where the intersections on the left hand side happen on $\tilde \CS_{U_v} $, and the intersection on the right hand side happens on $\CN_{U, \hat\CO_{H_w}^\ur}$.
Let $\BA_E^{s,v,\times}$ be the group of ideles away $|X|_s$ and $v$. Let $K_v\subset (U_E)_v $ be the stabilizer of $C_i$'s. Let $\hat \CO_{E,s}$ be the product of complete local ring of $\CO_E$ at split places.
\begin{lem}\label{admj0}For $t \in \BA_E^{s,v,\times}K_v\hat \CO_{E,s} \BA_F^\times $, the sum of the supersingular components in $\hat t $ is
$$\sum_{i=1}^n a_i[ C_i ,t^v] .$$
\end{lem}
\begin{proof}
We prove that \begin{equation*}\hat t=\overline t+\sum_{i=1}^n a_i[ C_i ,t^v] - (C_{\ord,1},{\mathrm{Nm}} (t^v)). \end{equation*}
We verify Definition \ref{admext} (1) for $ \hat t$ and supersingular components. For ordinary components, the verification of (1) is similar. The verification of Definition \ref{admext} (2) is similar and easier.
By the assumption that $t_v\in (U_E)_v$, we have $Z_{t_v}=Z_1$. Thus we only need to prove that for every $[A,h^v]\in \CV^{\mathrm{sing}}$, the equation \begin{equation} \sum_{\gamma \in t^v\tilde U^v(h^v)^{-1}\cap B^\times} ( Z_1+\sum_{i=1}^n a_i C_i )\cdot \gamma A = (C_{\ord,1},{\mathrm{Nm}} (t^v))\cdot [A,h^v] \label{ext2} \end{equation} holds.
Note that each $U_x$, $x\in |X|-\{\infty\}$, has the form \eqref{orbint1K}, and $U_\infty$ is generated by $\varpi_\infty$ and a subgroup of the form \eqref{orbint1K} or is $\BB_\infty^\times$.
By Assumption \ref{asmpe0} and a direct computation, we have $t^v\tilde U^v =\tilde U^vt^v$.
Thus \eqref{ext2} is equivalent to
\begin{equation} \sum_{\gamma \in \tilde U^v (h^v(t^v)^{-1})^{-1}\cap B^\times} ( Z_1 +\sum_{i=1}^n a_i C_i )\cdot \gamma A = (C_{\ord,1},{\mathrm{Nm}} (t^v))\cdot [A,h^v] . \label{ext3} \end{equation}
Claim: $$(C_{\ord,1},{\mathrm{Nm}} (t^v)))\cdot [A,h^v ] =(C_{\ord,1},1))\cdot [A,h^vt^{v,-1} ].$$Then \eqref{ext3} is implied by \eqref{ext1} by choosing $D=A$ and $g^v=h^vt^{v,-1}$. The lemma follows.
To prove the claim, we use the following description of the restriction of $\CV^\ord$ to $ \hat \CN_U $.
The morphism \eqref {Detuv} induces a morphism $$\tilde \CS_{U_v}^0 \to \Spec \CO_{F_{U_v}} \hat\otimes_{\hat\CO_{F_v}^\ur}\hat \CO_{H_w}^\ur\simeq \Spec \hat \CO_{H_w}^\ur\times \CO_{F_v}^\times/\det(U_v) .$$
Here the last isomorphism is due to that $ F_{U_v}\subset \hat H_w^\ur$ which comes from the class field theory.
Let $\tilde \CS_{U_v}^{00}$ be the preimage of $\Spec \hat \CO_{H_w}^\ur\times\{1\}$.
By \cite[Appendice 8, Proposition]{Car}, the non-exceptional irreducible components of the special fiber of
$\tilde \CS_{U_v}^{00}$ are indexed by $ B_v\backslash \GL_2(F_v)/U_v$ (this fact is similar to the second paragraph of \ref{special fibers}). Thus
there is a vertical divisor $V$ of $\tilde \CS_{U_v}^{00}$ which does not contain any exceptional curve such that
for every $b\in \BB^{v,\times}$, the restriction $ (C_{\ord,1},{\mathrm{Nm}} (b))$ to $\hat \CN_{U_v} $ is $[V, b] $ under the isomorphism
\eqref{wawa}. Then the claim follows from a direct computation (again we use that $t^v\tilde U^v =\tilde U^vt^v$).
\end{proof}
Lemma \ref{admj0} is improved as follows.
\begin{lem}\label{admj}For $t \in \BA_E^{ \times} $, the sum of the supersingular components in $\hat t $ is
$$\sum_{i=1}^n a_i[ t_vC_i ,t^v] .$$
\end{lem}
\begin{proof}Choose a subgroup $I $ of the groups of ideles at split places, such that $ \Gal(H/E) $ is the direct sum of image of $\BA_E^{s,v,\times}K_v\hat \CO_{E,s} \BA_F^\times $ and the image of $I$ in $ \Gal(H/E) $ via the reciprocity map. Let $H'$ be the fixed subfield of the image of $I$. Define $\CN_{U}'$ in the same way as $\CN_{U}$, but with $H$ replaced by $H'$. Then $\CN_{U,\hat\CO_{H_w}^\ur}=\CN'_{U,\hat\CO_{H_w}^\ur}$. For $t$ as in Lemma \ref{admj0}, define $\hat t'$ on $\CN_{U}'$ to be the admissible extension of $t$. Then $\hat t$ and $\hat t'$ have the same base change to $\CN_{U,\hat\CO_{H_w}^\ur}=\CN'_{U,\hat\CO_{H_w}^\ur}$, since $\CZ$ is defined over $\CM_U$. Note that Galois action keeps admissible $\CZ$-extensions, again since $\CZ$ is defined over $\CM_U$. Applying the action of $I$ on $\CN_{U}'$ to $\hat t' $ with $t$ as in Lemma \ref{admj0}, the lemma follows.
\end{proof}
\begin{rmk} In \cite[8.5.1]{YZZ}, the computation of admissible extension is missed.
\end{rmk}
\subsubsection{Proof of Proposition \ref{nonsplitj}}\label{cadmissible extensions} Let $l=\sum _{i=1}^na_i l_{C_i}$. By Lemma \ref{heightpullpack'}, we have \begin {align*} j_v(t_1g,t_2)&=\sum_{\delta\in B^\times} \sum_{i=1}^na_il_{t_{2,v}C_i}( \delta , t_{1,v} g_v ) 1_{\tilde U^v}((t_2^{v}) ^{-1} \delta t_1g^v )\\ &=\sum_{\delta\in B^\times} l( t_{2,v}^{-1}\delta^{-1} t_{1,v} ,g_v ) 1_{\tilde U^v}((t_2^{v}) ^{-1} \delta^{-1} t_1g^v ). \end{align*}
Similar to Proposition \ref{regwell}, we have the following expression of $j(f)_v$. Let \begin{equation} j(\delta,f_v):=\int_{ E^\times_v/F_v^\times}\int_{ E^\times_v} \sum _{g_v\in \BB_v^\times/U_v}f_{v}(g_v) l(t_{2,v}^{-1} \delta^{-1} t_{1,v} ,g_v )
\Omega_v^{-1}( t_{2,v})\Omega_v(t_{1,v}) dt_{2,v}d t_{1,v}\label{1129}\end{equation} which is well-defined by Lemma \ref{9123} (1). \begin{prop} \label{regwell1}
Under Assumption \ref{fvan} and Assumption \ref{asmpe0}, we have
\begin{equation*} j(f)_v= \sum_{\delta\in E^\times\backslash B_\reg ^\times/E^\times} \CO _{\Xi_\infty}(\delta,f^v) j(\delta,f_v). \end{equation*}
\end{prop}
Define a function $\overline {f_v}$ on $B_v^\times$ by \begin{equation}\overline {f_v}(h):=
\sum _{g_v\in \BB_v^\times/U_v}f_{v}(g_v) l( h^{-1} ,g_v ) .\label{barf}\end{equation}
By Lemma \ref{9123} (1), $\overline {f_v}\in C_c^\infty(B_v^\times)$. By \eqref{1129}, $j(\delta ,f_v)=\CO (\delta ,\overline {f_v}) .$
This finishes the proof of Proposition \ref{nonsplitj}. \subsection{ Superspecial case}\label{Superspecial case} Let $v\in{\mathrm{Ram}}-\{\infty\}$. Then $\BB_v$ is a division algebra.
In particular, $v$ is not split in $E$. Let $B=B(v)$ be the $v$-nearby quaternion algebra of $\BB.$ Then $B_v\simeq \RM_{2,F_v}$. Let $n$ be the level of the principal congruence subgroup $U_v\subset \BB_v^\times$.
\subsubsection{Formal models of $\CN_{U ,\CO_{H_w}}$ and the multiplicity function}
Let $\Omega_v$ be rigid analytic Drinfeld's upper half plane over $ {F_v}$, $\hat \Omega_v$ be Deligne's formal model of $\Omega_v$ over $\CO_{F_v}$. Then $\hat \Omega_v \hat \otimes \hat \CO_{F_v}^\ur $ is the deformation space of special height 4 formal $\CO_{F_v}$-modules (see \cite{Dridomain}).
Let $\Sigma_n$ be the $n$-th covering of $ \Omega_v \hat \otimes_{F_v} \hat F_v ^\ur $. Then $\Sigma_n$ admits a natural $B_v^\times\times\BB_v^\times$-action (see \cite{Dridomain}).
\begin{prop} \label{CDunif'} There is an isomorphism of rigid analytic spaces over $F_v$:
$$ M_{U} ^\an \simeq B ^\times\backslash \Sigma_n \times \BB^{v,\times}/\tilde U ^v.$$ \end{prop}
\begin{proof}When $U_\infty=\BB_\infty^\times$, this is proved in \cite[Theorem 8.3]{Hau}.
The general case can be obtained by applying \cite[Proposition 4.28]{Spi} to Proposition \ref{riguni}. \end{proof}
Let $\hat \Sigma_n$ be minimal desingularization the normalization of $ \hat \Omega_v \hat \otimes \hat \CO_{F_v}^\ur $ in the
rigid analytic space $\Sigma_n \hat \otimes_{ F_v } H_w $. Let $\hat \CN_{U }$ be the formal completion of $ \CN_{U,\CO_{H_w} }$ along its special fiber.
\begin{cor} \label{CDunif''}
There is an isomorphism of formal schemes over $\CO_{H_w}$:
$$ \hat\CN_{U} \simeq B ^\times\backslash \hat \Sigma_n \times \BB^{v,\times}/\tilde U ^v.$$ \end{cor}
Now we consider $\CO_{\overline v}$-points.
Define
\begin{equation*}\CH_{U_v}:= B_v^\times\times_{E_v^\times}\BB_v^\times/U_v \end{equation*} and let $B_v^\times$ acts on $\CH_{U_v}$ by left multiplication.
We define a $B_v^\times$-equivariant map $\CH_{U_v}\hookrightarrow \hat \Sigma_n(\CO_{\overline v}) $
as follows. Here $\hat \Sigma_n$ is regarded as a formal scheme over $\CO_{H_w}$.
Let $\hat P_0 \in \hat\CN_{U}(\CO_{\overline v})$ be the image of the Zariski closure of $P_0$ (see \ref{CM points}). Then up to the choice of the data defining the isomorphism in Corollary \ref{CDunif''}, there exists $ \hat z_0 \in \hat \Sigma_n(\CO_{\overline v}) $, fixed by the image of the diagonal embedding
$E_v^\times\hookrightarrow B_v^\times\times \BB_v^\times$,
such that under this isomorphism
$\hat P_0=[\hat z_0,1] .$
Then define $\CH_{U_v}\hookrightarrow \hat \Sigma_n(\CO_{\overline v}) $
by $(g_1,g_2)\mapsto (g_1,g_2)\cdot \hat z_0. $
\begin{defn} \label{mf2}Define the multiplicity function $m_v$ on $ \CH_{U_v}-\{(1,1)\}$ as follows: for $(g_1 ,g_2 )\in \CH_{U_v} $
and $(g_1,g_2)\neq (1,1)$, let $m_v(g_1 ,g_2 )$
to be the intersection number of the images of the
points $(g_1,g_2)$ and $(1,1)$ in $ \hat \Sigma_n (\CO_{\overline v}) $.
\end{defn}
Similar to Lemma \ref{vdetun1}, we have the following nonvanishing condition on $m_v$ by the determinant construction in \cite[IV]{Gen}. \begin{lem}\label{vdetun"}The multiplicty function $m_v(g_1,g_2)\neq 0$ only if $ \det(g_1)\det(g_2) \in \det(U_v)$.
\end{lem}
Let $\hat z_0'\in \hat \Omega_v (\CO_{\overline v})$ be the image of $\hat z_0$ by the composition of $\hat\Sigma_n \to \hat \Omega_v \hat \otimes \hat \CO_{F_v}^\ur \to \hat \Omega_v .$ It is the base change of a point $ \hat z_0'\in \hat \Omega_v (\hat \CO_{E_v}^\ur).$
Then the composition of
\begin{align}\CH_{U_v}\hookrightarrow\hat \Sigma_n (\CO_{\overline v})\to\hat \Omega_v (\hat \CO_{E_v}^\ur) \label{929}
\end{align} is given by
$(g_1,g_2)\mapsto g_1\hat z_0'. $
\subsubsection{Special fibers of $\hat \Omega_v$ } \label{stabilizer}
The irreducible components of the special fiber of $\hat\Omega_v \hat \otimes \hat \CO_{F_v}^\ur$, now regarded as a formal scheme over $ \hat \CO_{F_v}^\ur$, are $\BP^1_{\overline {k(v)}}$'s
parametrized by the set of dilation classes of $\CO_{F_v}$-lattices of $F_v^2$. Moreover, $B_v^\times$ acts on the special fibers in the same way as it acts on the lattices via an isomorphism $B_v^\times\simeq \GL_2(F_v)$. Assume that
$E_v/F_v$ is unramified.
Then the reduction of
of $\hat z_0'$ is a smooth point of the special fiber of $\hat\Omega_v \hat \otimes \hat \CO_{F_v}^\ur$, so is only in one irreducible component. Let $K_v$ be the maximal compact subgroup of $B_v^\times$ such that $F_v^\times K_v$ is the stabilizer of this irreducible component.
Each two $\BP^1_{\overline {k(v)}}$'s in the special fiber of $\hat\Omega_v \hat \otimes \hat \CO_{F_v}^\ur$
intersect at an ordinary double point. The double points in the special fiber of $\hat\Omega_v \hat \otimes \hat \CO_{F_v}^\ur$ one-to-one correspond to \textit{non-ordered} pairs of dilation classes of adjacent lattices.
Assume that $E_v/F_v$ is ramified. Then $$\hat z_0'\in \hat \Omega_v( \hat \CO_{E_v}^\ur)-\hat \Omega_v( \hat \CO_{F_v}^\ur).$$
In fact, the generic fiber of $\hat z_0'$ in $ \Omega_v( \hat E_v^\ur)$ is a fixed point of $E_v^\times\hookrightarrow B_v^\times$.
A direct computation on Drinfeld's upper half plane shows that the generic fiber of $\hat z_0'$ is not defined over $\hat F_v^\ur$.
Thus the reduction of
$\hat z_0' $ is a double point. Let $S_v$ be the stabilizer of this double point. Let $s\in B_v^\times $ which
switches the two dilation classes of adjacent lattices (such $s$ exists and is unique up to $F_v^\times$). Let $K_v\subset B_v^\times$ be the a maximal compact subgroup such that $F_v^\times K_v$ is the stabilizer of $[L_0]$ or $[L_0']$. Then the group $S_v$ is generated by $F_v^\times,s$ and $K_v$.
\begin{lem}\label{vdetun'}
We have the following necessary conditions for $m(g_1,g_2)\neq 0$:
\begin{itemize}
\item[(a)] when $E_v/F_v$ is unramified, then $g_1\in F_v^\times K_v$;
\item[(b)] when $E_v/F_v$ is ramified, then $g_1\in S_v.$
\end{itemize}
\end{lem}
\begin{proof}
Assume $n=0$ and $m(g_1,g_2)\neq 0$. Then the images of $(g_1,g_2)$ and $(1,1)$ under $\CH_{U_v}\hookrightarrow\hat \Sigma_0 (\CO_{\overline v}) $ have the same reduction. Recall that $\hat z_0', g_1 \hat z_0'$ are the images of $(1,1)$ and $(g_1,g_2)$ in $ \hat \Omega_v( \hat \CO_{F_v}^\ur)$ under the composition of \eqref{929}. Thus the reductions of
$\hat z_0', g_1 \hat z_0'$ are the same point. Assume that $E_v/F_v$ is unramified, then
the reductions of $\hat z_0', g_1 \hat z_0'$
are in the same irreducible component of the special fiber of $ \hat\Omega_v \hat \otimes \hat \CO_{F_v}^\ur $. This gives condition (a). Now let $E_v/F_v$ be ramified. Then the reductions of
$\hat z_0', g_1 \hat z_0'$ are the same double point.
This gives condition (2). \end{proof}
\subsubsection{Compute $i(f)_v$}\label{9.2.3} By Corollary \ref{CDunif''},
we can express $i_{\overline v}$ by $m_v$ as in Lemma \ref{heightpullpack}.
\begin{lem} Let $x,y\in CM_U $ be \textit{distinct} CM points. Assume $x=t_1g,y=t_2$ for $t_1,t_2\in E^\times(\BA_{F })$ and $g\in \BB ^\times$, then $$i_{\overline v}(x,y)=\sum_{\delta\in B^\times} m_v(t_{1,v}^{-1}\delta t_{2,v},g_v ^{-1}) 1_{\tilde U^v}(((t_1 g )^{-1} \delta t_2 )^v).$$ Moreover, this is a finite sum. \end{lem} The proof is the same as the one of Lemma \ref{heightpullpack}, except that here we use Lemma \ref{vdetun"} and \ref{vdetun'} to show that the support of $m_v(\cdot,g_2)$ for fixed $g_2$ is contained in a compact subset of $B_v^\times$.
\begin{prop}\label{regwell'}
Under Assumption \ref{fvan}, we have
\begin{equation*} i(f)_v= \sum_{\delta\in E^\times\backslash B_\reg ^\times/E^\times} \CO_{\Xi_\infty} (\delta,f^v) i(\delta,f_v).
\end{equation*}
\end{prop}
Here $ i(\delta,f_v)$ is defined as in {iterm2}.
The proof is the same as the one of Proposition \ref{regwell}, except that here we use the condition $n>0$,
Lemma \ref{vdetun"} and \ref{vdetun'}
to show that $$\inv_{E^\times}'(\{g_1\in (B_v^\times)_\reg :m_v(g_1 ,g_2 )\neq0\}) $$ is contained in a compact subset of $F_v$ for a fixed $g_2$.
\subsubsection{Compute the multiplicity function $m_v$} \label{9.2.4}
By Lemma \ref{vdetun"} , $m(g_1,1)$ is supported on the open subgroup $ B_{v,0}:=\{g_1\in B_v^\times: \det (g_1)\in \det U_v\}$ of $B_v^\times $ (and not defined on $ E_v^\times$).
Similar to Lemma \ref{U'}, we have the following result.
\begin{lem}\label{U''}There is an open compact subgroup $U'\subset B_{v,0}$ such that
\begin{itemize}
\item[(1)] as subgroups of $E_v^\times$, $U'\cap E_v^\times=U_v\cap E_v^\times$;
\item[(2)] the function $$m (g_1,1 ) - \frac{v (\inv_{E_v^\times}'(g_1)) }{2} 1_{U'}(g_1) $$
on $ B_{v,0} - E_v^\times$
can be extended to a locally constant function on $B_{v,0} $. \end{itemize}
\end{lem}
\subsubsection{Compute $j(f)_v$} \label{9.2.5}
The following proposition is the same as Proposition \ref{nonsplitj},
\begin{prop}\label{nonsplitj'} There exists $\overline {f_v}\in C_c^\infty(B_v^\times)$ such that
$$j(f)_v= \sum_{\delta\in E^\times\backslash B_\reg^\times/E^\times}\CO _{\Xi_\infty}(\delta,f^v) \CO (\delta ,\overline {f_v}) .$$ \end{prop}
Define a set $\CV$ of irreducible components of the special fiber of $\hat \Sigma_n\hat\otimes_{\CO_{F_v}}\hat \CO_{F_v}^\ur$ as follows. Consider the morphism $\hat \Sigma_n\to \hat\Omega_v\hat\otimes\hat \CO_{F_v}^\ur $ of formal schemes over
$\CO_{F_v}$. Its base change to $\hat \CO_{F_v}^\ur$ is
\begin{equation}\hat \Sigma_n\hat\otimes_{\CO_{F_v}}\hat \CO_{F_v}^\ur\to \hat\Omega_v\hat\otimes\hat \CO_{F_v}^\ur\hat\otimes_{\CO_{F_v}}\hat \CO_{F_v}^\ur\simeq
\hat\Omega_v\hat\otimes\hat \CO_{F_v}^\ur\times\hat \BZ. \label{1134}\end{equation}
Here $\hat \BZ$ is the profinite completion of $\BZ$, and we use the canonical isomorphism $\Gal (\hat F_v^\ur/F_v)\simeq \hat \BZ$ which maps the Frobenius map to $1$. Then the Galois action of $\Gal (\hat F_v^\ur/F_v)$ is given by the addition on $\hat \BZ$.
If $E_v/F_v$ is unramified and $n=0$, then let $\CV$ be the set of all irreducible components whose images via \eqref{1134} are not points.
Otherwise, let $\CV$ be the set of all irreducible components whose images are double points. For $C\in \CV$, let $S_C\subset B_v^\times$ be the stabilizer of the corresponding irreducible component or double point. Let $n_C$ be the image of $C$ in the factor $\hat \BZ$ in
the last term of \eqref{1134}. Let $$ \CV^{\mathrm{sp}} =B^\times\backslash \CV \times \BB^{v,\times}/\tilde U^v,$$ which is a subset of the set of irreducible components of $\CN_{\overline {k(w)}}$. For $C\in \CV$, $g\in\BB^{v,\times}$, let $[C,g]$ be the corresponding element in $\CV^{\mathrm{sp}}.$
\begin{defn} For $C\in \CV$, define a function $l_{ C}$ on $ \CH_{U_v}$
as follows. For $(g_1,g_2)\in \CH_{U_v}$, let $l_{ C}(g_1,g_2)$ be the intersection number of $C$ and the image of the
point $(g_1,g_2)$ in $\hat\Sigma_n(\CO_{\overline v})$.
\end{defn}
Let $\hat z_0$ be as above Definition \ref{mf2}. Let $n_{\hat z_0} $ be the image of ${\hat z_0} $ in the factor $\hat \BZ$ in
the last term of \eqref{1134}.
Similar to Lemma \ref{9123}, we have the following lemma.
\begin{lem}\label{lcq} The function $l_C$ satisfies the following properties:
\begin{itemize}
\item[(1)] $l_{C}(g_1,g_2)\neq 0$ only if
$ v(\det (g_1)\det (g_2))=n_C-n_{\hat z_0} $ and $g_1 \in S_C$;
\item[(2)] for $b\in B_v^\times $, $h \in \CH $, $l_{bC}(bh)=l_C(h)$;
\item[(3)] $l_C$ is locally constant. \end{itemize}
\end{lem}
\begin{proof} By \cite[\S 2 Theorem]{Dridomain}, the action of $B_v^\times\times \BB_v^\times$ on the factor $\hat \BZ$ in
the last term of \eqref{1134} is the addition by $ v(\det (g_1)\det (g_2))$. Then the first part of (1) follows. The proof of the second part of (1) is similar to the proof of Lemma \ref{vdetun'}. The proof of (2)(3) are similar to the proof of Lemma \ref{9123} (2)(3). \end{proof}
Similar to Lemma \ref{heightpullpack'}, we have the following lemma.
\begin{lem}\label{11.3.10} Let $g \in( \BB ^{v})^{\times}$ and $y\in CM_U \simeq B^\times\backslash B^\times\times_{E^\times}\BB_v^\times/U_v\times \BB^{v,\times}/\tilde U^v$, the intersection of $(C,g) $ and the Zariski closure of $y $ in $\CN_U \otimes _{\CO_{H_w}}\CO_{\overline v}$ is given by
$$\sum_{\delta\in B^\times} l_C ( \delta^{-1} ,y_v)1_{\tilde U^v}(g^{-1}\delta^{-1} y^v).$$
Moreover, this is a finite sum.
\end{lem}
Let $\sum_{i=1}^n a_i[C_i,1] ,$ where $ C_i\in \CV $, be the sum of the vertical components of the $\CZ $-admissible extension of $1\in CM_U$ which are contained in $\CV^{\mathrm{sp}}$.
\begin{lem}\label{admj'} For $t \in E^\times\backslash \BA_E^\times /\tilde U_E $, $\sum_{i=1}^n a_i[t_vC_i ,t^v] $ is the vertical part of the $\CZ $-admissible extension of $t\in CM_U$ which are contained in $\CV^{\mathrm{sp}}$. \end{lem}
The proof of Lemma \ref{admj'} is similar to the one of Lemma \ref{admj}.
Another more conceptual proof is to use the $\BB^\times$-action (this proof is not available for Lemma \ref{admj} since there the restriction of $\CZ$ on the local integral model is not compatible with the $\BB^\times$-action).
Similar to Proposition \ref{regwell1}, we have the following expression of $j(f)_v$. \begin{prop}
We have
\begin{equation*} j(f)_v= \sum_{\delta\in E^\times\backslash B_\reg ^\times/E^\times} \CO _{\Xi_\infty}(\delta,f^v) j(\delta,f_v), \end{equation*}
where $ j(\delta,f_v)$ is defined by the same formula as \eqref {1129}
\end{prop}
Define a function $\overline {f_v}$ on $(B_v^\times)$ by the same formula as \eqref{barf}. By Lemma \ref{lcq} (1), $\overline {f_v}\in C_c^\infty(B_v^\times)$.
Then Proposition \ref{nonsplitj'} follows.
\subsection{Ordinary case} \label{Ordinary case}
Let $v\in |X|_s$ be split in $E$. Let $i(f)_{v_n} $ and $j(f)_{v_n}$, $n=1,2$, be as in \ref{Decomposition of the height distribution}. \begin{prop}\label{splitI}
Under Assumption \ref{fvan}, $i(f)_{v_n} =0$.
\end{prop}
\begin{prop}\label{splitj}
Under Assumption \ref{fvan}, $j(f)_{v_n} =0$.
\end{prop}
Let $w_n$ be the restrictions of $ \overline {v_n}$ to $H$. The special fiber of $\CN_{U,\CO_{H_{w_n}}}$ is described in \ref{special fibers}. Since $v$ is split in $E$, the reductions of CM points in $\CN_{U,\CO_{H_{w_n}}}$ are ordinary points. Let $Y_v=N\backslash \BB_v^{\times}/ U_v$ where $N$ is a unipotent subgroup of $\BB_v^{\times}$ and $Y^v= \BB^{v,\times}/\tilde U^v$.
Then ordinary points in $\CN_{U, \overline { k(v)}} $ are parametrized by $E^\times \backslash Y_v\times Y^v$.
Indeed, this is a special case of the discussion in \cite[10.3, 10.4]{LLS}.
The reduction map from $CM_U\simeq E^\times\backslash \BB^\times/ \tilde U$ to the set of ordinary points (see Lemma \ref{lieord}) is induced by the natural map \begin{equation} \BB_v^{\times}/ U_v\to Y_v \label{BYv}.\end{equation}
\subsubsection{Compute $i(f)_v$}
Let $\hat\CN_{U }$ be the formal completion of $\CN_{U,\hat\CO_{H_{w_n}}^\ur}$ along the special fiber. For $y\in Y_v$, let $\cD_y$ be the formal completion of $\CN_{U,\hat\CO_{H_{w_n}}^\ur}$ at $[y,1]\in E^\times \backslash Y_v\times Y^v$. For $g_v\in \BB_v^{\times}/ U_v$, let $D_{g_v} $ be the image of $g_v1^v\in CM_U$ in $\hat\CN_{U }(\CO_{\overline {v_n}}).$ If $g_v$ has image $y$ in $Y_v$ via \eqref{BYv}, then $D_{g_v}\in \cD_y(\CO_{\bar v})$. Thus we have a map \begin{equation*}\BB_v/U_v\to \coprod _{y\in Y_c}\cD_y(\CO_{\overline {v_n}}). \end{equation*}
\begin{defn} Define the multiplicity function $m_{ {v_n}}$ on $\BB_v/U_v\times \BB_v/U_v$ as follows: for $(g_1 ,g_2 )\in \BB_v/U_v\times \BB_v/U_v $, let $m_{ {v_n}}(g_1 ,g_2 )$
be the intersection number of the image of the
points $g_1$ and $g_2$ in $ \coprod _{y\in Y_c}\cD_y(\CO_{\overline {v_n}})$.
\end{defn}
Similar to Lemma \ref{heightpullpack}, we have the following lemma.
\begin{lem}\label{heightpullpacksplit} Let $x,y\in CM_U:=E^\times\backslash \BB^\times/\tilde U$ represent two distinct CM points, then
$$i_{\overline {v_n}}(x,y)= \sum_{\delta\in E^\times} m_{ {v_n}}(x_v,\delta y_v) 1_{\tilde U^v}( (x^v)^{-1}\delta y^v) .$$
Moreover, this is a finite sum.
\end{lem}
Similar to Lemma \ref{innersum}, we have the following lemma.
\begin{lem} \label{innersum'} Let $V$ be an open compact subgroup of $\BA_{F,\mathrm{f}}^\times$,
$\phi$ be a function on $ \BA_E^\times $ which is $V$-invariant and $\Xi_\infty$-invariant. Then if either side of the equation \begin{align*} & \Vol(\tilde U/\Xi_\infty) \int_{E^\times\backslash \BA_E^\times / \BA_F^\times} \sum_{t\in F^\times\backslash \BA_F^\times/\Xi_\infty U}\sum_{x\in E^\times } \phi( xtt_2)d{t_2} \\
= &\int_{ E^\times(\BA_{F,\mathrm{f}}) }\int_{ E^\times(F_\infty)/\Xi_\infty } \phi( t_{2,f} t_{2,\infty})d{t_{2,\infty}}d{t_{2,f}}
\end{align*}
converges absolutely, the other side also converges absolutely,
and in this case the equation holds.
\end{lem}
Similarly to \eqref{1118},
we have \begin{align*}i(f)_{v_n} = &\int_{ E^\times\backslash \BA_E^\times / \BA_F^\times}\int_{ \BA_E^\times /\Xi_\infty } f( (t_{1} ^{v})^{-1} t_{2}^v ) \\ & \sum _{g_v\in \BB_v^\times/U_v}f(g_v)m_{ {v_n}}(t_{2,v} , t_{1,v} g_v ) d{t_{2,\infty}}\Omega^{-1}(t_2)\Omega(t_1)d{t_{2}} d{t_1},
\end{align*}
and the right hand side is absolutely convergent. Since $v\not \in S_{s,\reg,i}$ for $i=1$ or 2, by Assumption \ref{fvan} (1), $f( (t_{1} ^{v})^{-1} t_{2}^v )=0$. Thus Proposition \ref{splitI} follows.
\subsubsection{Compute $j(f)_v$}\label{11.4.3}
By Proposition \ref{123}, we have \begin{align}\CV ^\ord\simeq F ^\times\backslash \BA_F^\times/\det (\tilde U)\times B_v\backslash \GL_2(F_v)/U_v .\label{1234}\end{align}
Let $\BB^{v,\times}$ act on $ \CV ^\ord$ via $\det$ and multiplication on the first component.
\begin{defn} For $C\in \CV^\ord$, define a function $l_{ C,n}$ on $CM_U$
as follows. For $g\in CM_U $,
let $l_{ C,n}(g)$ be the intersection number of $C$ and the Zariski closure of $g$ in $ \CN_{U,{\CO_{\overline {v_n}}}}$.
\end{defn}
\begin{lem}\label{ord1}
(1) Let $v'\neq v $, $g\in \BB_{v'}^\times$ such that $\det g\in\det U_{v'}$.
Then $l_{ C,n}=l_{g\cdot C,n}$.
(2) Let $v'\in |X|-{\mathrm{Ram}}-\{v\}$ and $g\in \BB_{v'}^\times$.
Then $$l_{C,n}(Z(g)_{U,*}t)=|\tilde U g \tilde U/ \tilde U| l_{g^{-1} C ,n} (t).$$
\end{lem}
\begin{proof}(1) follows from definition. Now we prove (2).
Let $V=U\cap gUg^{-1}$, $V'= g^{-1}Ug\cap U$.
Let $L$ be a finite extension of $H$ over which all geometrically irreducible components of $M_{V}$ (and $M_{V'}$) are defined.
Let $u_n$ be the restriction of $\overline{v_n}$ to $L$.
Define $ \CN_{V,{\CO_{L_{u_n}}}}$ in the same way that $ \CN_{U,{\CO_{H_{w_n}}}}$ is defined. Then $ \CN_{V,{\CO_{L_{u_n}}}}$ is the minimal desingularization of the normalization of $ \CN_{U,{\CO_{H_{w_n}}}}$, also $\CM_{V,{\CO_{F_v}}}$, in the function field of $M_{V,L_n}$. Let $ \pi_g$ be the natural morphism from $\CN_{V,{\CO_{H_{w_n}}}}$ to $ \CN_{U,{\CO_{H_{w_n}}}}$. By \ref{mio}, $\pi_g$ finite \etale away from supersingular components and cusps. Similar conclusions hold for $ \CN_{V',{\CO_{L_{u_n}}}}$ and the natural morphism $ \pi_1$ from $ \CN_{V',{\CO_{H_{w_n}}}}$ to $\CN_{U,{\CO_{H_{w_n}}}}.$ Let $T_g:\CN_{V,{\CO_{H_{w_n}}}}\to\CN_{V',{\CO_{H_{w_n}}}}$ be the natural extension of the isomorphism $T_g$ between the generic fibers. Then the restriction of $ \pi_{1,*}T_{g,*} \pi_g^* $ to the generic fiber is $cZ(g)_{U,*}$ for a constant $c$. Note that the Zariski closure $\overline t$ of $t$ in $ \CN_{U,{\CO_{H_{w_n}}}}$ has reduction outside the supersingular components.
Thus
\begin{equation*}l_{C,n}(Z(g)_{U,*}t)=c\pi_{1,*}T_{g,*} \pi_g^*\overline { t} \cdot C =c\overline { t} \cdot \pi_{g,*}T_{g}^*\pi_1^* C .\end{equation*}
By the formation of \eqref{1234} (see the discussion above Proposition \ref{123}), it is easy to check that the support of $ \pi_{g,*}T_{g}^*\pi_1^* C $ is $g^{-1} C.$
Since $\pi_1$ and $\pi_g$ are finite \'{e}tale and the correspondence $Z(g^{-1})$ of the generic fibers has degree $|\tilde U g \tilde U/ \tilde U|$, we have
$$ \pi_{g,*}T_{g}^*\pi_1^* C =|\tilde U g \tilde U/ \tilde U| g^{-1} C.$$ Thus (2) follows. \end{proof}
\begin{prop}\label{splitj??}
Let $v'\in S_{s,{\mathrm{ave}}}$ such that $v'\neq v$.
Then $$ \sum_{g \in U_{v'}\backslash \BB_{v'}^{\times} /U_{v'}}f_{v'}(g )l_{C,n}(Z(g )_{U,*}t)=0.$$
\end{prop}
\begin{proof}
Let $h_{-m}$ be a fixed element in $ \BB_{v'}^{-m}$.
By Lemma \ref{ord1}, for every $m\in \BZ$, we have \begin{align*} \sum_{g \in U_{v'}\backslash \BB_{v'}^{m} /U_{v'}}f_{v'}(g )l_{C,n}(Z(g )_{U,*}t)=
\left( \sum_{g \in \BB_{v'}^m /U_{v'}} f_{v'}(g )\right) l_{h_{-m}C ,n}(t)
.\end{align*} Then inner sum of the right hand side is 0 by Assumption \ref{fvan} (2). The proposition follows. \end{proof}
From the definition of $j(f)_{v_n}$, Proposition \ref{splitj} is implied by Proposition \ref{splitj??}.
\subsection{The $\infty$ place}\label{theinf}
\subsubsection{Compute hyperbolic distances}
Extend $|\cdot|_\infty$ to $\BC_\infty$ and denote this extension by $|\cdot|$.
For $z\in \Omega_\infty(\BC_\infty)$, let $|z|_i:=\inf_{a\in F} |z-a|$ be the ``imaginary part" of $z$.
The ``hyperbolic" distance between $z_1,z_2\in \Omega$ is defined to be $$d(z_1,z_2):=\frac{|z_1-z_2|^2}{|z_1|_i|z_2|_i}.$$
The following lemma is easy to check.
\begin{lem}\label{152}
(1) For $\delta =\begin{bmatrix}a&b\\ c&d\end{bmatrix} \in \GL_2(F_\infty)$, we have
$|\delta z|_i= |\det \delta||z|_i/|cz+d|^2$
(2) The ``hyperbolic" distance is $ \GL_2(F_\infty)$-invariant, i.e. for every $g\in \GL_2(F_\infty)$ and $z_1,z_2\in \Omega$, we have $d(z_1,z_2)=d(gz_1,gz_2).$
\end{lem}
Let $z$ be a fixed point of an embedding $E_\infty^\times\subset \GL_2(F_\infty)$. \begin{lem}\label{minf}
Assume that $p>2$, or assume that $p=2$ and $E_\infty/F_\infty$ is unramified, then for $\delta \in \GL_2(F_\infty)$, the following equation holds: \begin{align}d(z,\delta z)=|\inv_{E_\infty^\times} ' (\delta)|.\label{1111}\end{align}
If $p\neq 2$ and $E_\infty/F_\infty$ is a ramified extension, let $\varpi_\infty'$
be a uniformizer of $E_\infty$, then for $\delta \in \GL_2(F_\infty)$, the following equation holds: \begin{align}d(z,\delta z)=|\varpi_\infty'|^2 |\inv_{E_\infty^\times} ' (\delta)|.\label{1112}\end{align}
\end{lem}\begin{proof}
Easy to see that the truth of this lemma does not depend on the embedding.
By Lemma \ref{152} (2)
we only need to prove
the lemma for one representative in the $E_\infty^\times\times E_\infty^\times$-orbit of $\delta$. Thus we choose a representative $\delta=a+bj$ (the notation is as in \ref {matchorb})
with $a=1$ or $b=1$ to simplify the computation. Applying Lemma \ref{152} (1), a direct computation gives the lemma.
\end{proof}
\begin{defn}
For $\delta \in D^\times-E^\times$, let
$$ m_\infty(\delta)=-\frac{\log_{q_\infty} d(z,\delta z)}{2}.$$
\end{defn}
\begin{prop}\label{heightpullpackinf} Suppose $U_\infty=\BB_\infty^\times. $ Let $x,y\in CM_U\simeq E^\times\backslash \BB_{\mathrm{f}}^\times/U$ representing two distinct CM points, then
$$i_{\bar \infty }(x,y) = \sum_{\delta\in D^\times -E^\times,\ d(z_0,\delta z_0)<1} m_\infty (\delta)1_U(x^{-1}\delta y ).$$
Moreover, this is a finite sum.
\end{prop}
\begin{proof} Use \cite[Proposition 4]{Tipp}
and the argument in \cite[8.1.1]{YZZ}.
Note that $x,y$ are defined over the maximal unramified extension of $E_v$ by the condition $U_\infty=\BB_\infty^\times$.
Thus the ramification index involved in \cite[Proposition 4]{Tipp} is cancelled by the normalization in the definition of $i_{\bar \infty }$. \end{proof}
\subsubsection{Compute $i(f)_\infty$ and $j(f)_\infty$}\label{ijinfty}
We compute $i(f)_\infty$ first. The results below can be proved via mild modifications of the computations in \ref{9.2.3}, \ref{9.2.4}. Let $f_\infty=1_{U_\infty}$ for the simplicity of notations. Regard $U_\infty\cap E_\infty^\times$ as a subgroup of $D_\infty^\times$ via the embedding $E_\infty^\times\hookrightarrow D_\infty^\times$.
\begin{defn}\label{iterm3}For $\delta\in D^\times_\reg$ and a function $m_\infty$ on $D_\infty^\times- U_\infty\cap E_\infty^\times$ invariant by $\Xi_\infty$, define the arithmetic orbital integral of the function $m_\infty$ weighted by $f_\infty$ to be
$$ i (\delta,f_\infty) := \int_{ E^\times_\infty/F_\infty^\times}\int_{ E^\times_\infty/\Xi_\infty} m_\infty( t_{1,\infty} ^{-1} \delta t_{2,\infty} )
\Omega_\infty^{-1}( t_{2,\infty})\Omega_\infty(t_{1,\infty}) dt_{2,\infty}d t_{1,\infty} . $$ \end{defn} Let $D_{\infty,0}:=\{g_1\in D_\infty^\times: \det (g_1)\in \det U_\infty\}$.
\begin{prop}\label{regwellinf}Under Assumption \ref{fvan}, we have
$$i (f)_\infty =
\sum_{\delta\in E^\times\backslash D^\times_{\reg }/E^\times} \CO( \delta,f ^\infty) i (\delta,f_\infty),$$
where $i (\delta,f_\infty)$ is the arithmetic orbital integral weighted by $f_\infty$ of a function $m_\infty$ which satisfies the following properties:
\begin{itemize}
\item[(1)] the support of $m_\infty$ is contained a compact modulo center subgroup of $D_{\infty,0}$;
\item[(2)] there is an open subgroup $U'\subset D_{\infty,0}$ which is compact modulo center such that $$U'\cap E_\infty^\times=U_\infty\cap E_\infty^\times$$ as subgroups of $E_\infty^\times$;
\item[(3)] the function $$m_\infty (g ) - \frac{v_{E_\infty}(\inv_{E_\infty^\times}'(g)) }{2} 1_{U'}(g)$$
on $ D_{\infty,0} - E_\infty^\times$
can be extended to a locally constant function on $D_{\infty,0}$.
\end{itemize}
In particular, $i (\delta,f_\infty)$ is a convergent integral, and $i (f)_\infty$ is a finite sum.
\end{prop}
For $j(f)_\infty$, we have the following result which can be obtained via a mild modification of the proof of Proposition \ref{nonsplitj'}. \begin{prop} \label{155}
Under Assumption \ref{fvan} and Assumption \ref{asmpe0}, there exists $\overline {f_\infty}\in C_c^\infty(D_\infty^\times/\Xi_\infty)$ such that
\begin{equation*} j(f)_\infty= \sum_{\delta\in E^\times\backslash D_\reg ^\times/E^\times} \CO( \delta,f ^\infty) \CO_{\Xi_\infty} (\delta ,\overline {f_\infty}) . \end{equation*}
\end{prop}
\section{Arithmetic smooth matching and arithmetic fundamental lemma }
\label{amafl}
Let $F$ be a local field.
For $\Phi\in C^\infty_c(\CS)$ and $x\in F^\times-\{1\}$,
we have defined the local orbital integral $\CO(s, x,\Phi)$
and its derivative $\CO'(0, x,\Phi)$ in \ref{local orbital integrals0}.
We want to compare these derivatives with orbital integrals of the local intersection multiplicity functions on $G_{\ep'}$ weighted by functions on $G_\ep$ coming from Definition \ref{iterm1} and \ref{iterm3}. In the application to the global setting, $G_\ep$ will be $\BB_v^\times$ and $G_{\ep'}$ will
be $B(v)_v^\times$ at a place $v\in |X| $ nonsplit in $E$.
\subsection{Compute $\CO'(0, x,\Phi)$ for arithmetic fundamental lemma}
In this and next subsection, we assume that $E/F$, $\psi$ and $\Omega$ (so $\omega$) are
unramified, and $x\in \varpi {\mathrm{Nm}}(E^\times)= \inv_{T_\varpi}(G_{\varpi,\reg})$.
Then $v(x)$ is odd.
Let $$h_c=\begin{bmatrix}\varpi^c&0\\0&1\end{bmatrix},$$ which is regarded as an element in $ G_1 \simeq \GL_{2,F}$ or $G\simeq \GL_{2,E}$.
\begin{eg}\label{derlocalint} Let $\Phi=1_{K\cap\CS } $ where $K$ is the standard maximal compact subgroup of $G$.
If $v(x)>0$, then $\CO'(0,x,\Phi)=\frac{v(x)+1}{2}(-\log q^2).$
If $v(x)<0$, then $\CO'(0,x,\Phi)=0.$
\end{eg}
\begin{eg}
Let $n>0
$ and be even. Let $\phi_n$ be the characteristic function of $Kh_n K\cap \CS$.
If $v(x)<0$, then $\CO'(0,x,\phi_n)=0.$
If $v(x)>0$, then $\CO'(0,x,\phi_n)= \xi^{n/2}(n+v(x))(-\log q^2).$
\end{eg}
\begin{eg}\label{dergeneralhecke'}
Let $\Phi$ be the characteristic function of the set of matrices $g\in \CS$ with integral entries such that $v (\det g)=n$.
Then
\begin{align*}\CO'(0,x,\Phi)
&= \sum _{0\leq c<n/2}\CO'(0,x,\phi_{n-2c}) \eta\omega^{-1} (\varpi^c)+\CO'(0,x,1_K)\eta\omega^{-1} (\varpi^{n/2}) .
\end{align*}
If $v(x)<0$, then $\CO'(0,x,\Phi )=0.$
If $v(x)>0$, then $\CO'(0,x,\Phi)= \xi^{n/2}\frac{v(x)+n+1}{2}(-\log q^2).$
\end{eg}
\subsection{Arithmetic fundamental lemma} \label{arithmetic fundamental lemmas}
Fix an isomorphism $ G_{ 1}\simeq \GL_{2,F}$ which maps $K_1$ to $\GL_2(\CO_F)$.
Then
$$ G_1=\coprod_{c\in \BZ_{\geq 0}} T _1 h_c K_1 .$$
Let $T_1^\circ\subset T_1\simeq E^\times$ be $\CO_E^\times$, then
\begin{equation}T_1^\circ h_c K_1 =K _1 h_c K _1. \label{THK}
\end{equation} We define the unramified multiplicity function as follows (see Lemma \ref{unramified multiplicity function}). \begin{defn} \label{unramified multiplicity function1}
The unramified multiplicity function $m(\delta,g)$
on $G_{\varpi} \times_{T_{\varpi} \simeq T_1 }G _1 $ is nonzero
only if $\det (\delta)\det(g)\in \CO_F^\times $.
In this case, let $g\in T_1 h_cK_1 $, then
\begin{itemize}
\item[(a)] if $c=0$ then $m(\delta,g)=\frac{v(\inv_{T_{\varpi}}'(\delta))+1}{2}$;
\item[(b)] if $c>0$ then $m(\delta,g)=q^{1-c}(q+1)^{-1}.$
\end{itemize} \end{defn}
The unramified arithmetic orbital integrals are define as follows (see Definition \ref{iterm1}).
\begin{defn}
Let $f\in C_c^\infty(G_1)$.
For $\delta\in G_\varpi$, define
$$ i (\delta,f ) := \int_{ T_\varpi/Z_\varpi}\int_{ T_\varpi} \sum _{g \in G_1 /K_1}f(g) m ( t_{1 } ^{-1} \delta t_{2} , g ^{-1} ) \Omega(t_1)\Omega^{-1}(t_2)
dt_{2 }d t_{1 } . $$ \end{defn}
We have the following arithmetic fundamental lemma for the full
spherical Hecke algebra (compare with \cite{Zha12}, which is only stated for the unit in the spherical Hecke algebra).
\begin{prop}[Arithmetic fundamental lemma] \label{AFLgeneral}
Let $f\in C_c^\infty(G)$ be bi-$K$-invariant, then for all $x\in F^\times-\{1\}$ with $v(x)$ odd, we have $$ i\left( \delta(x),{\mathrm{bc}}\left(\frac{\Vol(K_{H_0}) \Vol(K_1)}{\Vol(K)}f\right)\right) \cdot (-2 \log q)=O'(0,x,\Phi_{ f}).$$
\end{prop}
By the method in \cite[Section 3]{JLR}, it is enough to prove the following proposition.
\begin{prop} \label{AFLgeneral1} Let $n\geq 0$ and be even.
Let $f$ be the characteristic function of the set of matrices $g\in G_1$ with integral entries such that $v(\det g)=n$,
$\Phi$ be the characteristic function of the set of matrices $g\in \CS$ with integral entries such that $v(\det g)=n$.
Let $x\in F^\times-\{1\}$ with $v(x)$ odd, then we have $$2 i( \delta(x), f) \cdot (- \log q)=O'(0,x,\Phi).$$
\end{prop}
\begin{proof}
The case $n=0$, i.e. $f=1_{K_1}$, $\Phi=1_{K_1\cap \CS}$.
By definition $$i( \delta(x),1_{K_1}) =\int_{ T_\varpi/Z_\varpi}\int_{ T_\varpi}
m ( t_{1} ^{-1} \delta(x) t_{2} , 1) dt_{2} dt_{1} .$$
If $v(x)<0$, then $v(\det \delta(x) )=v(x)$ is odd. Since $\det t=t\bar t$ is even for every $t\in E^\times$ as $E/F$ is unramified, $\det (t_{1} ^{-1} \delta(x) t_{2} ) \not\in \CO_F^\times$. By Definition \ref{unramified multiplicity function1}, $m ( t_{1} ^{-1} \delta(x) t_{2},1)=0$. Thus $I( 1_{K_1},\delta(x))=0.$
If $v(x)>0$, then $v(\det \delta(x) )=1$. For each $ t_{1}$, we have $$\Vol(\{t_{2}\in T_\varpi :\det ( t_{1} ^{-1} \delta(x) t_{2} )\in\CO_F^\times\}) =\Vol( \CO_E^\times).$$
So $$i( \delta(x),1_{K_1}) = \frac{v(x)+1}{2}\Vol(E^\times/F^\times)\Vol( \CO_E^\times) =\frac{v(x)+1}{2} .$$ By Example \ref{derlocalint}, we proved the proposition in this case.
The case $n>0$.
Let $f'= 1_{\varpi^{(n-c)/2} K_1 h_c K_1}$. By \eqref{THK} and Definition \ref {unramified multiplicity function1}, we have
$$i( \delta(x),f') = [ \varpi^{(n-c)/2} T_1^\circ h_c K_1:K_1]\int_{ T_\varpi/Z_\varpi}\int_{ T_\varpi}
m ( t_{1} ^{-1} \delta(x) t_{2} , (\varpi^{(n-c)/2} h_c)^{-1})\Omega(t_1)\Omega^{-1}(t_2)
dt_{2 }d t_{1 } .$$
If $v(x)<0$, $v(\det t_{1} ^{-1} \delta(x) t_{2})$ is odd. Since $n$ is even, by Definition \ref{unramified multiplicity function1}, $m ( t_{1} ^{-1} \delta(x) t_{2} , (\varpi^{(n-c)/2} h_c)^{-1}) =0$.
Let $v(x)>0$, then for every $t_1,$ we have $$\Vol\{t_2:v(\det t_{1} ^{-1} \delta(x) t_{2})=n\}=\Vol( \CO_E^\times).$$ So if $c=0$, then
$$i( \delta(x),f') =\Vol(E^\times/F^\times)\Vol( \CO_E^\times) \frac{v(x)+1}{2} \xi^{n/2}= \frac{v(x)+1}{2} \xi^{n/2};$$
if $c>0$, then
$$i( \delta(x),f') =q^{c-1}(1+q) \Vol(E^\times/F^\times)\Vol( \CO_E^\times) q^{1-c}(q+1)^{-1}\xi^{n/2}=\xi^{n/2}.$$
Thus $$i( \delta(x),f)=\sum_{0\leq c\leq n/2} i( 1_{\varpi^{(n-c)/2} K_1 h_c K_1},\delta(x))= \frac{v(x)+1+n}{2}\xi^{n/2}.$$
By Example \ref{dergeneralhecke'}, we proved the proposition in this case.
\end{proof}
\subsection{Compute $\CO'(0, x,\Phi)$ for arithmetic smooth matching}
We use Lemma \ref{intwell} to compute $\CO'(0,x,\Phi)$.
Let $ K_{l,\xi,n}, K_{l,\xi,n}'$ be as in \eqref{orbintK1st} and \eqref{orbintK'}.
\begin{eg} \label{fepm1'}Let $l, n$ be large enough such that $\Omega(1+\fp_E^n)=1$ and $\eta(-\tr(\xi)+\tr(\fp_E^l))=\eta(-\tr(\xi))$.
Then for $x\in F^\times-\{1\}$, $\CO'(0,x,1_{ K_{l,\xi,n}\cap \CS})=0$ and $\CO'(0, x,1_{ K_{ l,\xi,n}'\cap \CS})=0$ unless $v(x)\geq n++v_E(\tr(\xi))$.
In this case, we have $$\CO'(x, 1_{ K_{l,\xi,n}\cap \CS})=\eta(-x\tr(\xi))\Vol(1+\fp_E^{n })\Vol(-\tr(\xi)+\tr(\fp_E^l))( v_E(x)-v_E(\tr(\xi))\log q_E$$
and $$ \CO'(0, x,1_{ K_{ l,\xi,n}'\cap \CS})=
\eta(-\tr(\xi)) \Vol^{\times}(1+\fp_E^{n })\Vol^{\times}(-\tr(\xi)+\tr(\fp_E^l))\cdot \Omega(-1)(-v_E(\tr(\xi))\log q_E.$$ \end {eg}
\begin{lem}\label{expmat'}For $f\in C_c^\infty(G)$ given in Proposition \ref{expmat}, we can further require that for $x\in \ep'{\mathrm{Nm}} E^\times-\{1\}$, if $v_E(x)\geq 2 m+v_F(\ep)$, then
$$\CO'(0,x, \Phi_f)=\frac{1}{2}\Vol(1+\fp_E^{m })\Vol(E^\times/F^\times) ( - v_E(x)\log q_E),$$
otherwise $\CO'(0,x, \Phi_f)=0$. \end{lem} \begin{proof} Apply
the example above to the explicit constructions in Lemma \ref{822}, \ref{823}.
\end{proof}
\subsection{Arithmetic smooth matching }\label{multiplicity functions and arithmetic smooth matching}
Let $\ep,\ep'$ be two the representatives of
$F^\times/{\mathrm{Nm}}(E^\times)$ fixed in Assumption \ref{asmpep}. We summarize the properties of the multiplicity functions $m(g)=m(g, 1)$ obtained in Lemma \ref{vdetun1}, \ref{U'}, \ref{vdetun"}, \ref{vdetun'}, and \ref{U''}.
Let $U $ be an open compact subgroup of $G_\ep$, and let $G_{\ep',0}:=\{g \in G_{\ep'}: \det (g )\in \det U\}.$
\begin{defn} \label{local multiplicity function}
A special multiplicity function of level $U$ is a function $m$ on $G_{\ep'}- T_{\ep'}$ supported on
$G_{\ep',0}- T_{\ep'}$
satisfying the following conditions:
\begin{itemize}
\item[(a)] there exists an open compact subgroup $U' \subset G_{\ep',0}$
such that $U'\cap T_{\ep'}=U\cap T_\ep$ as subgroups of $E^\times\simeq T_{\ep'}\simeq T_{\ep}$, and the function $$m ( g ) -\frac{v(\inv_{T_{\ep'}}' (g)) }{2} 1_{U'}(g) $$ on $G_{\ep',0}- T_\ep$
can be extended to a locally constant function on $G_{\ep',0} $;
\item[(b)]
if $\ep'\neq1$, then $m(g)\neq 0$ only if $ \det (g) \in \det (U )$;
\item[(c)] if $\ep'=1$,
then $m(g)\neq 0$ only if
$g\in K_{\ep'} $. \end{itemize}
\end{defn}
We fix a special multiplicity function $m$ of level $U$.
The arithmetic orbital integrals of $m$ weighted by $1_U$ are defined as follows (following Definition \ref{iterm1} and \ref{iterm3}).
\begin{defn}\label{I(f,delta)} Let $\delta\in G_{\ep,\reg}$. Define
$$i( \delta,1_U) := \int_{ T_{\ep'}/Z_{\ep'}}\int_{T_{\ep'} }
m ( t_{1} ^{-1} \delta t_{2} ) \Omega^{-1}(t_2) \Omega(t_1)dt_{2}t_{1}. $$
\end{defn}
\begin{prop}[Arithmetic smooth matching for $i$-part]\label{smoothmatching}
Let $m$ be a large enough positive integer. Let
$U=K_{\ep,m}$ and $ f$ be as in Proposition \ref{expmat}
which purely matches $1_U$.
There exists $ \overline f \in C_c^\infty(G_{\ep'})$ such that for $x\in F^\times-\ep{\mathrm{Nm}}(E^\times)-\{1\}$, the following equation holds:
\begin{equation*} i( \delta(x), 1_U) \cdot (-2\log q_F ) = \CO'(0, x ,\Phi_f )+\CO ( x, \bar f ). \end{equation*}
\end{prop}
\begin{proof}
By Proposition \ref {Proposition 2.4, Proposition 3.3Jac862}, it is enough to prove the following statements:
\begin{itemize}
\item[(1)] $i( \delta(x),1_U) $ is $\Omega (a)$ times a constant for $x=\ep a \bar a$ near $\infty$;
\item[(2)] $i( \delta(x),1_U)=0 $ for $x$ near 1;
\item[(3)] $ \CO'(0,\gamma(x),f) =0$ for $x$ near $\infty$ and $x$ near 1;
\item[(4)] $ \CO'(0,\gamma(x),f)- i( \delta(x), 1_U) \cdot (-2\log q_F )$ is a constant for $x$ near $0$.
\end{itemize}
(1) follows from Definition \ref{local multiplicity function} (b) (c) and Proposition \ref{Proposition 2.4, Proposition 3.3Jac862} (4).
Here we use the fact that $G_{\ep'}/Z_{\ep'}$ is compact if $\ep'\neq 1$.
(2) is already indicated in the proof of Proposition \ref{regwell} and \ref{regwell'}.
If $\ep=1$, then $1\not \in \ep' {\mathrm{Nm}} (E^\times).$ So only need to consider the case $\ep'=1$. By Definition \ref{local multiplicity function} (c), only need to prove that $t_1^{-1}\delta(x)t_2\not \in K_{\ep'}\cap G_{\ep',0} $ for $x$ near 1. This follows from the facts that
$\inv_{T_{\ep'}}' (t_1^{-1}\delta(x)t_2)\to \infty$ for $x\to 1$ and $\inv_{T_{\ep'}}' ( K_{\ep'} ) $ is a compact subset of $F$.
(3) follows from the explicit computations in Lemma \ref{expmat'}.
(4) We first compute $ i( \delta(x),1_U)$ for $x$ near $0$.
Fix $N$ large enough such that $K_{\ep',N}\subset U'$. Let $v(x)$ large enough such that $ \delta(x) \in K_{\ep',N}$.
Then $t^{-1} \delta(x) t \in K_{\ep',N}$ for every $t\in T_{\ep'} $.
Since $t^{-1}\delta(x) s=t^{-1} \delta(x) t t^{-1}s$, it is in $U'$ if and only if $t^{-1}s\in U'$, i.e. $s\in t(U'\cap E^\times)$.
So $$\Vol\{s\in E^\times:t\delta(x) s\in U' \}=\Vol(U'\cap T_{\ep'} )=\Vol(U\cap T_{\ep'} ) =\Vol(1+\fp_E^m).$$
Also note that for $t^{-1}s\in U'\cap T_{\ep'} $, $\Omega(t^{-1}s)=1$.
Since $v(\inv_{T_{\ep'}}' (\delta(x))) =v(x)$ for $x$ near $0$,
by Definition \ref{local multiplicity function} (a) and Proposition \ref{Proposition 2.4, Proposition 3.3Jac862} (3), we have $$i( \delta(x),1_U)=\Vol (E^\times/F^\times) \Vol(1+\fp_E^m) \frac{v(x)}{2} +C$$
for $x$ near 0, where $C$ is a constant. Compared with Lemma \ref{expmat'}, (4) follows.
\end{proof}
When in the situation of \ref{theinf}, we have a modification of Proposition \ref{smoothmatching1}.
Suppose $\ep\neq 1$.
Let $U= K_{\ep,n}\varpi^\BZ$, $\Xi=U\cap Z_\ep$. Define the multiplicity function $m(g)$ by the properties given in Proposition \ref{regwellinf}. Define the arithmetic orbital integral $i( \delta,1_U)$ of $m$ weighted by $1_U$ as in Definition \ref{iterm3}.
\begin{prop}[Arithmetic smooth matching for $i$-part]\label{smoothmatching1}
Let $m$ be a large enough positive integer. Let
$U=K_{\ep,m}\varpi^\BZ$. Let $ f$ be as in Proposition \ref{expmat1}.
Then $f$ purely matches $1_U$ and
there exists $ \overline f \in C_c^\infty(G_{\ep'}/\Xi)$ such that for $x\in F^\times-\ep{\mathrm{Nm}}(E^\times)-\{1\}$, the following equation holds:
\begin{equation*} i( \delta(x), 1_U \cdot (- 2\log q _F) = \CO'(0, x ,\Phi_f )+\CO _\Xi( x, \bar f ). \end{equation*}
\end{prop}
\begin{proof} The proof is similar to the proof of Proposition \ref{smoothmatching}.
\end{proof}
\section{Proof of Theorem \ref{GZ}}\label{Proof of Theorem} In Section \ref{Height distributions, Rational Representations, and Abelian varieties}, we have reduced Theorem \ref{GZ} to Theorem
\ref{GZdis'}.
Now we choose $f_v$'s in Theorem \ref{GZdis'} as follows. Our notations are as in the situation of Theorem \ref{GZ} and
\ref{GZdis'}.
Let $S\subset |X| $ be a finite set which contains
\begin{itemize} \item[(1)] all ramified places of $\BB$, $E/F$, $\psi $, or $\pi$ and all places below ramified places of $\Omega$;
\item[(2)]
a set $S_{s,\reg} \subset |X|_s$ of cardinality $\geq 2$;
\item[(3)]
a set $S_{s,{\mathrm{ave}}} \subset |X|_s-S_{s,\reg} $ of cardinality $\geq 2$ over which $\pi $ is unramified. \end{itemize}
Here $|X|_s\subset |X|$ is the subset of places split in $E$.
The sets $S_{s,\reg}$ and $S_{s,{\mathrm{ave}}}$ are not necessary disjoint from the set of places in (1).
\begin{lem}\label{splitplacesave} For $v\in S_{s,{\mathrm{ave}}}$, there exists $f_v$ satisfying Assumption \ref{fvan} (2) such that $$\alpha^\sharp_{\pi_v}(f_v)\neq 0.$$
\end{lem}
\begin{proof}Let $U_v =\GL_2(\CO_{F_v})$ and $g=\begin{bmatrix}\varpi_v^2&0 \\ 0&1\end{bmatrix}$. Let $$f_v=(q_v^2+q_v+1)1_{\varpi _vU_v}-1_{U_vg
U_v} . $$ Suppose the Satake parameters of $\pi$ are $q_v^{-s_1}$, $q_v^{-s_2}$ where $s_1,s_2$ are purely imaginary,
and let $W_0 \in W(\pi_v,\psi_v)$ be $U_v$-invariant. Then
$$\alpha^\sharp_{\pi_v}(f_v)= ((q_v^2+q_v+1)q_v^{-s_1-s_2}-(q_v^{-s_1-s_2}+q_v^{1-2s_1}+q_v^{1-2s_2}))\Vol({U_v})\frac{\lambda_{\pi_v}(W_0)\overline {\lambda_{\pi_v} ( W_0)}}{\pair{W_0,W_0}}.$$
Since $q_v^2+q_v> 2q_v$, $\alpha^\sharp_{\pi_v}(f_v)\neq 0$.
\end{proof}
For $v\in S_{s,{\mathrm{ave}}}$, choose $f_v$ as in Lemma \ref{splitplacesave}. For $v\in S_{s,\reg}$, choose $ f_v$ as in Lemma \ref{nontrisplit}. Then $\alpha^\sharp_{\pi_v}(f_v)\neq 0$ and $f$ satisfies Assumption \ref{fvan} (1).
For $v\in |X|_s\cap S-S_{s,\reg}-S_{s,{\mathrm{ave}}}$, choose an arbitrary $f_v$ such that $\alpha^\sharp_{\pi_v}(f_v)\neq 0$.
For all $v\in |X|_s\cap S$, choose $f'_v $ as in Lemma \ref{afneq00} (1). Then $f'_v$ purely matches $f_v$.
By Lemma \ref{afneq00} (2),
$f'$ satisfies Assumption \ref{freg}.
For $v\in S -|X|_s$,
let $f_v=1_{K_{\ep,m}}$ (take $U_v=K_{\ep,m}$) and $f_v'$ its matching function in Proposition \ref{expmat} and \ref{expmat1}.
Let $m$ be large enough, such that $\det (K_{\ep,m})\subset {\mathrm{Nm}}(E_v^\times)$ and
$\alpha^\sharp_{\pi_v}(f_v)\neq 0$
(see Lemma \ref{pipieta}).
\begin{defn} Functions $f^S\in C_c^\infty((\BB^\times)^S)$ and $f'^S\in C_c^\infty (G(\BA_E^S))$ are called matching spherical functions if
\begin{itemize}
\item[(1)] for $v\in |X|-S-|X|_s $, $ f'_v $ is spherical, and $f_v$ is the matching function of $f_v'$ on $B_v^\times $ given by Proposition \ref{FLgeneral};
\item[(2)] for $v\in |X|_s-S $, $ f'_v=(f_{1,v},f_{2,v}) $ is spherical,
and $f_v$ is the matching function of $f_v'$ on $B_v ^\times$ as in \eqref{splitplaces}. \end{itemize}
\end{defn} By the computations in Section \ref{Local intersection multiplicity} (summarized in Theorem \ref{summary}), Lemma \ref{decpure}, Proposition \ref {AFLgeneral}, Proposition \ref{smoothmatching} and \ref{smoothmatching1}, we have the following theorem.
\begin{thm} [Arithmetic relative trace formula identity] \label{jacrtf'} Let $f_S$, $f'_S$ be as above. There exists $\overline {f_v}\in C_c^\infty(B(v)_v^\times)$ for $v\in S-|X|_s$ and $\overline {f_\infty}\in C_c^\infty(D_\infty^\times/\Xi_\infty)$, such that for every pair of matching spherical functions $f^S$ and $f'^S$, the following equation holds:
$$2 H(f) =\CO' (0,f' )+\sum_{v\in S-|X|_s } \CO_{\Xi_\infty}(f^v\overline {f_v}).$$
\end{thm}
Here $\CO_{\Xi_\infty}(f^v\overline {f_v})$ is as in Definition \ref{COXi} for $B=B(v)$.
Now we prove Theorem \ref{GZdis'}. \begin{proof}[Proof of Theorem \ref{GZdis'}]
Let $\sigma$ be the base-change of
$\pi$ to $G=\GL_{2,E}$.
Suppose that the Jacquet-Langlands correspondence of $\pi$ to $\GL_{2,F}$ is not of the form $\pi_\xi$ as in the end of \ref{Local-global decomposition of periods 2}.
Then $\pi\not \simeq \pi\otimes\eta$, and $\sigma$ is cuspidal.
By the condition ${\mathrm{Ram}}=\Sigma(\pi,\Omega)$ and Theorem \ref{TSlocal},
the toric period associated to the Jacquet-Langlands correspondence of $\pi$ to $B(v)^\times$ is 0 for every $v\in S-|X|_s $. Thus by \eqref{COSigma}, \eqref{COSigma'}, and Corollary \ref{Eisterm'}, we have the following spectral decomposition of the arithmetic relative trace formula identity
$$\CO' _\sigma (0, f'_S f'^S )=2(H_\pi(f_Sf^S)+H_{\pi\otimes \eta}(f_Sf^S)).$$
If $f $ is supported on $\{g\in \BB^\times :\det (g)\in {\mathrm{Nm}} (\BA_E^\times )\}$, then $H_\pi(f_Sf^S)=H_{\pi\otimes \eta}(f_Sf^S)$. So
\begin{align}\CO'_\sigma (0, f'_Sf'^S)= 4H_\pi(f_Sf^S) .\label{COHpi}\end{align}
Suppose that the Jacquet-Langlands correspondence of $\pi$ to $\GL_{2,F}$ is the representation $\pi_\xi$ as in the end of \ref{Local-global decomposition of periods 2}. Then $\pi \simeq \pi\otimes\eta$, and $\sigma =\sigma_\xi $ which is defined in \ref{Local-global decomposition of periods 2}.
By the condition ${\mathrm{Ram}}=\Sigma(\pi,\Omega)$ and Theorem \ref{TSlocal},
the toric period associated to the Jacquet-Langlands correspondence of $\pi$ to $B(v)^\times$ is 0 for every $v\in S-|X|_s $. Thus by \eqref{COSigma}, \eqref{COSigma'}, and Corollary \ref{Eisterm'}, we have the following spectral decomposition of the arithmetic relative trace formula identity \begin{align} 2H_\pi(f)= \CO_{\sigma_\xi}'(0,f')+ \CO_{\sigma_{{\xi}^{-1}\omega_E^{-1}}}'(0,f')= 2 \CO_\sigma'(0,f') . \label{COHpid}\end{align}
Now we only need to deduce the equation \eqref{GZdiseq} in Theorem \ref{GZdis'} from \eqref{COHpi} and \eqref{COHpid} in these two cases. Consider the first case. Let $$I^\sharp_{ \sigma_v}(s, \cdot):=\frac{L (1,\pi_v,\ad)}{L (1/2+s,\pi_v,\Omega_v)L (2,1_{F_v})}I _{ \sigma_v}(s,\cdot).$$ Choose $$f'^S=\frac{1_{K^S}}{\Vol(K_{H_0}^S)\Vol(K'^S)},\ f^S= \frac{ 1_{K_1^S}}{ \Vol(K'^S)}= \frac{1_{K_1^S}}{ \Vol(K_1^S)} .$$ By Proposition \ref{CO1} and \eqref{COHpi}, we have
\begin{align*} 4H_\pi(f_Sf^S)= \CO_\sigma'(0,f'_Sf'^S)= \frac{d}{d s}|_{s=0}\left(\left(\prod_{v\in S}I^\sharp_{ \sigma_v} ( s, f'_v)\right) L_S(1,\eta) \frac{L (1/2+s,\pi,\Omega)L (2,1_F)}{L (1,\pi,\ad)}\right)\end{align*}
By the condition ${\mathrm{Ram}}=\Sigma(\pi,\Omega)$ and that ${\mathrm{Ram}}$ is of odd cardinality, we have $\vep(1/2,\pi,\Omega)=-1$. Thus $L(1/2,\pi,\Omega)=0$, and
\begin{align*} 4H_\pi(f) =\left(\prod_{v\in S}I^\sharp_{ \sigma_v} ( 0, f'_v)\right) L_S(1,\eta) \frac{L' (1/2 ,\pi,\Omega)L (2,1_F)}{L (1,\pi,\ad)}.\end{align*}
By Proposition \ref{localRTF2} and \ref{expmat}, the computation of $\vep$ (or $\gamma$)-factor in
\cite[2.5]{Tat}, and the fact that the product of the local root numbers of $\eta$ is 1, we have \begin{align} 4 H_\pi(f)= \frac{L' (1/2 ,\pi,\Omega)L (2,1_F)}{L (1,\pi,\ad)} \prod_{v\in |X|} \alpha_{\pi_v}^\sharp (f_v) . \label{4H}\end{align}
Recall that
\begin{equation*}H_\pi (f) = \frac{ [F^\times\backslash \BA_F^\times/\Xi]}{\Vol(\tilde U/\Xi )\Vol(M_U)} H_\pi^\sharp (f)\end{equation*}
(see \eqref{specdecomht0}), where $\Vol(M_U)=\deg L_U. $ Similar to \cite[(4.5.1)]{YZZ}, we have the following computation of the coefficient, which was promised below Corollary \ref{the constant}. \begin{lem}\label{the constant'} We have the following equation:
$$\frac{\Vol (\tilde U/\Xi )\Vol(M_U)}{ |F^\times\backslash \BA_F^\times/ \Xi|} =4.$$ \end{lem} Thus \eqref{GZdiseq} follows from \eqref{4H}, and Theorem \ref{GZdis'} is proved in this case. For the second case, only need to replace Proposition \ref{CO1} in the above reasoning by Proposition \ref{CO1d}. \end{proof}
\begin{proof}[Proof of Lemma \ref{the constant'}]We follow the proof of \cite[(4.5.1)]{YZZ}. Since the number on the left hand side of the equation is independent of the choice of $\tilde U$, we may assume that $U_\infty =\BB_\infty^\times$ and $U\cap D^\times$ is small enough so that it acts on $\Omega$ freely.
Then $$\frac{\Vol (\tilde U/\Xi )\Vol(M_U)}{ |F^\times\backslash \BA_F^\times/ \Xi|} =\Vol(\BB_\infty^\times/F_\infty^\times)\frac{\Vol (U)\Vol(M_U)}{\Vol(\Xi_U)|F^\times\backslash \BA_{F,\mathrm{f}}^\times/ \Xi_U|} .$$
Suppose $D$ is a division algebra. Let $ D^1 $ (resp. $U^1$) be the subgroup of $D$ (resp. $U$) of elements of norm 1. Then by the formula of Serre \cite{Ser}, the degree $\kappa$ of the canonical bundle of $(U^1\cap D^1)\backslash\Omega_\infty$ is $$2\frac {q_\infty-1}{\Vol(\SL_2(\CO_{F_\infty}))}\Vol((U^1\cap D^1)\backslash D_\infty^1).$$ By the simply-connectedness of $D^1$
and the strong approximation theorem for $D^1$, we have $$\Vol((U^1\cap D^1)\backslash D_\infty^1)\Vol(U^1)=\Vol(D^1\backslash D_\infty^1U^1)=1.$$ Thus $$\kappa=2\frac {q_\infty-1}{\Vol(\SL_2(\CO_{F_\infty}))\Vol(U^1)}.$$
Note that $$\Vol(M_U)=\deg L_U=\kappa |F^\times\backslash \BA_{F,\mathrm{f}}^\times/ \det (U)|,$$
where $|F^\times\backslash \BA_{F,\mathrm{f}}^\times/ \det (U)|$ is the number of geometrically connected component of $M_U$. Combining the last two equations and the easy fact that $$\frac{\Vol (U) }{\Vol(\Xi_U)|F^\times\backslash \BA_{F,\mathrm{f}}^\times/ \Xi_U|}=\frac{\Vol (U^1) }{ |F^\times\backslash \BA_{F,\mathrm{f}}^\times/ \det (U)|},$$ we have
$$\frac{\Vol (U)\Vol(M_U)}{\Vol(\Xi_U)|F^\times\backslash \BA_{F,\mathrm{f}}^\times/ \Xi_U|} =\frac{2(q_\infty-1)}{\Vol(\SL_2(\CO_{F_\infty}))}=\frac{2(q_\infty-1)\Vol(\CO_{F_\infty}^\times)}{\Vol(\GL_2(\CO_{F_\infty}))}. $$
Let $\CO_{ \BB_\infty}$ be the maximal order of $\BB_\infty$, then we have \begin{equation*}\Vol(\BB_\infty^\times/F_\infty^\times)=2\frac{\Vol (\CO_{ \BB_\infty}^\times)}{ \Vol(\CO_{F_\infty}^\times)}.
\end{equation*} Now the lemma follows from a direct computation which says that \begin{equation*}\Vol( \CO_{\BB_\infty}^\times)= \frac{\Vol(\GL_2(\CO_{F_\infty}))}{q_\infty -1}. \end{equation*}
If $D$ is the matrix algebra, apply the explicit formula in \cite[VII, Theorem 5.11]{Gek}. \end{proof}
\part{Application} \section{The Birch and Swinnerton-Dyer conjecture}
We apply Theorem \ref{GZ} to prove the Birch and Swinnerton-Dyer conjecture in the analytic rank 1 case. Indeed, we allow abelian varieties of $\GL_2$-type and twists by characters, see Theorem \ref{BSDTE} and Corollary \ref{BSDT0}.
\subsection{Abelian varieties of strict $\GL_2$-type}\label{GLT}
Let $A$ be an abelian variety over $F$ of strict $\GL_2$-type {\cite[3.2.1]{YZZ}}, i.e. $K:=\End(A)_\BQ$ is a finite field extension of $\BQ$ of degree $\dim A$ . Then $H^1(A_{F^\sep}, \BQ_l)$, $l\neq p$, is an irreducible representation of $G_F=\Gal(F^\sep/F)$ over $K\otimes_\BQ \BQ_l$ of rank 2.
For a continuous character $\chi$ of $G_F\to K^{\chi,\times}$, where $K^\chi$ is a finite extension of $K$, let $L(s,A, \chi)$ be the twisted $L$-function valued in $K^\chi\otimes_\BQ\BC$ defined as in \eqref{ldef0} \eqref{ldef}.
Then for all embeddings $K\hookrightarrow \BC$, the corresponding components of $L(s,A, \chi)$ have the same order at $s=1$, which is defined as $\ord _{s=1}L(s,A,\chi)$.
In other words, $L(s,A, \chi)$ is defined as $P(q^{-s})$ for a rational function $P$, and $\ord _{s=1}L(s,A,\chi)$ is the multiplicity of $q^{-1}$ as a root of $P$.
We consider the following twisted Birch and Swinnerton-Dyer conjecture.
\begin{conj}\label{BSD}
We have $$\rank_{K^\chi} (A(F^\sep) \otimes_{\BZ} \chi)^{G_F}=\ord _{s=1}L(s,A,\chi).$$
\end{conj}
Results of Tate \cite{Tat1}, Milne \cite{Mil} and Schneider \cite{Sch} imply the following partial result on Conjecture \ref{BSD}. \begin{thm}\label{BSDL}
We have $$\rank_{\BQ} (A(F) \otimes_{\BZ} \BQ) \leq \ord _{s=1}L(s,A).$$
\end{thm}
\begin{rmk} When $\chi$ is the trivial character, by the work of Tate \cite{Tat1}, Milne \cite{Mil}, Schneider \cite{Sch}, Kato and Trihan \cite{KT}, the full BSD conjecture for $A$ (which gives the leading term of the $L$-functions of $A$ at s = 1, see \cite{Tat1} or \cite{KT} for the explicit formulation) follows from Conjecture \ref{BSD}.
\end{rmk}
\subsection{Twisted abelian varieties}\label{TAV}
We assume that $$\End(A)=\CO_K$$ and $$K^\chi \text{ is generated by the values of }\chi$$ without loss of generality. (Indeed, it is obvious that Conjecture \ref{BSD} holds for $A$ and $K^\chi$ if and only if it holds for an abelian varieties isogenous to $A$ over $F$ and a finite extension of $K^\chi$.)
Under these two assumptions, the twist $A^\chi$ of $A$ by $\chi$ is defined as in \cite{MRS}, which is an abelian variety over $F$ of dimension $[K^\chi:K]\cdot \dim A$ with a natural action by $\CO_{K^\chi}$ (see \cite[Corollary 1.7]{MRS}). Moreover, by \cite[Theorem 2.2]{MRS}, we have
\begin{equation} L(s,A^\chi)=L(s,A,\chi).\label{Ltwist}\end{equation} and
\begin{equation}\label{RP}\rank_{K^\chi} \left (A(F^\sep) \otimes_{\BZ} \chi\right)^{G_F} =\rank_{K^\chi} \left( A^\chi (F^\sep) \right)^{G_F} . \end{equation}
\subsubsection{Determinant character}
Let $\mu_l:G_F \to \BZ_l$ be the $l$-adic cyclotomic character.
By the class field theory and the Weil-pairing on $A$, we have the following lemma. (The analog over number fields is stated in \cite{Rib})
\begin{lem} \label{rder}
There is a character $\omega:G_F\to K^\times$ of finite order
such that
$$\det_{K\otimes_\BQ \BQ_l}H^1(A_{F^\sep}, \BQ_l)= \omega \mu_l^{-1}.$$
\end{lem} Thus, by \cite[Theorem 2.2]{MRS}, we have \begin{equation} \det_{K^\chi \otimes_\BQ \BQ_l}H^1(A^\chi_{F^\sep}, \BQ_l)= \chi ^2\omega \mu_l^{-1}.\label{Ctwist}\end{equation}
\subsubsection{A direct application of Theorem \ref{GZ}}
Let $E/F$ be a separable quadratic extension and $\Omega$ a continuous character of $\Gal(F^\sep/E)$ valued in a finite extension $K'$ of $K$.
\begin{thm}\label{BSDTE}
Assume that
$\Omega|_{\BA_F^\times}=\omega $ where $\Omega$ and $\omega$ are regarded as Hecke characters via the reciprocity maps. If $\ord _{s=1}L(s,A_E,\Omega)=1$, then the counterpart of Conjecture \ref{BSD} for $A_E$ and $\Omega$ holds, i.e.,
$$\rank_{K'} (A(F^\sep) \otimes_{\BZ} \Omega)^{\Gal(F^\sep/E)}=1.$$
\end{thm}
\begin{proof} Since $\ep(1,A_E,\Omega)=-1$, the set $S$ of places $v$ where the local root numbers are not equal to $\Omega_v(-1)$ has odd cardinality.
Let $\BB$ be the incoherent quaternion algebra over $\BF$ which ramification set $S$.
Let $\infty\in S$ and let $\varpi_\infty$ be a uniformizer of $F_\infty$.
Choose $\chi$ such that $\chi^2\omega(\varpi_\infty)=1$ (see Lemma \ref{chi}).
Enlarge $K'$ such that it contains the field $K^\chi$ generated by the values of $\chi$. (The enlargement of $K'$ does not affect the truth of Theorem \ref{BSDTE}.)
Instead of $A$ and $\Omega$, we consider $A^\chi$ and $ \chi^{-1}_E \Omega$.
Apply Theorem \ref{GZ}, and choose test vectors $\phi$ and $\varphi$ such that the right hand side of \eqref{GZeq} in Theorem \ref{GZ} is nonzero (by the assumption that $\Omega|_{\BA_F^\times}=\omega $).
Then Theorem \ref{BSDTE}
holds
by (suitable adaptations of) Theorem \ref{BSDL}, \eqref{Ltwist}, \eqref{RP} and \eqref{Ctwist}.
\end{proof}
\subsection{Theorem without base change}
Assume that there is a place of $F$ at which $A$ does not have potential good reduction.
\begin{thm}\label{BSDT}
If
$\omega=1$
and $\ord _{s=1}L(s,A)=1$, then $\rank_{K} (A(E) \otimes_{\BZ} \BQ) =1$, i.e., Conjecture \ref{BSD} holds for $\chi=1$.
\end{thm}
In particular, the (untwisted) Birch and Swinnerton-Dyer conjecture holds for every elliptic curve with analytic rank 1. More precisely, Theorem \ref{BSDT} applies if the elliptic curve is not isotrivial (see \cite[11.4.1]{Ulm}).
For isotrivial elliptic curves, the Birch and Swinnerton-Dyer conjecture holds by a result of Tate \cite{Tat2} (see \cite[Theorem 12.2]{Ulm2}).
By \eqref{Ltwist}, \eqref{RP} and \eqref{Ctwist}, we have the following corollary of Theorem \ref{BSDT}.
\begin{cor}\label{BSDT0}
If $\chi^2=\omega^{-1}$ and $\ord _{s=1}L(s,A,\chi)=1$, then Conjecture \ref{BSD} holds.
\end{cor}
\subsection{Proof of Theorem \ref{BSDT}}
Let $\omega=1$
and $\ord _{s=1}L(s,A)=1$.
\begin{lem} The curve $X$ is geometrically connected, i.e., the composition $\overline{\BF_q} F$ is a field.
\end{lem}
\begin{proof} If the lemma if not true, the morphism $X\to \BF_q$ factors through a morphism $X\to \BF_{q^n}$
for $n>1$. Then $n$ divides $\ord _{s=1}L(s,A )=1$, which is a contradiction.
\end{proof}
If we have the nonvanishing of the $L$-function of a certain quadratic twist of $A$ at 1, we may apply Theorem \ref{BSDTE} directly. (Over number fields, the existence of such a quadratic twist is known, see \cite{Wal2}.)
We only need the following weaker result, which will be proved in the next subsection.
\begin{lem} \label{tlem}
There exists a positive integer $n$ and a separable quadratic field extension $E$ of $\BF_{q^n}F$ such that $$\ord _{s=1}L\left (s,A_{E} \right)=1.$$
\end{lem}
Now we prove Theorem \ref{BSDT}. Choose $n$ and $E$ as in Lemma \ref{tlem}.
By \cite[Theorem 4.5]{MRS}, the Weil restriction of $A_E$ to $F$ is isogenous to the direct sum of $A$
and the twists $A_\sigma$ for all nontrivial irreducible rational representations $\sigma$ of $\Gal(E/F)$.
Since $\ord _{s=1}L(s,A)=1$,
$\ord _{s=1}L(s,A_\sigma)=0$ for every nontrivial $\sigma$.
By Lemma \ref{tlem} and Theorem \ref{BSDTE}, $\rank_{\BQ} (A(E) \otimes_{\BZ} \BQ) =1.$ Then by Theorem \ref{BSDL}, $\rank_{\BQ} (A( F) \otimes_{\BZ} \BQ) =1.$
\subsection{Proof of Lemma \ref{tlem}}
Let $\iota: K^\chi\hookrightarrow \bar \BQ_l$ be an embedding, and let
\begin{equation}\rho:=H^1(A _{F^\sep}, \BQ_l)\otimes _{K \otimes \BQ_l,\iota} \bar \BQ_l.\label{riot}\end{equation}
We look for
a positive integer $n$ and a separable quadratic extension $E$ of $\BF_{q^n}F$ such that $$\ord _{s=1}L\left (s,\rho_{\BF_{q^n}F} \right)=1,\ L\left(1,\rho_{\BF_{q^n}F},\eta \right)\neq 0,$$
where $\eta$ is the quadratic Hecke character of $( \BF_{q^n}F)^\times$ associated to the quadratic extension $E/\BF_{q^n}F$.
\begin{lem} \label{verygood} Given a finite set $S$ of places and a quadratic character $\eta_v$ of $F_v^\times$ for each $v\in S$, there exists a quadratic Hecke character $\eta_0$ of $ F^\times$ such that $\ep(1,\rho,\eta_0)=1 $ and $\eta_{0,v}=\eta_v$ for $v\in S$. \end{lem} \begin{proof}By the Langlands correspondence, we consider the automorphic side. If $p\neq 2$, the lemma follows from (the proof of) \cite[Lemma 41]{Wal2}. We modify the proof of \cite[Lemma 41]{Wal2} in the case $p=2$ as follows. We only need to modify the proof of \cite[Proposition 16, b)]{Wal2}, which states that for a discrete series representation $\pi$ of $\PGL_2(C)$, where $C$ is a nonarchimedean local field, there exists a quadratic character $\chi$ of $C^\times$ such that $\ep(1/2, \pi\otimes \chi)=-\chi(-1)\ep(1/2, \pi) $.
The proof of \cite[Proposition 16, b)]{Wal2} involves taking sum over all quadratic characters $\chi$ of $C^\times$, which is not valid in characteristic 2. In characteristic 2, choose an open compact subgroup $U$ of $C^\times$ such that the set $Q$ of all quadratic characters of $C^\times/U$ has cardinality $|Q|>2$. Replacing the sum over all quadratic characters $\chi$ in \cite[Proof of Proposition 16, (1)]{Wal2} by the
sum of the characters in $Q$, we get the characteristic function of $UC^{\times,2}$ multiplied by $|Q|$. In the rest of the proof of \cite[Proposition 16, b)]{Wal2}, replace the kernel of the reduced norm map on the unit group of the division quaternion algebra over $C$ by the preimage of $U$ under the reduced norm map. Then \cite[Proposition 16, b)]{Wal2} follows. \end{proof}
\begin{rmk}\label{notgod} In the proof of Lemma \ref{verygood}, we used the place, say $v$, of $F$ at which $A$ does not have potential good reduction. Indeed, the local Langlands correspondence of $\rho_v$ is a discrete series representation. Moreover, this condition is necessary, see \cite[Lemma 41]{Wal2}. \end{rmk}
By the argument in \cite[11.3.1]{Ulm1}, there exists a positive integer $N$ such that for every $n$ coprime to $N$, $\ord _{s=1}L\left(s,\rho_{\BF_{q^n}F} \right)=1$.
When $p$ is odd and certain tame conditions are satisfied, the lemma follows from the results of Katz on twisted $L$-functions \cite{Kat1} as follows. Let $V$ be the moduli variety of functions on $X$ which satisfy the local conditions in \cite[5.0, 6.0, 6.1]{Kat1}. Points in $V(\BF_{q^n})$ give quadratic Hecke characters of $ (\BF_{q^n}F)^\times$ corresponding to separable quadratic extensions of $\BF_{q^n}F$ defined by the functions via the Kummer extension. And there is a lisse $\overline{\BQ_l}$-sheaf $\CG$ on $V$ such that the characteristic polynomials of the stalks give the $L$-functions of quadratic twists of $\rho_{\BF_{q^n}F}$. By Lemma \ref{verygood}, there exists a point in $V(\BF_q)$ corresponding to a quadratic Hecke character $\eta_0$ of $F^\times$ such that $\ep(1,\rho,\eta_0)=1 $. Then the monodromy of $\CG$ can only be the first two possibilities in \cite[8.3.2]{Kat1} by \cite[Theorem 8.3.8]{Kat1} about the emptiness of $X_{{\sign}+}$ in the case of the third possibility. Then Lemma \ref{tlem} follows from an application of Deligne's equidistribution theorem \cite[Theorem 8.3.2, Theorem 8.3.6]{Kat1}. Without the tame conditions, the monodromy of $\CG$ was computed in \cite[Theorem 8.2.1]{Kat2} using higher moments.
When $p=2$, in \cite[Chapter 8]{Kat2}, replacing the Kummer sheaf by the Artin-Schreier sheaf, the analog of \cite[Theorem 8.2.1]{Kat2} holds by the same proof. Moreover, Lemma \ref{verygood} and \cite[Theorem 8.3.8]{Kat1} do not depend on the tameness of $\rho$ or the characteristic $p$. Thus the same proof carries on to conclude Lemma \ref{tlem}.
\section*{Conflict of interest}
On behalf of all authors, the corresponding author states that there is no conflict of interest.
\section*{Data Availability Statement}
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
\end{document} | arXiv |
\begin{document}
\title{Fano hypersurfaces with no finite order birational automorphisms}
\thispagestyle{empty}
\begin{abstract} We use the specialization homomorphism for the birational automorphism group to study finite order birational automorphisms. For a family of varieties over a DVR, we prove that a birational automorphism of order coprime to the residue characteristic cannot specialize to the identity. As an application, we show that very general $n$-dimensional hypersurfaces of degree $d \geq 5 \lceil(n+3)/6 \rceil$ have no finite order birational automorphisms. \end{abstract}
The birational automorphism group of a variety $X$---denoted ${\mathrm{Bir}}(X)$---is one of the most natural birational invariants associated to $X$. For $X = {\mathbb{P}}^n_{{\mathbb C}}$, the \textit{Cremona group} ${\mathrm{Cr}}_n({\mathbb C}) = {\mathrm{Bir}}({\mathbb{P}}^n_{{\mathbb C}})$ is an object of classical and modern interest, and it is extremely interesting and complicated when $n\ge 2$. Beyond the case of projective space, it is natural to study the birational automorphism group of a smooth degree $d$ hypersurface $X\subset {\mathbb{P}}^{n+1}_{{\mathbb C}}$. In the general type case, if $n \geq 2$ and $d\ge n+3$, then $K_{X}$ is ample and Matsumura \cite{Matsumura63} showed that ${\mathrm{Bir}}(X)$ is equal to the automorphism group ${\mathrm{Aut}}(X)$. If $d=n+2$, in which case $X$ is Calabi--Yau (with Picard rank \(1\) if \(n \geq 3\)), then again \({\mathrm{Bir}}(X)={\mathrm{Aut}}(X)\) \cite{MatsusakaMumford64} (see also \cite[Lem A.1]{LS22}). However, if $d\le n+1$, in which case $X$ is Fano, very little is known about birational automorphisms in general once $n\ge 4$.
The most striking known result is the case of degree $d=n+1$ Fano hypersurfaces. To briefly summarize, there has been a great deal of work by many authors---including Fano, Segre, Iskovskikh, Manin, Pukhlikov, Corti, Cheltsov, de Fernex, Ein, Musta\c{t}\u{a}, and Zhuang---to show that if $n\ge 3$ and $d=n+1$, then any such smooth $X$ is birationally superrigid. As a consequence of their work, ${\mathrm{Bir}}(X) = {\mathrm{Aut}}(X)$ (see \cite{Kollar19} for a survey of the main ideas that were developed over time). In the case $d=n$, Pukhlikov used similar techniques to show that such hypersurfaces also satisfy ${\mathrm{Bir}}(X) = {\mathrm{Aut}}(X)$ once $n\ge 14$ \cite[Cor. 1]{Pukhlikov-index2}. For a smooth hypersurface $X$, having ${\mathrm{Bir}}(X) = {\mathrm{Aut}}(X)$ places strong constraints on the groups, as shown by Matsumura and Monsky \cite[Thm. 2 and Thm. 5]{MM63}: (1) if $n \geq 2$ and $d \geq 3$ (excluding the case $(n, d) = (2, 4)$), then ${\mathrm{Aut}}(X)$ is naturally identified with a finite subgroup of ${\mathrm{Aut}}({\mathbb{P}}^{n+1}_{{\mathbb C}}) = {\mathrm{PGL}}_{n+2}({\mathbb C})$, and (2) if $n \geq 2$, $d \geq 3$, and $X$ is very general, then ${\mathrm{Aut}}(X)$ is trivial. There seem to be few known restrictions on \({\mathrm{Bir}}\) when $d<n$.
We first prove a result about specializing finite order elements in the birational automorphism group (Proposition~\ref{prop:specialization-order-l}). For a family of varieties over a complex curve, this shows that a nontrivial finite order birational automorphism cannot specialize to the identity on the central fiber. We apply our result to hypersurfaces, but we believe that this specialization method will also be useful for studying the birational automorphism groups of other varieties.
By degenerating to a reducible hypersurface, our result will imply that if a very general non-ruled degree \(d\) hypersurface has no \(p\)-torsion in its birational automorphism group, then the same also holds in degree \(d+1\). By degenerating to positive characteristic---following the work of Koll\'{a}r \cite{Koll'ar-hypersurfaces}---we can control the torsion in ${\mathrm{Bir}}(X)$ for certain hypersurfaces in the Fano range.
\begin{Lthm}\label{thm:torsion-p^e} Let \(p\) be a prime and let \(n\) and \(d\) be integers; if $p=2$ further assume that $n$ is even. Let \(X\subset\mathbb P^{n+1}_{\mathbb C}\) be a very general hypersurface. If $d \geq p\left\lceil\frac{n+3}{p+1}\right\rceil$, then any finite order element in \({\mathrm{Bir}}(X)\) has order \(p^r\) for some \(r\). \end{Lthm} \noindent When $d\ge n+2$, Theorem~\ref{thm:torsion-p^e} is well known as $K_X$ is ample or trivial. When $d=n+1$, or when \(d=n\) and \(n\geq 14\), we use the known results on index one and two Fano hypersurfaces. So our contribution to Theorem~\ref{thm:torsion-p^e} is for Fano hypersurfaces of degree $d\le n$.
We achieve the largest range of degrees in which we can apply Theorem~\ref{thm:torsion-p^e} by choosing the smallest primes. This gives the following corollary.
\begin{Lcor}\label{cor:no-torsion} Let \(X\subset\mathbb P^{n+1}_{\mathbb C}\) be a very general degree \(d\) hypersurface. If either \begin{enumerate} \item \(d\geq 3\lceil\frac{n+3}{4}\rceil\) and \(n\) is even, or \item \(d\geq 5\lceil\frac{n+3}{6}\rceil\) and \(n\) is odd, \end{enumerate} Then \({\mathrm{Bir}} (X)\) has no elements of finite order. \end{Lcor}
The table below places our results in the context of previous work for some values of $(n, d)$. The number \(2\) means that any finite order element in \({\mathrm{Bir}}(X)\) has order a power of \(2\) (possibly order \(1\)) as a result of Theorem~\ref{thm:torsion-p^e}, and similarly for $3$.
\begin{table}[ht] \tiny \renewcommand{0.7}{0.7}
\begin{tabularx}{15cm}{c|YYYYYYYYYYYYYYYYYYYYYY} \tabsymbol{\small \emph{d}} & & & & & & & & & & & & & & & & & & & & & & \\
\textbf{21} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & $\ast$ & $\blacksquare$ & 3 & $\blacksquare$ \\
\textbf{20} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & $\ast$ & $\blacksquare$ & 2 & & 2 \\
\textbf{19} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & $\ast$ & $\blacksquare$ & 3 & 2 & & 2 \\
\textbf{18} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & $\ast$ & 3 & $\blacksquare$ & 3 & 2 & & 2 \\
\textbf{17} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & $\ast$ & 2 & & 2 & & & & \\
\textbf{16} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & $\ast$ & 3 & 2 & & 2 & & & & \\
\textbf{15} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & $\ast$ & $\blacksquare$ & 3 & 2 & & & & & & \\
\textbf{14} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & $\ast$ & & 2 & & 2 & & & & & & \\
\textbf{13} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & 3 & 2 & & & & & & & & & & \\
\textbf{12} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & $\blacksquare$ & 3 & 2 & & & & & & & & & & \\
\textbf{11} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & & 2 & & & & & & & & & & & & \\
\textbf{10} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & 2 & & 2 & & & & & & & & & & & & \\
\textbf{9} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & 3 & & & & & & & & & & & & & & & \\
\textbf{8} & $\cdot$ & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & 2 & & & & & & & & & & & & & & & & \\
\textbf{7} & $\cdot$ & $\cdot$ & $\cdot$ & $\diamond$ & & & & & & & & & & & & & & & & & & \\
\textbf{6} & $\cdot$ & $\cdot$ & $\diamond$ & 2 & & & & & & & & & & & & & & & & & & \\
\textbf{5} & $\cdot$ & $\diamond$ & & & & & & & & & & & & & & & & & & & & \\
\textbf{4} & $\diamond$ & & & & & & & & & & & & & & & & & & & & & \smash{\raisebox{-4pt}{\rlap{\qquad \small \emph{n}}}} \\ \hline \rule{0pt}{\normalbaselineskip} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} & \textbf{10} & \textbf{11} & \textbf{12} & \textbf{13} & \textbf{14} & \textbf{15} & \textbf{16} & \textbf{17} & \textbf{18} & \textbf{19} & \textbf{20} & \textbf{21} & \textbf{22} & \textbf{23} & \textbf{24} \end{tabularx} \caption*{$\blacksquare$ = Corollary~\ref{cor:no-torsion} \quad $\cdot$ = \text{non-Fano} \quad $\diamond$ = \text{Fano index } 1 \quad $\ast$ = \text{Pukhlikov's Fano index 2 results}} \end{table}
\noindent Restricting the possible orders of torsion elements places strong restrictions on the birational automorphism group. Since the Cremona group contains $p$-torsion for any $p$, Theorem~\ref{thm:torsion-p^e} with \(p=2\) if \(n\) is even and \(p=3\) if \(n\) is odd implies that \({\mathrm{Bir}}(X)\not\cong{\mathrm{Cr}}_n({\mathbb C})\) if \(d\geq 2\lceil\frac{n+3}{3}\rceil\) for \(n\) even and \(d\geq 3\lceil\frac{n+3}{4}\rceil\) for \(n\) odd. Remarkably, Cantat proved ${\mathrm{Bir}}(X)\not\cong{\mathrm{Cr}}_n({\mathbb C})$ whenever $X$ is \emph{any} irrational variety \cite[Thm.~C]{Cantat14}.
\begin{remark} The parity assumption on $n$ in Theorem~\ref{thm:torsion-p^e} comes from studying the singularities of odd dimensional double covers of hypersurfaces in characteristic \(2\) \cite[Thm.~C]{ChenStapleton-rational-endomorphisms}. At the moment, we cannot give an explicit resolution in this case. \end{remark}
In light of Corollary~\ref{cor:no-torsion}, which shows that \({\mathrm{Bir}}(X)\) contains no finite order elements, one might wonder how far apart this is from showing that \({\mathrm{Bir}}(X)=\{1\}\). (Recall that \({\mathrm{Aut}}(X)=\{1\}\) for these hypersurfaces.) There are a number of related works in this vein. Any finite order element of \({\mathrm{Bir}}(X)\) is regularizable, i.e. it is equivalent to a regular automorphism on a birational model of \(X\). For surfaces and for birationally rigid Fano threefolds, the regularizable automorphisms generate the birational automorphism group (e.g. \({\mathrm{Cr}}_2({\mathbb C})\) is generated by \({\mathrm{Aut}}({\mathbb{P}}^2)={\mathrm{PGL}}_3({\mathbb C})\) and the Cremona involution). Cheltsov has asked whether this holds in general \cite[Conj.~1.12]{Cheltsov04}. Recently, Lin and Shinder \cite{LS22} proved that this is false by showing that for \(n\geq 3\), \({\mathrm{Cr}}_n({\mathbb C})\) is not generated by (pseudo-)regularizable elements.
Throughout the paper we consider ${\mathrm{Bir}}$ as a group, not as a group scheme. However, for non-uniruled varieties Hanamura has several results on giving ${\mathrm{Bir}}$ a scheme structure \cite{Hanamura87, Hanamura88}.
\noindent\textbf{Notation.} $R$ will denote a DVR with field of fractions $K={\mathrm{Frac} \ } R$ and residue field $k$. We will write \(\eta\) for the generic point of ${\mathrm{Spec} \ } R$ and \(0\) for the closed point.
\noindent\textbf{Outline.} Let $X$ be a family over $R$ and let $Z \subset X_{0}$ be a component of the special fiber. In $\S 1$, we first identify a subgroup $\Xi_{\eta}(Z) \subset {\mathrm{Bir}}_K(X_{\eta})$, consisting of the birational automorphisms of the generic fiber $X_{\eta}$ that "specialize". We construct a specialization homomorphism \[ {\mathrm{sp}}_\eta \colon \Xi_{\eta}(Z) \rightarrow {\mathrm{Bir}}_{k}(Z). \] Next, we study torsion in the birational automorphism group in $\S 2$ and show that if $\ell$ is a positive integer that is invertible in $R$, then the kernel of the specialization map cannot contain birational automorphisms $\phi \in \Xi_{\eta}(Z)$ of order $\ell$ (see Proposition~\ref{prop:specialization-order-l}\eqref{item:mu_l-nontrivial-on-special-fiber}). In $\S 3$ we degenerate to characteristic $p > 0$ and take advantage of some nice properties that are satisfied by the special fiber $X_{0}$ to show that $\Xi_{\eta}(Z)$ coincides with ${\mathrm{Bir}}(X_{\eta})$. In particular, we use the fact (building on work of Koll\'{a}r \cite{Koll'ar-hypersurfaces} and of the first and third authors \cite{ChenStapleton-finite-BirX}) that certain $p$-cyclic covers in characteristic $p$ have no birational automorphisms. This is finally applied to families of hypersurfaces to prove Theorem~\ref{thm:torsion-p^e} and Corollary~\ref{cor:no-torsion}.
\noindent\textbf{Acknowledgements.} We are grateful to J\'er\'emy Blanc, Michel Brion, Serge Cantat, J\'anos Koll\'ar, Davesh Maulik, Aleksandr Pukhlikov, Evgeny Shinder, Burt Totaro, Ziquan Zhuang, and Susanna Zimmermann for helpful conversations. The first author would like to thank Professor Pietro Pirola and the University of Pavia for the opportunity to visit and their warm hospitality, during which parts of this paper were drafted.
\section{The specialization homomorphism for \texorpdfstring{${\mathrm{Bir}}$}{\texttwoinferior}}\label{sec:Specialization}
The specialization homomorphism was first defined by Matsusaka and Mumford (who attribute it to Artin) \cite{MatsusakaMumford64}, and it has also appeared in the literature for surfaces \cite[\S 3.1]{Persson77} \cite[\S 2]{LieblichMaulik18}. To our knowledge, it has not previously been applied to systematically study birational automorphisms.
\begin{definition}[{\cite[Thm I]{MatsusakaMumford64}}]\label{defn:specializes} Let \(X_R\) be an integral flat separated scheme over \(R\), and let \(Z\subset X_0\) be a reduced irreducible component that appears with multiplicity one in the central fiber. Let \(\phi\in{\mathrm{Bir}}_K(X_\eta)\) be a birational automorphism, and let \(\Gamma\subset X_R\times_R X_R\) be the closure of the graph of \(\phi\). We say \(\phi\) \emph{specializes to \(Z\)} if the special fiber \(\Gamma_0\) has a unique component that maps birationally to \(Z\) under both projections. \end{definition}
\begin{example} In the ruled setting, a birational automorphism of \(X_\eta\) need not specialize. For the automorphism \(x\mapsto\tfrac{t}{x}\) on the generic fiber of the constant family \({\mathbb{P}}^1_x\times\mathbb A^1_t\to\mathbb A^1_t\), the special fiber \(\Gamma_0\) has two irreducible components, each of which is contracted under one of the projections. \end{example}
\begin{definition}[{\cite[Def. 1.1, Def. 1.5]{ChenStapleton-rational-endomorphisms}}]\label{defn:sustained-modifications} A normal scheme \(X\) has \emph{(separably uni-)ruled modifications} if every exceptional divisor of every normal birational modification \(Y\to X\) is (separably uni-)ruled. A normal scheme $X_R$ has \textit{sustained (separably uni-) ruled modifications} if there exists a generically finite extension of DVRs $R\subset R'$ such that for every generically finite extension of DVRs $R'\subset S$, the normalization of $X_{S}$ has (separably uni-)ruled modifications. Here we fix an algebraic closure of \(K\), and the ring extension \(R\subset R'\) being generically finite means that \({\mathrm{Frac} \ } R'\) is a finite algebraic extension of \(K\). \end{definition}
\begin{proposition}[The specialization homomorphism]\label{prop:specialization-hom} Let $X_R$ be an integral flat separated scheme over \(R\) and \(Z\subset X_0\) a reduced irreducible component appearing with coefficient one in the special fiber. \begin{enumerate} \item\label{item:find-open} If $\phi$ is a birational automorphism of $X_\eta$ that specializes to $Z$, then there are open sets $U_1, U_2\subset X_R$ such that each \(U_i\) meets $Z$, $\phi$ gives an isomorphism between $U_1$ and $U_2$, and the restriction of $\phi$ to $X_{0}$ is an isomorphism: \[
\phi|_{X_0 \cap U_1} \colon Z\cap U_1\cong Z\cap U_2. \] \item\label{item:specialization-homomorphism} The set of birational automorphisms that specialize to \(Z\) forms a subgroup of \({\mathrm{Bir}}_K(X_\eta)\), which we denote \(\Xi_\eta(Z)\). There is a specialization group homomorphism: \[ {\mathrm{sp}}_\eta \colon \Xi_\eta(Z) {\rightarrow} {\mathrm{Bir}}_k(Z). \] \item\label{item:geometric-specialization} Assume \(X_\eta\) and \(Z\) are geometrically integral over \(K\) and \(k\), respectively. The group \(\Xi_{\overline{\eta}}(Z_{\overline{k}})\) is the colimit of \(\Xi_{\eta'}(Z_{k'})\) over generically finite extensions \(R\subset R'\) of DVRs; thus, there is an induced specialization homomorphism: \[ {\mathrm{sp}}_{\overline{\eta}} \colon \Xi_{\overline{\eta}}(Z_{\overline{k}}){\rightarrow} {\mathrm{Bir}}_{\overline{k}}(Z_{\overline{k}}). \] \item\label{item:Bir-specializes-nonruled} \cite[IV Ex.~1.17.3]{Koll'ar-rational-curves} Assume that \(X_R\) is proper and has (separably uni-)ruled modifications, and that $Z$ is the unique irreducible component of \(X_0\) that is not (separably uni-)ruled. Then every birational automorphism of \(X_\eta\) specializes to \(Z\). That is, \(\Xi_\eta(Z)={\mathrm{Bir}}_K(X_\eta)\). \item\label{item:specialization-sustained} In the setting of \eqref{item:Bir-specializes-nonruled}, assume furthermore that $X_R$ has sustained (separably uni-)ruled modifications; that \(X_\eta\) and \(Z\) are geometrically integral over $K$ and $k$, respectively; and that \(Z_{\overline{k}}\) is not (separably uni-)ruled over \(\overline{k}\). Let $R\subset R''$ be a generically finite extension of DVRs. Then \({\mathrm{Bir}}(X_\eta)\) is a subgroup of \({\mathrm{Bir}}(X_{\eta''})\), and there is a further generically finite extension \(R''\subset R'\) of DVRs such that the diagram commutes: \[ \begin{tikzcd} {\mathrm{Bir}}_K(X_\eta)\arrow[d]\arrow[r,"{\mathrm{sp}}_\eta"]& {\mathrm{Bir}}_k(Z)\arrow[d]\\ {\mathrm{Bir}}_{K'}(X_{\eta'})\arrow[r,"{\mathrm{sp}}_{\eta'}"]& {\mathrm{Bir}}_{k'}(Z_{k'}). \end{tikzcd} \] In particular, there is a homomorphism \[ {\mathrm{sp}}_{\overline{\eta}} \colon {\mathrm{Bir}}_{\overline{K}}(X_{\overline{\eta}}){\rightarrow} {\mathrm{Bir}}_{\overline{k}}(Z_{\overline{k}}). \] \end{enumerate} \end{proposition}
\begin{proof}
Let \(\tilde{\phi}\colon X_R\dashrightarrow X_R\) be the birational map over \(R\) obtained from the closure \(\Gamma\subset X_R\times_R X_R\) of the graph of \(\phi\). Let \(\Gamma_0'\) be the unique component of \(\Gamma_0\) mapping birationally to \(Z\) under both projections, \(U_1\subset X_R\) the largest open subset on which \(\tilde{\phi}\) is an isomorphism, and \(U_2=\tilde{\phi}(U_1)\). Each \(U_i\) meets \(Z\) by maximality, and \(\phi_0=\tilde{\phi}|_{Z}\) as rational maps. This proves~\eqref{item:find-open}.
For~\eqref{item:specialization-homomorphism}, it is clear that the identity on \(X_\eta\) specializes to the identity on \(Z\). If \(\phi\) specializes to \(Z\), then so does \(\phi^{-1}\) by exchanging the first and second projections. It remains to show that if \(\phi\) and \(\psi\) specialize to \(Z\), then so does \(\psi\circ\phi\), and that the specialization of the composition is the composition of the specializations. For this, let \(\phi,\psi\in\Xi_\eta(Z)\), and let \({\tilde{\phi}}|_{U_{1,\tilde{\phi}}}\colon U_{1,\tilde{\phi}}\to U_{2,\tilde{\phi}}\) and \({\tilde{\psi}}|_{U_{1,\tilde{\psi}}}\colon U_{1,\tilde{\psi}}\to U_{2,\tilde{\psi}}\) be morphisms defined on the largest open subsets on which \({\tilde{\phi}}\) and \({\tilde{\psi}}\), respectively, induce isomorphisms. Let \(U_2=U_{2,\tilde{\phi}}\cap U_{1,\tilde{\psi}}, U_1={\tilde{\phi}}^{-1}(U_2),\) and \(U_3={\tilde{\psi}}(U_2)\). Then each \(U_i\cap Z\neq\emptyset\), so the assertion follows from the fact that \({\tilde{\psi}}|_{U_2\cap Z}\circ{\tilde{\phi}}|_{U_1\cap Z}=({\tilde{\psi}}\circ{\tilde{\phi}})|_{U_1\cap Z}\).
For \eqref{item:geometric-specialization}, if \(R\subset R'\) is a generically finite extension of DVRs, then \(X_{\eta'}\) and \(Z'\coloneqq Z\otimes_k k'\) are both integral and \(Z'\) has coefficient one in the special fiber, so they satisfy the assumptions in Definition~\ref{defn:specializes}. If \(\Gamma\) is the closure of the graph of an element of \(\Xi_\eta(Z)\), then by assumption it has a unique component mapping birationally to \(Z\) under the projections. Therefore, the base change to \(k'\) gives a component of the special fiber of \(\Gamma\otimes_R R'\) birational to \(Z_{k'}\) under the projections, and there is a unique such component since \(\Gamma\otimes_R R'\to R'\) is flat. This proves that \(\Xi_\eta(Z)\) is a subgroup of \(\Xi_{\eta'}(Z')\).
Before showing \eqref{item:Bir-specializes-nonruled}, first suppose that \(Y_R\) and \(Y'_R\) are flat integral schemes over \({\mathrm{Spec} \ } (R)\) such that \(Y_R\) has (separably uni-)ruled modifications, every non-(separably uni-)ruled component of \(Y_0\) appears with coefficient one in \(Y_0\), \(Y'_R\) is proper, and \(Y'_0\) has a unique irreducible component \(Z'\) that is not (separably uni-)ruled. Then any birational map \(\phi\colon Y_\eta\dashrightarrow Y'_{\eta}\) induces a birational map \(\phi_0\colon Z\dashrightarrow Z'\) from some component \(Z=Z_{\phi}\) of \(Y_{0}\) that is not (separably uni-)ruled (c.f. \cite[IV Ex.~1.17]{Koll'ar-rational-curves}). For this claim, first observe that the assumption on the coefficients of \(Y_0\) implies that the local ring at the generic point of every non-(separably uni-)ruled component of \(Y_0\) is a DVR. Now let \(\Gamma\) be the closure of the graph of \(\phi\) in \(Y_{R}\times_R Y'_{R}\), and let \(\Gamma_0'\) be the unique component of \(\Gamma_0\) mapping birationally to \(Z'\). Since \(Y_{R}\) has (separably uni-)ruled modifications and \(Z'\) is not (separably uni-)ruled, then \(\Gamma_0'\) maps birationally to a component \(Z\) of \(Y_{0}\), so the composition \(\phi_0 \colon Z\dashrightarrow\Gamma_0'\dashrightarrow Z'\) is a birational map.
We will now apply this to $X_R = Y_R =X'_{R}$ and $Z = Z'$ to prove \eqref{item:Bir-specializes-nonruled}. Let \(U_1\subset X\) be the largest open subset on which \(\tilde{\phi}\) is an isomorphism, and let \(U_2=\tilde{\phi}(U_1)\). Note that \(\tilde{\phi}^{-1}(U_2\cap X_{0})=U_1\cap X_{0}\), so each \(U_i\) meets \(Z\) by maximality, and \(\phi_0=\tilde{\phi}|_{Z}\) as rational maps.
For \eqref{item:specialization-sustained}, let \(R\subset R'\) be as in Definition~\ref{defn:sustained-modifications}. After replacing \(R'\) by a localization of its integral closure in \(K'\otimes_K K''\) we may assume \(R\subset R''\subset R'\). Then \(X_{\eta'}\) and \(Z'\coloneqq Z_{k'}\) are integral, and \(Z'\) appears with coefficient one in the central fiber of \(X_{R'}\), so the local ring of \(X_{R'}\) at the generic point of \(Z'\) is a DVR. Thus, the normalization \(X_{R'}^\nu\to X_{R'}\) is an isomorphism at the generic point of \(Z'\), so on the special fiber there is a component \(W\) mapping birationally to \(Z'\). Now we apply \eqref{item:Bir-specializes-nonruled} to obtain a specialization map \[{\mathrm{Bir}}_{K'}(X_{\eta'})={\mathrm{Bir}}_{K'}(X_{\eta'}^\nu) \to {\mathrm{Bir}}_{k'}(W) = {\mathrm{Bir}}_{k'}(Z').\] \eqref{item:specialization-sustained} then follows from \eqref{item:geometric-specialization}. \end{proof}
Let \(X_R\) be a family of smooth proper varieties. The previous proposition describes how to specialize birational automorphisms, and one may wonder what the image of the subgroup \({\mathrm{Aut}}_K(X_\eta)\cap\Xi_\eta(Z) \subset {\mathrm{Bir}}_K(X_\eta)\) is in \({\mathrm{Bir}}_k(X_0)\). Let \(\phi \in {\mathrm{Aut}}_K (X_\eta) \cap \Xi_\eta(Z)\). If there is an ample divisor \(\mathcal L\) on \(X_\eta\) such that \(\mathcal L\) and \(\phi^*\mathcal L\) both extend to relatively ample divisors on the family \(X_R\), then a theorem of Matsusaka and Mumford shows that \(\phi\) extends to a (regular) automorphism \(\tilde{\phi}\in{\mathrm{Aut}}_R(X_R)\) and that \({\mathrm{sp}}_\eta(\phi)\) is a (regular) automorphism of \(X_0\) \cite[Cor.~1]{MatsusakaMumford64}. Without this additional assumption that \(\phi\) preserves an ample class, one may ask:
\begin{question} Is there a smooth proper family $X_R$ and an element $\phi \in {\mathrm{Aut}}(X_\eta)\cap \Xi_\eta(X_0)$ such that ${\mathrm{sp}}_\eta(\phi)\in {\mathrm{Bir}}(X_0)$ is not a regular automorphism? \end{question}
\noindent In the next section, we will give an example of a family of K3 surfaces and an element \(\iota\in{\mathrm{Aut}}_K(X_\eta)\) which does not extend to a regular automorphism in \({\mathrm{Aut}}_R(X_R)\) (Example~\ref{exmp:K3-example}). In our example \({\mathrm{Aut}}(X_0)={\mathrm{Bir}}(X_0)\), so \({\mathrm{sp}}_\eta(\iota)\) is still a regular automorphism of \(X_0\).
\section{Kernel of the specialization homomorphism} In this section, we study the kernel of the specialization homomorphism from \S\ref{sec:Specialization}. After regularizing an order \(\ell\) birational automorphism on a birational model of \(X_R\), our argument shows that any component of the special fiber fixed by the \({\mathbb Z}/\ell{\mathbb Z}\) group action must be a multiple component.
\begin{proposition}\label{prop:specialization-order-l} Let \(X_R\) be an integral flat separated scheme over \(R\) and \(Z\subset X_0\) an irreducible component. Let \(\phi\in\Xi_\eta(Z)\) be a birational automorphism of order $\ell$, for some integer $\ell > 1$. \begin{enumerate} \item\label{item:automorphism-on-open} There is an affine open $U\subset X_R$ meeting $Z$ on which \(\phi\) induces an automorphism over \(R\). \item\label{item:quotient-by-mu_l} If $\ell$ is invertible in $R$, then the quotient $U/\langle\phi\rangle$ exists and \((U/\langle\phi\rangle)_0=(U\cap Z)/\langle{\mathrm{sp}}_\eta(\phi)\rangle\). \item\label{item:mu_l-nontrivial-on-special-fiber} If $\ell$ is invertible in $R$, then \({\mathrm{sp}}_\eta(\phi)\) has order \(\ell\) in \({\mathrm{Bir}}_k(X_0)\). In particular, $\phi\not\in\ker({\mathrm{sp}}_\eta)$. \end{enumerate} \end{proposition}
\begin{proof} Let \(\tilde{\phi}\in{\mathrm{Bir}}_R(X_R)\) be induced by \(\phi\), and set \(U=\bigcap_{i=1}^{\ell-1}\tilde{\phi}^i(U')\), where \(U'\subset U_1\cap U_2\) is an affine open subset meeting \(Z\), and \(U_1\) and \(U_2\) are as in Proposition~\ref{prop:specialization-hom}\eqref{item:find-open}. This shows \eqref{item:automorphism-on-open}.
Now let \(U={\mathrm{Spec} \ } A\), and let \(\phi\in{\mathrm{Aut}}_R(A)\) denote the induced automorphism. We write \((-)^\phi\) to mean the submodule of \(\phi\)-invariant elements of an \(A\)-module. The quotient \({\mathrm{Spec} \ }(A^\phi)\) is integral and normal \cite[Thm.~4.16]{Moonen-AV}, and it remains to show that \((A^\phi)\otimes_R k \cong (A\otimes_R k)^\phi\).
Left exactness of \((-)^\phi\) implies \(A^\phi/(\pi A)^{\phi}\hookrightarrow(A\otimes_R k)^\phi\) is injective. Since \(\phi\) is an automorphism over \(R\) and \(R\to A\) is flat, we have \((\pi A)^\phi=\pi(A^\phi)\) and \(A^\phi/(\pi A)^\phi\cong(A^\phi)\otimes_R k\), where \(\pi\) is a uniformizer of \(R\). This shows injectivity of \((A^\phi)\otimes_R k\to(A\otimes_R k)^\phi\). For surjectivity, let \(a\in (A\otimes_R k)^\phi\) and let \(\tilde{a}\in A\) be a lift. Then \(\frac{1}{\ell}\sum_{i=0}^{\ell-1}\phi^i(\tilde{a})\) is an element of \(A^\phi\) mapping to \(a\in A\otimes_R k\). This shows \eqref{item:quotient-by-mu_l}.
For \eqref{item:mu_l-nontrivial-on-special-fiber}, let \(G=\langle\phi\rangle\subset {\mathrm{Aut}}_R(U)\). Since the quotient morphism \(q\colon U\to(U/G)\) is a morphism over \(R\), the pullback of the effective Cartier divisor \((U/G)_0\) on \(U/G\) is \(U_0\) \cite[\href{https://stacks.math.columbia.edu/tag/01WV}{Tag 01WV},\href{https://stacks.math.columbia.edu/tag/0C4U}{Tag 0C4U}]{stacks-project}. The projection formula \cite[Ch. 9 Prop. 2.11]{Liu-AG-book} yields \( q_*[q^*((U/G)_0)]=\ell[(U/G)_0]. \) The restriction of \(q\) to \(U_0\) is thus a finite morphism of degree \(\ell\), so the order of \({\mathrm{sp}}_\eta(\phi)\) in \({\mathrm{Bir}}(X_0)\) must be \(\ell\). \end{proof}
\begin{remark} Proposition~\ref{prop:specialization-order-l}\eqref{item:mu_l-nontrivial-on-special-fiber} shows that the kernel of the specialization homomorphism does not contain any torsion of order coprime to the characteristic of the residue field of \(R\). In some special cases, the kernel is even trivial: when the specialization homomorphism \({\mathrm{Pic}}(X_{\overline{\eta}})\to{\mathrm{Pic}}(X_{\overline{0}})\) is an isomorphism and \(\HH^0(X_{\overline{0}},\mathcal T_{X_{\overline{0}}})=0\), Lieblich and Maulik show using the Matsusaka--Mumford theorem \cite[Cor.~1]{MatsusakaMumford64} and a deformation theory argument that the specialization homomorphism is injective \cite[\S 2]{LieblichMaulik18}. \end{remark}
However, injectivity does not hold in general. We now give a series of examples exhibiting nontrivial elements in the kernel of the specialization map.
\begin{example}\label{exmp:cremona-example}
Let \(k\) be a field, and let \(P_1,P_2,Q_t\in\mathbb P^2(k)\) be points such that \(P_1,P_2,Q_t\) are not collinear for \(t\neq 0\), but \(P_1,P_2,Q_0\) lie on a common line \(L\). Denote the subscheme \(P_1 + P_2 + Q_t\) by \(Y_t\), and consider the linear system of conics with base locus \(Y_t\). For \(t\neq 0\) this defines the quadratic transformation with base locus \(P_1,P_2,Q_t\), but on the special fiber ${\mathbb{P}} \HH^0(\mathbb P^2,\mathcal I_{Y_0}(2))^{\vee}=L+|\mathcal O_{\mathbb P^2}(1)|$. For the family \(\mathbb P^2\times \mathbb A^1_t\to\mathbb A^1_t\) this gives an infinite order element \(\phi\) in the birational automorphism group of the generic fiber whose specialization is the identity. Explicitly, one can choose coordinates so that \(\phi\colon [x:y:z]\mapsto [x(x-ty): (x-tz)y: (x-ty)z]\). \end{example}
It is well known that birational automorphisms on K3 surfaces extend to regular automorphisms, so in the next example the specialization homomorphism is defined on ${\mathrm{Bir}} = {\mathrm{Aut}}$.
\begin{example}\label{exmp:K3-example} Let \(X\) be a complex K3 surface of Picard rank \(2\) obtained as the intersection of two divisors of type \((1,1)\) and \((2,2)\) in \({\mathbb{P}}^2\times{\mathbb{P}}^2\). There are two projections $p_{j} \colon X \rightarrow {\mathbb{P}}^{2}$ ($j = 1, 2$), which induce involutions $\iota_{j}$ on $X$. By \cite[Thm.~2.9]{Wehler88}, for a general such $X$ it is known that the automorphism group of $X$ is the free product ${\mathrm{Aut}}(X) \cong {\mathbb Z}/2{\mathbb Z} \ast {\mathbb Z}/2{\mathbb Z}$ generated by the involutions. On the other hand, there are special examples of such K3 surfaces where the involutions commute. In the coordinates $([X_{0}: X_{1}: X_{2}], [Y_{0}: Y_{1}: Y_{2}]) \in {\mathbb{P}}^{2} \times {\mathbb{P}}^{2}$, one may take the complete intersection $X_{0}$ given by the equations \[ \sum_{i,j\in\{0,1\}} a_{ij} X_{i}Y_{j} = 0 \quad \text{and} \quad \sum_{i,j\in\{0,1,2\}} b_{ij} X_{i}^{2} Y_{j}^{2} = 0, \] which is smooth for general coefficients $a_{ij}$ and $b_{ij}$. On \(X_0\) the covering involutions extend to (regular) involutions: \[ \iota_{1,0} \colon ([X_{0}: X_{1}: X_{2}] , [Y_{0}: Y_{1}: Y_{2}]) \mapsto ([X_{0}: X_{1}: (-1) \cdot X_{2}] , [Y_{0}: Y_{1}: Y_{2}]), \] \[ \iota_{2,0} \colon ([X_{0}: X_{1}: X_{2}] , [Y_{0}: Y_{1}: Y_{2}]) \mapsto ([X_{0}: X_{1}: X_{2}] , [Y_{0}: Y_{1}: (-1) \cdot Y_{2}]). \] By construction, these involutions on $X_{0}$ automatically commute. This shows that the birational automorphism $\iota_{1}\iota_{2}\iota_{1}\iota_{2}$ on the general K3 surface $X$ has infinite order but specializes to the identity on the special fiber $X_{0}$. On the special fiber, each projection \(p_j\) contracts the conic over \([0:0:1]\), so the Picard rank jumps and the covering involution does not extend to a regular involution on the family (c.f. \cite[Thm.~2.1]{LieblichMaulik18}).
One may exhibit similar behavior on K3 surfaces of type (2,2,2) in $({\mathbb{P}}^{1})^{3}$, see \cite[\S 3]{Wang-thesis} and \cite[Prop.~3.5]{Schaffler2018}. For an example with Enriques surfaces, see \cite{BarthPeters-Enriques-surfaces} (c.f. \cite[IV Ex.~1.17.4]{Koll'ar-rational-curves}).
\end{example}
\begin{example} In mixed characteristic \((0,p)\), the kernel of \({\mathrm{sp}}_\eta\) can contain \(p\)-torsion. It is not clear if this can be accounted for by considering an additional scheme structure on ${\mathrm{Bir}}(X)$. For instance, the group of \(p\)-torsion geometric points of an elliptic curve is isomorphic to \({\mathbb Z}/p{\mathbb Z}\times{\mathbb Z}/p{\mathbb Z}\) in characteristic \(0\), but is isomorphic to \({\mathbb Z}/p{\mathbb Z}\) or is trivial in characteristic \(p\), so translating by a $p$-torsion point that specializes to the identity gives such an example. Similarly one can construct examples by considering $\mu_p$ actions on a scheme in mixed characteristic $(0,p)$. This happens when considering $\mu_p$-covers of schemes, and it will be an important tool in the next section. \end{example}
\section{Applications to birational automorphisms of Fano hypersurfaces}
We now give the proofs of Theorem~\ref{thm:torsion-p^e} and Corollary~\ref{cor:no-torsion}. The key ingredients used are the specialization homomorphism for \({\mathrm{Bir}}\), a result of the first and third authors showing that certain \(p\)-cyclic covers in characteristic \(p\) have no birational automorphisms \cite[Cor. C]{ChenStapleton-finite-BirX}, and a construction of Mori \cite{Mori75} (see also \cite[V.5.14.4]{Koll'ar-rational-curves}) that allows us to degenerate from a hypersurface to a \(p\)-cyclic cover. We begin by recalling Mori's construction:
\begin{construction}\label{construction:mori} Let \(f,g\in R[x_0,\ldots,x_{n+1}]\) be homogeneous polynomials of degree \(pe\) and \(e\), respectively. Assume \(g^p-f\) is not uniformly \(0\). Let \(Z=(y^p-f=g-\pi y=0)\subset\mathbb P_R(1^{n+2};e)\). Then \(Z_\eta\) is isomorphic to the degree \(pe\) hypersurface \((g^p-\pi^pf=0)\subset\mathbb P^{n+1}_K\), and \(Z_0\) is isomorphic to a \(p\)-cyclic cover of the degree \(e\) hypersurface \((g=0)\subset\mathbb P^{n+1}_k\). \end{construction}
There are two different degenerations that are most useful in our case: \begin{itemize}
\item A $p$-cyclic cover in mixed characteristic $(0,p)$, and
\item Mori's construction in equicharacteristic \(0\). \end{itemize} By \cite[Thm.~C \& Ex.~1.7]{ChenStapleton-rational-endomorphisms}, these families have sustained separably uniruled modifications and sustained ruled modifications, respectively. Therefore we may apply Proposition~\ref{prop:specialization-hom}\eqref{item:specialization-sustained}.
\begin{proposition}\label{prop:BirtorsX-p} Let \(p\) be a prime and let \(n,e\geq 3\) be integers such that \((p-1)e\leq n-e\leq pe-3\). Furthermore, assume \(n\) is even if \(p=2\). If \(X\subset\mathbb P^{n+1}_{\mathbb C}\) is a very general hypersurface of degree \(pe\), then any finite order element of \({\mathrm{Bir}}(X)\) has order a power of $p$. \end{proposition}
\begin{proof} The inequalities in the statement of the proposition imply that over \(\overline{\mathbb F}_p\), a general \(p\)-cyclic cover of a degree \(e\) hypersurface in \(\mathbb P^{n+1}\) has trivial birational automorphism group by \cite[Cor.~C]{ChenStapleton-finite-BirX} and is not separably uniruled by \cite[Lem.~7]{Koll'ar-hypersurfaces}. So it follows from \cite[Thm.~C]{ChenStapleton-rational-endomorphisms}, Proposition~\ref{prop:specialization-hom}\eqref{item:specialization-sustained}, and Proposition~\ref{prop:specialization-order-l}\eqref{item:mu_l-nontrivial-on-special-fiber} that for a very general such \(p\)-cyclic cover \(Z\) over ${\mathbb C}$, ${\mathrm{Bir}}_{\mathbb C}(Z)$ only contains elements whose orders are \(p\)-powers. By Construction~\ref{construction:mori}, there is a family of degree \(pe\) hypersurfaces over a complex curve that degenerates to a general such \(p\)-cyclic cover. Since $Z$ is not ruled \cite[Prop.~5.12]{Koll'ar-rational-curves} and the total space has sustained ruled modifications \cite[Ex.~1.7]{ChenStapleton-rational-endomorphisms}, we may apply Proposition~\ref{prop:specialization-hom}\eqref{item:specialization-sustained}. Together with Proposition~\ref{prop:specialization-order-l}\eqref{item:mu_l-nontrivial-on-special-fiber} and the isomorphism between the geometric generic and very general fibers of the family \cite[Lem. 2.1]{Vial13}, this gives the result for a very general degree \(pe\) hypersurface over ${\mathbb C}$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:torsion-p^e}] Let \(e\coloneqq \lceil \frac{n+3}{p+1}\rceil\). We will first show the result for \(d=pe\). By the comment after Theorem~\ref{thm:torsion-p^e}, we may assume that $d \leq n$ (note that this implies \(n\geq 3p\)). The assumptions in the theorem then imply that \((p-1)e\leq n-e\leq pe-3\), so by Proposition~\ref{prop:BirtorsX-p} any torsion element in the birational automorphism group of a very general hypersurface of degree \(pe\) in \(\mathbb P^{n+1}_{\mathbb C}\) has order a power of $p$.
For \(d>pe\) we prove the result by induction, showing that the degree $d-1$ result implies the degree $d$ result. To start, consider a pencil of hypersurfaces spanned by a smooth degree $d$ hypersurface and a degree $d-1$ hypersurface union with a hyperplane. Assume that the union of all three is an snc divisor. Then the total space of the pencil is singular (as the dimension of the hypersurfaces is $\ge 3$) and admits a small resolution by blowing up the hyperplane in the central fiber. After this blowup, the localization of the family at the reducible fiber has reduced snc central fiber with two components birational to the original ones. Thus the localized family has sustained ruled modifications by \cite[Ex.~1.7]{ChenStapleton-rational-endomorphisms}.
By induction the only finite order birational automorphisms of a very general degree $d-1$ hypersurface have order a power of $p$. Moreover, it is not ruled by \cite[Thm.~2]{Koll'ar-hypersurfaces}, so we may apply Proposition~\ref{prop:specialization-hom}\eqref{item:specialization-sustained} to the above degeneration to prove the result in degree $d$. \end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:no-torsion}] Combine the results for the primes \(p=2,3\) in Theorem~\ref{thm:torsion-p^e} if \(n\) is even, and consider the primes \(p=3,5\) if \(n\) is odd. \end{proof}
\footnotesize{ \textsc{Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138} \\ \indent \textit{E-mail address:} \href{mailto:[email protected]}{[email protected]}
\textsc{Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109} \\ \indent \textit{E-mail address:} \href{mailto:[email protected]}{[email protected]}
\textsc{Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109} \\ \indent \textit{E-mail address:} \href{mailto:[email protected]}{[email protected]} }
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Question 37 IRR
If a project's net present value (NPV) is zero, then its internal rate of return (IRR) will be:
(a) Positive infinity (##+\infty##)
(b) Zero (0).
(c) Less than the project's required return.
(d) More than the project's required return.
(e) Equal to the project's required return.
Question 502 NPV, IRR, mutually exclusive projects
An investor owns an empty block of land that has local government approval to be developed into a petrol station, car wash or car park. The council will only allow a single development so the projects are mutually exclusive.
All of the development projects have the same risk and the required return of each is 10% pa. Each project has an immediate cost and once construction is finished in one year the land and development will be sold. The table below shows the estimated costs payable now, expected sale prices in one year and the internal rates of returns (IRR's).
Mutually Exclusive Projects
now ($) Sale price in
one year ($) IRR
(% pa)
Petrol station 9,000,000 11,000,000 22.22
Car wash 800,000 1,100,000 37.50
Car park 70,000 110,000 57.14
Which project should the investor accept?
(a) Petrol station.
(b) Car wash.
(c) Car park.
(d) None of the projects.
(e) All of the projects.
Question 250 NPV, Loan, arbitrage table
Your neighbour asks you for a loan of $100 and offers to pay you back $120 in one year.
You don't actually have any money right now, but you can borrow and lend from the bank at a rate of 10% pa. Rates are given as effective annual rates.
Assume that your neighbour will definitely pay you back. Ignore interest tax shields and transaction costs.
The Net Present Value (NPV) of lending to your neighbour is $9.09. Describe what you would do to actually receive a $9.09 cash flow right now with zero net cash flows in the future.
(a) Borrow $109.09 from the bank and lend $100 of it to your neighbour now.
(b) Borrow $100 from the bank and lend it to your neighbour now.
(c) Borrow $209.09 from the bank and lend $100 to your neighbour now.
(d) Borrow $120 from the bank and lend $100 of it to your neighbour now.
(e) Borrow $90.91 from the bank and lend it to your neighbour now.
Question 251 NPV
You have $100,000 in the bank. The bank pays interest at 10% pa, given as an effective annual rate.
You wish to consume an equal amount now (t=0) and in one year (t=1) and have nothing left in the bank at the end (t=1).
How much can you consume at each time?
(a) $57,619.0476
(b) $55,000
(c) $53,809.5238
(d) $52,380.9524
(e) $50,000
You wish to consume an equal amount now (t=0), in one year (t=1) and in two years (t=2), and still have $50,000 in the bank after that (t=2).
(b) $23,666.6667
(e) $16,666.6667
Question 60 pay back period
The required return of a project is 10%, given as an effective annual rate.
What is the payback period of the project in years?
Assume that the cash flows shown in the table are received smoothly over the year. So the $121 at time 2 is actually earned smoothly from t=1 to t=2.
Project Cash Flows
Time (yrs) Cash flow ($)
(a) 2.7355
(b) 2.3596
(c) 1.7355
(d) 1.2645
(e) 0.2645
Question 190 pay back period
A project has the following cash flows:
Normally cash flows are assumed to happen at the given time. But here, assume that the cash flows are received smoothly over the year. So the $500 at time 2 is actually earned smoothly from t=1 to t=2.
(a) -0.80
(b) 0.80
(c) 1.20
(d) 1.80
(e) 2.20
A project to build a toll road will take 3 years to complete, costing three payments of $50 million, paid at the start of each year (at times 0, 1, and 2).
After completion, the toll road will yield a constant $10 million at the end of each year forever with no costs. So the first payment will be at t=4.
The required return of the project is 10% pa given as an effective nominal rate. All cash flows are nominal.
What is the payback period?
(a) Negative since the NPV is negative.
(b) Zero since the project's internal rate of return is less than the required return.
(c) 15 years.
(d) 18 years.
(e) Infinite, since the project will never pay itself off.
Question 141 time calculation, APR, effective rate
You're trying to save enough money to buy your first car which costs $2,500. You can save $100 at the end of each month starting from now. You currently have no money at all. You just opened a bank account with an interest rate of 6% pa payable monthly.
How many months will it take to save enough money to buy the car? Assume that the price of the car will stay the same over time.
(a) 23.62
(b) 25.00
(c) 26.60
(d) 26.77
(e) 55.24
Question 254 time calculation, APR
Your main expense is fuel for your car which costs $100 per month. You just refueled, so you won't need any more fuel for another month (first payment at t=1 month).
You have $2,500 in a bank account which pays interest at a rate of 6% pa, payable monthly. Interest rates are not expected to change.
Assuming that you have no income, in how many months time will you not have enough money to fully refuel your car?
(a) In 23 months (t=23 months).
(b) In 24 months (t=24 months).
(c) In 25 months (t=25 months).
(d) In 26 months (t=26 months).
(e) In 27 months (t=27 months).
Question 32 time calculation, APR
You really want to go on a back packing trip to Europe when you finish university. Currently you have $1,500 in the bank. Bank interest rates are 8% pa, given as an APR compounding per month. If the holiday will cost $2,000, how long will it take for your bank account to reach that amount?
(a) -3.74 years
(b) 1.81 years
(c) 3.33 years
(d) 3.61 years
(e) 3.74 years
Question 204 time calculation, fully amortising loan, APR
You just signed up for a 30 year fully amortising mortgage loan with monthly payments of $1,500 per month. The interest rate is 9% pa which is not expected to change.
To your surprise, you can actually afford to pay $2,000 per month and your mortgage allows early repayments without fees. If you maintain these higher monthly payments, how long will it take to pay off your mortgage?
(a) 38.87 months, which is 3.24 yrs.
(b) 47.91 months, which is 3.99 yrs.
(c) 160.72 months, which is 13.39 yrs.
(d) 164.65 months, which is 13.72 yrs.
(e) None of the above.
You're trying to save enough money for a deposit to buy a house. You want to buy a house worth $400,000 and the bank requires a 20% deposit ($80,000) before it will give you a loan for the other $320,000 that you need.
You currently have no savings, but you just started working and can save $2,000 per month, with the first payment in one month from now. Bank interest rates on savings accounts are 4.8% pa with interest paid monthly and interest rates are not expected to change.
How long will it take to save the $80,000 deposit? Round your answer up to the nearest month.
(a) 27 months (t=27 months).
(b) 38 months (t=38 months).
(c) 40 months (t=40 months).
(d) 43 months (t=43 months).
(e) 79 months (t=79 months).
Question 333 DDM, time calculation
When using the dividend discount model, care must be taken to avoid using a nominal dividend growth rate that exceeds the country's nominal GDP growth rate. Otherwise the firm is forecast to take over the country since it grows faster than the average business forever.
Suppose a firm's nominal dividend grows at 10% pa forever, and nominal GDP growth is 5% pa forever. The firm's total dividends are currently $1 billion (t=0). The country's GDP is currently $1,000 billion (t=0).
In approximately how many years will the company's total dividends be as large as the country's GDP?
(a) 1,443 years
(b) 1,199 years
(c) 955 years
(d) 674 years
(e) 148 years
Question 44 NPV
The required return of a project is 10%, given as an effective annual rate. Assume that the cash flows shown in the table are paid all at once at the given point in time.
What is the Net Present Value (NPV) of the project?
(a) -100
(b) 0
(c) 10
(d) 21
(e) 121
Question 500 NPV, IRR
The below graph shows a project's net present value (NPV) against its annual discount rate.
For what discount rate or range of discount rates would you accept and commence the project?
All answer choices are given as approximations from reading off the graph.
(a) From 0 to 10% pa.
(b) From 0 to 5% pa.
(c) At 5.5% pa.
(d) From 6 to 20% pa.
(e) From 0 to 20% pa.
Question 501 NPV, IRR, pay back period
Which of the following statements is NOT correct?
(a) When the project's discount rate is 18% pa, the NPV is approximately -$30m.
(b) The payback period is infinite, the project never pays itself off.
(c) The addition of the project's cash flows, ignoring the time value of money, is approximately $20m.
(d) The project's IRR is approximately 5.5% pa.
(e) As the discount rate rises, the NPV falls.
Question 489 NPV, IRR, pay back period, DDM
A firm is considering a business project which costs $11m now and is expected to pay a constant $1m at the end of every year forever.
Assume that the initial $11m cost is funded using the firm's existing cash so no new equity or debt will be raised. The cost of capital is 10% pa.
Which of the following statements about net present value (NPV), internal rate of return (IRR) and payback period is NOT correct?
(a) The NPV is negative $1m.
(b) The IRR is 9.09% pa, less than the 10% cost of capital.
(c) The payback period is infinite, the project will never pay itself off.
(d) The project should be rejected.
(e) If the project is accepted then the market value of the firm's assets will fall by $1m.
Question 532 mutually exclusive projects, NPV, IRR
An investor owns a whole level of an old office building which is currently worth $1 million. There are three mutually exclusive projects that can be started by the investor. The office building level can be:
Rented out to a tenant for one year at $0.1m paid immediately, and then sold for $0.99m in one year.
Refurbished into more modern commercial office rooms at a cost of $1m now, and then sold for $2.4m when the refurbishment is finished in one year.
Converted into residential apartments at a cost of $2m now, and then sold for $3.4m when the conversion is finished in one year.
All of the development projects have the same risk so the required return of each is 10% pa. The table below shows the estimated cash flows and internal rates of returns (IRR's).
Project Cash flow
now ($) Cash flow in
Rent then sell as is -900,000 990,000 10
Refurbishment into modern offices -2,000,000 2,400,000 20
Conversion into residential apartments -3,000,000 3,400,000 13.33
(a) Rent then sell as is.
(b) Refurbishment into modern offices.
(c) Conversion into residential apartments.
(d) All of the above.
(e) Any of the above.
Question 579 price gains and returns over time, time calculation, effective rate
How many years will it take for an asset's price to double if the price grows by 10% pa?
(a) 1.8182 years
(b) 3.3219 years
(c) 7.2725 years
(d) 11.5267 years
(e) 13.7504 years
How many years will it take for an asset's price to quadruple (be four times as big, say from $1 to $4) if the price grows by 15% pa?
(d) 9.919 years
Question 476 income and capital returns, idiom
The saying "buy low, sell high" suggests that investors should make a:
(a) Positive income return.
(b) Positive capital return.
(c) Negative income return.
(d) Negative capital return.
(e) Positive total return.
Question 490 expected and historical returns, accounting ratio
Which of the following is NOT a synonym of 'required return'?
(a) total required yield
(b) cost of capital
(c) discount rate
(d) opportunity cost of capital
(e) accounting rate of return
Question 478 income and capital returns
Total cash flows can be broken into income and capital cash flows. What is the name given to the income cash flow from owning shares?
(a) Dividends.
(b) Rent.
(c) Coupons.
(d) Loan payments.
(e) Capital gains.
Which of the following equations is NOT equal to the total return of an asset?
Let ##p_0## be the current price, ##p_1## the expected price in one year and ##c_1## the expected income in one year.
(a) ##r_\text{total} = \dfrac{c_1+p_1-p_0}{p_0} ##
(b) ##r_\text{total} = \dfrac{c_1+p_1}{p_0} - 1##
(c) ##r_\text{total} = \dfrac{c_1}{p_0} + \dfrac{p_1-p_0}{p_0}##
(d) ##r_\text{total} = \dfrac{c_1}{p_0} + \dfrac{p_1}{p_0} ##
(e) ##r_\text{total} = \dfrac{c_1}{p_0} + \dfrac{p_1}{p_0} - 1##
An asset's total expected return over the next year is given by:
###r_\text{total} = \dfrac{c_1+p_1-p_0}{p_0} ###
Where ##p_0## is the current price, ##c_1## is the expected income in one year and ##p_1## is the expected price in one year. The total return can be split into the income return and the capital return.
Which of the following is the expected capital return?
(a) ##c_1##
(b) ##p_1-p_0##
(c) ##\dfrac{c_1}{p_0} ##
(d) ##\dfrac{p_1}{p_0} -1##
(e) ##\dfrac{p_1}{p_0} ##
A stock was bought for $8 and paid a dividend of $0.50 one year later (at t=1 year). Just after the dividend was paid, the stock price was $7 (at t=1 year).
What were the total, capital and dividend returns given as effective annual rates? The choices are given in the same order:
##r_\text{total}##, ##r_\text{capital}##, ##r_\text{dividend}##.
(a) 0.0625, -0.0625, -0.125.
(b) 0.0625, 0.125, -0.0625.
(c) -0.0625, 0.0625, -0.125.
(d) -0.0625, -0.125, 0.0625.
(e) -0.125, -0.1875, 0.0625.
A share was bought for $30 (at t=0) and paid its annual dividend of $6 one year later (at t=1).
Just after the dividend was paid, the share price fell to $27 (at t=1). What were the total, capital and income returns given as effective annual rates?
The choices are given in the same order:
##r_\text{total}## , ##r_\text{capital}## , ##r_\text{dividend}##.
(a) -0.1, -0.3, 0.2.
(b) -0.1, 0.1, -0.2.
(c) 0.1, -0.1, 0.2.
(d) 0.1, 0.2, -0.1.
(e) 0.2, 0.1, -0.1.
Question 21 income and capital returns, bond pricing
A fixed coupon bond was bought for $90 and paid its annual coupon of $3 one year later (at t=1 year). Just after the coupon was paid, the bond price was $92 (at t=1 year). What was the total return, capital return and income return? Calculate your answers as effective annual rates.
The choices are given in the same order: ## r_\text{total},r_\text{capital},r_\text{income} ##.
(a) -0.0556, -0.0222, -0.0333
(b) 0.0222, -0.0111, 0.0333.
(c) 0.0333, 0.0556, 0.0222.
(d) 0.0556, 0.0222, 0.0333.
(e) 0.0556, 0.0333, 0.0222.
Question 404 income and capital returns, real estate
One and a half years ago Frank bought a house for $600,000. Now it's worth only $500,000, based on recent similar sales in the area.
The expected total return on Frank's residential property is 7% pa.
He rents his house out for $1,600 per month, paid in advance. Every 12 months he plans to increase the rental payments.
The present value of 12 months of rental payments is $18,617.27.
The future value of 12 months of rental payments one year in the future is $19,920.48.
What is the expected annual rental yield of the property? Ignore the costs of renting such as maintenance, real estate agent fees and so on.
(a) 3.1029%
(b) 3.3201%
(c) 3.7235%
(d) 3.9841%
(e) 7%
Question 278 inflation, real and nominal returns and cash flows
Imagine that the interest rate on your savings account was 1% per year and inflation was 2% per year.
After one year, would you be able to buy , exactly the as or than today with the money in this account?
Question 295 inflation, real and nominal returns and cash flows, NPV
When valuing assets using discounted cash flow (net present value) methods, it is important to consider inflation. To properly deal with inflation:
(I) Discount nominal cash flows by nominal discount rates.
(II) Discount nominal cash flows by real discount rates.
(III) Discount real cash flows by nominal discount rates.
(IV) Discount real cash flows by real discount rates.
Which of the above statements is or are correct?
(a) I only.
(b) III only.
(c) IV only.
(d) I and IV only.
(e) II and III only.
Question 353 income and capital returns, inflation, real and nominal returns and cash flows, real estate
A residential investment property has an expected nominal total return of 6% pa and nominal capital return of 3% pa.
Inflation is expected to be 2% pa. All rates are given as effective annual rates.
What are the property's expected real total, capital and income returns? The answer choices below are given in the same order.
(a) 3.9216%, 2.9412%, 0.9804%.
(b) 3.9216%, 0.9804%, 2.9412%.
(c) 3.9216%, 0.9804%, 0.9804%.
(d) 1.9804%, 1.0000%, 0.9804%.
(e) 1.9608%, 0.9804%, 0.9804%.
(a) 5.8824%, 0.9804%, 4.902%.
Question 155 inflation, real and nominal returns and cash flows, Loan, effective rate conversion
You are a banker about to grant a 2 year loan to a customer. The loan's principal and interest will be repaid in a single payment at maturity, sometimes called a zero-coupon loan, discount loan or bullet loan.
You require a real return of 6% pa over the two years, given as an effective annual rate. Inflation is expected to be 2% this year and 4% next year, both given as effective annual rates.
You judge that the customer can afford to pay back $1,000,000 in 2 years, given as a nominal cash flow. How much should you lend to her right now?
(a) $838,907.00
(b) $838,986.09
(c) $841,754.97
(d) $889,996.44
(e) $944,108.22
Question 473 market capitalisation of equity
The below screenshot of Commonwealth Bank of Australia's (CBA) details were taken from the Google Finance website on 7 Nov 2014. Some information has been deliberately blanked out.
What was CBA's market capitalisation of equity?
(a) $431.18 billion
(b) $429 billion
(c) $134.07 billion
(d) $8.44 billion
(e) $3.21 billion
The below screenshot of Microsoft's (MSFT) details were taken from the Google Finance website on 28 Nov 2014. Some information has been deliberately blanked out.
What was MSFT's market capitalisation of equity?
(a) $395.11 million
(b) $21.01 billion
(d) $393.95 billion
(e) $1.02935 trillion
Question 467 book and market values
Which of the following statements about book and market equity is NOT correct?
(a) The market value of equity of a listed company's common stock is equal to the number of common shares multiplied by the share price.
(b) The book value of equity is the sum of contributed equity, retained profits and reserves.
(c) A company's book value of equity is recorded in its income statement, also known as the 'profit and loss' or the 'statement of financial performance'.
(d) A new company's market value of equity equals its book value of equity the moment that its shares are first sold. From then on, the market value changes continuously but the book value which is recorded at historical cost tends to only change due to retained profits.
(e) To buy all of the firm's shares, generally a price close to the market value of equity will have to be paid.
Question 444 investment decision, corporate financial decision theory
The investment decision primarily affects which part of a business?
(a) Assets.
(b) Liabilities and owner's equity.
(c) Current assets and current liabilities.
(d) Dividends and buy backs.
(e) Net income, also known as earnings or net profit after tax.
Question 446 working capital decision, corporate financial decision theory
The working capital decision primarily affects which part of a business?
Question 445 financing decision, corporate financial decision theory
The financing decision primarily affects which part of a business?
Question 447 payout policy, corporate financial decision theory
Payout policy is most closely related to which part of a business?
Question 443 corporate financial decision theory, investment decision, financing decision, working capital decision, payout policy
Business people make lots of important decisions. Which of the following is the most important long term decision?
(a) Investment decision.
(b) Financing decision.
(c) Working capital decision.
(d) Payout policy decision.
(e) Capital or labour decision.
Question 221 credit risk
You're considering making an investment in a particular company. They have preference shares, ordinary shares, senior debt and junior debt.
Which is the safest investment? Which will give the highest returns?
(a) Junior debt is the safest. Preference shares will have the highest returns.
(b) Preference shares are the safest. Ordinary shares will have the highest returns.
(c) Senior debt is the safest. Ordinary shares will have the highest returns.
(d) Junior debt is the safest. Ordinary shares will have the highest returns.
(e) Senior debt is the safest. Junior debt will have the highest returns.
Question 120 credit risk, payout policy
A newly floated farming company is financed with senior bonds, junior bonds, cumulative non-voting preferred stock and common stock. The new company has no retained profits and due to floods it was unable to record any revenues this year, leading to a loss. The firm is not bankrupt yet since it still has substantial contributed equity (same as paid-up capital).
On which securities must it pay interest or dividend payments in this terrible financial year?
(a) Preferred stock only.
(b) The senior and junior bonds only.
(c) Common stock only.
(d) The senior and junior bonds and the preferred stock.
(e) No payments on any security is required since the firm made a loss.
Question 466 limited liability, business structure
Which business structure or structures have the advantage of limited liability for equity investors?
(a) Sole traders.
(b) Partnerships.
(c) Corporations.
(e) None of the above
Question 452 limited liability, expected and historical returns
What is the lowest and highest expected share price and expected return from owning shares in a company over a finite period of time?
Let the current share price be ##p_0##, the expected future share price be ##p_1##, the expected future dividend be ##d_1## and the expected return be ##r##. Define the expected return as:
##r=\dfrac{p_1-p_0+d_1}{p_0} ##
The answer choices are stated using inequalities. As an example, the first answer choice "(a) ##0≤p<∞## and ##0≤r< 1##", states that the share price must be larger than or equal to zero and less than positive infinity, and that the return must be larger than or equal to zero and less than one.
(a) ##0≤p<∞## and ##0≤r< 1##
(b) ##0≤p<∞## and ##-1≤r< ∞##
(c) ##0≤p<∞## and ##0≤r< ∞##
(d) ##0≤p<∞## and ##-∞≤r< ∞##
(e) ##-∞<p<∞## and ##-∞<r< ∞##
Question 542 price gains and returns over time, IRR, NPV, income and capital returns, effective return
For an asset price to double every 10 years, what must be the expected future capital return, given as an effective annual rate?
(a) 0.2
(b) 0.116123
(c) 0.082037
(d) 0.071773
(e) 0.06054
Question 525 income and capital returns, real and nominal returns and cash flows, inflation
Which of the following statements about cash in the form of notes and coins is NOT correct? Assume that inflation is positive.
Notes and coins:
(a) Pay no income cash flow.
(b) Have a nominal total return of zero.
(c) Have a nominal capital return of zero.
(d) Have a nominal income return of zero.
(e) Have a real total return of zero.
Question 526 real and nominal returns and cash flows, inflation, no explanation
How can a nominal cash flow be precisely converted into a real cash flow?
(a) ##C_\text{real, t}=C_\text{nominal,t}.(1+r_\text{inflation})^t##
(b) ##C_\text{real,t}=\dfrac{C_\text{nominal,t}}{(1+r_\text{inflation})^t} ##
(c) ##C_\text{real,t}=\dfrac{C_\text{nominal,t}}{r_\text{inflation}} ##
(d) ##C_\text{real,t}=C_\text{nominal,t}.r_\text{inflation} ##
(e) ##C_\text{real,t}=C_\text{nominal,t}.r_\text{inflation}.t##
What is the present value of a real payment of $500 in 2 years? The nominal discount rate is 7% pa and the inflation rate is 4% pa.
(a) $472.3557
(b) $471.298
(c) $436.7194
(d) $435.7415
(e) $405.8112
Question 531 bankruptcy or insolvency, capital structure, risk, limited liability
Who is most in danger of being personally bankrupt? Assume that all of their businesses' assets are highly liquid and can therefore be sold immediately.
(a) Alice has $6,000 cash, owes $10,000 credit card debt due immediately and 100% owns a sole tradership business with assets worth $10,000 and liabilities of $3,000.
(b) Billy has $10,000 cash, owes $6,000 credit card debt due immediately and 100% owns a corporate business with assets worth $3,000 and liabilities of $10,000.
(c) Carla has $6,000 cash, owes $10,000 credit card debt due immediately and 100% owns a corporate business with assets worth $10,000 and liabilities of $3,000.
(d) Darren has $10,000 cash, owes $6,000 credit card debt due immediately and 100% owns a sole tradership business with assets worth $3,000 and liabilities of $10,000.
(e) Ernie has $1,000 cash, lent $3,000 to his friend, and doesn't have any personal debt or own any businesses.
Question 515 corporate financial decision theory, idiom
The expression 'you have to spend money to make money' relates to which business decision?
(e) Diversification decision.
Question 2 NPV, Annuity
Katya offers to pay you $10 at the end of every year for the next 5 years (t=1,2,3,4,5) if you pay her $50 now (t=0). You can borrow and lend from the bank at an interest rate of 10% pa, given as an effective annual rate.
Ignore credit risk.
Will you or Katya's deal?
Question 288 Annuity
There are many ways to write the ordinary annuity formula.
Which of the following is NOT equal to the ordinary annuity formula?
(a) ##V_0 = \dfrac{C_1}{r} \left(1-\dfrac{1}{(1+r)^T} \right) ##
(b) ##V_0 = \dfrac{C_1 \left(1-\dfrac{1}{(1+r)^T} \right)}{r}##
(c) ##V_0 = C_1\dfrac{\left(1-(1+r)^{-T} \right)}{r}##
(d) ##V_0 = C_1 \left(1-(1+r)^{-T} \right) r^{-1}##
(e) ##V_0 = C_1 \left(r^{-1}-(1+r)^{-T-1} \right)##
Question 137 NPV, Annuity
The following cash flows are expected:
10 yearly payments of $60, with the first payment in 3 years from now (first payment at t=3).
1 payment of $400 in 5 years and 6 months (t=5.5) from now.
What is the NPV of the cash flows if the discount rate is 10% given as an effective annual rate?
(a) $513.80
(b) $525.36
(c) $541.50
(d) $605.48
(e) $704.69
Your friend overheard that you need some cash and asks if you would like to borrow some money. She can lend you $5,000 now (t=0), and in return she wants you to pay her back $1,000 in two years (t=2) and every year after that for the next 5 years, so there will be 6 payments of $1,000 from t=2 to t=7 inclusive.
What is the net present value (NPV) of borrowing from your friend?
Assume that banks loan funds at interest rates of 10% pa, given as an effective annual rate.
(a) -$1,000
(b) $209.2132
(d) $1,040.6721
(e) $1,400.611
Question 58 NPV, inflation, real and nominal returns and cash flows, Annuity
A project to build a toll bridge will take two years to complete, costing three payments of $100 million at the start of each year for the next three years, that is at t=0, 1 and 2.
After completion, the toll bridge will yield a constant $50 million at the end of each year for the next 10 years. So the first payment will be at t=3 and the last at t=12. After the last payment at t=12, the bridge will be given to the government.
The required return of the project is 21% pa given as an effective annual nominal rate.
All cash flows are real and the expected inflation rate is 10% pa given as an effective annual rate. Ignore taxes.
The Net Present Value is:
(a) -$112,496,484
(b) -$32,260,693
(c) -$19,645,987
(d) $5,222,533
(e) $200,000,000
Some countries' interest rates are so low that they're zero.
If interest rates are 0% pa and are expected to stay at that level for the foreseeable future, what is the most that you would be prepared to pay a bank now if it offered to pay you $10 at the end of every year for the next 5 years?
In other words, what is the present value of five $10 payments at time 1, 2, 3, 4 and 5 if interest rates are 0% pa?
(a) $0
(b) $10
(c) $50
(d) Positive infinity
(e) Priceless
Question 479 perpetuity with growth, DDM, NPV
Discounted cash flow (DCF) valuation prices assets by finding the present value of the asset's future cash flows. The single cash flow, annuity, and perpetuity equations are very useful for this.
Which of the following equations is the 'perpetuity with growth' equation?
(a) ##V_0=\dfrac{C_t}{(1+r)^t} ##
(b) ##V_0=\dfrac{C_1}{r}.\left(1-\dfrac{1}{(1+r)^T} \right)= \sum\limits_{t=1}^T \left( \dfrac{C_t}{(1+r)^t} \right) ##
(c) ##V_0=\dfrac{C_1}{r-g}.\left(1-\left(\dfrac{1+g}{1+r}\right)^T \right)= \sum\limits_{t=1}^T \left( \dfrac{C_t.(1+g)^t}{(1+r)^t} \right) ##
(d) ##V_0=\dfrac{C_1}{r} = \sum\limits_{t=1}^\infty \left( \dfrac{C_t}{(1+r)^t} \right) ##
(e) ##V_0=\dfrac{C_1}{r-g} = \sum\limits_{t=1}^\infty \left( \dfrac{C_t.(1+g)^t}{(1+r)^t} \right) ##
Question 3 DDM, income and capital returns
The following equation is called the Dividend Discount Model (DDM), Gordon Growth Model or the perpetuity with growth formula: ### P_0 = \frac{ C_1 }{ r - g } ###
What is ##g##? The value ##g## is the long term expected:
(a) Income return of the stock.
(b) Capital return of the stock.
(c) Total return of the stock.
(d) Dividend yield of the stock.
(e) Price-earnings ratio of the stock.
Question 451 DDM
The first payment of a constant perpetual annual cash flow is received at time 5. Let this cash flow be ##C_5## and the required return be ##r##.
So there will be equal annual cash flows at time 5, 6, 7 and so on forever, and all of the cash flows will be equal so ##C_5 = C_6 = C_7 = ...##
When the perpetuity formula is used to value this stream of cash flows, it will give a value (V) at time:
(a) 0, so ##V_0=\dfrac{C_5}{r}##
(b) 1, so ##V_1=\dfrac{C_5}{r}##
(c) 4, so ##V_4=\dfrac{C_5}{r}##
(d) 5, so ##V_5=\dfrac{C_5}{r}##
(e) 6, so ##V_6=\dfrac{C_5}{r}##
Question 28 DDM, income and capital returns
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
### P_{0} = \frac{C_1}{r_{\text{eff}} - g_{\text{eff}}} ###
What would you call the expression ## C_1/P_0 ##?
(a) The expected total return of the stock.
(b) The expected income return of the stock.
(c) The expected capital return of the stock.
(d) The expected growth rate of the dividend.
(e) The expected growth rate of the stock price.
A stock just paid its annual dividend of $9. The share price is $60. The required return of the stock is 10% pa as an effective annual rate.
What is the implied growth rate of the dividend per year?
(a) -0.8565
(b) -0.0500
(c) -0.0435
###P_0=\frac{d_1}{r-g}###
A stock pays dividends annually. It just paid a dividend, but the next dividend (##d_1##) will be paid in one year.
According to the DDM, what is the correct formula for the expected price of the stock in 2.5 years?
(a) ##P_{2.5}=P_0 (1+g)^{2.5} ##
(b) ##P_{2.5}=P_0 (1+r)^{2.5} ##
(c) ##P_{2.5}=P_0 (1+g)^2 (1+r)^{0.5} ##
(d) ##P_{2.5}=P_0 (1+r)^2 (1+g)^{0.5} ##
(e) ##P_{2.5}=P_0 (1+r)^3 (1+g)^{-0.5} ##
Question 289 DDM, expected and historical returns, ROE
In the dividend discount model:
###P_0 = \dfrac{C_1}{r-g}###
The return ##r## is supposed to be the:
(a) Expected future total return of the market price of equity.
(b) Expected future total return of the book price of equity.
(c) Actual historical total return on the market price of equity.
(d) Actual historical total return on the book price of equity.
(e) Actual historical return on equity (ROE) defined as (Net Income / Owners Equity).
Question 352 income and capital returns, DDM, real estate
Two years ago Fred bought a house for $300,000.
Now it's worth $500,000, based on recent similar sales in the area.
Fred's residential property has an expected total return of 8% pa.
The future value of 12 months of rental payments one year ahead is $25,027.77.
What is the expected annual growth rate of the rental payments? In other words, by what percentage increase will Fred have to raise the monthly rent by each year to sustain the expected annual total return of 8%?
(a) -0.3426%
(b) 0%
(e) 3.3652%
A share just paid its semi-annual dividend of $10. The dividend is expected to grow at 2% every 6 months forever. This 2% growth rate is an effective 6 month rate. Therefore the next dividend will be $10.20 in six months. The required return of the stock 10% pa, given as an effective annual rate.
What is the price of the share now?
Question 36 DDM, perpetuity with growth
A stock pays annual dividends which are expected to continue forever. It just paid a dividend of $10. The growth rate in the dividend is 2% pa. You estimate that the stock's required return is 10% pa. Both the discount rate and growth rate are given as effective annual rates. Using the dividend discount model, what will be the share price?
(a) $127.5
(b) $125
(c) $102
(e) $100
A stock is expected to pay the following dividends:
Cash Flows of a Stock
Time (yrs) 0 1 2 3 4 ...
Dividend ($) 0.00 1.00 1.05 1.10 1.15 ...
After year 4, the annual dividend will grow in perpetuity at 5% pa, so;
the dividend at t=5 will be $1.15(1+0.05),
the dividend at t=6 will be $1.15(1+0.05)^2, and so on.
The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates. What is the current price of the stock?
(a) $25.6033
(b) $20.7476
(c) $20.0000
(d) $19.8835
(e) $18.3126
The following is the Dividend Discount Model (DDM) used to price stocks:
### P_0 = \frac{d_1}{r-g} ###
Assume that the assumptions of the DDM hold and that the time period is measured in years.
Which of the following is equal to the expected dividend in 3 years, ## d_3 ##?
(a) ## d_1(1+g)^3 ##
(b) ## P_3(r-g) ##
(c) ## d_2(1+g)^2 ##
(d) ## P_0(1+g)^3(r-g) ##
(e) ## P_0(1+g)^2(r-g) ##
Question 148 DDM, income and capital returns
### p_0 = \frac{d_1}{r - g} ###
Which expression is NOT equal to the expected dividend yield?
(a) ## r-g ##
(b) ## \dfrac{d_1}{p_0} ##
(c) ## \dfrac{d_5}{p_4} ##
(d) ## \dfrac{d_5(1+g)^2}{p_6} ##
(e) ## \dfrac{d_3}{p_0(1+r)^2} ##
###p_0=\frac{d_1}{r_\text{eff}-g_\text{eff}}###
Which expression is NOT equal to the expected capital return?
(a) ## g_\text{eff} ##
(b) ## \dfrac{p_1}{p_0} -1 ##
(c) ## \dfrac{d_5}{d_4} -1 ##
(d) ## \dfrac{d_1}{p_0} - 1 ##
(e) ## \dfrac{p_1-p_0}{p_0} ##
A fairly valued share's current price is $4 and it has a total required return of 30%. Dividends are paid annually and next year's dividend is expected to be $1. After that, dividends are expected to grow by 5% pa in perpetuity. All rates are effective annual returns.
What is the expected dividend income paid at the end of the second year (t=2) and what is the expected capital gain from just after the first dividend (t=1) to just after the second dividend (t=2)? The answers are given in the same order, the dividend and then the capital gain.
(a) $1.3, $0.26
(b) $1.25, $0.25
(c) $1.1025, $0.2205
(d) $1.05, $0.21
(e) $1, $0.2
Question 51 DDM
A stock pays semi-annual dividends. It just paid a dividend of $10. The growth rate in the dividend is 1% every 6 months, given as an effective 6 month rate. You estimate that the stock's required return is 21% pa, as an effective annual rate.
Using the dividend discount model, what will be the share price?
(a) $50.00
(b) $50.50
Question 488 income and capital returns, payout policy, payout ratio, DDM
Two companies BigDiv and ZeroDiv are exactly the same except for their dividend payouts.
BigDiv pays large dividends and ZeroDiv doesn't pay any dividends.
Currently the two firms have the same earnings, assets, number of shares, share price, expected total return and risk.
Assume a perfect world with no taxes, no transaction costs, no asymmetric information and that all assets including business projects are fairly priced and therefore zero-NPV.
All things remaining equal, which of the following statements is NOT correct?
(a) BigDiv is expected to have a lower capital return than ZeroDiv in the future.
(b) BigDiv is expected to have a lower total return than ZeroDiv in the future.
(c) ZeroDiv's assets are likely to grow faster than BigDiv's.
(d) ZeroDiv's share price will increase faster than BigDiv's.
(e) BigDiv currently has a higher payout ratio than ZeroDiv.
Question 498 NPV, Annuity, perpetuity with growth, multi stage growth model
A business project is expected to cost $100 now (t=0), then pay $10 at the end of the third (t=3), fourth, fifth and sixth years, and then grow by 5% pa every year forever. So the cash flow will be $10.5 at the end of the seventh year (t=7), then $11.025 at the end of the eighth year (t=8) and so on perpetually. The total required return is 10℅ pa.
Which of the following formulas will NOT give the correct net present value of the project?
(a) ##-100+ \dfrac{ \dfrac{10}{0.1} \left(1-\dfrac{1}{(1+0.1)^3} \right)}{(1+0.1)^2} +\dfrac{\left(\dfrac{10}{0.1-0.05}\right)}{(1+0.1)^5} ##
(b) ##-100+ \dfrac{10}{(1+0.1)^3} +\dfrac{10}{(1+0.1)^4} +\dfrac{10}{(1+0.1)^5} +\dfrac{\left(\dfrac{10}{0.1-0.05}\right)}{(1+0.1)^5} ##
(c) ##-100+ \dfrac{ \dfrac{10}{0.1} \left(1-\dfrac{1}{(1+0.1)^4} \right)}{(1+0.1)^2} +\dfrac{\left(\dfrac{10.5}{0.1-0.05}\right)}{(1+0.1)^6} ##
(d) ##-100+ \dfrac{10}{(1+0.1)^3} +\dfrac{10}{(1+0.1)^4} +\dfrac{10}{(1+0.1)^5} +\dfrac{10}{(1+0.1)^6} +\dfrac{\left(\dfrac{10.5}{0.1-0.05}\right)}{(1+0.1)^6} ##
(e) ##-100+ \dfrac{ \dfrac{10}{0.1} \left(1-\dfrac{1}{(1+0.1)^3} \right)}{(1+0.1)^3} +\dfrac{\left(\dfrac{10}{0.1-0.05}\right)}{(1+0.1)^5} ##
Question 505 equivalent annual cash flow
A low-quality second-hand car can be bought now for $1,000 and will last for 1 year before it will be scrapped for nothing.
A high-quality second-hand car can be bought now for $4,900 and it will last for 5 years before it will be scrapped for nothing.
What is the equivalent annual cost of each car? Assume a discount rate of 10% pa, given as an effective annual rate.
The answer choices are given as the equivalent annual cost of the low-quality car and then the high quality car.
(a) $100, $490
(b) $909.09, $608.5
(c) $1,000, $980
(d) $1,000, $1578.3
(e) $1,100, $1,292.61
Question 180 equivalent annual cash flow, inflation, real and nominal returns and cash flows
Details of two different types of light bulbs are given below:
Low-energy light bulbs cost $3.50, have a life of nine years, and use about $1.60 of electricity a year, paid at the end of each year.
Conventional light bulbs cost only $0.50, but last only about a year and use about $6.60 of energy a year, paid at the end of each year.
The real discount rate is 5%, given as an effective annual rate. Assume that all cash flows are real. The inflation rate is 3% given as an effective annual rate.
Find the Equivalent Annual Cost (EAC) of the low-energy and conventional light bulbs. The below choices are listed in that order.
(a) 1.4873, 6.7857
(b) 1.6525, 6.7857
(c) 2.1415, 7.1250
(d) 14.8725, 6.7857
(e) 2.0924, 7.1250
You're advising your superstar client 40-cent who is weighing up buying a private jet or a luxury yacht. 40-cent is just as happy with either, but he wants to go with the more cost-effective option. These are the cash flows of the two options:
The private jet can be bought for $6m now, which will cost $12,000 per month in fuel, piloting and airport costs, payable at the end of each month. The jet will last for 12 years.
Or the luxury yacht can be bought for $4m now, which will cost $20,000 per month in fuel, crew and berthing costs, payable at the end of each month. The yacht will last for 20 years.
What's unusual about 40-cent is that he is so famous that he will actually be able to sell his jet or yacht for the same price as it was bought since the next generation of superstar musicians will buy it from him as a status symbol.
Bank interest rates are 10% pa, given as an effective annual rate. You can assume that 40-cent will live for another 60 years and that when the jet or yacht's life is at an end, he will buy a new one with the same details as above.
Would you advise 40-cent to buy the or the ?
Note that the effective monthly rate is ##r_\text{eff monthly}=(1+0.1)^{1/12}-1=0.00797414##
Question 215 equivalent annual cash flow, effective rate conversion
You're about to buy a car. These are the cash flows of the two different cars that you can buy:
You can buy an old car for $5,000 now, for which you will have to buy $90 of fuel at the end of each week from the date of purchase. The old car will last for 3 years, at which point you will sell the old car for $500.
Or you can buy a new car for $14,000 now for which you will have to buy $50 of fuel at the end of each week from the date of purchase. The new car will last for 4 years, at which point you will sell the new car for $1,000.
Bank interest rates are 10% pa, given as an effective annual rate. Assume that there are exactly 52 weeks in a year. Ignore taxes and environmental and pollution factors.
Should you buy the or the ?
Question 348 PE ratio, Multiples valuation
Estimate the US bank JP Morgan's share price using a price earnings (PE) multiples approach with the following assumptions and figures only:
The major US banks JP Morgan Chase (JPM), Citi Group (C) and Wells Fargo (WFC) are comparable companies;
JP Morgan Chase's historical earnings per share (EPS) is $4.37;
Citi Group's share price is $50.05 and historical EPS is $4.26;
Wells Fargo's share price is $48.98 and historical EPS is $3.89.
Note: Figures sourced from Google Finance on 24 March 2014.
(e) $2.7849
Estimate the Chinese bank ICBC's share price using a backward-looking price earnings (PE) multiples approach with the following assumptions and figures only. Note that the renminbi (RMB) is the Chinese currency, also known as the yuan (CNY).
The 4 major Chinese banks ICBC, China Construction Bank (CCB), Bank of China (BOC) and Agricultural Bank of China (ABC) are comparable companies;
ICBC 's historical earnings per share (EPS) is RMB 0.74;
CCB's backward-looking PE ratio is 4.59;
BOC 's backward-looking PE ratio is 4.78;
ABC's backward-looking PE ratio is also 4.78;
Note: Figures sourced from Google Finance on 25 March 2014. Share prices are from the Shanghai stock exchange.
(a) RMB 6.4595
(b) RMB 6.3739
(c) RMB 6.3311
(d) RMB 3.4903
(e) RMB 3.3966
Question 341 Multiples valuation, PE ratio
Estimate Microsoft's (MSFT) share price using a price earnings (PE) multiples approach with the following assumptions and figures only:
Apple, Google and Microsoft are comparable companies,
Apple's (AAPL) share price is $526.24 and historical EPS is $40.32.
Google's (GOOG) share price is $1,215.65 and historical EPS is $36.23.
Micrsoft's (MSFT) historical earnings per share (EPS) is $2.71.
Source: Google Finance 28 Feb 2014.
(c) $30.83
(d) $28.25
(e) $8.60
Which firms tend to have low forward-looking price-earnings (PE) ratios?
Only consider firms with positive earnings, disregard firms with negative earnings and therefore negative PE ratios.
(a) Illiquid small private companies.
(b) High growth technology firms.
(c) Firms expected to have temporarily low earnings over the next year, but with higher earnings later.
(d) Firms with a very low level of systematic risk.
(e) Firms whose assets include a very large proportion of cash.
Private equity firms are known to buy medium sized private companies operating in the same industry, merge them together into a larger company, and then sell it off in a public float (initial public offering, IPO).
If medium-sized private companies trade at PE ratios of 5 and larger listed companies trade at PE ratios of 15, what return can be achieved from this strategy?
Assume that:
The medium-sized companies can be bought, merged and sold in an IPO instantaneously.
There are no costs of finding, valuing, merging and restructuring the medium sized companies. Also, there is no competition to buy the medium-sized companies from other private equity firms.
The large merged firm's earnings are the sum of the medium firms' earnings.
The only reason for the difference in medium and large firm's PE ratios is due to the illiquidity of the medium firms' shares.
Return is defined as: ##r_{0→1} = (p_1-p_0+c_1)/p_0## , where time zero is just before the merger and time one is just after.
(a) 300%.
(b) 200%
(c) 33.33%
(d) 30%
(e) 20%
Question 290 APR, effective rate, debt terminology
Which of the below statements about effective rates and annualised percentage rates (APR's) is NOT correct?
(a) An effective annual rate could be called: "a yearly rate compounding per year".
(b) An APR compounding monthly could be called: "a yearly rate compounding per month".
(c) An effective monthly rate could be called: "a yearly rate compounding per month".
(d) An APR compounding daily could be called: "a yearly rate compounding per day".
(e) An effective 2-year rate could be called: "a 2-year rate compounding every 2 years".
Which of the following statements about effective rates and annualised percentage rates (APR's) is NOT correct?
(a) Effective rates compound once over their time period. So an effective monthly rate compounds once per month.
(b) APR's compound more than once per year. So an APR compounding monthly compounds 12 times per year. The exception is an APR that compounds annually (once per year) which is the same thing as an effective annual rate.
(c) To convert an effective rate to an APR, multiply the effective rate by the number of time periods in one year. So an effective monthly rate multiplied by 12 is equal to an APR compounding monthly.
(d) To convert an APR compounding monthly to an effective monthly rate, divide the APR by the number of months in one year (12).
(e) To convert an APR compounding monthly to an effective weekly rate, divide the APR by the number of weeks in one year (approximately 52).
Question 16 credit card, APR, effective rate
A credit card offers an interest rate of 18% pa, compounding monthly.
Find the effective monthly rate, effective annual rate and the effective daily rate. Assume that there are 365 days in a year.
All answers are given in the same order:
### r_\text{eff monthly} , r_\text{eff yearly} , r_\text{eff daily} ###
(a) 0.0072, 0.09, 0.0002.
(b) 0.0139, 0.18, 0.0005.
(d) 0.015, 0.1956, 0.0005.
(e) 0.015, 0.1956, 0.006.
Question 131 APR, effective rate
Calculate the effective annual rates of the following three APR's:
A credit card offering an interest rate of 18% pa, compounding monthly.
A bond offering a yield of 6% pa, compounding semi-annually.
An annual dividend-paying stock offering a return of 10% pa compounding annually.
##r_\text{credit card, eff yrly}##, ##r_\text{bond, eff yrly}##, ##r_\text{stock, eff yrly}##
(a) 0.1956, 0.0609, 0.1.
(b) 0.015, 0.09, 0.1.
(e) 6.2876, 0.1236, 0.1.
Question 64 inflation, real and nominal returns and cash flows, APR, effective rate
In Germany, nominal yields on semi-annual coupon paying Government Bonds with 2 years until maturity are currently 0.04% pa.
The inflation rate is currently 1.4% pa, given as an APR compounding per quarter. The inflation rate is not expected to change over the next 2 years.
What is the real yield on these bonds, given as an APR compounding every 6 months?
(a) -1.3529627%
(b) -0.4977348%
(c) 0.4977348%
(d) 1.3529627%
(e) 1.3621776%
Question 265 APR, Annuity
On his 20th birthday, a man makes a resolution. He will deposit $30 into a bank account at the end of every month starting from now, which is the start of the month. So the first payment will be in one month. He will write in his will that when he dies the money in the account should be given to charity.
The bank account pays interest at 6% pa compounding monthly, which is not expected to change.
If the man lives for another 60 years, how much money will be in the bank account if he dies just after making his last (720th) payment?
(c) $21,600.00
(d) $15,993.85
(e) $5,834.58
Question 128 debt terminology, needs refinement
An 'interest payment' is the same thing as a 'coupon payment'. or ?
Question 129 debt terminology
An 'interest rate' is the same thing as a 'coupon rate'. or ?
An 'interest rate' is the same thing as a 'yield'. or ?
Which of the following statements is NOT equivalent to the yield on debt?
Assume that the debt being referred to is fairly priced, but do not assume that it's priced at par.
(a) Debt coupon rate.
(b) Required return on debt.
(c) Total return on debt.
(d) Opportunity cost of debt.
(e) Cost of debt capital.
Which of the following statements is NOT correct? Borrowers:
(a) Receive cash at the start and promise to pay cash in the future, as set out in the debt contract.
(b) Are debtors.
(c) Owe money.
(d) Are funded by debt.
(e) Buy debt.
Which of the following statements is NOT correct? Lenders:
(a) Are long debt.
(b) Invest in debt.
(c) Are owed money.
(d) Provide debt funding.
(e) Have debt liabilities.
Question 19 fully amortising loan, APR
You want to buy an apartment priced at $300,000. You have saved a deposit of $30,000. The bank has agreed to lend you the $270,000 as a fully amortising loan with a term of 25 years. The interest rate is 12% pa and is not expected to change.
What will be your monthly payments? Remember that mortgage loan payments are paid in arrears (at the end of the month).
(a) 900
(b) 2,700
(c) 2,722.1
(d) 2,843.71
(e) 34,424.99
You want to buy an apartment worth $500,000. You have saved a deposit of $50,000. The bank has agreed to lend you the $450,000 as a fully amortising mortgage loan with a term of 25 years. The interest rate is 6% pa and is not expected to change.
What will be your monthly payments?
(a) 1,500.00
(b) 2,250.00
(c) 2,855.79
Question 134 fully amortising loan, APR
You want to buy an apartment worth $400,000. You have saved a deposit of $80,000. The bank has agreed to lend you the $320,000 as a fully amortising mortgage loan with a term of 30 years. The interest rate is 6% pa and is not expected to change. What will be your monthly payments?
(b) $1,600.00
(c) $1,885.99
(d) $1,918.56
(e) $23,247.65
You want to buy an apartment priced at $500,000. You have saved a deposit of $50,000. The bank has agreed to lend you the $450,000 as a fully amortising loan with a term of 30 years. The interest rate is 6% pa and is not expected to change. What will be your monthly payments?
(a) $32,692.01
How much did you borrow? After 5 years, how much will be owing on the mortgage? The interest rate is still 9% and is not expected to change.
(a) 246,567.70, 93,351.63
(b) 246,567.70, 235,741.91
(c) 248,563.73, 96,346.75
(d) 248,563.73, 238,323.24
(e) 256,580.38, 245,314.97
How much did you borrow? After 10 years, how much will be owing on the mortgage? The interest rate is still 9% and is not expected to change.
(a) 184,925.77, 164,313.82
(c) 186,422.80, 166,717.43
You just agreed to a 30 year fully amortising mortgage loan with monthly payments of $2,500. The interest rate is 9% pa which is not expected to change.
How much did you borrow? After 10 years, how much will be owing on the mortgage? The interest rate is still 9% and is not expected to change. The below choices are given in the same order.
(a) $320,725.47, $284,977.19
(b) $310,704.66, $277,862.39
(c) $310,704.66, $197,354.23
(d) $308,209.62, $273,856.37
(e) $308,209.62, $192,529.73
You want to buy a house priced at $400,000. You have saved a deposit of $40,000. The bank has agreed to lend you $360,000 as a fully amortising loan with a term of 30 years. The interest rate is 8% pa payable monthly and is not expected to change.
(a) $1,000
(b) $1,106.6497
(c) $2,400
(e) $2,641.5525
Question 42 interest only loan
You just signed up for a 30 year interest-only mortgage with monthly payments of $3,000 per month. The interest rate is 6% pa which is not expected to change.
How much did you borrow? After 15 years, just after the 180th payment at that time, how much will be owing on the mortgage? The interest rate is still 6% and is not expected to change. Remember that the mortgage is interest-only and that mortgage payments are paid in arrears (at the end of the month).
Question 107 interest only loan
You want to buy an apartment worth $300,000. You have saved a deposit of $60,000.
The bank has agreed to lend you $240,000 as an interest only mortgage loan with a term of 30 years. The interest rate is 6% pa and is not expected to change. What will be your monthly payments?
(a) 17,435.74
(e) 666.67
You want to buy an apartment priced at $500,000. You have saved a deposit of $50,000. The bank has agreed to lend you the $450,000 as an interest only loan with a term of 30 years. The interest rate is 6% pa and is not expected to change. What will be your monthly payments?
(a) $ 1,250.00
(b) $ 2,250.00
(c) $ 2,652.17
(d) $ 2,697.98
(e) $ 32,692.01
A prospective home buyer can afford to pay $2,000 per month in mortgage loan repayments. The central bank recently lowered its policy rate by 0.25%, and residential home lenders cut their mortgage loan rates from 4.74% to 4.49%.
How much more can the prospective home buyer borrow now that interest rates are 4.49% rather than 4.74%? Give your answer as a proportional increase over the original amount he could borrow (##V_\text{before}##), so:
###\text{Proportional increase} = \frac{V_\text{after}-V_\text{before}}{V_\text{before}} ###
Interest rates are expected to be constant over the life of the loan.
Loans are interest-only and have a life of 30 years.
Mortgage loan payments are made every month in arrears and all interest rates are given as annualised percentage rates compounding per month.
(a) 0.055679
Question 459 interest only loan, inflation
In Australia in the 1980's, inflation was around 8% pa, and residential mortgage loan interest rates were around 14%.
In 2013, inflation was around 2.5% pa, and residential mortgage loan interest rates were around 4.5%.
If a person can afford constant mortgage loan payments of $2,000 per month, how much more can they borrow when interest rates are 4.5% pa compared with 14.0% pa?
Give your answer as a proportional increase over the amount you could borrow when interest rates were high ##(V_\text{high rates})##, so:
###\text{Proportional increase} = \dfrac{V_\text{low rates}-V_\text{high rates}}{V_\text{high rates}} ###
Mortgage loan payments are made every month in arrears and all interest rates are given as annualised percentage rates (APR's) compounding per month.
(a) 0.095
(e) 2.111111
Question 509 bond pricing
Calculate the price of a newly issued ten year bond with a face value of $100, a yield of 8% pa and a fixed coupon rate of 6% pa, paid annually. So there's only one coupon per year, paid in arrears every year.
Calculate the price of a newly issued ten year bond with a face value of $100, a yield of 8% pa and a fixed coupon rate of 6% pa, paid semi-annually. So there are two coupons per year, paid in arrears every six months.
Question 11 bond pricing
For a price of $100, Vera will sell you a 2 year bond paying semi-annual coupons of 10% pa. The face value of the bond is $100. Other bonds with similar risk, maturity and coupon characteristics trade at a yield of 8% pa.
Would you like to her bond or politely ?
For a price of $95, Nicole will sell you a 10 year bond paying semi-annual coupons of 8% pa. The face value of the bond is $100. Other bonds with the same risk, maturity and coupon characteristics trade at a yield of 8% pa.
Would you like to the bond or politely ?
Question 23 bond pricing, premium par and discount bonds
Bonds X and Y are issued by the same US company. Both bonds yield 10% pa, and they have the same face value ($100), maturity, seniority, and payment frequency.
The only difference is that bond X and Y's coupon rates are 8 and 12% pa respectively. Which of the following statements is true?
(a) Bonds X and Y are premium bonds.
(b) Bonds X and Y are discount bonds.
(c) Bond X is a discount bond but bond Y is a premium bond.
(d) Bond X is a premium bond but bond Y is a discount bond.
(e) Bonds X and Y are par bonds.
A two year Government bond has a face value of $100, a yield of 0.5% and a fixed coupon rate of 0.5%, paid semi-annually. What is its price?
Question 48 IRR, NPV, bond pricing, premium par and discount bonds, market efficiency
The theory of fixed interest bond pricing is an application of the theory of Net Present Value (NPV). Also, a 'fairly priced' asset is not over- or under-priced. Buying or selling a fairly priced asset has an NPV of zero.
Considering this, which of the following statements is NOT correct?
(a) The internal rate of return (IRR) of buying a fairly priced bond is equal to the bond's yield.
(b) The Present Value of a fairly priced bond's coupons and face value is equal to its price.
(c) If a fairly priced bond's required return rises, its price will fall.
(d) Fairly priced premium bonds' yields are less than their coupon rates, prices are more than their face values, and the NPV of buying them is therefore positive.
(e) The NPV of buying a fairly priced bond is zero.
A two year Government bond has a face value of $100, a yield of 2.5% pa and a fixed coupon rate of 0.5% pa, paid semi-annually. What is its price?
(a) 90.6421
(b) 95.1524
(c) 95.2055
(d) 96.1219
(e) 103.9751
Question 63 bond pricing, NPV, market efficiency
(a) The internal rate of return (IRR) of buying a bond is equal to the bond's yield.
(c) If the required return of a bond falls, its price will fall.
(d) Fairly priced discount bonds' yield is more than the coupon rate, price is less than face value, and the NPV of buying them is zero.
A bond maturing in 10 years has a coupon rate of 4% pa, paid semi-annually. The bond's yield is currently 6% pa. The face value of the bond is $100. What is its price?
(e) $85.12
Question 138 bond pricing, premium par and discount bonds
Bonds A and B are issued by the same Australian company. Both bonds yield 7% pa, and they have the same face value ($100), maturity, seniority, and payment frequency.
The only difference is that bond A pays coupons of 10% pa and bond B pays coupons of 5% pa. Which of the following statements is true about the bonds' prices?
(a) The prices of bonds A and B will be more than $100.
(b) The prices of bonds A and B will be less than $100.
(c) Bond A will have a price more than $100, and bond B will have a price less than $100.
(d) Bond A will have a price less than $100, and bond B will have a price more than $100.
(e) Bonds A and B will both have a price of $100.
Bonds X and Y are issued by different companies, but they both pay a semi-annual coupon of 10% pa and they have the same face value ($100) and maturity (3 years).
The only difference is that bond X and Y's yields are 8 and 12% pa respectively. Which of the following statements is true?
(e) Bonds X and Y have the same price.
A three year bond has a fixed coupon rate of 12% pa, paid semi-annually. The bond's yield is currently 6% pa. The face value is $100. What is its price?
Bonds X and Y are issued by different companies, but they both pay a semi-annual coupon of 10% pa and they have the same face value ($100), maturity (3 years) and yield (10%) as each other.
Which of the following statements is true?
A four year bond has a face value of $100, a yield of 6% and a fixed coupon rate of 12%, paid semi-annually. What is its price?
(d) 111.1513
Which one of the following bonds is trading at a discount?
(a) a ten-year bond with a $4000 face value whose yield to maturity is 6.0% and coupon rate is 6.5% paid semi-annually.
(b) a 6-year bond with a principal of $40,000 and a price of $45,000.
(c) a 15-year bond with a $10,000 face value whose yield to maturity is 8.0% and coupon rate is 10.0% paid semi-annually.
(d) a two-year bond with a $50,000 face value whose yield to maturity is 5.2% and coupon rate is 5.2% paid semi-annually.
(e) None of the above bonds are discount bonds.
Question 179 bond pricing, capital raising
A firm wishes to raise $20 million now. They will issue 8% pa semi-annual coupon bonds that will mature in 5 years and have a face value of $100 each. Bond yields are 6% pa, given as an APR compounding every 6 months, and the yield curve is flat.
How many bonds should the firm issue?
(a) 140,202
(b) 184,280
(c) 184,460
(d) 186,881
(e) 200,000
A five year bond has a face value of $100, a yield of 12% and a fixed coupon rate of 6%, paid semi-annually.
What is the bond's price?
(e) 87.3629
Which one of the following bonds is trading at par?
(d) a two-year bond with a $50,000 face value whose yield to maturity is 5.2% compounding semi-annually which has a price of $50,000.
(e) None of the above bonds are trading at par.
A firm wishes to raise $8 million now. They will issue 7% pa semi-annual coupon bonds that will mature in 10 years and have a face value of $100 each. Bond yields are 10% pa, given as an APR compounding every 6 months, and the yield curve is flat.
(b) 98,393
(c) 90,480
(d) 80,000
(e) 64,039
Which one of the following bonds is trading at a premium?
(a) a ten-year bond with a $4,000 face value whose yield to maturity is 6.0% and coupon rate is 5.9% paid semi-annually.
(b) a fifteen-year bond with a $10,000 face value whose yield to maturity is 8.0% and coupon rate is 7.8% paid semi-annually.
(c) a five-year bond with a $2,000 face value whose yield to maturity is 7.0% and coupon rate is 7.2% paid semi-annually.
(e) None of the above bonds are premium bonds.
A firm wishes to raise $10 million now. They will issue 6% pa semi-annual coupon bonds that will mature in 8 years and have a face value of $1,000 each. Bond yields are 10% pa, given as an APR compounding every 6 months, and the yield curve is flat.
(a) 9,022.2 bonds
(b) 10,000.0 bonds
(c) 11,484.5 bonds
(d) 12,712.9 bonds
(e) 12,767.4 bonds
A four year bond has a face value of $100, a yield of 9% and a fixed coupon rate of 6%, paid semi-annually. What is its price?
(c) $72.592
A 10 year bond has a face value of $100, a yield of 6% pa and a fixed coupon rate of 8% pa, paid semi-annually. What is its price?
(d) $126.628
A 30 year Japanese government bond was just issued at par with a yield of 1.7% pa. The fixed coupon payments are semi-annual. The bond has a face value of $100.
Six months later, just after the first coupon is paid, the yield of the bond increases to 2% pa. What is the bond's new price?
Question 328 bond pricing, APR
A 10 year Australian government bond was just issued at par with a yield of 3.9% pa. The fixed coupon payments are semi-annual. The bond has a face value of $1,000.
Six months later, just after the first coupon is paid, the yield of the bond decreases to 3.65% pa. What is the bond's new price?
(c) $1,033.8330
Bonds X and Y are issued by the same US company. Both bonds yield 6% pa, and they have the same face value ($100), maturity, seniority, and payment frequency.
The only difference is that bond X pays coupons of 8% pa and bond Y pays coupons of 12% pa. Which of the following statements is true?
Below are some statements about loans and bonds. The first descriptive sentence is correct. But one of the second sentences about the loans' or bonds' prices is not correct. Which statement is NOT correct? Assume that interest rates are positive.
Note that coupons or interest payments are the periodic payments made throughout a bond or loan's life. The face or par value of a bond or loan is the amount paid at the end when the debt matures.
(a) A bullet loan has no interest payments but it does have a face value. Therefore it's a discount loan.
(b) A fully amortising loan has interest payments but does not have a face value. Therefore it's a premium loan.
(c) An interest only loan has interest payments and its price and face value are equal. Therefore it's a par loan.
(d) A zero coupon bond has no coupon payments but it does have a face value. Therefore it's a premium bond.
(e) A balloon loan has interest payments and its face value is more than its price. Therefore it's a discount loan.
Question 581 APR, effective rate, effective rate conversion
A home loan company advertises an interest rate of 6% pa, payable monthly. Which of the following statements about the interest rate is NOT correct? All rates are given to four decimal places.
(a) The APR compounding monthly is 6.0000% per annum.
(b) The effective monthly rate is 0.5000% per month.
(c) The effective annual rate is 6.1678% per annum.
(d) The effective 6 month rate is 3.0000% per six months.
(e) The APR compounding semi-annually is 6.0755% per annum.
Question 616 idiom, debt terminology, bond pricing
"Buy low, sell high" is a phrase commonly heard in financial markets. It states that traders should try to buy assets at low prices and sell at high prices.
Traders in the fixed-coupon bond markets often quote promised bond yields rather than prices. Fixed-coupon bond traders should try to:
(a) Buy at low yields, sell at high yields.
(b) Buy at high yields, sell at low yields.
(c) Buy at high yields, sell at high yields.
(d) Buy at low yields, sell at low yields.
(e) There is no preferable yield to buy or sell fixed-coupon debt.
Question 35 bond pricing, zero coupon bond, term structure of interest rates, forward interest rate
A European company just issued two bonds, a
1 year zero coupon bond at a yield of 8% pa, and a
2 year zero coupon bond at a yield of 10% pa.
What is the company's forward rate over the second year (from t=1 to t=2)? Give your answer as an effective annual rate, which is how the above bond yields are quoted.
An Australian company just issued two bonds:
A 1 year zero coupon bond at a yield of 8% pa, and
A 2 year zero coupon bond at a yield of 10% pa.
What is the forward rate on the company's debt from years 1 to 2? Give your answer as an APR compounding every 6 months, which is how the above bond yields are quoted.
(a) 6.01%
(b) 6.02%
(c) 9.20%
(d) 12.02%
(e) 18.40%
(a) $1,006.25
The Australian Federal Government lends money to domestic students to pay for their university education. This is known as the Higher Education Contribution Scheme (HECS). The nominal interest rate on the HECS loan is set equal to the consumer price index (CPI) inflation rate. The interest is capitalised every year, which means that the interest is added to the principal. The interest and principal does not need to be repaid by students until they finish study and begin working.
Which of the following statements about HECS loans is NOT correct?
(a) The real interest rate is zero.
(b) The nominal amount owing on the loan will increase at the inflation rate over time, if there are no extra payments or borrowings.
(c) The real amount owing on the loan in today's money will remain the same over time, if there are no extra payments or borrowings.
(d) The interest rate on the HECS loan advantages rich domestic students because they are able to pay off their debt sooner and avoid paying as much HECS interest.
(e) The interest rate on the HECS loan advantages all domestic students because the alternative of borrowing from the bank to pay for education would attract a higher interest rate.
Question 729 book and market values, balance sheet, no explanation
If a firm makes a profit and pays no dividends, which of the following accounts will increase?
(a) Asset revaluation reserve.
(b) Foreign currency translation reserve.
(c) Retained earnings, also known as retained profits.
(d) Contributed equity, also known as paid up capital.
(e) General reserve.
Question 731 DDM, income and capital returns, no explanation
In the dividend discount model (DDM), share prices fall when dividends are paid. Let the high price before the fall be called the peak, and the low price after the fall be called the trough.
###P_0=\dfrac{C_1}{r-g}###
Which of the following statements about the DDM is NOT correct?
(a) In between dividends, the stock price is expected to grow by the total return 'r'.
(b) From trough to trough, the stock price is expected to grow by the capital return 'g'.
(c) From peak to peak, the stock price is expected to grow by the capital return 'g'.
(d) Dividends are expected to grow by the total return 'r'.
(e) If the stock's dividends are re-invested by using the cash to buy more stocks, then the growth rate of the shareholder's wealth will be the total return 'r'.
Question 503 DDM, NPV, stock pricing
A share currently worth $100 is expected to pay a constant dividend of $4 for the next 5 years with the first dividend in one year (t=1) and the last in 5 years (t=5).
The total required return is 10% pa.
What do you expected the share price to be in 5 years, just after the dividend at that time has been paid?
(a) $100
Question 173 CFFA
Find Candys Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.
Candys Corp
Income Statement for
year ending 30th June 2013
$m
COGS 50
Operating expense 10
Depreciation 20
Interest expense 10
Income before tax 110
Tax at 30% 33
Net income 77
as at 30th June 2013 2012
$m $m
Current assets 220 180
Cost 300 340
Accumul. depr. 60 40
Carrying amount 240 300
Total assets 460 480
Current liabilities 175 190
Non-current liabilities 135 130
Owners' equity
Retained earnings 50 60
Contributed equity 100 100
Total L and OE 460 480
Note: all figures are given in millions of dollars ($m).
(b) 182
(c) 112
(e) 52
Why is Capital Expenditure (CapEx) subtracted in the Cash Flow From Assets (CFFA) formula?
###CFFA=NI+Depr-CapEx - \Delta NWC+IntExp###
(a) CapEx is added in the Net Income (NI) equation so it needs subtracting in the CFFA equation.
(b) CapEx is a financing cash flow that needs to be ignored. Therefore it should be subtracted.
(c) CapEx is not a cash flow, it's a non-cash expense made up by accountants that needs to be subtracted.
(d) CapEx is subtracted to account for the net cash spent on capital assets.
(e) CapEx is subtracted because it's too hard to predict, therefore we exclude it.
Cash Flow From Assets (CFFA) can be defined as:
(a) Cash available to distribute to creditors and stockholders.
(b) Cash flow to creditors minus cash flow to stockholders.
(c) Net income (or earnings) plus depreciation plus interest expense.
(d) Net income minus the increase in net working capital.
(e) Net income minus net capital spending minus the increase in net working capital.
A firm has forecast its Cash Flow From Assets (CFFA) for this year and management is worried that it is too low. Which one of the following actions will lead to a higher CFFA for this year (t=0 to 1)? Only consider cash flows this year. Do not consider cash flows after one year, or the change in the NPV of the firm. Consider each action in isolation.
(a) Buy less land, buildings and trucks than what was planned. Assume that this has no impact on revenue.
(b) Pay less cash to creditors by refinancing the firm's existing coupon bonds with zero-coupon bonds that require no interest payments. Assume that there are no transaction costs and that both types of bonds have the same yield to maturity.
(c) Change the depreciation method used for tax purposes from diminishing value to straight line, so less depreciation occurs this year and more occurs in later years. Assume that the government's tax department allow this.
(d) Buying more inventory than was planned, so there is an increase in net working capital. Assume that there is no increase in sales.
(e) Raising new equity through a rights issue. Assume that all of the money raised is spent on new capital assets such as land and trucks, but they will be fitted out and delivered in one year so no new cash will be earned from them.
Question 349 CFFA, depreciation tax shield
Which one of the following will decrease net income (NI) but increase cash flow from assets (CFFA) in this year for a tax-paying firm, all else remaining constant?
###NI = (Rev-COGS-FC-Depr-IntExp).(1-t_c )### ###CFFA=NI+Depr-CapEx - \Delta NWC+IntExp###
(a) An increase in revenue (Rev).
(b) An increase in rent expense (part of fixed costs, FC).
(c) An increase in depreciation expense (Depr).
(d) An decrease in net working capital (ΔNWC).
(e) An increase in dividends.
Find Sidebar Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.
Sidebar Corp
COGS 100
Rent expense 22
Taxable Income 210
Taxes at 30% 63
Net income 147
Inventory 70 50
Trade debtors 11 16
Rent paid in advance 4 3
PPE 700 680
Trade creditors 11 19
Bond liabilities 400 390
Retained profits 154 120
The cash flow from assets was:
(a) $138m
(b) $142m
(c) $143m
(d) $172m
(e) $176m
Which one of the following will have no effect on net income (NI) but decrease cash flow from assets (CFFA or FFCF) in this year for a tax-paying firm, all else remaining constant?
###NI=(Rev-COGS-FC-Depr-IntExp).(1-t_c )### ###CFFA=NI+Depr-CapEx - ΔNWC+IntExp###
(b) An increase in rent expense (a type of recurring fixed cost, FC).
(d) An increase in inventories (a current asset).
(e) An decrease in interest expense (IntExp).
Find Ching-A-Lings Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.
Ching-A-Lings Corp
Taxable Income 30
Taxes at 30% 9
Trade debtors 14 2
Trade creditors 4 10
(a) $43m
(b) $31m
(c) $23m
(d) $11m
(e) $1m
Over the next year, the management of an unlevered company plans to:
Make $5m in sales, $1.9m in net income and $2m in equity free cash flow (EFCF).
Pay dividends of $1m.
Complete a $1.3m share buy-back.
All amounts are received and paid at the end of the year so you can ignore the time value of money.
The firm has sufficient retained profits to legally pay the dividend and complete the buy back.
The firm plans to run a very tight ship, with no excess cash above operating requirements currently or over the next year.
How much new equity financing will the company need? In other words, what is the value of new shares that will need to be issued?
(a) $2m
(b) $1m
(c) $0.4m
(d) $0.3m
(e) No new shares need to be issued, the firm will be sufficiently financed.
Read the following financial statements and calculate the firm's free cash flow over the 2014 financial year.
UBar Corp
Gas expense 8
EBIT 60
Interest expense 0
Cash 30 29
Accounts receivable 5 7
Pre-paid rent expense 1 0
Trade payables 20 18
Accrued gas expense 3 2
Non-current liabilities 0 0
Asset revaluation reserve 5 0
The firm's free cash flow over the 2014 financial year was:
(a) $51
(d) $61
(e) $62
Find Trademark Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.
Trademark Corp
Operating expense 5
Income before tax 30
Tax at 30% 9
Current assets 120 80
Carrying amount 90 100
Current liabilities 75 65
Non-current liabilities 75 55
Contributed equity 50 50
(a) -19
(b) 21
Find UniBar Corp's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.
UniBar Corp
Net income 7
Current liabilities 110 60
(a) 12
(c) -8
(d) -18
Find Piano Bar's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.
Find World Bar's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.
World Bar
Retained earnings 100 100
Note: all figures above and below are given in millions of dollars ($m).
(e) -20
Find Scubar Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.
Scubar Corp
Trade creditors 10 8
(e) $74m
Question 485 capital budgeting, opportunity cost, sunk cost
A young lady is trying to decide if she should attend university or not.
The young lady's parents say that she must attend university because otherwise all of her hard work studying and attending school during her childhood was a waste.
What's the correct way to classify this item from a capital budgeting perspective when trying to decide whether to attend university?
The hard work studying at school in her childhood should be classified as:
(a) A sunk cost.
(b) An opportunity cost.
(c) A negative side effect.
(d) A positive side effect.
(e) A depreciation expense.
A young lady is trying to decide if she should attend university. Her friends say that she should go to university because she is more likely to meet a clever young man than if she begins full time work straight away.
What's the correct way to classify this item from a capital budgeting perspective when trying to find the Net Present Value of going to university rather than working?
The opportunity to meet a desirable future spouse should be classified as:
A man is thinking about taking a day off from his casual painting job to relax.
He just woke up early in the morning and he's about to call his boss to say that he won't be coming in to work.
But he's thinking about the hours that he could work today (in the future) which are:
(d) A capital expense.
A man has taken a day off from his casual painting job to relax.
It's the end of the day and he's thinking about the hours that he could have spent working (in the past) which are now:
Question 512 capital budgeting, CFFA
Find the cash flow from assets (CFFA) of the following project.
Project life 2 years
Initial investment in equipment $6m
Depreciation of equipment per year for tax purposes $1m
Unit sales per year 4m
Sale price per unit $8
Variable cost per unit $3
Fixed costs per year, paid at the end of each year $1.5m
Tax rate 30%
Note 1: The equipment will have a book value of $4m at the end of the project for tax purposes. However, the equipment is expected to fetch $0.9 million when it is sold at t=2.
Note 2: Due to the project, the firm will have to purchase $0.8m of inventory initially, which it will sell at t=1. The firm will buy another $0.8m at t=1 and sell it all again at t=2 with zero inventory left. The project will have no effect on the firm's current liabilities.
Find the project's CFFA at time zero, one and two. Answers are given in millions of dollars ($m).
(a) -6, 12.25, 16.68
(b) -6.8, 13.25, 14.05
(c) -6.8, 13.25, 15.88
(d) -6.8, 13.25, 18.51
(e) -6.8, 13.25, 17.71
Question 377 leverage, capital structure
Issuing debt doesn't give away control of the firm because debt holders can't cast votes to determine the company's affairs, such as at the annual general meeting (AGM), and can't appoint directors to the board. or ?
Question 379 leverage, capital structure, payout policy
Companies must pay interest and principal payments to debt-holders. They're compulsory. But companies are not forced to pay dividends to share holders. or ?
Question 94 leverage, capital structure, real estate
Your friend just bought a house for $400,000. He financed it using a $320,000 mortgage loan and a deposit of $80,000.
In the context of residential housing and mortgages, the 'equity' tied up in the value of a person's house is the value of the house less the value of the mortgage. So the initial equity your friend has in his house is $80,000. Let this amount be E, let the value of the mortgage be D and the value of the house be V. So ##V=D+E##.
If house prices suddenly fall by 10%, what would be your friend's percentage change in equity (E)? Assume that the value of the mortgage is unchanged and that no income (rent) was received from the house during the short time over which house prices fell.
### r_{0\rightarrow1}=\frac{p_1-p_0+c_1}{p_0} ###
where ##r_{0-1}## is the return (percentage change) of an asset with price ##p_0## initially, ##p_1## one period later, and paying a cash flow of ##c_1## at time ##t=1##.
(a) -100%
(b) -50%
(c) -12.5%
(d) -10%
(e) -8%
Question 301 leverage, capital structure, real estate
Your friend just bought a house for $1,000,000. He financed it using a $900,000 mortgage loan and a deposit of $100,000.
In the context of residential housing and mortgages, the 'equity' or 'net wealth' tied up in a house is the value of the house less the value of the mortgage loan. Assuming that your friend's only asset is his house, his net wealth is $100,000.
If house prices suddenly fall by 15%, what would be your friend's percentage change in net wealth?
No income (rent) was received from the house during the short time over which house prices fell.
Your friend will not declare bankruptcy, he will always pay off his debts.
(a) -1,000%
(b) -150%
(c) -100%
(e) -10%
Question 67 CFFA, interest tax shield
Here are the Net Income (NI) and Cash Flow From Assets (CFFA) equations:
###NI=(Rev-COGS-FC-Depr-IntExp).(1-t_c)###
###CFFA=NI+Depr-CapEx - \varDelta NWC+IntExp###
What is the formula for calculating annual interest expense (IntExp) which is used in the equations above?
Select one of the following answers. Note that D is the value of debt which is constant through time, and ##r_D## is the cost of debt.
(a) ##D(1+r_D)##
(b) ##D/(1+r_D) ##
(c) ##D.r_D ##
(d) ##D / r_D##
(e) ##NI.r_D##
Question 223 CFFA, interest tax shield
Which one of the following will increase the Cash Flow From Assets in this year for a tax-paying firm, all else remaining constant?
(a) An increase in net capital spending.
(b) A decrease in the cash flow to creditors.
(c) An increase in interest expense.
(d) An increase in net working capital.
(e) A decrease in dividends paid.
(a) An increase in revenue (##Rev##).
(b) An decrease in revenue (##Rev##).
(c) An increase in rent expense (part of fixed costs, ##FC##).
(d) An increase in interest expense (##IntExp##).
Question 68 WACC, CFFA, capital budgeting
A manufacturing company is considering a new project in the more risky services industry. The cash flows from assets (CFFA) are estimated for the new project, with interest expense excluded from the calculations. To get the levered value of the project, what should these unlevered cash flows be discounted by?
Assume that the manufacturing firm has a target debt-to-assets ratio that it sticks to.
(a) The manufacturing firm's before-tax WACC.
(b) The manufacturing firm's after-tax WACC.
(c) A services firm's before-tax WACC, assuming that the services firm has the same debt-to-assets ratio as the manufacturing firm.
(d) A services firm's after-tax WACC, assuming that the services firm has the same debt-to-assets ratio as the manufacturing firm.
(e) The services firm's levered cost of equity.
Question 89 WACC, CFFA, interest tax shield
A retail furniture company buys furniture wholesale and distributes it through its retail stores. The owner believes that she has some good ideas for making stylish new furniture. She is considering a project to buy a factory and employ workers to manufacture the new furniture she's designed. Furniture manufacturing has more systematic risk than furniture retailing.
Her furniture retailing firm's after-tax WACC is 20%. Furniture manufacturing firms have an after-tax WACC of 30%. Both firms are optimally geared. Assume a classical tax system.
Which method(s) will give the correct valuation of the new furniture-making project? Select the most correct answer.
(a) Discount the project's unlevered CFFA by the furniture manufacturing firms' 30% WACC after tax.
(b) Discount the project's unlevered CFFA by the company's 20% WACC after tax.
(c) Discount the project's levered CFFA by the company's 20% WACC after tax.
(d) Discount the project's levered CFFA by the furniture manufacturing firms' 30% WACC after tax.
(e) The methods outlined in answers (a) and (c) will give the same valuations, both are correct.
Question 113 WACC, CFFA, capital budgeting
The US firm Google operates in the online advertising business. In 2011 Google bought Motorola Mobility which manufactures mobile phones.
Assume the following:
Google had a 10% after-tax weighted average cost of capital (WACC) before it bought Motorola.
Motorola had a 20% after-tax WACC before it merged with Google.
Google and Motorola have the same level of gearing.
Both companies operate in a classical tax system.
You are a manager at Motorola. You must value a project for making mobile phones. Which method(s) will give the correct valuation of the mobile phone manufacturing project? Select the most correct answer.
The mobile phone manufacturing project's:
(a) Unlevered CFFA should be discounted by Google's 10% WACC after tax.
(b) Unlevered CFFA should be discounted by Motorola's 20% WACC after tax.
(c) Levered CFFA should be discounted by Google's 10% WACC after tax.
(d) Levered CFFA should be discounted by Motorola's 20% WACC after tax.
(e) Unlevered CFFA by 15%, the average of Google and Motorola's WACC after tax.
Question 69 interest tax shield, capital structure, leverage, WACC
Which statement about risk, required return and capital structure is the most correct?
(a) The before-tax cost of debt is less than the before-tax cost of equity. Therefore debt is a cheaper form of financing than equity so companies should try to finance their projects with debt only.
(b) Debt makes a firm's equity more risky. Therefore the higher the amount of debt, the higher the cost of equity.
(c) The more debt a firm has, the higher its tax shields. Therefore firms should seek to have as much debt and as little equity as possible.
(d) The more debt, the lower the firm's after tax WACC. The after tax WACC is the discount rate that discounts the firm's cash flows, so the lower it is the more the firm is worth. Therefore firms should try to make their after tax WACC as low as possible by using as much debt as possible.
(e) The less debt, the lower the chance of bankruptcy. Therefore firms should try to pay off all of their debt so that they are financed by equity only.
Question 78 WACC, capital structure
A company issues a large amount of bonds to raise money for new projects of similar risk to the company's existing projects. The net present value (NPV) of the new projects is positive but small. Assume a classical tax system. Which statement is NOT correct?
(a) The debt-to-assets (D/V) ratio will increase.
(b) The debt-to-equity ratio (D/E) will increase.
(c) Firm value is likely to have increased due to the higher amount of interest tax shields, assuming that there will not be any costs of financial distress.
(d) The company's after-tax WACC is likely to have decreased.
(e) The company's before-tax WACC is likely to have decreased.
Question 84 WACC, capital structure, capital budgeting
A firm is considering a new project of similar risk to the current risk of the firm. This project will expand its existing business. The cash flows of the project have been calculated assuming that there is no interest expense. In other words, the cash flows assume that the project is all-equity financed.
In fact the firm has a target debt-to-equity ratio of 1, so the project will be financed with 50% debt and 50% equity. To find the levered value of the firm's assets, what discount rate should be applied to the project's unlevered cash flows? Assume a classical tax system.
(a) The required return on equity, ##r_E##
(b) The required return on debt, ##r_D##
(c) The after-tax required return on debt, ##r_D.(1-t_c)##
(d) The after-tax WACC, ##\text{WACC after tax}=\frac{D}{V_L}.r_D.(1-t_c )+\frac{E_L}{V_L}.r_E##
(e) The pre-tax WACC, ##\text{WACC before tax}=\frac{D}{V_L}.r_D+\frac{E_L}{V_L}.r_E##
Question 99 capital structure, interest tax shield, Miller and Modigliani, trade off theory of capital structure
A firm changes its capital structure by issuing a large amount of debt and using the funds to repurchase shares. Its assets are unchanged.
The firm and individual investors can borrow at the same rate and have the same tax rates.
The firm's debt and shares are fairly priced and the shares are repurchased at the market price, not at a premium.
There are no market frictions relating to debt such as asymmetric information or transaction costs.
Shareholders wealth is measured in terms of utiliity. Shareholders are wealth-maximising and risk-averse. They have a preferred level of overall leverage. Before the firm's capital restructure all shareholders were optimally levered.
According to Miller and Modigliani's theory, which statement is correct?
(a) The firm's share price and shareholder wealth will both decrease. This is because the firm will have more debt and therefore more risk so the discount rate applied to its cash flows will be higher, decreasing the value of the firm and therefore the value of the firm's equity and share price.
(b) The firm's share price and shareholder wealth will both increase. This is because the firm will have more debt which will amplify the returns of equity investors. This will mean that returns on equity can be much higher and investors will pay a premium for this, leading to an increase in the stock price.
(c) The firm's share price and shareholder wealth will both increase since it has more debt and therefore more tax shields.
(d) The firm's share price will increase due to the higher value of tax shields. But shareholder wealth will remain unchanged because capital structure is irrelevant when investors can use home-made leverage to create tax-shields themselves.
(e) The firm's share price and shareholder wealth will both increase. This is because the cost of debt is cheaper than equity, leading to a lower (before and after tax) WACC. This lower WACC will lead to a higher value of the firm and a higher share price.
Question 115 capital structure, leverage, WACC
A firm has a debt-to-assets ratio of 50%. The firm then issues a large amount of debt to raise money for new projects of similar risk to the company's existing projects. Assume a classical tax system. Which statement is correct?
(a) The debt-to-assets (D/V) ratio will decrease.
(b) The debt-to-equity ratio (D/E) will decrease.
(c) The firm's cost of equity will decrease.
(d) The company's after-tax WACC will decrease.
(e) The company's before-tax WACC will decrease.
Question 121 capital structure, leverage, financial distress, interest tax shield
Fill in the missing words in the following sentence:
All things remaining equal, as a firm's amount of debt funding falls, benefits of interest tax shields __________ and the costs of financial distress __________.
(a) Fall, fall.
(b) Fall, rises.
(c) Rise, fall.
(d) Rise, rise.
(e) Remain unchanged, remain unchanged.
Question 337 capital structure, interest tax shield, leverage, real and nominal returns and cash flows, multi stage growth model
A fast-growing firm is suitable for valuation using a multi-stage growth model.
It's nominal unlevered cash flow from assets (##CFFA_U##) at the end of this year (t=1) is expected to be $1 million. After that it is expected to grow at a rate of:
12% pa for the next two years (from t=1 to 3),
5% over the fourth year (from t=3 to 4), and
-1% forever after that (from t=4 onwards). Note that this is a negative one percent growth rate.
The nominal WACC after tax is 9.5% pa and is not expected to change.
The nominal WACC before tax is 10% pa and is not expected to change.
The firm has a target debt-to-equity ratio that it plans to maintain.
The inflation rate is 3% pa.
All rates are given as nominal effective annual rates.
What is the levered value of this fast growing firm's assets?
(a) $13.19m
(b) $12.36m
(c) $11.77m
(d) $11.53m
(e) $11.20m
Question 411 WACC, capital structure
A firm plans to issue equity and use the cash raised to pay off its debt. No assets will be bought or sold. Ignore the costs of financial distress.
Which of the following statements is NOT correct, all things remaining equal?
(a) The firm's WACC before tax will rise.
(b) The firm's WACC after tax will rise.
(c) The firm's required return on equity will be lower.
(d) The firm's net income will be higher.
(e) The firm's free cash flow will be lower.
Question 559 variance, standard deviation, covariance, correlation
Which of the following statements about standard statistical mathematics notation is NOT correct?
(a) The arithmetic average of variable X is represented by ##\bar{X}##.
(b) The standard deviation of variable X is represented by ##\sigma_X##.
(c) The variance of variable X is represented by ##\sigma_X^2##.
(d) The covariance between variables X and Y is represented by ##\sigma_{X,Y}^2##.
(e) The correlation between variables X and Y is represented by ##\rho_{X,Y}##.
Question 236 diversification, correlation, risk
Diversification in a portfolio of two assets works best when the correlation between their returns is:
(a) -1
(b) -0.5
(c) 0
(d) 0.5
(e) 1
Question 111 portfolio risk, correlation
All things remaining equal, the variance of a portfolio of two positively-weighted stocks rises as:
(a) The correlation between the stocks' returns rise.
(b) The correlation between the stocks' returns decline.
(c) The portfolio standard deviation declines.
(d) Both stocks' individual variances decline.
(e) Both stocks' individual standard deviations decline.
Question 83 portfolio risk, standard deviation
Stock Expected
return Standard
deviation Correlation ##(\rho_{A,B})## Dollars
A 0.1 0.4 0.5 60
B 0.2 0.6 140
What is the standard deviation (not variance) of the above portfolio?
Question 285 covariance, portfolio risk
Two risky stocks A and B comprise an equal-weighted portfolio. The correlation between the stocks' returns is 70%.
If the variance of stock A increases but the:
Prices and expected returns of each stock stays the same,
Variance of stock B's returns stays the same,
Correlation of returns between the stocks stays the same.
(a) The variance of the portfolio will increase.
(b) The standard deviation of the portfolio will increase.
(c) The covariance of returns between stocks A and B will stay the same.
(d) The portfolio return will stay the same.
(e) The portfolio value will stay the same.
Question 293 covariance, correlation, portfolio risk
All things remaining equal, the higher the correlation of returns between two stocks:
(a) The more diversification is possible when those stocks are combined in a portfolio.
(b) The lower the variance of returns of an equally-weighted portfolio of those stocks.
(c) The lower the volatility of returns of an equal-weighted portfolio of those stocks.
(d) The higher the covariance between those stocks' returns.
(e) The more likely that when one stock has a positive return, the other has a negative return.
Question 557 portfolio weights, portfolio return
An investor wants to make a portfolio of two stocks A and B with a target expected portfolio return of 6% pa.
Stock A has an expected return of 5% pa.
Stock B has an expected return of 10% pa.
What portfolio weights should the investor have in stocks A and B respectively?
(a) 80%, 20%
(b) 60%, 40%
(c) 40%, 60%
(d) 20%, 80%
(e) 20%, 20%
Question 556 portfolio risk, portfolio return, standard deviation
An investor wants to make a portfolio of two stocks A and B with a target expected portfolio return of 12% pa.
Stock A has an expected return of 10% pa and a standard deviation of 20% pa.
Stock B has an expected return of 15% pa and a standard deviation of 30% pa.
The correlation coefficient between stock A and B's expected returns is 70%.
What will be the annual standard deviation of the portfolio with this 12% pa target return?
(a) 24.28168% pa
(b) 24% pa
(c) 22.126907% pa
(d) 19.697716% pa
(e) 16.970563% pa
Question 563 correlation
What is the correlation of a variable X with itself?
The corr(X, X) or ##\rho_{X,X}## equals:
(a) var(X) or ##\sigma_X^2##
(b) sd(X) or ##\sigma_X##
(d) 0
(e) Mathematically undefined
Question 703 utility, risk aversion, utility function, gamble
Mr Blue, Miss Red and Mrs Green are people with different utility functions.
Each person has $500 of initial wealth. A coin toss game is offered to each person at a casino where the player can win or lose $500. Each player can flip a coin and if they flip heads, they receive $500. If they flip tails then they will lose $500. Which of the following statements is NOT correct?
(a) All people prefer more rather than less wealth which is rational.
(b) Mr Blue is risk averse, Miss Red is risk neutral and Mrs Green is risk loving.
(c) Mr Blue's certainty equivalent of the gamble is $225. This is less than his current $500 which is why he would dislike the gamble.
(d) Miss Red's certainty equivalent of the gamble is $500. This is the same as her current $500 which is why she would be indifferent to gambling.
(e) Mrs Green's certainty equivalent of the gamble is $793.70. This is more than her current $500 which is why she would like the gamble.
(a) All people would appear rational to an economist since they prefer more wealth to less.
(b) Mrs Green and Miss Red would appear unusual to an economist since they are not risk averse.
(c) Mr Blue's certainty equivalent of the gamble is $64. This is less than his current wealth of $256 which is why he would refuse the gamble.
(d) Miss Red's certainty equivalent of the gamble is $256. This is the same as her current wealth of $256 which is why she would be indifferent to playing or not.
(e) Mrs Green's certainty equivalent of the gamble is $512. This is more than her current wealth of $256 which is why she would love to play.
Each person has $50 of initial wealth. A coin toss game is offered to each person at a casino where the player can win or lose $50. Each player can flip a coin and if they flip heads, they receive $50. If they flip tails then they will lose $50. Which of the following statements is NOT correct?
(a) Mr Blue would enjoy the gamble.
(b) Miss Red would be indifferent to gambling or not.
(c) Mrs Green would dislike the gamble.
(d) Mr Blue's certainty equivalent of the risky gamble is $70.71. This is more than his current wealth which is why he would like to gamble.
(e) Miss Red's certainty equivalent of the risky gamble is $50. This is the same as her current wealth which is why she is indifferent to gambling or not.
Question 80 CAPM, risk, diversification
Diversification is achieved by investing in a large amount of stocks. What type of risk is reduced by diversification?
(a) Idiosyncratic risk.
(b) Systematic risk.
(c) Both idiosyncratic and systematic risk.
(d) Market risk.
(e) Beta risk.
Question 112 CAPM, risk
According to the theory of the Capital Asset Pricing Model (CAPM), total risk can be broken into two components, systematic risk and idiosyncratic risk. Which of the following events would be considered a systematic, undiversifiable event according to the theory of the CAPM?
(a) A decrease in house prices in one city.
(b) An increase in mining industry tax rates.
(c) An increase in corporate tax rates.
(d) A case of fraud at a major retailer.
(e) A poor earnings announcement from a major firm.
Question 326 CAPM
A fairly priced stock has an expected return equal to the market's. Treasury bonds yield 5% pa and the market portfolio's expected return is 10% pa. What is the stock's beta?
(a) 0
(b) 0.5
Question 110 CAPM, SML, NPV
The security market line (SML) shows the relationship between beta and expected return.
Investment projects that plot above the SML would have:
(a) A positive NPV.
(b) A zero NPV.
(c) A negative NPV.
(d) A large amount of diversifiable risk.
(e) Zero diversifiable risk.
Question 71 CAPM, risk
Stock A has a beta of 0.5 and stock B has a beta of 1. Which statement is NOT correct?
(a) Stock A has less systematic risk than stock B, so stock A's return should be less than stock B's.
(b) Stock B has the same systematic risk as the market, so its return should be the same as the market's.
(c) Stock B has the same beta as the market, so it also has the same total risk as the market.
(d) If stock A and B were combined in a portfolio with weights of 50% each, the beta of the portfolio would be 0.75.
(e) Stocks A and B have more systematic risk than the risk free security (government bonds) so their return should be greater than the risk free rate.
Question 93 correlation, CAPM, systematic risk
A stock's correlation with the market portfolio increases while its total risk is unchanged. What will happen to the stock's expected return and systematic risk?
(a) The stock will have a higher return and higher systematic risk.
(b) The stock will have a lower return and higher systematic risk.
(c) The stock will have a higher return and lower systematic risk.
(d) The stock will have a lower return and lower systematic risk.
(e) The stock's return and systematic risk will be unchanged.
Question 627 CAPM, SML, NPV, Jensens alpha
Assets A, B, M and ##r_f## are shown on the graphs above. Asset M is the market portfolio and ##r_f## is the risk free yield on government bonds. Which of the below statements is NOT correct?
(a) Asset A has a Jensen's alpha of 4.5% pa.
(b) Asset A is under-priced.
(c) The NPV of buying asset A is zero.
(d) Asset B has a Jensen's alpha of zero.
(e) Asset B is fairly priced.
Question 628 CAPM, SML, risk, no explanation
Assets A, B, M and ##r_f## are shown on the graphs above. Asset M is the market portfolio and ##r_f## is the risk free yield on government bonds. Assume that investors can borrow and lend at the risk free rate. Which of the below statements is NOT correct?
(a) Asset A has the same systematic risk as asset B.
(b) Asset A has more total variance than asset B.
(c) Asset B has zero idiosyncratic risk. Asset B must be a portfolio of half the market portfolio and half government bonds.
(d) If risk-averse investors were forced to invest all of their wealth in a single risky asset, so they could not diversify, every investor would logically choose asset A over the other three assets.
(e) Assets M and B have the highest Sharpe ratios, which is defined as the gradient of the capital allocation line (CAL) from the government bonds through the asset on the graph of expected return versus total standard deviation.
Question 672 CAPM, beta
A stock has a beta of 1.5. The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates.
What do you think will be the stock's expected return over the next year, given as an effective annual rate?
(a) 5% pa
(b) 7.5% pa
(c) 10% pa
(d) 12.5% pa
(e) 20% pa
Question 418 capital budgeting, NPV, interest tax shield, WACC, CFFA, CAPM
Project life 1 year
Depreciation of equipment per year $8m
Expected sale price of equipment at end of project 0
Sale price per unit $10
Fixed costs per year, paid at the end of each year $2m
Interest expense in first year (at t=1) $0.562m
Corporate tax rate 30%
Government treasury bond yield 5%
Bank loan debt yield 9%
Market portfolio return 10%
Covariance of levered equity returns with market 0.32
Variance of market portfolio returns 0.16
Firm's and project's debt-to-equity ratio 50%
Due to the project, current assets will increase by $6m now (t=0) and fall by $6m at the end (t=1). Current liabilities will not be affected.
The debt-to-equity ratio will be kept constant throughout the life of the project. The amount of interest expense at the end of each period has been correctly calculated to maintain this constant debt-to-equity ratio.
Millions are represented by 'm'.
All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year.
All rates and cash flows are real. The inflation rate is 2% pa. All rates are given as effective annual rates.
The project is undertaken by a firm, not an individual.
(a) $5.772m
(b) $4.979m
(c) $4.959m
(d) $4.733m
(e) $4.584m
Question 242 technical analysis, market efficiency
Select the most correct statement from the following.
'Chartists', also known as 'technical traders', believe that:
(a) Markets are weak-form efficient.
(b) Markets are semi-strong-form efficient.
(c) Past prices cannot be used to predict future prices.
(d) Past returns can be used to predict future returns.
(e) Stock prices reflect all publically available information.
Question 243 fundamental analysis, market efficiency
Fundamentalists who analyse company financial reports and news announcements (but who don't have inside information) will make positive abnormal returns if:
(a) Markets are weak and semi-strong form efficient but strong-form inefficient.
(b) Markets are weak form efficient but semi-strong and strong-form inefficient.
(c) Technical traders make positive excess returns.
(d) Chartists make negative excess returns.
(e) Insiders make negative excess returns.
Question 100 market efficiency, technical analysis, joint hypothesis problem
A company selling charting and technical analysis software claims that independent academic studies have shown that its software makes significantly positive abnormal returns. Assuming the claim is true, which statement(s) are correct?
(I) Weak form market efficiency is broken.
(II) Semi-strong form market efficiency is broken.
(III) Strong form market efficiency is broken.
(IV) The asset pricing model used to measure the abnormal returns (such as the CAPM) had mis-specification error so the returns may not be abnormal but rather fair for the level of risk.
Select the most correct response:
(a) Only III is true.
(b) Only II and III are true.
(c) Only I, II and III are true.
(d) Only IV is true.
(e) Either I, II and III are true, or IV is true, or they are all true.
Question 119 market efficiency, fundamental analysis, joint hypothesis problem
Your friend claims that by reading 'The Economist' magazine's economic news articles, she can identify shares that will have positive abnormal expected returns over the next 2 years. Assuming that her claim is true, which statement(s) are correct?
(iv) The asset pricing model used to measure the abnormal returns (such as the CAPM) is either wrong (mis-specification error) or is measured using the wrong inputs (data errors) so the returns may not be abnormal but rather fair for the level of risk.
(a) Only (iii) is true.
(b) Only (ii) and (iii) are true.
(c) Only (i), (ii) and (iii) are true.
(d) Either (ii) and (iii) are true, or (iv) is true, or (ii), (iii) and (iv) are true.
(e) Either (i), (ii) and (iii) are true, or (iv) is true, or all are true.
Question 338 market efficiency, CAPM, opportunity cost, technical analysis
A man inherits $500,000 worth of shares.
He believes that by learning the secrets of trading, keeping up with the financial news and doing complex trend analysis with charts that he can quit his job and become a self-employed day trader in the equities markets.
What is the expected gain from doing this over the first year? Measure the net gain in wealth received at the end of this first year due to the decision to become a day trader. Assume the following:
He earns $60,000 pa in his current job, paid in a lump sum at the end of each year.
He enjoys examining share price graphs and day trading just as much as he enjoys his current job.
Stock markets are weak form and semi-strong form efficient.
He has no inside information.
He makes 1 trade every day and there are 250 trading days in the year. Trading costs are $20 per trade. His broker invoices him for the trading costs at the end of the year.
The shares that he currently owns and the shares that he intends to trade have the same level of systematic risk as the market portfolio.
The market portfolio's expected return is 10% pa.
Measure the net gain over the first year as an expected wealth increase at the end of the year.
(a) $110,000
(c) $45,000
(d) -$15,000
(e) -$65,000
Question 105 NPV, risk, market efficiency
A person is thinking about borrowing $100 from the bank at 7% pa and investing it in shares with an expected return of 10% pa. One year later the person will sell the shares and pay back the loan in full. Both the loan and the shares are fairly priced.
What is the Net Present Value (NPV) of this one year investment? Note that you are asked to find the present value (##V_0##), not the value in one year (##V_1##).
(b) $3
(c) $2.8037
(d) $2.7273
(e) $0
Question 339 bond pricing, inflation, market efficiency, income and capital returns
Economic statistics released this morning were a surprise: they show a strong chance of consumer price inflation (CPI) reaching 5% pa over the next 2 years.
This is much higher than the previous forecast of 3% pa.
A vanilla fixed-coupon 2-year risk-free government bond was issued at par this morning, just before the economic news was released.
What is the expected change in bond price after the economic news this morning, and in the next 2 years? Assume that:
Inflation remains at 5% over the next 2 years.
Investors demand a constant real bond yield.
The bond price falls by the (after-tax) value of the coupon the night before the ex-coupon date, as in real life.
(a) Today the price would have increased significantly.
Over the next 2 years, the bond price is expected to increase, measured on each ex-coupon date.
(b) Today the price would have increased significantly.
Over the next 2 years, the bond price is expected to be unchanged, measured on each ex-coupon date.
(c) Today the price would have been unchanged.
(d) Today the price would have decreased significantly.
(e) Today the price would have decreased significantly.
Question 340 market efficiency, opportunity cost
A managed fund charges fees based on the amount of money that you keep with them. The fee is 2% of the start-of-year amount, but it is paid at the end of every year.
This fee is charged regardless of whether the fund makes gains or losses on your money.
The fund offers to invest your money in shares which have an expected return of 10% pa before fees.
You are thinking of investing $100,000 in the fund and keeping it there for 40 years when you plan to retire.
What is the Net Present Value (NPV) of investing your money in the fund? Note that the question is not asking how much money you will have in 40 years, it is asking: what is the NPV of investing in the fund? Assume that:
The fund has no private information.
Markets are weak and semi-strong form efficient.
The fund's transaction costs are negligible.
The cost and trouble of investing your money in shares by yourself, without the managed fund, is negligible.
(b) -$20,000.00
(c) -$48,000.17
(d) -$51,999.83
(e) -$80,000.00
Question 416 real estate, market efficiency, income and capital returns, DDM, CAPM
A residential real estate investor believes that house prices will grow at a rate of 5% pa and that rents will grow by 2% pa forever.
All rates are given as nominal effective annual returns. Assume that:
His forecast is true.
Real estate is and always will be fairly priced and the capital asset pricing model (CAPM) is true.
Ignore all costs such as taxes, agent fees, maintenance and so on.
All rental income cash flow is paid out to the owner, so there is no re-investment and therefore no additions or improvements made to the property.
The non-monetary benefits of owning real estate and renting remain constant.
Which one of the following statements is NOT correct? Over time:
(a) The rental yield will fall and approach zero.
(b) The total return will fall and approach the capital return (5% pa).
(c) One or all of the following must fall: the systematic risk of real estate, the risk free rate or the market risk premium.
(d) If the country's nominal wealth growth rate is 4% pa and the nominal real estate growth rate is 5% pa then real estate will approach 100% of the country's wealth over time.
(e) If the country's nominal gross domestic production (GDP) growth rate is 4% pa and the nominal real estate rent growth rate is 2% pa then real estate rent will approach 100% of the country's GDP over time.
Question 464 mispriced asset, NPV, DDM, market efficiency
A company advertises an investment costing $1,000 which they say is underpriced. They say that it has an expected total return of 15% pa, but a required return of only 10% pa. Assume that there are no dividend payments so the entire 15% total return is all capital return.
Assuming that the company's statements are correct, what is the NPV of buying the investment if the 15% return lasts for the next 100 years (t=0 to 100), then reverts to 10% pa after that time? Also, what is the NPV of the investment if the 15% return lasts forever?
In both cases, assume that the required return of 10% remains constant. All returns are given as effective annual rates.
The answer choices below are given in the same order (15% for 100 years, and 15% forever):
(a) $0, $0
(b) $1,977.19, $2,000
(c) $2,977.19, $3,000
(d) $499.96, $500
(e) $84,214.9, Infinite
Question 621 market efficiency, technical analysis, no explanation
Technical traders:
(a) Believe that asset prices follow a random walk.
(b) Believe that markets are weak form efficient
(c) Are pessimists, they believe that they cannot beat the market
(d) Base their investment decisions on past publicly available news
(e) Believe that the theory of weak form market efficiency is broken.
Question 623 market efficiency, no explanation
The efficient markets hypothesis (EMH) and no-arbitrage pricing theory is most closely related to which of the following concepts?
(a) Competition.
(b) Opportunity costs.
(c) Separation of the investment and financing decisions.
(d) Diversification.
(e) The stand-alone principle, incremental cash flows and sunk costs.
A company advertises an investment costing $1,000 which they say is underpriced. They say that it has an expected total return of 15% pa, but a required return of only 10% pa. Of the 15% pa total expected return, the dividend yield is expected to always be 7% pa and rest is the capital yield.
Assuming that the company's statements are correct, what is the NPV of buying the investment if the 15% total return lasts for the next 100 years (t=0 to 100), then reverts to 10% after that time? Also, what is the NPV of the investment if the 15% return lasts forever?
In both cases, assume that the required return of 10% remains constant, the dividends can only be re-invested at 10% pa and all returns are given as effective annual rates.
(a) 84,214.90, Infinite
(b) 2,521.12, 3,000
(c) 2,100.93, 2,500
(d) 1,249.38, 1,333.33
(e) 0, 0
Question 569 personal tax
The average weekly earnings of an Australian adult worker before tax was $1,542.40 per week in November 2014 according to the Australian Bureau of Statistics. Therefore average annual earnings before tax were $80,204.80 assuming 52 weeks per year. Personal income tax rates published by the Australian Tax Office are reproduced for the 2014-2015 financial year in the table below:
Tax on this income
0 – $18,200 Nil
$18,201 – $37,000 19c for each $1 over $18,200
$37,001 – $80,000 $3,572 plus 32.5c for each $1 over $37,000
$80,001 – $180,000 $17,547 plus 37c for each $1 over $80,000
$180,001 and over $54,547 plus 45c for each $1 over $180,000
The above rates do not include the Medicare levy of 2%. Exclude the Medicare levy from your calculations
How much personal income tax would you have to pay per year if you earned $80,204.80 per annum before-tax?
(b) $29,675.78
Question 448 franking credit, personal tax on dividends, imputation tax system
A small private company has a single shareholder. This year the firm earned a $100 profit before tax. All of the firm's after tax profits will be paid out as dividends to the owner.
The corporate tax rate is 30% and the sole shareholder's personal marginal tax rate is 45%.
The Australian imputation tax system applies because the company generates all of its income in Australia and pays corporate tax to the Australian Tax Office. Therefore all of the company's dividends are fully franked. The sole shareholder is an Australian for tax purposes and can therefore use the franking credits to offset his personal income tax liability.
What will be the personal tax payable by the shareholder and the corporate tax payable by the company?
(a) Personal tax of $6.43 and corporate tax of $45.
(b) Personal tax of $15 and corporate tax of $30.
(c) Personal tax of $16.5 and corporate tax of $45.
(d) Personal tax of $31.5 and corporate tax of $30.
(e) Personal tax of $45 and corporate tax of $0.
Question 309 stock pricing, ex dividend date
A company announces that it will pay a dividend, as the market expected. The company's shares trade on the stock exchange which is open from 10am in the morning to 4pm in the afternoon each weekday. When would the share price be expected to fall by the amount of the dividend? Ignore taxes.
The share price is expected to fall during the:
(a) Day of the payment date, between the payment date's morning opening price and afternoon closing price.
(b) Night before the payment date, between the previous day's afternoon closing price and the payment date's morning opening price.
(c) Day of the ex-dividend date, between the ex-dividend date's morning opening price and afternoon closing price.
(d) Night before the ex-dividend date, between the last with-dividend date's afternoon closing price and the ex-dividend date's morning opening price.
(e) Day of the last with-dividend date, between the with-dividend date's morning opening price and afternoon closing price.
Question 454 NPV, capital structure, capital budgeting
A mining firm has just discovered a new mine. So far the news has been kept a secret.
The net present value of digging the mine and selling the minerals is $250 million, but $500 million of new equity and $300 million of new bonds will need to be issued to fund the project and buy the necessary plant and equipment.
The firm will release the news of the discovery and equity and bond raising to shareholders simultaneously in the same announcement. The shares and bonds will be issued shortly after.
Once the announcement is made and the new shares and bonds are issued, what is the expected increase in the value of the firm's assets ##(\Delta V)##, market capitalisation of debt ##(\Delta D)## and market cap of equity ##(\Delta E)##? Assume that markets are semi-strong form efficient.
The triangle symbol ##\Delta## is the Greek letter capital delta which means change or increase in mathematics.
Ignore the benefit of interest tax shields from having more debt.
Remember: ##\Delta V = \Delta D+ \Delta E##
(a) ##\Delta V = 250m##, ##ΔD = 300m##, ##ΔE= 250##
(b) ##\Delta V = 250m##, ##ΔD = 300m##, ##ΔE= 750##
(c) ##\Delta V = 400m##, ##ΔD = 300m##, ##ΔE= -250##
(d) ##\Delta V = 1,050m##, ##ΔD = 300m##, ##ΔE= 750##
(e) ##\Delta V = 1,050m##, ##ΔD = 300m##, ##ΔE= 250##
Question 625 dividend re-investment plan, capital raising
Which of the following statements about dividend re-investment plans (DRP's) is NOT correct?
(a) DRP's are voluntary, shareholders only participate if they choose.
(b) DRP's increase the number of shares.
(c) The number of shares issued to a shareholder participating in a DRP is usually calculated as their total dividends owed, divided by the allocation share price which is usually close to the current market share price.
(d) DRP's do not incur brokerage costs for the shareholder. This is unlike the case where the shareholder uses the cash dividend to buy more shares herself.
(e) If all shareholders participated in a company's DRP, the company would not pay any dividends and the firm's share price would not fall due to the cash dividend or the DRP.
Question 214 rights issue
In late 2003 the listed bank ANZ announced a 2-for-11 rights issue to fund the takeover of New Zealand bank NBNZ. Below is the chronology of events:
23/10/2003. Share price closes at $18.30.
24/10/2003. 2-for-11 rights issue announced at a subscription price of $13. The proceeds of the rights issue will be used to acquire New Zealand bank NBNZ. Trading halt announced in morning before market opens.
28/10/2003. Trading halt lifted. Last (and only) day that shares trade cum-rights. Share price opens at $18.00 and closes at $18.14.
29/10/2003. Shares trade ex-rights.
All things remaining equal, what would you expect ANZ's stock price to open at on the first day that it trades ex-rights (29/10/2003)? Ignore the time value of money since time is negligibly short. Also ignore taxes.
Question 708 continuously compounding rate, continuously compounding rate conversion
Convert a 10% continuously compounded annual rate ##(r_\text{cc annual})## into an effective annual rate ##(r_\text{eff annual})##. The equivalent effective annual rate is:
(a) 230.258509% pa
(b) 10.536052% pa
(e) 9.531018% pa
Question 767 idiom, corporate financial decision theory, no explanation
The sayings "Don't cry over spilt milk", "Don't regret the things that you can't change" and "What's done is done" are most closely related to which financial concept?
(e) Sunk costs.
Question 768 accounting terminology, book and market values, no explanation
Accountants and finance professionals have lots of names for the same things which can be quite confusing.
Which of the following groups of items are NOT synonyms?
(a) Revenue, sales, turn over.
(b) Paid up capital, contributed equity.
(c) Shares, stock, equity.
(d) Net income, earnings, net profit after tax, the bottom line.
(e) Market capitalisation of equity, book value of equity.
Question 769 short selling, idiom, no explanation
"Buy low, sell high" is a well-known saying. It suggests that investors should buy low then sell high, in that order.
How would you re-phrase that saying to describe short selling?
(a) Buy high, then sell low.
(b) Buy low, then sell high.
(c) Sell high, then buy low.
(d) Sell low, then buy high.
(e) Sell high, then buy high.
Question 770 expected and historical returns, income and capital returns, coupon rate, bond pricing, no explanation
Which of the following statements is NOT correct? Assume that all things remain equal. So for example, don't assume that just because a company's dividends and profit rise that its required return will also rise, assume the required return stays the same.
(a) If a company's dividend (and profit and free cash flow to equity) rises, its share price will rise.
(b) If a fixed coupon bond's yield to maturity rises, its price will rise.
(c) If a residential property's rent revenue rises, its price will rise.
(d) If a patent's royalty revenue rises, its price will also rise.
(e) If a software asset's licensing revenue rises, its price will also rise.
Question 771 debt terminology, interest expense, interest tax shield, credit risk, no explanation
You deposit money into a bank account. Which of the following statements about this deposit is NOT correct?
(a) You have a debt asset.
(b) The bank sold you its promise to pay back interest and principal payments.
(c) The bank is exposed to your credit risk, it's afraid that you'll default on your debt.
(d) The interest income you're paid is taxable income for you.
(e) The interest expense that the bank pays is tax-deductible for the bank.
Question 772 interest tax shield, capital structure, leverage
A firm issues debt and uses the funds to buy back equity. Assume that there are no costs of financial distress or transactions costs. Which of the following statements about interest tax shields is NOT correct?
(a) Higher debt leads to higher interest expense.
(b) Higher interest expense leads to lower profit before tax, following on from above.
(c) Lower profit before tax leads to lower tax payments, following on from above.
(d) Lower tax payments lead to higher cash flow from assets, following on from above.
(e) Lower profit after tax leads to a lower share price, following on from above.
Question 773 CFFA, WACC, interest tax shield, DDM
Use the below information to value a levered company with constant annual perpetual cash flows from assets. The next cash flow will be generated in one year from now, so a perpetuity can be used to value this firm. Both the cash flow from assets including and excluding interest tax shields are constant (but not equal to each other).
Data on a Levered Firm with Perpetual Cash Flows
Item abbreviation Value Item full name
##\text{CFFA}_\text{U}## $48.5m Cash flow from assets excluding interest tax shields (unlevered)
##\text{CFFA}_\text{L}## $50m Cash flow from assets including interest tax shields (levered)
##g## 0% pa Growth rate of cash flow from assets, levered and unlevered
##\text{WACC}_\text{BeforeTax}## 10% pa Weighted average cost of capital before tax
##\text{WACC}_\text{AfterTax}## 9.7% pa Weighted average cost of capital after tax
##r_\text{D}## 5% pa Cost of debt
##r_\text{EL}## 11.25% pa Cost of levered equity
##D/V_L## 20% pa Debt to assets ratio, where the asset value includes tax shields
##t_c## 30% Corporate tax rate
What is the value of the levered firm including interest tax shields?
(a) $515.464m
(d) $444.444m
(e) $431.111m
Question 774 leverage, WACC, real estate
One year ago you bought a $1,000,000 house partly funded using a mortgage loan. The loan size was $800,000 and the other $200,000 was your wealth or 'equity' in the house asset.
The interest rate on the home loan was 4% pa.
Over the year, the house produced a net rental yield of 2% pa and a capital gain of 2.5% pa.
Assuming that all cash flows (interest payments and net rental payments) were paid and received at the end of the year, and all rates are given as effective annual rates, what was the total return on your wealth over the past year?
(a) 0.4808% pa
(b) 2% pa
(c) 4.5% pa
(d) 6.5% pa
(e) 8.5% pa
Hint: Remember that wealth in this context is your equity (E) in the house asset (V = D+E) which is funded by the loan (D) and your deposit or equity (E).
Question 775 utility, utility function
Below is a graph of 3 peoples' utility functions, Mr Blue (U=W^(1/2) ), Miss Red (U=W/10) and Mrs Green (U=W^2/1000). Assume that each of them currently have $50 of wealth.
Which of the following statements about them is NOT correct?
(a) Mr Blue would prefer to invest his wealth in a well diversified portfolio of stocks rather than a single stock, assuming that all stocks had the same total risk and return.
(b) Mrs Green would prefer to invest her wealth in a single stock rather than a well diversified portfolio of stocks, assuming that all stocks had the same total risk and return.
(c) The popularity of insurance only makes sense if people are similar to Mr Blue.
(d) CAPM theory only makes sense if people are similar to Miss Red.
(e) The popularity of casino gambling and lottery tickets only make sense if people are similar to Mrs Green.
The market's expected total return is 10% pa and the risk free rate is 5% pa, both given as effective annual rates.
A stock has a beta of 0.5.
In the last 5 minutes, the federal government unexpectedly raised taxes. Over this time the share market fell by 3%. The risk free rate was unchanged.
What do you think was the stock's historical return over the last 5 minutes, given as an effective 5 minute rate?
(a) -1%
(b) -1.5%
(c) -2.5%
(d) -3%
(e) -7.5%
The 'time value of money' is most closely related to which of the following concepts?
(a) Competition: Firms in competitive markets earn zero economic profit.
(b) Opportunity cost: The cost of the next best alternative foregone should be subtracted.
(d) Diversification: Risks can often be reduced by pooling them together.
(e) Sunk costs: Costs that cannot be recouped should be ignored.
(a) Buy debt.
(b) Own debt.
(d) Write debt.
(e) Have debt assets.
Question 657 systematic and idiosyncratic risk, CAPM, no explanation
A stock's required total return will decrease when its:
(a) Systematic risk increases.
(b) Idiosyncratic risk increases.
(c) Total risk increases.
(d) Systematic risk decreases.
(e) Idiosyncratic risk decreases.
Question 658 CFFA, income statement, balance sheet, no explanation
To value a business's assets, the free cash flow of the firm (FCFF, also called CFFA) needs to be calculated. This requires figures from the firm's income statement and balance sheet. For what figures is the income statement needed? Note that the income statement is sometimes also called the profit and loss, P&L, or statement of financial performance.
(a) Net income, depreciation and interest expense.
(b) Depreciation and capital expenditure.
(d) Current assets, current liabilities and capital expenditure.
(e) Current assets, current liabilities and depreciation expense.
Question 659 APR, effective rate, effective rate conversion, no explanation
A home loan company advertises an interest rate of 9% pa, payable monthly. Which of the following statements about the interest rate is NOT correct? All rates are given with an accuracy of 4 decimal places.
(a) The APR compounding monthly is 9.000% pa.
(b) The effective monthly rate is 0.7500% pa.
(c) The effective annual rate is 9.3807% pa.
(d) The effective 6 month rate is 4.5000% pa.
(e) The APR compounding semi-annually is 9.1704% pa.
Question 660 fully amortising loan, interest only loan, APR
How much more can you borrow using an interest-only loan compared to a 25-year fully amortising loan if interest rates are 6% pa compounding per month and are not expected to change? If it makes it easier, assume that you can afford to pay $2,000 per month on either loan. Express your answer as a proportional increase using the following formula:
###\text{Proportional Increase} = \dfrac{V_\text{0,interest only}}{V_\text{0,fully amortising}} - 1###
(a) 77.6034%
(b) 30.3779%
(c) 28.8603%
(d) 22.3966%
Question 661 systematic and idiosyncratic risk, CAPM
A stock's total standard deviation of returns is 20% pa. The market portfolio's total standard deviation of returns is 15% pa. The beta of the stock is 0.8.
What is the stock's diversifiable standard deviation?
(a) 16% pa
(c) 8% pa
(d) 5% pa
(e) 4% pa
Which of the following interest rate labels does NOT make sense?
(a) Annualised percentage rate compounding per month.
(b) Effective monthly rate compounding per year.
(c) Annualised percentage rate compounding per year.
(d) Effective annual rate compounding per year.
(e) Annualised percentage rate compounding semi-annually.
Question 663 leverage, accounting ratio, no explanation
A firm has a debt-to-assets ratio of 20%. What is its debt-to-equity ratio?
(a) 20%
(b) 25%
(c) 60%
What is the present value of real payments of $100 every year forever, with the first payment in one year? The nominal discount rate is 7% pa and the inflation rate is 4% pa.
(a) $3,466.6667
Question 665 stock split
A company conducts a 10 for 3 stock split. What is the percentage increase in the stock price and the number of shares outstanding? The answers are given in the same order.
(a) -76.92%, 333.33%
(b) -70%, 333.33%
(c) -70%, 233.33%
(d) -57.14%, 233.33%
(e) 233.33%, -70%
Question 666 rights issue, capital raising
A company conducts a 2 for 3 rights issue at a subscription price of $8 when the pre-announcement stock price was $9. Assume that all investors use their rights to buy those extra shares.
What is the percentage increase in the stock price and the number of shares outstanding? The answers are given in the same order.
(a) -60%, 150%
(b) -42.22%, 150%
(c) -40%, 66.67%
(d) -22.22%, 66.67%
(e) -4.44%, 66.67%
Question 669 beta, CAPM, risk
Which of the following is NOT a valid method for estimating the beta of a company's stock? Assume that markets are efficient, a long history of past data is available, the stock possesses idiosyncratic and market risk. The variances and standard deviations below denote total risks.
(a) ##Β_E=\dfrac{cov(r_E,r_M )}{var(r_M)}##
(b) ##Β_E=\dfrac{correl(r_E,r_M ).sd(r_E)}{sd(r_M)} ##
(c) ##Β_E=\dfrac{sd(r_E)}{sd(r_M)}##, since ##var(r_E)=β_E^2.var(r_M)##
(d) ##Β_E=\dfrac{r_E-r_f}{r_M-r_f }##, since ##r_E=r_f+Β_E. (r_M-r_f )##
(e) ##Β_E= \left(B_V - \dfrac{D}{V}.Β_D \right).\dfrac{V}{E}##, since ##B_V=\dfrac{E}{V}.Β_E+\dfrac{D}{V}.Β_D ##
Question 617 systematic and idiosyncratic risk, risk, CAPM
A stock's required total return will increase when its:
To value a business's assets, the free cash flow of the firm (FCFF, also called CFFA) needs to be calculated. This requires figures from the firm's income statement and balance sheet. For what figures is the balance sheet needed? Note that the balance sheet is sometimes also called the statement of financial position.
(c) Current assets, current liabilities and cost of goods sold (COGS).
Question 622 expected and historical returns, risk
An economy has only two investable assets: stocks and cash.
Stocks had a historical nominal average total return of negative two percent per annum (-2% pa) over the last 20 years. Stocks are liquid and actively traded. Stock returns are variable, they have risk.
Cash is riskless and has a nominal constant return of zero percent per annum (0% pa), which it had in the past and will have in the future. Cash can be kept safely at zero cost. Cash can be converted into shares and vice versa at zero cost.
The nominal total return of the shares over the next year is expected to be:
(a) Less than or equal to negative two percent per annum ##(r_\text{shares} \leq -0.02)##.
(b) Exactly negative two percent per annum ##(r_\text{shares} = -0.02)##.
(c) More than or equal to negative two percent per annum ##(r_\text{shares} \geq -0.02)##.
(d) Less than or equal to zero percent per annum ##(r_\text{shares} \leq 0)##.
(e) More than or equal to zero percent per annum ##(r_\text{shares} \geq 0)##.
Copyright © 2014 Keith Woodward | CommonCrawl |
Metabelian group
In mathematics, a metabelian group is a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian normal subgroup A such that the quotient group G/A is abelian.
Subgroups of metabelian groups are metabelian, as are images of metabelian groups over group homomorphisms.
Metabelian groups are solvable. In fact, they are precisely the solvable groups of derived length at most 2.
Examples
• Any dihedral group is metabelian, as it has a cyclic normal subgroup of index 2. More generally, any generalized dihedral group is metabelian, as it has an abelian normal subgroup of index 2.
• If F is a field, the group of affine maps $x\mapsto ax+b$ (where a ≠ 0) acting on F is metabelian. Here the abelian normal subgroup is the group of pure translations $x\mapsto x+b$, and the abelian quotient group is isomorphic to the group of homotheties $x\mapsto ax$. If F is a finite field with q elements, this metabelian group is of order q(q − 1).
• The group of direct isometries of the Euclidean plane is metabelian. This is similar to the above example, as the elements are again affine maps. The translations of the plane form an abelian normal subgroup of the group, and the corresponding quotient is the circle group.
• The finite Heisenberg group H3,p of order p3 is metabelian. The same is true for any Heisenberg group defined over a ring (group of upper-triangular 3 × 3 matrices with entries in a commutative ring).
• All nilpotent groups of class 3 or less are metabelian.
• The lamplighter group is metabelian.
• All groups of order p5 are metabelian (for prime p).[1]
• All groups of order less than 24 are metabelian.
In contrast to this last example, the symmetric group S4 of order 24 is not metabelian, as its commutator subgroup is the non-abelian alternating group A4.
References
1. MSE
• Robinson, Derek J.S. (1996), A Course in the Theory of Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6
External links
• Ryan Wisnesky, Solvable groups (subsection Metabelian Groups)
• Groupprops, The Group Properties Wiki Metabelian group
| Wikipedia |
Momentum of a photon equals Planck's constant over wavelength
A common identity in Quantum Mechanics is relation between the momentum of a photon and its wavelength:
$$p = \frac{h}{\lambda}$$
The identity is discussed here, for example:
https://en.wikipedia.org/wiki/Matter_wave
Apparently, this is the identity rearranged by de Broglie to give the wavelength of the wave nature of a particle. But where does this identity come from in the first place? I have seen some quite "hand-wavy" ways of deriving this using $E=mc^2$, but it seems quite strange having to rely on relativity to obtain this identity. Or is it exactly what we must do? This seems to be a quite fundamental identity in Quantum Mechanics, so I would like to understand its justification as well as possible. I've been told that light having momentum is an idea present in classical mechanics as well and was known much before quantization of light and photons were discovered.
quantum-mechanics photons momentum wave-particle-duality wavelength
exp ikx
S. RotosS. Rotos
$\begingroup$ drphysics.com/syllabus/energy/energy.html This thought experiment is one that I've seen in many places, and is used to derive the energy-momentum relation. But it also uses the identity $E=pc$. Isn't this circular? $\endgroup$ – S. Rotos Jun 5 '18 at 18:06
$\begingroup$ Possible duplicate? physics.stackexchange.com/questions/12545/… $\endgroup$ – jacob1729 Jun 5 '18 at 18:27
$\begingroup$ @S.Rotos is your question "How does one get the de Broglie relation without using relativity?". I agree it only relies on quantum mechanics, I may type up an answer later if I have time. $\endgroup$ – jacob1729 Jun 5 '18 at 18:28
$\begingroup$ A work that uses $E=mc^2$ on photons is using $m$ to denote relativistic mass, but in modern treatments $m$ always means rest mass. Relativistic mass is a deprecated concept because it can be misleading and lead to errors if you aren't careful. So no wonder the derivation you saw looked hand-wavey. $\endgroup$ – PM 2Ring Jun 5 '18 at 18:40
$\begingroup$ This is a postulate and more or less the definition of h by Planck (E=hf was Planck's original) so you won't find a "derivation" of it. Experiments force you to come up with this equality, really classical mechanics and optics are quite fine by themselves without it up until modern age physics. I think you should search for historical texts about Planck's problems and ideas first maybe? $\endgroup$ – BjornW Oct 25 '18 at 1:10
It seems like your are not satisfied by answers involving axioms. I think that you instead want to know the motivation behind the axiom beyond just saying that it works. I am not sure if my answer is the original motivation, but I think it can be viewed as a good motivation for the validity of $p= \frac{h}{\lambda}$. While other answers do a great job at going into the theory, I will tackle the question using more of an experimental motivation.
We will first start with the double slit experiment. This experiment is usually first introduced as evidence of the wave-like nature of light, where light emanating from one slit interferes with light emanating from the other (of course a different interpretation is found if we send single photons through the slits and the same interference pattern arises, but I digress). However, this experiment also works with electrons. You get an interference pattern consistent with treating the electrons as waves with wavelength $$\lambda=\frac hp$$
You get maxima in intensity such that $$\sin\theta_n=\frac{n\lambda}{d}$$
Where $\theta$ is the angle formed by the central maximum, the slit, and the maximum in question, $d$ is the slit separation, and $n$ is an integer.
This would then be a way to experimentally motivate/verify this relationship between momentum and wavelength for matter, but what about photons? The double slit experiment does not give us a way to validate $p=\frac h\lambda$ (that I know of. Maybe you could determine the radiation pressure on the detector?). Let's look at a different experiment.
We know that the energy of a photon from special relativity is $$E=pc$$
So, if our momentum relation is true, it must be that $$E=\frac{hc}{\lambda}=hf$$ which is something that can be verified experimentally to be true. The photoelectric effect is one such experiment we could do, where shining light onto a material causes electrons or other charge carriers to become emitted from that material. The higher the frequency of the light, the more energetic the electrons coming from the material are, and the maximum kinetic energy of an electron can be shown to follow $K_{max}=h(f-f_0)$ where $f$ is the frequency of the light and $f_0$ is the material-dependent threshold frequency (i.e. we need $f>f_0$).
I know that my answer does not get to a fundamental explanation of this relation in question, but I hope it shows why one would want it to be a fundamental idea that holds true when formulating QM. If you want a more fundamental explanation, then I will edit or remove this answer due to some pretty good fundamental answers already here.
Aaron StevensAaron Stevens
$\begingroup$ @AaronStevens I like your answer... its a bit vague, but it makes contact with reality (i.e. experiment) rather than just delving deeper into theoretical abstractness. Also I am utterly baffled by tparker's absurd claim that your answer is "entirely classical." Your answer very clearly relies on quantum behavior as you begin by discussing the wave-behavior of electrons, which is a fundamentally quantum mechanical phenomenon. $\endgroup$ – user105620 Oct 25 '18 at 5:37
$\begingroup$ I like your answer as well. But if I did a double slit experiment with electrons, how would I define their momentum? Would I use the classical momentum $p = mv$, utilizing known mass of the electron and their speed in the particular experiment? $\endgroup$ – S. Rotos Oct 26 '18 at 17:01
$\begingroup$ @SRotos Yes that is exactly right. Unless you were working with relativistic electrons. But the idea is the same. It's just the momentum they have when you fire them at the slits. $\endgroup$ – Aaron Stevens Oct 27 '18 at 1:30
Glossing over lots of subtleties:
A fundamental axiom of quantum mechanics is the canonical commutation relation $[\hat{X}, \hat{P}] = i \hbar$. In this position basis, this becomes $\hat{X} \to x$ and $\hat{P} \to -i \hbar \frac{\partial}{\partial x}$ (modulo lots of technical details involving the Stone-von Neumann theorem, etc.).
Another fundamental axiom of quantum mechanics is that states with definite values of a physical observable must be eigenstates of the corresponding Hermitian operator. So a particle with momentum $p$ is described by a wave function $|p\rangle$ satisfying $\hat{P} |p\rangle = p |p\rangle$. (Whether we can legitimately talk about the wavefunction of a massless relativistic particle is another subtlety that I'll gloss over.)
Putting this together, we have that in the position basis $$-i \hbar \frac{\partial \psi}{\partial x} = p \psi(x) \implies \psi(x) \propto e^{i p x / \hbar}.$$ So the wavefunction is spatially periodic with period $\lambda = 2 \pi \hbar / p = h / p$, so $p = h / \lambda$. This "derivation" works equally well whether or not the particle is massive or massless.
tparkertparker
For convenience, let $k=2\pi/\lambda$ and $\omega=2\pi f$. Here $k$ is called the wavevector and $\omega$ is a version of the frequency that is in units of radians per second rather than oscillations per second.
Then we have the following two completely analogous relationships:
$$p=\hbar k $$
$$E=\hbar \omega .$$
The analogy holds because in relativity, momentum is to space as energy is to time.
If you assume $p=\hbar k$, then there are straightforward arguments that lead to $E=\hbar \omega$. If you assume $E=\hbar \omega$, there are similar aguments that get you to $p=\hbar k$. They're not independent of each other. If you believe in one, and you believe in relativity, then you have to believe in the other.
These are fundamental relationships that hold true in all of quantum mechanics. They're not just true for photons, they're true for electrons and baseballs.
With "why" questions like this, you have to decide what you want to take as a fundamental assumption. There are treatments of quantum mechanics that take various sets of axioms. Depending on what set of axioms you choose, these relations could be derived or they could be axioms. If someone tells you they have a proof of one of these relationships, you should ask them what assumptions they started from, and then ask yourself whether you find the assumptions more solid than these relations. Are the assumptions more intuitively reasonable? Better verified by experiment? Aesthetically preferable?
Ben CrowellBen Crowell
$\begingroup$ Thank you for the answer but I can't really accept it because yes, there are certain things that are axioms and not really derived so to speak, but there still must be some kind of intuition or justification. They don't just appear out of nowhere. $\endgroup$ – S. Rotos Jun 6 '18 at 15:42
$\begingroup$ I do not agree with that last statement, but my reply was too wordy for a comment, so I upgraded it as an answer below ^^ $\endgroup$ – Barbaud Julien Oct 26 '18 at 2:15
I'll build on tparker's answer in a way that emphasizes the great generality of the relationship between momentum and wavelength.
In classical physics, a very general result called Noether's theorem can be used as the foundation for a definition of momentum. The inputs to Noether's theorem are:
the action principle — loosely translated, this says that if one physical entity influences another, then they must both influence each other;
any continuous symmetry — such as rotational symmetry or time-translation symmetry.
Noether's theorem says that these inputs imply the existence of a conservation law associated with the given symmetry. For example, rotational symmetry leads to the conservation of angular momentum, and time-translation symmetry leads to the conservation of energy. These connections may be regarded as the definitions of angular momentum and of energy, respectively.
If the symmetry is symmetry under translations in space, meaning roughly that the laws of physics are the same in all places, then the resulting conservation law is the conservation of momentum — that is, the total momentum of the system. This connection may be regarded as the definition of momentum.
In a model that includes the electromagnetic field, this definition of momentum includes a contribution from the electromagnetic field — and from anything else that participates in the action principle by influencing (and being influenced by) other entities.
In quantum physics, these same general connections come with another twist: for each of these symmetries, we have an operator that generates those symmetries (more detail given below), and this operator is quantum theory's representation of the corresponding conserved quantity. In particular, the momentum operator generates translations in space. More precisely, this is the total momentum operator, which generates translations of the whole physical system in space. This operator is a basic ingredient in any quantum system whose laws are the same in all places. This is true in both non-relativistic quantum mechanics and in relativistic quantum field theory. Although the concept of a massless particle does involve relativity, the connection between momentum and space-translation symmetry does not rely on relativity.
Now, as promised, here's more detail about what it means to say that the momentum operator "generates translations in space." As in tparker's answer, let $\hat P$ denote any single component of the momentum operator, which generates translations in that one direction in space. The answer by tparker already illustrated this nicely in the case of single-particle quantum mechanics. For another example, I'll consider how a massless photon is described in the quantum model of the electromagnetic field. In this model, instead of having an operator $\hat X$ for the position of a single particle, we have field-operators like $\hat E(x)$ and $\hat B(x)$ representing the electric and magnetic fields. These operators parameterized by the location $x$ in space. I'm omitting their vector indices to avoid cluttering the equations.
Now, a photon is a particle that, mathematically, is created by applying an appropriate linear combination of $\hat E(x)$ and $\hat B(x)$ to the vacuum state. Such a single-photon state may be written in the form $$ |1\rangle = \int dx\ \big(f(x)\hat E(x) + g(x) \hat B(x)\big)|0\rangle $$ where $|0\rangle$ is the vacuum state and where $f$ and $g$ are appropriate complex-valued functions of the spatial coordinate $x$. Given any such single-photon state, we can translate the photon in space by an amount $a$ by applying the operator $\exp(i\hat P a/\hbar)$, like this: \begin{align*} \exp\big(i\hat P a/\hbar\big)|1\rangle &= \int dx\ \big(f(x)\hat E(x+a) + g(x) \hat B(x+a)\big)|0\rangle \\ &= \int dx\ \big(f(x-a)\hat E(x) + g(x-a) \hat B(x)\big)|0\rangle. \end{align*} The second step follows simply by changing the integration variable. The first step follows from \begin{align*} \exp\big(i\hat P a/\hbar\big)\hat E(x) &= \hat E(x+a)\exp\big(i\hat P a/\hbar\big) \\ \exp\big(i\hat P a/\hbar\big)\hat B(x) &= \hat B(x+a)\exp\big(i\hat P a/\hbar\big) \end{align*} which is what it means to say that $\hat P$ generates translations, together with $$ \hat P\,|0\rangle = 0 \hskip1cm \Rightarrow \hskip1cm \exp\big(i\hat P a/\hbar\big)\,|0\rangle = |0\rangle, $$ which says that the vacuum state is invariant under translations. Since $\hat P$ is also the momentum operator by definition (as in the Noether's-theorem perspective described above), saying that a photon has a single momentum $p$ is equivalent to saying that the state $|1\rangle$ satisfies $$ \hat P\,|1\rangle = p\,|1\rangle. $$ (By the way, the other equation $\hat P\,|0\rangle=0$ shown above says that the vacuum state has zero momentum.) This implies $$ \exp\big(i\hat P a/\hbar\big)\,|1\rangle = \exp\big(i p a/\hbar\big)\,|1\rangle. $$ By itself, this is inconclusive, because in a single-photon state, there is nothing else for the photon to interact with that could reveal its wavelength. However, the same principles still apply when we consider a photon in the context of some kind of interferometer, and then the fact that the photon's phase oscillates like $ \exp(i p a/\hbar)$ has observable consequences. In particular, translating the photon through a distance $a$ such that $pa/\hbar = 2\pi$ is the same as multiplying its state by $\exp(2\pi i)=1$. In other words, its wavelength is $$ \lambda=2\pi\frac{\hbar}{p} = \frac{h}{p}. $$ Although the idea of a massless photon does rely on relativity, the idea that the momentum and wavelength of a particle are related in this way does not. This relationship follows from the very general fact that the momentum operator generates translations in space — illustrated here using a model of the elecromagnetic field, and illustrated by tparker using single-particle quantum mechanics.
Chiral AnomalyChiral Anomaly
Ok, this is not a real direct answer to the question, but just a reaction to one of OP's comments. My remark was too long to fit in a comment, so I put it in an answer, sorry for that. S. rotos said he had a problem with one of the answers because : "yes, there are certain things that are axioms and not really derived so to speak, but there still must be some kind of intuition or justification. They don't just appear out of nowhere"
But I believe, they sometimes (often ?) do !
Actually, if we follow the historical train of thought on this matter, we should remember Planck who was working to solve the UV catastrophe. He was desperatly trying to describe black-body radiation through statistical mechanics. Out of idea, he tried out the hypothesis that radiations were emitted in discrete bundles of energy E=hf. An idea that came (almost) straight up out of nowhere, as he admitted himself (at least, with absolutely no physical justification behind). He did not credit any physical meaning to it and considered it as mere mathematical trickery.
Einstein, acknowledging how well Planck's result was describing the experimental results, later declared that there was indeed a physical meaning in all this. He interpreted it as the statement that light could also behave as a discrete particle with a given energy. Many consider this idea to be the beginning of quantum mechanics. De Broglie later took back this idea and mirrored it : he said that if a "wave" such as light could be described as a "particle" then a "particle" like an electron could be described as a "wave". This equivalence is done through the famous relationship we are talking about, and that can be considered as a natural consequence of the E=hf relation, as it has been explained in another answer
So as you see, we can pretty well say that the hypothesis "E=hf" did come out of nowhere ! Definitely not an intuitive statement : it went against all intuitions of that time. Just an hypothesis that was working so damn well that we tried to put some sense into it... And came up with quantum mechanics. I believe it is something that you have to take first as a mathematical trick, later confirmed by experimental fact.
Trying to find an intuitive principle for something that, in its very core, is as counter-intuitive as QM is, imo, a desperate attempt. All those "paradoxes" and insane behaviours at the quantum scale have to come from something that is at least a little bit fucked up, wouldn't you agree ?
The idea that physics should be intuitive is something I generally disagree with. If it was, we would live on a flat earth, with a sun circling all around. That's intuition right there :D
Barbaud JulienBarbaud Julien
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Quantum momentum (De Broglie)
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A probabilistic approach for calibration time reduction in hybrid EEG–fTCD brain–computer interfaces
Aya Khalaf1 &
Murat Akcakaya1
Generally, brain–computer interfaces (BCIs) require calibration before usage to ensure efficient performance. Therefore, each BCI user has to attend a certain number of calibration sessions to be able to use the system. However, such calibration requirements may be difficult to fulfill especially for patients with disabilities. In this paper, we introduce a probabilistic transfer learning approach to reduce the calibration requirements of our EEG–fTCD hybrid BCI designed using motor imagery (MI) and flickering mental rotation (MR)/word generation (WG) paradigms. The proposed approach identifies the top similar datasets from previous BCI users to a small training dataset collected from a current BCI user and uses these datasets to augment the training data of the current BCI user. To achieve such an aim, EEG and fTCD feature vectors of each trial were projected into scalar scores using support vector machines. EEG and fTCD class conditional distributions were learnt separately using the scores of each class. Bhattacharyya distance was used to identify similarities between class conditional distributions obtained using training trials of the current BCI user and those obtained using trials of previous users.
Experimental results showed that the performance obtained using the proposed transfer learning approach outperforms the performance obtained without transfer learning for both MI and flickering MR/WG paradigms. In particular, it was found that the calibration requirements can be reduced by at least 60.43% for the MI paradigm, while at most a reduction of 17.31% can be achieved for the MR/WG paradigm.
Data collected using the MI paradigm show better generalization across subjects.
Noninvasive BCIs are designed mainly to help individuals with limited speech and physical abilities (LSPA) due to neurological deficits to communicate with the surrounding environment without any surgical interventions [1]. Due to its low cost, high temporal resolution, and portability, EEG is the most common neuroimaging modality used to design noninvasive BCIs [1, 2]. However, performance of EEG BCIs degrades due to the low EEG signal-to-noise ratio (SNR) and nonstationarities existing in the EEG signals due to the background brain activity.
Multimodal BCIs have been recently studied to overcome the limitations of EEG-based BCIs and to improve their performance [3]. The most commonly used modality with EEG for hybrid BCI design is functional near-infrared spectroscopy (fNIRS). However, fNIRS lacks the high temporal resolution required for real-time BCI applications [4]. Previously, we have proved that functional transcranial Doppler (fTCD) can be used simultaneously with EEG to design an efficient hybrid EEG–fTCD BCI that outperforms all EEG–fNIRS systems in literature in terms of accuracy and speed [5, 6].
Generally, before being used by each individual, a BCI requires calibration to ensure that it can identify user intent with sufficient accuracy in a reasonable amount of time. Moreover, since the BCI performance is directly proportional to the amount of available training data, each BCI user has to attend a certain number of calibration sessions which may be burdensome for individuals with LSPA.
One potential solution for such a problem is combining data from different BCI users to calibrate the system for a certain user. However, the statistical distribution of the data varies across subjects and even across sessions within the same subject [7]. This limits the transferability of training data across sessions and subjects. The concept of transfer learning focuses on developing algorithms that can improve learning capacity so that the prediction model either learns faster or better on a given dataset through exposure to other datasets [8]. Recently, two categories of transfer learning methods have been studied including domain adaptation and rule adaptation methods [7]. Rule adaptation methods require learning a decision boundary for each subject separately. The decision boundary is considered as a random variable. The distribution of this random variable is found using the decision boundaries estimated based on datasets collected from previous subjects. However, for rule adaptation methods to be efficient, a high number of datasets are needed to estimate the distribution of the decision boundary.
In contrast, domain adaptation approaches have been extensively used for BCIs' applications. These approaches aim at finding a common structure in the data such that one decision boundary can be generalized across subjects. Finding a common structure can be performed either by finding a linear transformation where the data are invariant across all individuals [9] or using similarity measures to find the most similar datasets to the dataset under test [10]. Data alignment is one of the most common domain adaptation approaches. In a recent study, Zanini et al. [11] proposed aligning the covariance matrices of all sessions and subjects in the Riemannian space to center these covariance matrices with respect to a reference covariance matrix. However, the Riemannian distance (geodesic) computation in the Riemannian space is computationally expensive and unstable. To overcome these limitations, He et al. [12] introduced a method in which, instead of aligning the covariance matrices, they aligned the EEG trials within each session/subject in the Euclidean space based on a reference covariance matrix. However, these alignment methods require high-dimensional covariance matrix estimation which is highly dependent on the number of the available training trials and cannot be performed efficiently with a few number of training trials. In another study, Azab et al. [13] measured the similarity across different feature spaces in which the features were obtained using subject-specific CSP. The measured similarity was used to add a regularization parameter to the objective function of a logistic regression classifier such that the classification parameters are as similar as possible to the parameters of the previous BCI users whose feature spaces are similar to that of the current user. However, this transfer learning approach is not robust for BCIs employing CSP features because extracting features through applying subject-specific CSP yields different feature spaces for different subjects.
Deep learning approaches have been also used to transfer knowledge across different BCI users. For instance, Fahimi et al. [14] used a convolutional neural network (CNN) to learn a general model based on the data from a group of subjects. For a new BCI user, the general CNN model is updated based on a subset of data collected from that new user. In other studies, simultaneous training of an autoencoder and adversarial network was used to learn subject-invariant representations [15, 16]. However, such deep learning approaches require a large number of trials for training.
In this paper, we propose a domain adaptation-based transfer learning approach to reduce the calibration requirements of our hybrid EEG–fTCD BCI utilizing both MI and flickering MR/WG paradigms through transferring BCI training experience. To evaluate the performance of the proposed approach, we formulated 3 binary selection problems for each presentation paradigm including right arm MI versus baseline, left arm MI versus baseline, right versus left arm MI, MR versus baseline, WG versus baseline, and MR versus WG. Common spatial pattern (CSP) and wavelet decomposition were used to extract features from EEG and fTCD data collected using MI paradigm while template matching and wavelet decomposition were used to extract features from EEG and fTCD data of flickering MR/WG paradigm.
To apply transfer learning, similarity between the EEG and fTCD data of the current BCI user and those of the previous users has to be measured. To achieve such aim, we reduced feature vectors of EEG and fTCD data of each trial into scalar SVM scores to learn EEG and fTCD class conditional distributions. Similarities across participants were identified based on these class conditional distributions. In particular, we computed Bhattacharyya distance between the class conditional distributions obtained using the training data of the current BCI user and class conditional distributions obtained using datasets collected from the previous BCI users. After identifying the top similar datasets, we combined the training trials of the current user with trials of these top similar datasets to form a training set that can be used to calibrate the BCI system.
Using the new training set, we evaluated the performance of the system through assessing the test trials of the current BCI user. As mentioned above, for MI paradigm, CSP and wavelet features were extracted while template matching and wavelet features were considered in case of MR/WG paradigm. A probabilistic fusion approach was used to combine EEG and fTCD evidences which were obtained through reducing EEG and fTCD feature vectors of each trial into scalar SVM scores.
For both MI and flickering MR/WG paradigms, to evaluate the effectiveness of the proposed TL approach, for each binary selection problem, we reported the average accuracies and ITRs across participants obtained using different training set sizes. Moreover, we compare these accuracies/ITRs with those obtained without transfer learning (NTL). Figures 1, 2, 3, 4, 5 and 6 reflect the impact of the amount of data available to train a prediction model on the accuracy/ITR that can be obtained with and without transfer learning. In particular, the x-axis shows the number of training trials, ranging from 10 to 90 trials, used to train a prediction model, while the y axis shows the average accuracy/ITR across participants corresponding to those training trials.
Average accuracy (a) and average ITR (b) as a function of the number of training trials for right MI versus baseline problem
Average accuracy (a) and average ITR (b) as a function of the number of training trials for left MI versus baseline problem
Average accuracy (a) and average ITR (b) as a function of the number of training trials for right MI versus left MI problem
Average accuracy (a) and average ITR (b) as a function of the number of training trials for MR versus baseline problem
Average accuracy (a) and average ITR (b) as a function of the number of training trials MR versus WG problem
Average accuracy (a) and average ITR (b) as a function of the number of training trials for WG versus baseline problem
For both TL and NTL cases, at each training set size, a classifier is trained, and its performance is evaluated for each participant at trial lengths of 1, 2…10 s. The maximum accuracy/ITR at each training set size is reported regardless of the corresponding trial length. The average accuracy/ITR is computed across all participants at different training set sizes. Therefore, in terms of calibration requirements, comparing the best possible performances obtained for TL and NTL cases are not entirely fair since these performances are not evaluated at the same calibration length. In particular, calibration length is not only a function of the number of training trials, but also a function of trial length which varies depending on when maximum accuracy/ITR could be achieved. Therefore, as seen in Figs. 1, 2, 3, 4, 5 and 6, to ensure fair comparison, in addition to reporting the best possible TL and NTL performances, we evaluated the performance of NTL at the same trial lengths that yield the maximum possible TL performance. In addition, we evaluated the performance of TL and the same trial lengths that yield the maximum NTL performance.
MI paradigm
As seen in Figs. 1, 2 and 3, TL performance evaluated at the trial lengths that yield the maximum NTL performance is similar to maximum NTL performance while the performance of NTL at the same trial lengths that yield the maximum possible TL performance is significantly worse than the maximum TL performance. Disregarding differences in trial length, average accuracies obtained using TL are significantly higher than those obtained without transfer learning (NTL) as shown in Figs. 1, 2 and 3. Moreover, in terms of ITRs, it can be also noted that TL provides the highest ITRs compared to NTL case.
In addition, we observed that, when TL is employed, using only 10 training trials, average accuracies of 80.58%, 75.29%, and 69.16% can be achieved for right MI versus baseline, left MI versus baseline, and right MI versus left MI, while for NTL case, the average accuracies that can be obtained using 10 training trials are 56.63%, 58.14%, and 60.21%, respectively. In terms of ITRs, at 10 training trials, it can be noted that right MI versus baseline, left MI versus baseline, and right MI versus left MI achieved average ITRs of 2.34, 2.13, and 2.98 bits/min, respectively, compared to 1.51, 0.54, and 1.74 bits/min obtained for NTL case.
Using 90% of the available data for training which corresponds to 90 training trials, TL achieved accuracies of 98.89%, 98.00%, and 94.67% and ITRs of 16.5, 20.51, and 11.3 bits/min for right MI versus baseline, left MI versus baseline, and right MI versus left MI, respectively, compared to of accuracies of 80.00%, 78.33%, and 76.67% and ITRs of 7.83, 7.04, and 6.27 bits/min achieved without TL.
Using (15), we found that the calibration requirements for MI paradigm can be reduced by 80.00%, 60.43%, and 81.99% for right MI versus baseline, left MI versus baseline, and right MI versus left MI, respectively.
Flickering MR/WG paradigm
Figures 4, 5 and 6 show that TL performance evaluated at the trial lengths yielding the maximum NTL performance is comparable to maximum NTL performance while the performance of NTL at the same trial lengths yielding the maximum possible TL performance is significantly worse than maximum TL performance. Disregarding trial length, for the 3 binary selection problems, average accuracy and ITR trends obtained using TL are significantly higher than those obtained without transfer learning (NTL) especially at smaller training set sizes as shown in Figs. 4, 5 and 6. However, for WG versus baseline problem, we observed that ITRs obtained using TL outperform those obtained without TL for training set sizes < 50 trials.
We observed also that when the training set size drops to 10 trials, transfer learning provides an improvement in the accuracy by approximately 11%, 5%, and 7% for MR versus baseline, WG versus baseline, and MR versus WG. In terms of ITRs, at 10 training trials, 1, 0.37, and 0.71 bits/min were obtained for MR versus baseline, WG versus baseline, and MR versus WG using TL, while without TL, 0.28, 0.29, and 0.17 bits/min were achieved for the same classification problems.
Using 90 training trials, TL achieved 82.83%, 79.09%, and 80.00% average accuracies and 8.13, 10.66, and 15.28 bits/min average ITRs MR for versus baseline, WG versus baseline, and MR versus WG, respectively, while NTL obtained 75.76%, 80.52%, and 69.97% average accuracies and 6.83, 11.13, 6.55 bits/min average ITRs for the same classification problems.
Using (15), we found that the calibration requirements for flickering MR/WG paradigm can be reduced by 17.31% and 12.96% for MR versus baseline and MR versus WG, respectively, while for WG versus baseline, TL approach only boosted the performance accuracy without reducing the calibration requirement.
For MI paradigm, it can be concluded that, using 10 training trials, TL can improve the average performance accuracy by 9–24% for the 3 binary selection problems compared to NTL case, while using 100% of the available training data (90 trials), performance of NTL case can be enhanced by 18–20% for the 3 classification problems. Moreover, ITRs obtained using TL at 10 training trials are 1.8–2.90 times the ITRs obtained without TL, while at 90 training trials, ITRs of TL case are 1.5–3.94 times the ITRs obtained without TL.
As for MR/WG paradigm, at 10 training trials, improvements ranging from 5 to 11% in average accuracy as well as ITRs that are 1.28–4.18 times ITRs of NTL case can be achieved for MR versus baseline and MR versus WG. At 90 training trials, performance can be enhanced by 7–10% average accuracy with 1.19–2.33 times ITRs of NTL case. However, there is no improvement in performance for WG versus baseline problem when using 100% of the available training data.
Comparing the average accuracies and ITRs obtained using both paradigms as well as their average accuracy and ITR improvements compared to NTL case especially at 10 training trials, it can be concluded that the proposed transfer learning algorithm is more efficient when used with MI paradigm. Therefore, TL can be used to reduce the calibration requirements of the system while maintaining sufficient performance that is comparable to NTL performance with a higher number of training trials. For instance, given only 10 training trials from the current BCI user who uses MI paradigm, accuracies ranging from 70 to 80% can be achieved for the 3 classification problems when using the proposed transfer learning approach. This corresponds to a maximum of 100 s calibration length.
Considering the trade-off between the calibration length and the corresponding BCI performance, it is the BCI designer's decision to choose the optimal number of trials to be recorded from each BCI user to calibrate the system. Given that the proposed transfer learning approach has significantly reduced the calibration requirements of the MI-based hybrid BCI by at least 60.43%, we believe that our proposed approach gives more flexibility to the BCI designers to control and reduce the calibration requirements of the system which is an important criterion especially when the BCI is intended to be used by patients with disabilities.
In this paper, aiming at decreasing the calibration requirements of our hybrid EEG–fTCD BCI as well as improving its performance, we propose a transfer learning approach that identifies the top similar datasets to the current BCI user and combines the trials from these datasets as well as few training trials from the current user to train a classifier that can predict the test trials of that user with high accuracy. To achieve such aim, EEG and fTCD feature vectors of each trial were projected into two scalar SVM scores. EEG and fTCD class conditional distributions were learnt separately using the scores of each class. Bhattacharyya distance was used to identify similarities between class conditional distributions obtained using training trials of the current BCI user and those obtained using trials of previous BCI users. Experimental results showed that the performance obtained using the proposed transfer learning approach outperforms the performance obtained without transfer learning for both MI and flickering MR/WG paradigm. However, comparing performance improvement achieved for both paradigms, it can be noted that the proposed transfer learning algorithm is more efficient when used with MI paradigm. In particular, average accuracies and ITRs of 80.58%, 75.29%, and 69.16% and 2.34, 2.13, and 2.98 bits/min can be achieved for right MI versus baseline, left MI versus baseline, and right MI versus left MI using 10% of the available data which corresponds to a calibration length of 100 s. Moreover, it was found that the calibration requirements of MI paradigm can be reduced by at least 60.43% when using the proposed transfer learning approach.
A g.tec EEG system was employed for EEG data acquisition using 16 EEG electrodes positioned at locations Fp1, Fp2, F3, F4, Fz, Fc1, Fc2, Cz, C1, C2, Cp3, Cp4, P1, P2, P5, and P6. Reference electrode was placed over left mastoid. The collected data were sampled with 256 samples/s sampling rate. Moreover, data were filtered using the g.tec amplifier's bandpass filter (corner frequencies: 2 and 62 Hz) and the amplifier's notch filter with 58 and 62 Hz corner frequencies.
fTCD data collection was performed using a SONARA TCD system with two 2 MHz transducers placed on the right and left sides of the transtemporal window which is located above the zygomatic arch [17]. Given that middle cerebral arteries (MCAs) are responsible of approximately 80% of brain blood perfusion [18], the fTCD depth was set to the depth of the mid-point of the MCAs which is 50 mm [19].
Presentation paradigms
We designed two different presentation paradigms to be used with the proposed hybrid BCI. The first paradigm employed motor imagery (MI) tasks while the other paradigm used flickering mental rotation (MR) and word generation (WG) tasks as shown in Fig. 7. For both paradigms, while acquiring EEG and fTCD simultaneously, two tasks and a fixation cross that represents the baseline were presented on the screen. Total of 150 trials were presented to each user and during each trial, a vertical arrow randomly selected one of the three visual icons representing the two tasks and the baseline. The vertical arrow pointed to the selected icon for 10 s and the user was asked to perform the task identified by that arrow until the arrow points to another visual icon.
Stimulus presentation for our motor imagery EEG–fTCD BCI (a) and the proposed flickering MR/WG hybrid BCI (b)
During the MI-based presentation scheme, a basic MI task was presented to the users as shown in Fig. 7a. In particular, a horizontal white arrow that points to the right represented right arm MI while a horizontal white arrow that points to the left represented left arm MI. The baseline was represented by the fixation cross shown in the middle [20].
During MR/WG presentation paradigm, since MR and WG tasks are known to be differentiated using fTCD only, to make them differentiable in terms of EEG, the visual icons of MR and WG tasks were textured with a flickering checkerboard pattern as seen in Fig. 7b and they flickered at 7 Hz and 17 Hz, respectively, to induce different SSVEPs in EEG [21]. During WG task, the user was asked to silently generate words starting with the letter shown on the screen while during MR task, the user was given two 3D shapes and was asked to mentally rotate one of these shapes and decide if they were identical or mirrored.
The local Institutional Review Board (IRB) of University of Pittsburgh approved all the study procedures (IRB number: PRO16080475). All the subjects were consented before starting the experiment. A total of 21 healthy individuals participated in this study. In particular, to assess flickering MR/WG paradigm, data were collected from 11 individuals (3 females and 8 males) with ages ranging from 25 to 32 years while, to test MI paradigm, data were collected from 10 subjects (4 males and 6 females) with ages ranging from 23 to 32 years. None of the subjects participated in the study had a history of heart murmurs, concussions, migraines, strokes, or any brain-related injuries. Each subject attended one session that included 150 trials and each trial lasted for 10 s.
In this section, we describe our feature extraction approaches applied to EEG and fTCD signals collected using both MI and flickering MR/WG paradigm.
We employed common spatial pattern (CSP) to analyze EEG MI data [6]. CSP is known to be an efficient feature extraction technique for MI-based EEG BCIs as it can extract EEG spatial patterns that characterize different MI tasks [22]. CSP aims at finding a linear transformation that changes the variance of the observations representing two different classes such that the two classes are more separable [23]. More specifically, CSP learns the optimal spatial filters that result in maximizing the variance of one class in a certain direction, and in the mean time, minimize the variance of the second class in the same direction [24]. These filters can be found by solving the following optimization problem:
$$\text{max} _{W} {\text{tr}}\;W^{T} \varSigma_{c} W$$
$${\text{s}}.{\text{t}}. \, W^{T} \left( {\varSigma_{\left( + \right)} + \varSigma_{\left( - \right)} } \right)W = 1,$$
where \(\varSigma_{c}\) is the average trial covariance for class \(c \epsilon \left\{ { + , - } \right\}\) and \(W\) the transformation matrix.
Assume each trial data are represented as a matrix \(R^{N \times T}\) where \(N\) is the number of EEG electrodes and \(T\) represents the number of the samples for each electrode. Sample covariance of each trial \(m\) can be calculated as follows:
$$S_{m} = \frac{{{\text{RR}}^{T} }}{{{\text{tr}}\left( {{\text{RR}}^{T} } \right)}}.$$
Using (2), the average trial covariance can be calculated as given below
$$\sum_{c} = \frac{1}{M}\mathop \sum \limits_{m = 1}^{M} S_{m} ,$$
where \(M\) is the number of trials belonging to class \(c\).
(1) is solved through simultaneously diagonalizing the covariance matrices \(\varSigma_{c}\) which can be represented as follows:
$$\begin{aligned} W^{{}} \varSigma_{\left( + \right)} W = \varLambda_{\left( + \right)} \hfill \\ W^{T} \varSigma_{\left( - \right)} W = \varLambda_{\left( - \right)} , \hfill \\ {\text{s}}.{\text{t}}. \, \varLambda_{\left( + \right)} + \varLambda_{\left( - \right)} = I \hfill \\ \end{aligned}$$
where \(\varLambda_{c}\) is a diagonal matrix with eigenvalues \(\lambda_{j}^{c} ,\)\(j = 1,2,3, \ldots N\) on its diagonal.
Solution of (4) is similar to the solution of the generalized eigenvalue problem below:
$$\varSigma_{\left( + \right)} w_{j} = \lambda \varSigma_{\left( - \right)} w_{j}$$
where \(w_{j}\) is the \(j{\text{th}}\) generalized eigenvector and \(\lambda = \frac{{\lambda_{j}^{\left( + \right)} }}{{\lambda_{j}^{\left( - \right)} }}\). (4) is satisfied when the transformation matrix is equivalent to \(W = \left[ {w_{1} ,w_{2} , \ldots w_{N} } \right]\) and \(\lambda_{j}^{c}\) is given by
$$\lambda_{j}^{c} = w_{j}^{T} \varSigma_{c} w_{j}$$
where \(\lambda_{j}^{c}\) are the elements on diagonal of \(\varLambda_{c}\). \(\lambda_{j}^{\left( + \right)} + \lambda_{j}^{\left( - \right)} = 1\), since \(\varLambda_{\left( + \right)} + \varLambda_{\left( - \right)} = I\).
It can be noted that a higher value of \(\lambda_{j}^{\left( + \right)}\) will result in a higher variance in the data representing class \(c = +\) when filtered using \(w_{j}\). Given that a high value of \(\lambda_{j}^{\left( + \right)}\) results in a low \(\lambda_{j}^{\left( - \right)}\) value, when filtering the data of class \(c = -\) using \(w_{j}\), a low variance will be obtained. In this study, we solved 3 binary MI selection problems by considering different numbers of eigenvectors. More specifically, MI EEG data were spatially filtered using 1, 2, …., and 8 eigenvectors from both ends of the transformation matrix \(\left( W \right)\). For each trial, log variance of each filtered signal was computed and considered as a feature.
MR/WG paradigm
As explained before in our previous study, we used template matching to extract features from EEG data [5]. More specifically, for each class, since each trial is represented by 16 EEG segments collected from 16 electrodes, we extract 16 templates corresponding to the 16 EEG electrodes by averaging EEG training trials over each electrode. To extract features representing each trial, cross-correlations between the segments of that trial and the corresponding 16 templates representing each class were calculated. Maximum cross-correlation score across each of 16 cross-correlations was considered as a feature resulting in a total of 16 features. Given that the problems of interest are binary classification problems, the feature vector representing each trial contained a total of 32 features.
fTCD
5-level wavelet decomposition [25] was used to analyze the two fTCD data segments of each trial with Daubechies 4 mother wavelet. To decrease the fTCD feature vector dimensions, instead of considering each wavelet coefficient as a feature, we calculated statistical features for each of the 6 wavelet bands resulting from the wavelet analysis. These features included mean, variance, skewness, and kurtosis [26, 27]. Therefore, each trial was represented by 24 features for each fTCD data segment and a total of 48 features.
Feature selection and projection
Wilcoxon rank-sum test [28] with a p value of 0.05 was employed for the selection of the significant features from both EEG and fTCD feature vectors of MR/WG paradigm while it was used to select only fTCD significant features of MI paradigm. As for MI EEG, the feature vector representing a certain trial was composed of \(2f\) features obtained through transforming the data of the trial using \(f =\) 1, 2, …., and 8 eigenvectors from both ends of the transformation matrix \(\left( W \right)\).
EEG and fTCD feature vectors of each trial were then projected separately into 2 SVM scalar scores (EEG and fTCD evidences). To evaluate the performance of both MI and MR/WG paradigms, these evidences were combined under the Bayesian fusion approach explained in "Bayesian fusion and decision making" section. Performance of the MI hybrid system was evaluated using \(2f\) (2, 4,…, and 16) EEG CSP features. The highest performance measures obtained with and without transfer learning were reported and compared in the results section while for MR/WG paradigm, performance measures with and without transfer learning were calculated and compared only at p value of 0.05.
Bayesian fusion and decision making
We developed a Bayesian fusion approach of EEG and fTCD evidences to infer user intent at a given trial considering three different assumptions [5, 6]. Under assumption (\(A1\)), EEG and fTCD evidences are assumed to be jointly distributed while under assumption (\(A2\)), EEG and fTCD evidences are assumed to be independent. Under assumption (\(A3\)), evidences of EEG and fTCD are assumed to be independent, but they contribute unequally toward taking a right decision. For each binary selection problem, tenfold cross validation was used to define training and testing trials. Our previous work showed that the best performance was achieved under assumption \(A3\) for MI paradigm; therefore, for MI, we utilized the assumption A3 in this paper [6]. On the other hand, for flickering WG/MR paradigm, A2 and \(A3\) both had high performance without any statistically significant differences [5]. However, \(A3\) is more computationally complex compared to \(A2\); therefore, for WG/MR paradigm, we performed probabilistic fusion under assumption \(A2\) [5].
Given that \(N\) trials are introduced to each participant, these trials are represented by a set of EEG and fTCD evidences \(Y = \left\{ {y_{1} , \ldots y_{N} } \right\}\) where \(y_{k} = \left\{ {e_{k} , f_{k} } \right\}\), \(e_{k}\) and \(f_{k}\) are EEG and fTCD evidences of a test trial \(k\). User intent \(x_{k}\) for the test trial \(k\) can be inferred through joint state estimation using EEG and fTCD evidences which can be represented as follows:
$$\widehat{{x_{k} }} = \arg \mathop {\text{max} }\limits_{{x_{k} }} p\left( {x_{k} |Y = y_{k} } \right)$$
where \(p(x_{k} |Y)\) is the state posterior distribution conditioned on the observations \(Y\). Using Bayes rule, (7) can be rewritten as
$$\widehat{{x_{k} }} = \arg \mathop {\text{max} }\limits_{{x_{k} }} p(Y = y_{k} |x_{k} ) p\left( {x_{k} } \right)$$
where \(p(Y|x_{k} )\) is the state conditional distribution of the measurements \(Y\) and \(p\left( {x_{k} } \right)\) is the prior distribution of user intent \(x_{k}\). Since the trials are randomized, \(p\left( {x_{k} } \right)\) is assumed to be uniform. Therefore, (8) can be reduced to
$$\widehat{{x_{k} }} = \arg \mathop {\text{max} }\limits_{{x_{k} }} p(Y = y_{k} |x_{k} ).$$
\(p(Y|x_{k} )\) of each class can be estimated using the EEG and fTCD evidences computed for the training trials. To infer user intent at a test trial \(k\), Eq. (9) is solved at \(Y = y_{k} .\) Here, the distributions \(p(Y|x_{k} = 1)\) and \(p(Y|x_{k} = 2)\) are evaluated under two assumptions as explained below.
Assumption 2: independent distributions
Here, the evidences of EEG and fTCD, conditioned on \(x_{k}\), are assumed to be independent. Therefore, (9) can be rewritten as
$$\widehat{{x_{k} }} = \arg ,\mathop {\text{max} }\limits_{{x_{k} }} p(e = e_{k} |x_{k} )p(f = f_{k} |x_{k} )$$
where \(p(e|x_{k} )\) and \(p(f|x_{k} )\) are the distributions of EEG and fTCD evidences conditioned on the state \(x_{k}\) respectively. To find \(p(e|x_{k} )\) and \(p(f|x_{k} )\), kernel density estimation (KDE) with Gaussian kernel was employed using evidences of EEG and fTCD of the training trials. Kernel bandwidth was computed using Silverman's rule of thumb [29]. \(e_{k}\) and \(f_{k}\) are plugged in (10) to infer the user intent of a test trial \(k\) where the user intent \(x_{k}\) that maximizes the likelihood is selected.
Assumption 3: weighted independent distributions
Here, we assume that evidences of EEG and fTCD are independent, but they contribute unequally toward taking a right decision. Therefore, we propose weighting \(p(e|x_{k} )\) and \(p(f|x_{k} )\) conditional distributions with weights of \(\alpha\) and \(1 - \alpha\), respectively. (9) can be rewritten as
$$\widehat{{x_{k} }} = \arg ,\mathop {\text{max} }\limits_{{x_{k} }} p(e = e_{k} |x_{k} )^{\alpha } p(f = f_{k} |x_{k} )^{1 - \alpha }$$
where \(\alpha\) is a weighting factor ranging from 0 to 1. \(p(e|x_{k} )\) and \(p(f|x_{k} )\) are computed as mentioned in "Assumption 2: independent distributions" section. Finding the optimal α value is performed through applying a grid search over \(\alpha\) values ranging between 0 and 1 with a step of 0.01.
Transfer learning algorithm
With the aim of decreasing calibration requirements and improving the performance of the hybrid system, we propose a transfer learning approach that identifies the top similar datasets collected from previous BCI users to a training dataset collected from a current BCI user and uses these datasets to augment the training data of the current BCI user. The proposed transfer learning approach is intended to be used for both MI and flickering MR/WG paradigms. Therefore, the performance of the proposed approach was tested using the 6 binary selection problems of both paradigms.
Similarity measure
To apply transfer learning to a certain binary selection problem, for each dataset from previous BCI users, EEG and fTCD feature vectors of trials corresponding to that problem were projected into scalar SVM scores. Therefore, each trial was represented by a scalar EEG SVM score and a scalar fTCD SVM score. Using KDE, 2 EEG class conditional distributions and 2 fTCD class conditional distributions were learnt from these scores. KDE was performed using Gaussian kernel. EEG and fTCD class conditional distributions of the current BCI user were also estimated using his/her training trials.
To measure the similarity between the class conditional distributions of the current BCI user and those of the previous users, Bhattacharyya distance [30], given by (12), was used since it is a symmetric measure that can be applied to general distributions especially if these distributions are diverging from normal distributions and it provides bounds on Bayesian misclassification probability, which overall fits very well to our approach of making Bayesian decisions on binary classification problems using the estimated density functions.
$$d = - \ln \mathop \sum \limits_{i = 1}^{N} P_{i} Q_{i} ,$$
where \(P\) and \(Q\) are 2 probability distributions and \(N\) is the number of points composing each distribution.
Bhattacharyya distance between EEG class conditional distribution of class \(i \left( {i = 1, 2} \right)\) and the corresponding EEG class conditional distribution of the current BCI user was calculated. Bhattacharyya distance was also calculated between the fTCD class conditional distributions of each previous BCI user and the current BCI user. Sum of these 4 distances (2 EEG distances and 2 fTCD distances) represented the total distance between the current BCI user and a certain previous BCI user.
Proposed transfer learning algorithm
The proposed transfer learning approach is described in detail in Figs. 8 and 9. Given a group of previous BCI users where each user is represented by one dataset, the objective is to find the most similar datasets to the training dataset of the current BCI user and to combine the trials from these datasets with small number of training trials from the current user to train a classifier that can predict the labels of the test trials of that user with high accuracy. In particular, for each binary selection problem, the dataset of the current user was divided into training and testing sets. Initially, given that each binary selection problem is represented by 100 trials, we used the first 10 trials from the current BCI user for training the prediction model and the remaining 90 trials for testing. As seen in Fig. 8, features are extracted from training trials of the current user as well as the trials corresponding to the binary problem of interest from each of the previous BCI users. Extracted EEG and fTCD features vary depending on the paradigm used for data collection. In particular, CSP and wavelet decomposition were used to extract features from the data of the MI paradigm while template matching and wavelet decomposition were used to extract features from the data of the flickering MR/WG paradigm as explained in "Feature extraction" section. After applying the feature selection step detailed in "Feature selection and projection" section, EEG and fTCD feature vectors of each trial were projected into 2 scalar SVM scores.
Identification of the top similar datasets using the proposed transfer learning approach
Testing phase of the proposed transfer learning approach
For each class within the binary selection problem of interest, we learnt class conditional distributions of the EEG and fTCD scores obtained from SVM projection as seen in Fig. 8. Distance between class conditional distributions of the current BCI user and those of each of the previous BCI users was computed as explained in "Similarity measure" section. To identify the top similar datasets, these distances were sorted ascendingly. At this point, it was required to decide on how many similar datasets should be considered to train the classifier besides the training trials from the current BCI user. Here, we considered a maximum of 3 datasets to be combined with the training trials of the current BCI user. Through crossvalidation, the number of top similar datasets that maximize the performance accuracy when combined with the training trials of the current user was chosen to be used later to predict test trials of the current BCI user as shown in Fig. 9. Here, for each participant, we used up to 3 datasets to be used for transfer learning. However, the maximum number of datasets could be increased or decreased depending on the needs of the designers. Moreover, the presented framework could be used to identify person-specific maximum number of datasets. For future versions of this algorithm, instead of using a maximum of 3 datasets to be combined with the training trials of the current BCI, such number can be optimized for each subject separately by means of model order selection techniques [31].
To study the impact of the training set size (from the current BCI user) on the performance of the proposed transfer learning approach, we applied the proposed approach on training sets of size ranging from 10 to 90 trials which corresponds to test sets of size ranging from 90 to 10 trials.
For both MI and flickering MR/WG paradigms, to assess the significance of the transfer learning (TL) compared to the no transfer learning case (NTL), for each participant, accuracy and information transfer rate (ITR) [32] were calculated and compared at different number of training trials from the current BCI user. In particular, at every number of training trials, accuracy and ITR were calculated at time points 1, 2….,10 s. For each number of training trials, maximum accuracy and ITR across the 10-s trial length were reported for TL and NTL cases. ITR can be calculated as follows:
$$B = \log_{2} \left( N \right) + P\log_{2} \left( P \right) + \left( {1 - P} \right)\log_{2} \left( {\frac{1 - P}{N - 1}} \right)$$
where N represents the number of BCI selections, P represents the classification accuracy, and B is the information transfer rate per trial.
To compute the reduction in calibration requirements for each binary problem when using TL compared to NTL case, at each training set size, we formed a vector containing performance accuracies obtained for all participant at that training set size. We statistically compared the accuracy vectors of TL at training set sizes of 10, 20…,90 with accuracy vector obtained for NTL case at maximum training set size (90 trials). Initially, at 10 training trials, we performed one-sided Wilcoxon signed rank test between the accuracy vector of TL with 10 training trials and NTL accuracy vector at 90 training trials. Such statistical comparison is repeated with TL applied at bigger training set sizes until there is no statistically significant difference between the performance of TL and the performance of NTL at 90 trials. The number of trials \(N\) at which that statistical insignificance occurs is used in (14) to compute percentage of reduction.
$${\text{Reduction}}\% = \frac{1}{P}\mathop \sum \limits_{i = 1}^{P} \frac{{{\text{Calibration length}}_{\text{NTL}} \left( i \right) - {\text{Calibration}}\;{\text{length}}_{\text{TL}} \left( i \right)}}{{{\text{Calibration length}}_{\text{NTL}} \left( i \right)}} \times 100\% .$$
Equation (14) is equivalent to
$${\text{Reduction\% }} = \frac{1}{P}\mathop \sum \limits_{i = 1}^{P} \frac{{N \times {\text{Trial length}}_{N} \left( i \right)_{\text{NTL}} - m \times {\text{Trial length}}_{m} \left( i \right)_{\text{TL}} }}{{N \times {\text{Trial length}}_{N} \left( i \right)_{\text{NTL}} }} \times 100{\text{\% }}$$
where \(N\) is the maximum number of training trials (\(N\) = 90) from the current BCI user and \(m\) is the minimum number of trials at which TL performance is at least equivalent to NTL performance where \(m\) ranges from 10 to 90 trials.
To guarantee that TL will improve or at least achieve the same average performance accuracy obtained for the NTL case, we checked if the TL average performance accuracy at \(m\) training trials was similar to or outperforms the average performance accuracy of NTL case at 90 training trials. If this condition is not satisfied, we consider statistical comparisons at training set sizes \(> m\) until this condition is satisfied.
The datasets used and/or analysed during the current study are available on reasonable request.
BCI:
Brain–computer interface
CSP:
Common spatial pattern
fNIRS:
Functional near-infrared spectroscopy
fTCD:
Functional transcranial Doppler
ITR:
Information transfer rate
LSPA:
Limited speech and physical abilities
MR:
Mental rotation
NTL:
No transfer learning
SNR:
TL:
Transfer learning
WG:
Word generation
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We would like to thank Swanson School of Engineering for funding this study.
This study is supported by Swanson School of Engineering startup funds.
Electrical and Computer Engineering Department, University of Pittsburgh, Pittsburgh, PA, USA
Aya Khalaf & Murat Akcakaya
Aya Khalaf
Murat Akcakaya
AK analyzed the data and wrote the paper. MA supervised the data analysis and revised the manuscript. Both authors read and approved the final manuscript.
Correspondence to Aya Khalaf.
All research procedures employed in this study were approved by University of Pittsburgh IRB (IRB number: PRO16080475).
Khalaf, A., Akcakaya, M. A probabilistic approach for calibration time reduction in hybrid EEG–fTCD brain–computer interfaces. BioMed Eng OnLine 19, 23 (2020). https://doi.org/10.1186/s12938-020-00765-4
Hybrid brain–computer interfaces
Template matching
Wavelet decomposition
Probabilistic fusion | CommonCrawl |
\begin{document}
\title{A lower bound on seller revenue in single buyer monopoly
auctions}
\author{Omer Tamuz\footnote{Weizmann Institute, Rehovot 76100, Israel}} \maketitle
\begin{abstract}
We consider a monopoly seller who optimally auctions a single object
to a single potential buyer, with a known distribution of
valuations. We show that a tight lower bound on the seller's
expected revenue is $1/e$ times the geometric expectation of the
buyer's valuation, and that this bound is uniquely achieved for the
equal revenue distribution. We show also that when the valuation's
expectation and geometric expectation are close, then the seller's
expected revenue is close to the expected valuation. \end{abstract}
\section{Introduction} Consider a monopoly seller, selling a single object to a single potential buyer. We assume that the buyer has a valuation for the object which is unknown to the seller, and that the seller's uncertainty is quantified by a probability distribution, from which it believes the buyer picks its valuation.
Assuming that the seller wishes to maximize its expected revenue, Myerson~\cite{myerson1981optimal} shows that the optimal incentive compatible mechanism involves a simple one-time offer: the seller (optimally) chooses a price and offers the buyer to buy the object for this price; the assumption is that the buyer accepts the offer if its valuation exceeds this price. Myerson's seminal paper has become a classical result in auction theory, with numerous follow-up studies. A survey of this literature is beyond the scope of this paper (see, e.g.,~\cite{krishna2009auction,klemperer1999auction}).
The expected seller revenue is an important, simple intrinsic characteristic of the valuation distribution. A natural question is its relation with various other properties of the distribution. For example, can seller revenue be bounded given such characterizations of the valuation as its expectation, entropy, etc.? An immediate upper bound on seller revenue is the buyer's expected valuation. In fact, the seller can extract the buyer's expected valuation only if the seller knows the buyer's valuation exactly - i.e., the distribution over valuations is a point mass.
Lower bounds on seller revenue are important in the study of approximations to Myerson auctions (see., e.g., Hartline and Roughgarden~\cite{hartline2009simple}, Daskalakis and Pierrakos~\cite{daskalakis2011simple}). A general lower bound on the seller's revenue is known when the distribution of the buyer's valuation has a monotone hazard rate; in this case, the seller's expected revenue is at least $1/e$ times the expected valuation (see Hartline, Mirrokni and Sundararajan~\cite{hartline2008optimal}, as well as Dhangwatnotai, Roughgarden and Yan~\cite{dhangwatnotai2010revenue}).
This bound does not hold in general: as an extreme example, the equal revenue distribution discussed below has infinite expectation but finite seller revenue. The family of monotone hazard rate distributions does not include many important distributions such the Pareto distribution or other power law distributions, or in fact any distribution that doesn't have a very thin tail, vanishing at least exponentially. The above mentioned lower bound for monotone hazard rate distributions does not apply to these distributions, and indeed it seems that the literature lacks any similar, general lower bounds on seller revenue.
The {\em geometric expectation} of a positive random variable $X$ is $\geo{X} = \exp(\E{\log X})$ (see, e.g.,~\cite{paolella2006fundamental}). We show that a general lower bound on the seller's expected revenue is $1/e$ times the geometric expectation of the valuation. Equivalently, the (natural) logarithm of the expected seller revenue is greater than or equal to the expectation of the logarithm of the valuation, minus one. This bound holds for any distribution of positive valuations. Notably, the {\em
regularity} condition, which often appears in the context of Myerson auctions, is not required here. This result is a new and perhaps unexpected connection between two natural properties of distributions: the geometric expectation and expected seller revenue.
We show that this bound is tight in the following sense: for a fixed value of the geometric mean, there is a unique cumulative distribution function (CDF) of the buyer's valuation for which the bound is achieved; this distribution is the equal revenue distribution, with CDF of the form $F(v) = 1-c/v$ for $v \geq c$. This distribution is ``special'' in the context of single buyer Myerson auctions, as it is the only one where seller revenue is identical for all prices.
The ratio between expected valuation and expected seller revenue is a natural measure of the uncertainty of the valuation distribution. Also, the discrepancy between the geometric expectation and the (arithmetic) expectation of a positive random variable is a well known measure of its dispersion. Hence, when the ratio between the expectations is close to one, one would expect the amount of uncertainty to be low and therefore seller revenue to be close to the expected valuation. We show that this is indeed the case: when the buyer's valuation has finite expectation, and the geometric expectation is within a factor of $1-\delta$ of the expectation, then seller revenue is within a factor of $1-2^{4/3}\delta^{1/3}$ of the expected valuation. Similarly, it is easy to show that when the variance of the valuation approaches zero then seller revenue also approaches the expected valuation.
\section{Definitions and results} We consider a seller who wishes to sell a single object to a single potential buyer. The buyer has a valuation $V$ for the object which is picked from a distribution with CDF $F$, i.e. $F(v) = \P{V \leq v}$.
We assume that $V$ is positive, so that $\P{V \leq 0} = 0$ or $F(0) = 0$. We otherwise make no assumptions on the distribution of $V$; it may be atomic or non-atomic, have or not have an expectation, etc.
The seller offers the object to the buyer for a fixed price $p$. The buyer accepts the offer if $p < V$, in which case the seller's revenue is $p$. Otherwise, i.e., if $p \geq V$, then the seller's revenue is 0. Thus, the seller's expected revenue for price $p$, which we denote by $\uup{p}{V}$, is given by \begin{align}
\label{eq:u-p}
\uup{p}{V} = p\P{p < V} = p(1-F(p)). \end{align} We define \begin{align}
\label{eq:u-s-max}
\uu{V} = \sup_p\uup{p}{V} = \sup_p p(1-F(p)). \end{align} When this supremum is achieved for some price $p$ then $\uu{V}$ is the seller's maximal expected revenue, achieved in the optimal Myerson auction with price $p$.
We define the {\em geometric expectation} (see, e.g.,~\cite{paolella2006fundamental}) of a positive real random variable $X$ by $\geo{X} = \exp\left(\E{\log X}\right)$. Note that $\geo{X} \leq \E{X}$ by Jensen's inequality, and that equality is achieved only for point mass distributions, i.e., when the buyer's valuation is some fixed number. Note that likewise $\uu{V} \leq \E{V}$, again with equality only for point mass distributions.
The equal revenue distribution with parameter $c$ has the following CDF: \begin{align}
\label{eq:tight}
\Phi_c(p) = \begin{cases}0&p \leq c\\ 1-\frac{c}{p}& p>c\end{cases}. \end{align} It is called ``equal revenue'' because if $V_c$ has CDF $\Phi_c$ then $\uup{p}{V_c} = \uu{V_c}$ for all $p \geq c$.
Our main result is the following theorem. \begin{theorem}\label{thmGeometricLowerBound}
Let $V$ be a positive random variable. Then $\uu{V} \geq
\frac{1}{e}\geo{V}$, with equality if and only if $V$ has the equal
revenue CDF $\Phi_c$ with $c = \uu{V}$. \end{theorem} \begin{proof}
Let $V$ be a positive random variable with CDF $F$. By
Eq.~\ref{eq:u-s-max} we have that
\begin{align}
\label{eq:basic}
\log \uu{V} \geq \log p + \log(1-F(p))
\end{align}
for all $p$. We now take the expectation of both sides with respect
to $p \sim F$:
\begin{align}
\label{eq:expectations}
\int_0^\infty \log \uu{V} dF(p) \geq \int_0^\infty \log p \,dF(p) +
\int_0^\infty\log(1-F(p)) dF(p).
\end{align}
Since $\uu{V}$ is a constant then the l.h.s.\ equals $\log \uu{V}$. The first
addend on the r.h.s.\ is simply $\E{\log V}$. The second is $\E{\log
(1-F(V))}$; note that $F(V)$ is distributed uniformly on $[0,1]$,
and that therefore
\begin{align*}
\E{\log (1-F(V))} = \int_0^1\log(1-x)dx = -1.
\end{align*}
Hence Eq.~\ref{eq:expectations} becomes:
\begin{align*}
\log \uu{V} \geq \E{\log V} - 1,
\end{align*}
and
\begin{align*}
\uu{V} \geq \frac{1}{e}\exp(\E{\log V}) = \frac{1}{e}\geo{V}.
\end{align*}
To see that $\uu{V}=\frac{1}{e}\geo{V}$ only for the equal revenue
distribution with parameter $\uu{V}$, note that we have equality in
Eq.~\ref{eq:basic} for all $p$ in the support of $F$ if and only if
$F=\Phi_c$ for some $c$, and that therefore we have equality in
Eq.~\ref{eq:expectations} if and only if $F=\Phi_c$ for some
$c$. Finally, a simple calculation yields that $c=\uu{V}$. \end{proof}
Note that this proof in fact demonstrates a stronger statement, namely that the expected revenue is at least $\frac{1}{e}\geo{V}$ for a seller picking a random price from the distribution of $V$. Dhangwatnotai, Roughgarden and Yan~\cite{dhangwatnotai2010revenue} use similar ideas to show lower bounds on revenue, for valuation distributions with monotone hazard rates.
We next show that when the geometric expectation approaches the (arithmetic) expectation then the seller revenue also approaches the expectation.
\begin{theorem}
Let $V$ be a positive random variable with finite expectation, and
let $\geo{V}=(1-\delta)\E{V}$. Then $\uu{V} \geq
\left(1-2^{4/3}\delta^{1/3}\right)\E{V}$. \end{theorem} \begin{proof}
Let $V$ be a positive random variable with finite expectation, and
denote $1-\delta = \frac{\geo{V}}{\E{V}}$. We normalize $V$ so that
$\E{V} = 1$, and prove the claim by showing that $\uu{V} \geq
1-2^{4/3}\delta^{1/3}$.
Consider the random variable $V-1-\log V$. Since $\E{V}=1$, we have
that $\E{V-1-\log V} = -\log \geo{V} = -\log(1-\delta)$. Since $x -
1 \geq \log x$ for all $x>0$, then $V-1-\log V$ is
non-negative. Hence by Markov's inequality
\begin{align*}
\P{V-1-\log V \geq -k \log (1-\delta)} \leq \frac{1}{k},
\end{align*}
or
\begin{align}
\label{eq:concentration}
\P{Ve^{1-V} \leq (1-\delta)^k} \leq \frac{1}{k}.
\end{align}
This inequality is a concentration result, showing that when
$\delta$ is small then $Ve^{1-V}$ is unlikely to be much less than
one. However, for our end we require a concentration result on $V$
rather than on $Ve^{1-V}$; that will enable us to show that the
seller can sell with high probability for a price close to the
arithmetic expectation. To this end, we will use the {\em Lambert
$W$ function}, which is defined at $x$ as the solution of the
equation $W(x)e^{W(x)} = x$. We use it to solve the inequality of
Eq.~\ref{eq:concentration} and arrive at
\begin{align*}
\P{V \leq -W\left(-(1-\delta)^k/e\right)} \leq \frac{1}{k},
\end{align*}
which is the concentration result we needed: $V$ is unlikely to be
small when $\delta$ is small. It follows that by setting the price
at $-W\left(-(1-\delta)^k/e\right)$, the seller sells with
probability at least $1-1/k$, and so
\begin{align*}
\uu{V} \geq -W\left(-(1-\delta)^k/e\right) \cdot \Big(1-1/k\Big).
\end{align*}
Now, an upper bound on $W$ is the
following~\cite{corless1996lambertw}:
\begin{align*}
W(x) \leq -1+\sqrt{2(ex+1)},
\end{align*}
and so
\begin{align*}
\uu{V} \geq \Big(1-\sqrt{2(1-(1-\delta)^k}\Big) \cdot
\Big(1-1/k\Big)
\geq \Big(1-\sqrt{2\delta k}\Big) \cdot
\Big(1-1/k\Big).
\end{align*}
Setting $k=(2\delta)^{-1/3}$ we get
\begin{align*}
\uu{V}
\geq \Big(1-(2\delta)^{1/2} (2\delta)^{-1/6}\Big) \cdot \Big(1-(2\delta)^{1/3}\Big) \geq 1-2(2\delta)^{1/3}.
\end{align*} \end{proof}
\section{Open questions} It may very well be possible to show tighter {\em upper} bounds for $\uu{V}$, using continuous entropy. For example, let $V$ have expectation $1$ and entropy at least $1$. Then $\uu{V}$ is at most $1/e$: in fact, it is equal to $1/e$ since, by maximum entropy arguments, there is only one distribution on $\R^+$ (the exponential with expectation $1$) that satisfies both conditions, and for this distribution $\uu{V}=1/e$.
One could hope that it is likewise possible to prove upper bounds on $\uu{V}$, given that $V$ has expectation $1$ and entropy at least $h < 1$; intuitively, the entropy constraint should force $V$ to spread rather than concentrate around its expectation, decreasing the seller's expected revenue.
\section{Acknowledgments} We would like to thank Elchanan Mossel for commenting on a preliminary version of this paper. We owe a debt of gratitude to the anonymous reviewer who helped us improve the paper significantly through many helpful suggestions.
This research is supported by ISF grant 1300/08, and by a Google Europe Fellowship in Social Computing.
\end{document} | arXiv |
What is the size of Australia's sexual minority population?
Tom Wilson ORCID: orcid.org/0000-0001-8812-75561,
Jeromey Temple1,
Anthony Lyons2,3 &
Fiona Shalley4
The aim is to present updated estimates of the size of Australia's sexual minority adult population (gay, lesbian, bisexual, and other sexual minority identities). No estimate of this population is currently available from the Australian Bureau of Statistics, and very little is available from other sources. We obtained data on sexual minority identities from three data collections of two national surveys of recent years. Combining averaged prevalence rates from these surveys with official Estimated Resident Population data, we produce estimates of Australia's sexual minority population for recent years.
According to percentages averaged across the three survey datasets, 3.6% of males and 3.4% of females described themselves with a minority sexual identity. When applied to Estimated Resident Populations, this gives a sexual minority population at ages 18 + in Australia of 599,500 in 2011 and 651,800 in 2016. Population estimates were also produced by sex and broad age group, revealing larger numbers and higher sexual minority percentages in the younger age groups, and smaller numbers and percentages in the oldest age group. Separate population estimates were also prepared for lesbian, gay, bisexual, and other sexual minority identities.
How many people in Australia identify themselves as lesbian, gay, bisexual or an alternative sexual minority orientation (e.g., queer, pansexual)? The question is difficult to answer because the Australian Bureau of Statistics (ABS) does not publish population estimates which include a sexual identity breakdown, nor does it directly collect data on sexual identity in the census or its continuous large-scale surveys which would permit such estimates to be easily calculated. The availability of population statistics from other sources is extremely limited; only a handful of academic studies have attempted to estimate the size of Australia's sexual minority population [1,2,3,4].
Despite this paucity of data, the value of population statistics by sexual identity has been increasingly recognised in recent years [5, 6]. Population estimates on sexual minorities can provide visibility and voice to those communities. They may assist in combating misinformation and stereotypes [7]. Population numbers can inform the likely demand for specialised goods and services aimed at sexual minorities. They provide the denominators for demographic rates and indicators which enable the health and wellbeing of sexual minorities to be monitored [8]. Sexual identity population statistics should also be useful in light of legislative requirements. For example, the federal Sex Discrimination Act [9] prohibits discrimination on the basis of sexual orientation, and the Aged Care Act [10] mentions "lesbian, gay, bisexual, transgender and intersex people" as a special needs group.
This paper updates and extends the sexual minority population estimates for Australia calculated previously [3]. It reports proportions of the population identifying as a sexual minority from reliable national surveys. It then presents population estimates for the sexual minority adult population of Australia in 2011 and 2016, including by age group, and by sexual identity (gay, lesbian, bisexual, and other sexual minority identities).
Data on the proportions of the population with a specific sexual identity were sourced from three data collections from two representative national household surveys, namely the General Social Survey (GSS), and waves 12 and 16 of the Household, Income And Labour Dynamics in Australia (HILDA) Survey. Other large surveys also ask about sexual identity [11,12,13], but were not considered for this study because they cover only part of the Australian population. The GSS was undertaken by the ABS between March and June 2014 [14] with face to face computer-assisted interviewing. It achieved a sample of about 13,000 people aged 15 years and over in households (i.e., excluding institutional accommodation). HILDA is an ongoing national longitudinal study which began data collection in 2001 [15]. About 17,000 people in households are interviewed every year using both face-to-face computer-assisted interviewing and a self-completion questionnaire (which contains the sexual identity questions). We made use of data from waves 12 and 16, conducted in 2012 and 2016 respectively, when sexual identity questions were asked.
The questions on sexual identity from the surveys are reproduced in Additional file 1: Figure S1. It is important to note that responses to these questions refer to reported sexual identity, not sexual attraction or sexual behaviour. There can be quite large variations in population numbers depending on which aspect of sexual orientation is being considered [16]. Importantly, this is reported sexual identity; people who are uncomfortable disclosing a minority sexual identity may not respond to the question or may report a different sexual identity.
The 2011 and 2016 estimated resident populations (ERPs) of Australia by sex and age group were obtained from the ABS [17]. These two years were chosen because the ERPs for these years are based on 2011 and 2016 census counts and likely to be more accurate than those for non-census years, and they are close to the reference dates of the surveys.
Sexual identity proportions were calculated from the weighted number of adults in each sexual identity category in all three datasets. The proportions were calculated by sex for individual sexual identity categories (gay, lesbian, bisexual, and other), and for the total sexual minority population—defined as the sum of those four categories. Proportions were also calculated for the total sexual minority population by broad age group and sex. We do not present proportions by sex, age group and individual sexual identity categories as the variability around the point estimates increases significantly. The don't know, not stated/refused responses were included in the denominators of the proportions.
Population estimates by sexual identity were calculated by taking the proportion of the population in each sexual identity category derived from the surveys and multiplying them by the published ERPs of Australia for 2011 and 2016. They were prepared in three steps. First, an estimate of the total sexual minority population aged 18 + by sex was calculated. Given a lack of information to suggest that any one survey dataset was more reliable than the others, we weighted all proportions equally. Thus, the sexual minority (\(M\)) population (\(P\)) aged 18 + by sex (\(s\)) was calculated as:
$${P}_{s,18+}^{M} ={ P}_{s,18+}^{ERP} \frac{1}{3}\left({p}_{s,18+}^{M,GSS}+{p}_{s,18+}^{M,HILDA-12}+{p}_{s,18+}^{M,HILDA-16}\right),$$
where \(ERP\) is the official estimated resident population, \(p\) denotes the proportion of the population, and \(GSS\), \(HILDA-12\), and \(HILDA-16\) refer to the three survey datasets with the HILDA labels including the survey wave number.
Second, estimates of the total sexual minority population by age groups 18–24, 25–34, 35–44, 45–54, 55–64 and 65 + were calculated. Preliminary (\(pr\)) estimates were calculated as:
$${P}_{s,a}^{M}\left[pr\right]={ P}_{s,a}^{ERP} \frac{1}{3}\left({p}_{s,a}^{M,GSS}+{p}_{s,a}^{M,HILDA-12}+{p}_{s,a}^{M,HILDA-16}\right),$$
where \(a\) refers to age group. Then a small constraining adjustment was made to ensure these age-specific estimates summed to the overall 18 + estimate:
$${P}_{s,a}^{M} ={P}_{s,a}^{M}\left[pr\right] \frac{{P}_{s,18+}^{M}}{\sum_{a}{P}_{s,a}^{M}\left[pr\right]}$$
Third, estimates of the 18 + population by sex by individual sexual identity category were calculated:
$${P}_{s,18+}^{m}\left[pr\right]={ P}_{s,18+}^{ERP} \frac{1}{3}\left({p}_{s,18+}^{m,GSS}+{p}_{s,18+}^{m,HILDA-12}+{p}_{s,18+}^{m,HILDA-16}\right),$$
where \(m\) refers to gay/lesbian, bisexual or other. As before, a slight adjustment was required to ensure consistency with the overall sexual minority estimate:
$${P}_{s,18+}^{m} ={P}_{s,18+}^{m}\left[pr\right] \frac{{P}_{s,18+}^{M}}{\sum_{m}{P}_{s,18+}^{m}\left[pr\right]}.$$
The distribution of the adult population of Australia across sexual identity categories from the three survey datasets is shown in Table 1. The total sexual minority population varies from just under 3% according to the GSS to just over 4% in HILDA wave 16, with slightly higher percentages for females than males in the GSS but not HILDA. For males, the percentage of the population identifying as gay is higher than the bisexual percentage, while for females the bisexual percentages are higher than those for lesbian in the two HILDA datasets. Interestingly, the HILDA survey results from wave 16 indicate an increase in the share of the population identifying as a sexual minority from four years earlier. Although it is not possible to determine a trend from the limited data available, an increase over time would be consistent with recent evidence from the USA and UK [18, 19]. Complicating the analysis is the fact that the percentages for heterosexual and don't know/not stated/refused differ noticeably between HILDA and the GSS. This may be related to differences in the survey mode and list of available responses in the three surveys. Relative standard errors for the data in this table are provided in the Additional file 1.
Table 1 Percentage of Australia's adult population by sexual identity and sex
Table 2 presents the percentage of the population identifying as a sexual minority by sex and age group. Relative standard errors are also provided in the Additional file 1. There is a strong relationship between sexual minority identity and age in the GSS results whereby percentages decline with increasing age, but the relationship is less distinct in the HILDA data, especially for males. Overall, percentages are highest in the younger 18–24 and 25–34 age groups, and lowest in the 65 + age group. Amongst those aged 18–24, the percentages reach as high as 7.5% for females and 5.7% for males, while in the 65 + age group all percentages are below the population averages for each gender.
Table 2 Percentage of Australia's adult population identifying as a sexual minority by age group
Population estimates for Australia's sexual minority populations in 2011 and 2016 are shown in Table 3. The total sexual minority population of Australia aged 18 + is estimated to have been 599,500 in 2011 with slightly fewer females (296,400) than males (303,100); by 2016 it is estimated to have grown to about 651,800 (323,500 females and 328,300 males). The population is young compared to the Australian population overall, with close to half (46%) aged 18–34. The numbers in the 65 + age group are relatively small—about 63,900 in 2011 and 76,600 in 2016. The population aged 18 + identifying as lesbian/gay is estimated to have been about 286,400 in 2016 (44% of the sexual minority population), with 215,600 as bisexual (33%), and 149,700 (23%) as other. For males, the gay population was larger than the bisexual population (182,100 and 77,900 respectively), while for females the opposite was the case (104,400 lesbian and 137,800 bisexual).
Table 3 The estimated sexual minority population of Australia, 2011 and 2016
This paper has presented new estimates of Australia's sexual minority population based on official Estimated Resident Populations and representative surveys which collect information on sexual identity. Our study shows that Australia's sexual minority population reached about 651,700 in 2016, representing 3.5% of the adult population, a little higher than the 3.2% estimated previously [3] due to the inclusion of HILDA wave 16 data. Equivalent percentages for other countries in recent years include 3.5% for New Zealand [20], 2.5% [21] and 2.9% for the UK [19], and 4.1% for the US [22], though these statistics are not strictly comparable due to differences in questions and survey modes. We hope that the new population estimates (Table 3) will prove useful for various policy, planning and research activities.
The sexual identity population estimates reported in this paper are probably as good as possible given the available data sources, and limitations of the population estimates are listed below. The accuracy and detail of population estimates will only be enhanced if sexual identity is included in the quinquennial census or a very large-scale national survey, such as the ABS Monthly Population Survey [23].
One of the most important findings of this study is the higher proportion of younger people reporting a minority sexual identity. This suggests that a cohort effect may be at work. Sexual identities and the willingness to disclose one's identity can be influenced by the social attitudes and legal environment of the time when each cohort passes through their formative years. Older cohorts have spent much of their lives during a time when social acceptance was lower than today [24], and this might still influence how some of them report their identity. This cohort effect may have an important role in the proportions of people reporting a sexual minority identity in future surveys. Those young cohorts with 5–8% sexual minority identities may well maintain their identities as they get older in the future, and in the current accepting environment younger cohorts replacing them are likely to report similar, or perhaps higher, percentages. If this occurs, then Australia's known sexual minority population will increase rapidly over the coming decades, and the estimates will need regular updating.
This study contains several limitations.
We assumed that the sexual minority percentages obtained from surveys undertaken between 2012 and 2016 were valid for creating 2011 and 2016 sexual minority populations. If the trend in identifying as a sexual minority is increasing, then the 2011 population might be slightly over-estimated and the 2016 population slightly under-estimated.
The survey data were collected using different survey modes and with slightly different wording in the sexual identity question, so they are not perfectly comparable.
The various residual categories of don't know, not stated, and refused need careful consideration. They vary substantially between surveys and their interpretation is not straightforward. Sexual minority percentages would be slightly higher if they were excluded from denominators.
The scope of all surveys excluded institutional accommodation which may have led to a small amount of bias.
Our population estimates only refer to those who reported a sexual minority identity. It is therefore a 'revealed' population which excludes those who do not wish to disclose their sexuality (in the survey, at least).
Finally, the sexual minority population estimates are approximate. They are based on ABS estimated resident populations, which are good quality data, but also weighted survey data based on fairly small samples of sexual minority individuals.
An Excel file of sexual minority population estimates is available from the corresponding author on request.
ABS:
ERP:
Estimated resident population
GSS:
General Social Survey
HILDA:
Household, Income and Labour Dynamics in Australia
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Callander D, Mooney-Somers J, Keen P, Guy R, Duck T, Bavinton BR, Grulich AE, Holt M, Prestage G. Australian 'gayborhoods' and 'lesborhoods': a new method for estimating the number and prevalence of adult gay men and lesbian women living in each Australian postcode. Int J Geogr Inf Sci. 2020. https://doi.org/10.1080/13658816.2019.1709973.
Office for National Statistics (ONS) Measuring sexual identity: an evaluation report. 2010. https://webarchive.nationalarchives.gov.uk/20151014015853/http://www.ons.gov.uk/ons/rel/ethnicity/measuring-sexual-identity---evaluation-report/2010/index.html. Accessed 24 Sept 2010
LGBTI Health Alliance. Joint statement in support of LGBTI inclusion in the 2021 Census (2019) https://lgbtihealth.org.au/joint-statement-in-support-of-lgbti-inclusion-in-the-2021-census/. Accessed 24 Mar 2020
Gates GJ. LGBT identity: a demographer's perspective. Loy LA L Rev. 2012;45:693–714.
Perales F. The health and wellbeing of Australian lesbian, gay and bisexual people: a systematic assessment using a longitudinal national sample. Aust NZ J Publ Heal. 2019;43:281–7.
Sex Discrimination Act 1984 (Cth). https://www.legislation.gov.au/Details/C2014C00002. Accessed 5 Feb 2020
Aged Care Act 1997 (Cth). https://www.legislation.gov.au/Details/C2020C00054. Accessed 25 Jan 2020
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SA Health. South Australia Population Health Survey Questionnaire 2020. 2020. https://www.sahealth.sa.gov.au/wps/wcm/connect/f88b47b7-ca05-4ea4-898c-1581d47bc249/SAPHS+2020_Public+Document.pdf?MOD=AJPERES&CACHEID=ROOTWORKSPACE-f88b47b7-ca05-4ea4-898c-1581d47bc249-naxHPpI. Accessed 21 Feb 2020
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Wilkins R, Laß I, Butterworth P, Esperanza V-T. The household, income and labour dynamics in Australia survey: selected findings from waves 1 to 17. Melbourne: Melbourne Institute; 2019.
Richters J, Altman D, Badcock PB, Smith AMA, de Visser RO, Grulich AE, Rissel C, Simpson JM. Sexual identity, sexual attraction and sexual experience: the Second Australian Study of Health and Relationships. Sex Health. 2014;11:451–60.
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This paper uses unit record data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey. The HILDA Project was initiated and is funded by the Australian Government Department of Social Services (DSS) and is managed by the Melbourne Institute of Applied Economic and Social Research (Melbourne Institute). The findings and views reported in this paper, however, are those of the authors and should not be attributed to the DSS or the Melbourne Institute.
TW and JT were supported by the Australian Research Council Centre of Excellence in Population Ageing Research (Project number CE1101029).
Melbourne School of Population and Global Health, The University of Melbourne, Melbourne, VIC, Australia
Tom Wilson & Jeromey Temple
Australian Research Centre in Sex, Health and Society, La Trobe University, Melbourne, VIC, Australia
Anthony Lyons
School of Psychology and Public Health, La Trobe University, Melbourne, VIC, Australia
Northern Institute, Charles Darwin University, Darwin, NT, Australia
Fiona Shalley
Jeromey Temple
TW designed the study, undertook analysis, and wrote the first draft of the manuscript. JT did statistical analysis and contributed to the manuscript. FS participated in the acquisition of data, analysis and writing of the manuscript. AL contributed to the analysis and writing of the manuscript. All authors read and approved the final manuscript.
Correspondence to Tom Wilson.
Ethics approval for this project was granted by the Melbourne School of Population and Global Health Human Ethics Advisory Group (ID 2056346.1).
All authors read and approved the final version of the manuscript.
. Figure S1: Sexual identity questions in HILDA and the GSS. Table S1: Relative standard errors for sexual identity percentages in Table 1. Table S2: Relative standard errors for the sexual minority percentages in Table 2.
Wilson, T., Temple, J., Lyons, A. et al. What is the size of Australia's sexual minority population?. BMC Res Notes 13, 535 (2020). https://doi.org/10.1186/s13104-020-05383-w
Sexual minority | CommonCrawl |
Muriel Glauert
Muriel Glauert (née Barker) (7 May 1892 – 23 December 1949) was a British mathematician who made significant contributions to early advances in aerodynamics.
Muriel Glauert
Born7 May 1892
Nottingham
Died23 December 1949
Alma materNewnham College
OccupationAeronautical engineering
EmployerRoyal Aircraft Establishment Farnborough
Early life and education
Muriel Barker was born in Nottingham, the daughter of a textile manufacturer, and attended Nottingham Girls' High School, where she won prizes for her achievements in German, maths and chemistry.[1] She attended Newnham College, Cambridge, from 1912 to 1915 and completed the mathematical tripos, although this was awarded by London University, as Cambridge was yet to award degrees to women.[2]
Career at the Royal Aircraft Establishment
Barker taught in Liverpool before joining the Royal Aircraft Establishment (RAE) in Farnborough in 1918 as a researcher. Her first publication in her early career at Farnborough was on theoretical streamlines for the flow over an aerofoil. In 1919 she went to study at Bryn Mawr for a year and then undertook postgraduate studies in aeronautics at Cambridge. In August 1922 she published her paper 'On the use of very small pitot-tubes for measuring wind velocity' in the Proceedings of the Royal Society.[3] A pitot tube is a slender tube with two holes used to calculate speed through the air or water, used by both ships and aeroplanes. Barker was the first researcher to demonstrate that the difference between the pitot tube's reading and the static pressure is proportional to the flow speed rather than to its square.[1] In the same year she returned to the RAE, and became engaged to, and later married, the aerodynamicist Hermann Glauert, Principal Scientific Officer at the RAE.
Later career
After her husband was killed in an accident in 1934, Barker later became Examiner in Mathematics for the London and Cambridge and Joint Northern Universities.[4] In 1940 she published a final academic paper, which looked at the capture of raindrops by a cylinder and an aerofoil moving at uniform speed, a problem of ongoing concern due to ice formation, for example, on aeroplane wings in flight.[1]
Personal life
She was married to Hermann Glauert, who died in 1934. They had three children: Michael, and twins Audrey and Richard. Muriel Glauert died in 1949 and was buried alongside her husband.[5]
References
1. Baker, Nina C. "Magnificent Women".{{cite web}}: CS1 maint: url-status (link)
2. Knight, Karen Lovejoy (2018). A.C. Pigou and the 'Marshallian' Thought Style: A Study in the Philosophy and Mathematics Underlying Cambridge Economics (Palgrave Studies in the History of Economic Thought). 67: Palgrave Macmillan.{{cite book}}: CS1 maint: location (link)
3. Barker, Muriel; Taylor, Geoffrey Ingram (1 August 1922). "On the use of very small pitot-tubes for measuring wind velocity". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 101 (712): 435–445. Bibcode:1922RSPSA.101..435B. doi:10.1098/rspa.1922.0055.
4. "Graces Guide".{{cite web}}: CS1 maint: url-status (link)
5. "Hermann Glauert grave monument details at Ship Lane Cemetery, Farnborough, Hampshire,England". www.gravestonephotos.com. Retrieved 21 March 2021.
| Wikipedia |
\begin{document}
\title{On linear periods}
\date{} \maketitle
\begin{abstract} Let $\pi'$ be a cuspidal automorphic representation of $\GL_{2n}(\BA)$, which is assumed to be the Jacquet-Langlands transfer from a cuspidal automorphic representation $\pi$ of $\GL_{2m}(D)(\BA)$, where $D$ is a division algebra so that $\GL_{2m}(D)$ is an inner form of $\GL_{2n}$. In this paper, we consider the relation between linear periods on $\pi$ and $\pi'$. We conjecture that the non-vanishing of the linear period on $\pi$ would imply the non-vanishing of that on $\pi'$. We illustrate an approach using a relative trace formula towards this conjecture, and prove the existence of smooth transfer over non-archimedean local fields.
\end{abstract}
\section{Introduction}\label{section. intro}
\paragraph{Goal of this article} Let $k$ be a number field, $\BA$ its ring of adeles, and $D$ a central division algebra over $k$ of index $d$, that is, $\dim_kD=d^2$. Let $\FG=\GL_{2m}(D)$, viewed as an algebraic group over $k$, which is an inner form of $\FG'=\GL_{2n}$ with $n=md$. Let $\pi$ be an irreducible cuspidal automorphic representation of $\FG(\BA)$, and $\pi'$ the irreducible automorphic representation of $\FG'(\BA)$ associated to $\pi$ by the Jacquet-Langlands correspondence, which is assumed to be cuspidal. For the Jacquet-Langlands correspondence involving general linear group and its inner forms, we refer to \cite{dkv}, \cite{ba} and \cite{br} for more details. The main purpose of this paper is to investigate a relation between certain automorphic periods under the Jacquet-Langlands correspondence.
To be more precise, let $\FZ$ be the center of $\FG$, which is identified with the center $\FZ'$ of $\FG'$ via the obvious identifications of $\FZ$ and $\FZ'$ with $\BG_m$ over $k$. Let $\FH=\GL_m(D)\times\GL_m(D)$ (resp. $\FH'=\GL_n\times\GL_n$) be embedded into $\FG$ (resp. $\FG'$) diagonally. The periods considered in this paper are given by $$\ell(\phi):=\int_{\FH(k)\FZ(\BA)\bs\FH(\BA)}\phi(h)\ {\mathrm{d}} h, \quad \phi\in\pi,$$ and $$\ell'(\varphi):=\int_{\FH'(k)\FZ'(\BA)\bs\FH'(\BA)}\varphi(h)\ {\mathrm{d}} h, \quad \varphi\in\pi'.$$ We call them {\em linear periods}. In the context of general linear groups (hence applying to $(\FG',\FH')$ above), this notion was introduced by \cite{fj}. We say that $\pi$
is $\FH$-distinguished or has a linear period if $\ell|_\pi\neq0$. Of course, as a special case we get an analogous definition in the context of $(\FG',\FH')$. Conjecturally, such a period has a close relation with an L-value. For instance, it was shown in \cite{fj} that $\pi'$ is $\FH'$-distinguished if and only if the L-value $L^S(\frac{1}{2},\pi'){\operatorname{res}}_{s=1}L^S(s,\pi',\wedge^2)$ is nonzero, using an integral representation of the L-function $L^S(s_1,\pi')\cdot L^S(s_2,\pi',\wedge^2)$. What happens if $\pi$ is $\FH$-distinguished? The partial L-functions attached to $\pi$ and $\pi'$ should be the same, while there is no integral representation for the ones associated to $\pi$. However, since $\FH$ is an inner form of $\FH'$, it is natural to make the following conjecture.
\begin{conj}\label{conj. main} If $\pi$ is $\FH$-distinguished, then $\pi'$ is $\FH'$-distinguished. \end{conj}
\begin{remark} As pointed out by D. Prasad, the converse of the above conjecture should also hold. In other words, if $\pi'$ is $\FH'$-distinguished, then $\pi$ should be $\FH$-distinguished too. Moreover, Conjecture 2 of \cite{pt} may be viewed as the local analog of Conjecture \ref{conj. main} together with its converse. \end{remark}
In this paper, we illustrate an approach towards this conjecture using a relative trace formula. One of the key steps in this approach is establishing the existence of smooth transfer over the non-archimedean places. This is accomplished by Theorem \ref{thm. main1}. Note that there is no need to prove the fundamental lemma, since $(\FG(k_v),\FH(k_v))\simeq(\FG'(k_v),\FH'(k_v))$ for almost all places $v$. Roughly speaking, let $v$ be a finite place of $k$. Then the smooth transfer at $v$ is a ``transfer" $\lambda_v$ from $\CC_c^\infty(\FG(k_v))$ to $\CC_c^\infty(\FG'(k_v))$ (in fact a map from $\CC_c^\infty(\FG(k_v))$ to a suitable quotient of $\CC_c^\infty(\FG'(k_v))$) such that for any $f\in\CC_c^\infty(\FG(k_v))$, the orbital integrals of $f$ and $\lambda_v(f)$ ``match''. Here the orbits are those of $\FH\times\FH$ on $\FG$ (resp., $\FH'\times\FH'$ on $\FG'$) by left and right translation. See Section \ref{section. smooth transfer} for a precise definition.
Our proof of the existence of smooth transfer is mainly inspired by Wei Zhang's work \cite{zh1} on the smooth transfer conjecture for the Jacquet-Rallis relative trace formula towards the global Gan-Gross-Prasad conjecture for unitary groups, and by Waldspurger's work \cite{wa95} \cite{wa97} on endoscopic transfer which inspired \cite{zh1}. The first step is to reduce the question of the existence of smooth transfer to Lie algebras, that is, to linearize the question. The second step is to show that, roughly speaking, the Fourier transform commutes with smooth transfer. We will use a global method to show such a property.
\paragraph{Some related work} Conjecture \ref{conj. main} is motivated by the conjecture of H. Jacquet and K. Martin \cite{jm} on Shalika periods. We briefly recall it. Now let $d=n$ and $\FG=\GL_2(D)$. Denote by $\FS$ the Shalika subgroup of $\FG$. To review its definition, consider the parabolic subgroup $\FP={\mathbf{M}}\FN$ of $\FG$, where ${\mathbf{M}}\simeq D^\times\times D^\times$ is the obvious Levi subgroup and $\FN\simeq D$ is the unipotent radical. Let $\psi$ be a nontrivial character of $\BA/k$, which defines a nondegenerate character (still denoted by $\psi$) of $\FN(k)\bs\FN(\BA)$ given by $\psi(x):=\psi(\tr_D(x))$ for $x\in\FN(\BA)\simeq D(\BA)$, where $\tr_D$ is the reduced trace map on $D$. Then its stabilizer in $\FP$ is the Shalika subgroup $\FS=\FL\FN$, where $\FL$ is $\Delta D^\times$ (i.e., $D^\times$ embedded diagonally in ${\mathbf{M}}\simeq D^\times\times D^\times$). We can extend $\psi$ to a character of $\FS(k)\bs\FS(\BA)$ by $\psi(l\cdot n)=\psi(n)$ for $l\in\FL(\BA)$ and $n\in\FN(\BA)$. One can define the Shalika subgroup $\FS'$ of $\FG'$ similarly, where the corresponding parabolic subgroup is $\FP'={\mathbf{M}}'\FN'$ with Levi factor ${\mathbf{M}}'\simeq\GL_n\times\GL_n$. Then the Shalika period $\CS$ is a linear form on $\pi$ given by $$\CS(\phi)=\int_{\FS(k)\bs\FS(\BA)}\phi(u)\psi^{-1}(u)\ {\mathrm{d}} u,$$ and the Shalika period $\CS'$ on $\pi'$ is defined similarly. In \cite{jm}, Jacquet and Martin conjectured that if $\pi$ is distinguished with respect to $\CS$ then $\pi'$ is also distinguished with respect to $\CS'$. Under some hypotheses, using relative trace formulae, Jacquet and Martin showed that this is true if $n=2$. However, they did not prove the smooth transfer for the full space $\CC_c^\infty(\FG(k_v))$ of Bruhat-Schwartz functions. Of course, if one aims to completely prove this conjecture using the method of the relative trace formula, one has to show the existence of smooth transfer for the full space $\CC_c^\infty(\FG(k_v))$. In the case $n=2$, this conjecture (together with its converse) was completely proved by W. T. Gan and S. Takeda \cite{gt} using theta correspondence. However, this method cannot be generalized to the higher rank cases. Separately, D. Jiang, C. Nien and Y. Qin \cite{jnq} proved this conjecture, under some conditions, for general $n$ using the method of automorphic descent.
There is a relation between the linear period and the Shalika period on $\pi'$. In fact, by the criterion for $\FH'$-distinction from \cite{fj} recalled earlier, $\FH'$-distinction implies $\FS'$-distinction, since $\pi'$ is $\FS'$-distinguished if and only if the exterior L-function $L(s,\wedge^2,\pi')$ has a simple pole at $s=1$. Locally, it was shown in \cite{jr} that if $\pi'_v$ is $\FS'(k_v)$-distinguished then it was $\FH'(k_v)$-distinguished, and it is conjectured that if $\pi'_v$ is generic then $\FS'(k_v)$-distinction is equivalent to $\FH'(k_v)$-distinction. Recently, Gan \cite{ga} proved this local conjecture using local theta correspondence for dual pairs of type II. Therefore one can ask whether there are such relations between linear and Shalika periods on $\pi$, both globally and locally. Such a conjectural relation together with the conjecture of Jacquet and Martin motivates Conjecture \ref{conj. main}.
As we have said before, our proof of the existence of smooth transfer is inspired by \cite{zh1} and \cite{wa97}. However, there are still some significant differences between our method and that of either of \cite{zh1} or \cite{wa97}. It is fair to say that ours is a combination of theirs. We follow \cite{zh1} in reducing the question of smooth transfer at the level of groups to showing Theorem \ref{thm. fourier}, namely, the assertion that the Fourier transform commutes with smooth transfer (up to an explicit constant). However, we could not follow \cite{zh1} for the rest of the proof, since the absence of a suitable partial Fourier transform in our situation meant that the inductive arguments in \cite[\S4]{zh1} could not be applied. We follow \cite{wa97} in using a global method to prove Theorem \ref{thm. fourier}. This requires us to study harmonic analysis on the corresponding $p$-adic symmetric spaces, and prove several results analogous to ones appearing in \cite{wa95} and \cite{wa97}, and others that are analogues of more classical results in \cite{hc1} and \cite{hc}. We just state these results and explain them briefly, since they are direct generalizations of those that have been proved in the case of $(\FG',\FH')$ in \cite{zh}.
In \cite{zh}, we studied the relation between similar periods (involving an additional twist with a character) for the symmetric pairs $(\FG',\FH')$ and $(\FG,\FH)$, where $(\FG',\FH')$ is as before and $(\FG,\FH)=(\GL_n(D),\GL_n(k')$ with $D$ being a quaternion algebra over $k$ and $k'$ being a quadratic field extension of $k$ included in $D$. However, in \cite{zh}, we could prove only ``half'' of the property that the Fourier transform commutes with smooth transfer, due to the fact that there are ``fewer'' regular semisimple orbits associated to the pair $(\FG,\FH)$ than to $(\FG',\FH')$. We encounter a similar problem in this paper, though, fortunately, it turns out that this hurdle can be circumvented. The point is that we have a nice description for the orbits of $(\FG',\FH')$ that can be matched with ones of $(\FG,\FH)$.
Sometimes the existence of smooth transfer for functions belonging to a proper subspace of $\CC_c^\infty(\FG(k_v))$ suffices to prove partial results towards Conjecture \ref{conj. main}. This is the case, for instance, in the work of Jacquet-Martin \cite{jm}.
\paragraph{Structure of this article} In \S\ref{section. trace formula}, we introduce the relative trace formulae considered in this paper, which are natural for the conjecture concerned. The contents of this section are more or less routine and informal. The main purpose of this section is to show the motivation for the study of smooth transfer.
To factor the global linear periods into local ones, we need to study the property of multiplicity one for the symmetric pair $(\FG(k_v),\FH(k_v))$ at each place $v$ of $k$, or, in other words, to study the space $\Hom_{\FH(k_v)}(\pi_v,\BC)$ for any irreducible admissible representation $\pi_v$ of $\FG(k_v)$. If $\dim\Hom_{\FH(k_v)}(\pi_v,\BC)\leq 1$ for each irreducible admissible representation $\pi_v$ of $\FG(k_v)$ we call $(\FG(k_v),\FH(k_v))$ a Gelfand pair. We have not been able to show $(\FG(k_v),\FH(k_v))$ to be a Gelfand pair, but we can show that it satisfies a weaker variant of that property, which is enough for our purpose of factoring the global period. In \S\ref{section. multiplicity one}, we systematically follow the approach developed by \cite{ag1} to study questions of this kind, i.e. using generalized Harish-Chandra descent to study $\FH(k_v)\times\FH(k_v)$-invariant distributions on $\FG(k_v)$. This will also be important further into the paper (\S\ref{section. smooth transfer} and \S\ref{section. local orbital integrals}), while studying smooth transfer.
In \S\ref{section. smooth transfer}, we introduce the notion of smooth transfer explicitly, both for groups and Lie algebras. After several reduction steps, we show the existence of smooth transfer (Theorem \ref{thm. main1}) assuming Theorem \ref{thm. fourier} on the commutativity of Fourier transform with transfer. Theorem \ref{thm. fourier} is proved in \S\ref{section. local orbital integrals}.
The main aim of \S\ref{section. local orbital integrals} is to prove Theorem \ref{thm. fourier}. We first recall some results on $p$-adic harmonic analysis developed in \cite{zh} and give their generalizations to our situation. With the aid of these results, we prove Theorem \ref{thm. fourier} at the end of this section.
\section{Notations and conventions}\label{section. notation}
\paragraph{Actions of algebraic groups} Let $k$ be a number field or a $p$-adic field. Let $\pi$ be an action of a reductive group ${\mathbf{M}}$ on a smooth affine variety $\FX$, all defined over $k$. Write $M={\mathbf{M}}(k)$ and $X=\FX(k)$. Recall that for $x\in X$, we say that $x$ is ${\mathbf{M}}$-semisimple or $M$-semisimple if ${\mathbf{M}} x$ is Zariski closed in $\FX$ (or, equivalently, if $k$ is $p$-adic, $M x$ is closed in $X$ for the analytic topology). We say $x$ is ${\mathbf{M}}$-regular or $M$-regular if the stabilizer ${\mathbf{M}}_x$ of $x$ has minimal dimension. We denote by $\FX_{\mathrm{rs}}(k)$ or $X_{\mathrm{rs}}$ (resp. $\FX_{\mathrm{ss}}(k)$ or $X_{\mathrm{ss}}$) the set of $M$-regular semisimple (resp. $M$-semisimple) elements in $X$. If $k$ is $p$-adic, we call an algebraic automorphism $\tau$ of $\FX$ ${\mathbf{M}}$-admissible if (i) $\tau$ normalizes $\pi(M)$ and $\tau^2\in\pi(M)$, (ii) for any closed $M$-orbit $O\subset X$, $\tau(O)=O$.
\paragraph{Analysis on $\ell$-spaces} Now let $k$ be a $p$-adic field. For a locally compact totally disconnected topological space $X$, we denote by $\CC_c^\infty(X)$ the space of locally constant and compactly supported $\BC$-valued functions on $X$, and by $\CD(X)$ the space of distributions on $X$, namely, the dual of $\CC_c^\infty(X)$. If there is an action of an $\ell$-group $M$ on $X$, we denote by $\CD(X)^M$ the subspace of $\CD(X)$ consisting of $M$-invariant distributions.
\paragraph{Fourier transform and Weil index} Now suppose that $k$ is a local field of characteristic 0. Let $X$ be a finite dimensional vector space over $k$ with the natural topology induced from that of $k$, $\psi$ a nontrivial continuous additive character of $k$, and $q$ a nondegenerate quadratic form on $X$. We always equip $X$ with the self-dual Haar measure with respect to the bi-character $\psi(q(\cdot,\cdot))$. Define the Fourier transform $f\mapsto\widehat{f}$ of the space $\CS(X)$ of Bruhat-Schwartz functions on $X$ by $$\widehat{f}(x)=\int_X f(y)\psi(q(x,y))\ {\mathrm{d}} y.$$ Then $\hat{\hat{f}}(x)=f(-x)$. We write $\gamma_\psi(q)$ for the Weil index associated to $q$ and $\psi$, which is an 8th root of unity. For the definition and some properties of the Weil index, see \cite{we}.
\section{Relative trace formulae}\label{section. trace formula} Let $(\FG,\FH)$ and $(\FG',\FH')$ be as defined in \S\ref{section. intro}. Fix a Haar measure on $\FZ(\BA)$. For $f\in\CC_c^\infty(\FG(\BA))$, define the kernel function $$K_f(x,y)=\int_{\FZ(k)\bs\FZ(\BA)}\sum_{\gamma\in\FG(k)} f(zx^{-1}\gamma y)\ {\mathrm{d}} z.$$ We consider the partially defined distribution on $\FG(\BA)$ $$I(f)=\int_{\FH(k)\FZ(\BA)\bs\FH(\BA)}\int_{\FH(k)\FZ(\BA)\bs\FH(\BA)} K_f(h_1,h_2)\ {\mathrm{d}} h_1\ {\mathrm{d}} h_2,$$ defined on the subspace of all $f\in\CC_c^\infty(\FG(\BA))$ such that the above expression is absolutely convergent. Choose the Haar measure on $\FZ'(\BA)$ to be compatible with that on $\FZ(\BA)$. For $f'\in\CC_c^\infty(\FG'(\BA))$ we define the kernel function $K_{f'}(x,y)$ similarly. In the same way, we obtain a partially defined distribution $J(\cdot)$ on $\FG'(\BA)$. The art of relative trace formula is to compare $I(\cdot)$ with $J(\cdot)$.
Very informally, we have two ways to decompose $I(f)$ - the so called spectral expansion and the so called geometric expansion. On the spectral side, spherical characters $I_\pi(f)$ associated to irreducible cuspidal representations $\pi$ of $\FG(\BA)$ are involved. On the geometric side, orbital integrals $I_\gamma(f)$ associated to $\FH(k)\times\FH(k)$-regular semisimple orbits $\gamma$ in $\FG(k)$ are involved. We give precise definitions of $I_\pi(f)$ and $I_\gamma(f)$ below. Similarly, $J(f')$ can be decomposed in these two ways.
Fix a Haar measure on $\FH(\BA)$. If $\pi$ is an irreducible cuspidal representation of $\FG(\BA)$, let $$K_{\pi,f}(x,y)=\sum_\varphi\left(\pi(f)\varphi\right)(x)\overline{\varphi(y)},$$ where $\varphi$ runs over an orthonormal basis for the space of $\pi$. Define the spherical character $I_\pi$ to be $$I_\pi(f)=\int_{\FH(k)\FZ(\BA)\bs\FH(\BA)}\int_{\FH(k)\FZ(\BA)\bs\FH(\BA)} K_{\pi,f}(h_1,h_2)\ {\mathrm{d}} h_1\ {\mathrm{d}} h_2,$$ where $f\in\CC_c^\infty(\FG(\BA))$. Both $K_{\pi,f}(x,y)$ and $I_\pi(f)$ are well defined, and we refer the reader to \cite[\S5]{hh} for a detailed explanation. Thus, by definition, we have $$I_\pi(f)=\sum_\varphi\ell\left(\pi(f)\varphi\right)\overline{\ell(\varphi)}.$$ Notice that $I_\pi$ is a distribution of positive type. In other words, if $f=f_1*f_1^*$ where $f_1^*(g):=\bar{f}_1(g^{-1})$, then $$I_\pi(f)=\sum_\varphi\ell\left(\pi(f_1)\varphi\right) \overline{\ell(\pi(f_1)\varphi)}.$$ Hence $\pi$ is $\FH$-distinguished if and only if $I_\pi$ is nonzero as a distribution on $\FG(\BA)$. Therefore, the spectral expansion of $I(f)$ in terms of $I_\pi(f)$ can reflect the property of $\FH$-distinction. Similarly, we define the spherical character $J_{\pi'}(f')$ associated to an irreducible cuspidal representation $\pi'$ of $\FG'(\BA)$.
It is believed that, if we can compare the distributions $I$ and $J$ in some ways, $I_\pi$ and $J_{\pi'}$ are closely related, where $\pi'$ is the Jacquet-Langlands correspondence of $\pi$. To compare $I$ with $J$, we consider their geometric expansions.
If $\gamma\in\FG(k)$ is $\FH\times\FH$-regular semisimple, we fix a Haar measure ${\mathrm{d}}_\gamma h$ on $\FH_\gamma(\BA)$, where $$\FH_\gamma=\{ (h_1,h_2)\in\FH\times\FH;\ h_1\gamma h_2^{-1}=\gamma\}$$ is the stabilizer of $\gamma$ under the action of $\FH\times\FH$. For $f\in\CC_c^\infty(\FG(\BA))$, define the orbital integral of $f$ at $\gamma$ to be $$I_\gamma(f)=\int_{\FH_\gamma(\BA)\bs(\FH(\BA)\times\FH(\BA))} f(h_1^{-1}\gamma h_2)\ {\mathrm{d}} h_1\ {\mathrm{d}} h_2.$$ This is well defined, since the semisimple orbit is closed in $\FG(\BA)$ and therefore the above integral is absolutely convergent. We fix a Haar measure ${\mathrm{d}} h_v$ on $\FH(k_v)$ at each place $v$ of $k$ so that $\vol(\FH(\CO_{k_v}))=1$ for each unramified place $v$ and set ${\mathrm{d}} h=\prod_{v}{\mathrm{d}} h_v$. We also fix a Haar measure ${\mathrm{d}}_\gamma h_v$ on $\FH_\gamma(k_v)$ at each place $v$ of $k$ so that $\vol(\FH_\gamma(\CO_{k_v}))=1$ for each unramified place $v$ and set ${\mathrm{d}}_\gamma h=\prod_v{\mathrm{d}}_\gamma h_v$. If $f=\otimes'f_v$ is a pure tensor, we have \begin{equation}\label{equation. global and local integrals 1} I_\gamma(f)=\prod_v\int_{\FH_\gamma(k_v)\bs(\FH(k_v)\times\FH(k_v))} f_v(h_1^{-1}\gamma h_2){\mathrm{d}} h_1\ {\mathrm{d}} h_2,\end{equation} since the integrals in the product are absolutely convergent and equal to 1 at almost all places (cf. \cite[Proposition 12.21]{ge}). For $f_v\in\CC_c^\infty(\FG(k_v))$, set $$O_\gamma(f_v)=\int_{\FH_\gamma(k_v)\bs(\FH(k_v)\times\FH(k_v))} f_v(h_1^{-1}\gamma h_2){\mathrm{d}} h_1\ {\mathrm{d}} h_2.$$ Of course, the discussions above contain the case $(\FG',\FH')$. We define $J_\delta(f')$ for $\FH'\times\FH'$-regular semisimple elements $\delta\in\FG'(k)$ and $f'\in\CC_c^\infty(\FG'(\BA))$, and define $O_\delta(f'_v)$ for $f'_v\in\CC_c^\infty(\FG'(k_v))$ in the same way. Thus we have the relation \begin{equation}\label{equation. global and local integrals 2} J_\delta(f')=\prod_v O_\delta(f'_v)\end{equation} if $f'=\otimes'f'_v$.
From now on, when we say ``regular semisimple'', we mean ``$\FH\times\FH$-regular semisimple'' or ``$\FH'\times\FH'$-regular semisimple'' if there is no confusion. If $\gamma\in\FG(k)$ is regular semisimple, there exists a regular semisimple $\delta\in\FG'(k)$ matching $\gamma$ (see Proposition \ref{prop. match orbits} for more details). It turns out that $\FH_\gamma$ is isomorphic to $\FH'_\delta$ (see Remark \ref{matching stabilizer}). We equip $\FH'_\delta(\BA)$ with the Haar measure compatible with that on $\FH_\gamma(\BA)$. To compare the regular parts of the distributions $I$ with $J$ on the geometric side, we need to show the following conjecture on smooth transfer.
\begin{conj}\label{conj. global smooth transfer} For each $f$ in $\CC_c^\infty(\FG(\BA))$ there exists $f'$ in $\CC_c^\infty(\FG'(\BA))$ such that for each $\delta\in\FG'(k)_{\mathrm{rs}}$ $$\begin{array}{lll}J_{\delta}(f')=\left\{\begin{array}{ll} \begin{aligned}I_\gamma(f),&\quad \mathrm{if\ there\ exists\ } \gamma\in \FG(k)_{\mathrm{rs}}\ \mathrm{ matching\ }\delta,\\ 0,&\quad \mathrm{otherwise}. \end{aligned} \end{array}\right. \end{array}$$ \end{conj}
\begin{thm}\label{thm. global smooth transfer} Suppose that $D$ is split at all archimedean places. Then Conjecture \ref{conj. global smooth transfer} holds. \end{thm}
\begin{proof} This is a direct consequence of the relations (\ref{equation. global and local integrals 1}) and (\ref{equation. global and local integrals 2}) between global and local orbital integrals, and Theorem \ref{thm. main1} on the existence of smooth transfer at the non-archimedean places. \end{proof}
\section{Multiplicity one}\label{section. multiplicity one} The global linear period $\ell$ belongs to the space $\Hom_{\FH(\BA)}(\pi,\BC)$. To factor it into local ones, we need to study the space $\Hom_{\FH(k_v)}(\pi_v,\BC)$ for each place $v$ of $k$. We expect the so-called multiplicity one property to hold at each place $v$, that is, if $\pi_v$ is an irreducible admissible representation of $\FG(k_v)$, then $\dim\Hom_{\FH(k_v)}(\pi_v,\BC)\leq1$. If $(\FG(k_v),\FH(k_v))$ satisfies this multiplicity one property, we call it a Gelfand pair. It was proved in \cite{jr} in the non-archimedean case and in \cite{ag1} in the archimedean case that $(\FG'(k_v),\FH'(k_v))$ is a Gelfand pair. When $m=1$, $v$ is non-archimedean and $D$ is a general division algebra, $(\FG(k_v),\FH(k_v))$ is also a Gelfand pair, as was proved by Prasad \cite{pr}. We have not been able to show $(\FG(k_v),\FH(k_v))$ to be a Gelfand pair for general $m$ and $D$. However, we can prove a weaker result which is enough for us to factor the global period $\ell$, namely Proposition \ref{prop. multiplicity} and Corollary \ref{cor. multiplicity one for unitary repn}.
From now on, and until the end of the paper, we fix a $p$-adic field $F$. We follow the same line as that of \cite{ag1} where an effective way to prove results like multiplicity one for symmetric pairs is systematically studied, and we refer the reader to \cite{ag1} for more details.
\paragraph{Symmetric pairs} Now let $D$ be a division algebra over $F$ of index $d$. Let $\FG$ and $\FH$ be as defined in \S\ref{section. intro} associated to $D$, both viewed as algebraic groups over $F$ now. Write $G=\FG(F)$ and $H=\FH(F)$. Let $\epsilon=\begin{pmatrix}{\bf1}_m&0\\0&-{\bf1}_m\end{pmatrix}$ and define an involution $\theta$ on $\FG$ by $\theta(g)=\epsilon g\epsilon$. Then $\FH=\FG^\theta$, that is, $(\FG,\FH,\theta)$ is a symmetric pair. When there is no confusion, we write $(\FG,\FH)$ instead of $(\FG,\FH,\theta)$ for simplicity. Let $\iota$ be the anti-involution on $\FG$ defined by $\iota(g)=\theta(g^{-1})$. Write $\FG^\iota=\{g\in\FG;\ \iota(g)=g\}$ and define a symmetrization map $$s:\FG\lra\FG^\iota,\ s(g)=g\iota(g).$$ Via this map we can identify the $p$-adic symmetric space $S=G/H$ with its image in $\FG^\iota(F)$. Since $H^1(F,\FH)$ is trivial, we also have $S=(\FG/\FH)(F)$.
Let $\theta$ act by its differential on $\fg=\Lie(\FG)$. Write $\fh=\Lie(\FH)$. Thus, $\fh=\{X\in\fg;\ \theta(X)=X\}$. Write $\fs=\{X\in\fg;\ \theta(X)=-X\}$. $\fs$ can be viewed as a sort of ``Lie algebra" for $\FG/\FH$, on which $\FH$ acts by adjoint action. This action can be described more concretely as follows. It is easy to see that $\fs\simeq\fg\fl_m(D)\oplus\fg\fl_m(D)$, and modulo this isomorphism, the action of $\FH$ on $\fs$ is given by $$(h_1,h_2)\cdot(X_1,X_2)=(h_1X_1h_2^{-1},h_2X_2h_1^{-1})$$ for $(h_1,h_2)\in\FH$ and $(X_1,X_2)\in\fs$.
We fix the nondegenerate symmetric bilinear form $\pair{\ ,\ }$ on $\fg(F)$ defined by $$\pair{X,Y}=\tr(XY),\quad\mathrm{for}\ X,Y\in\fg(F),$$ where $\tr$ is the reduced trace map on $\fg\fl_{2m}(D)$, identified with the space $\End_D(D^{2m})$. Notice that the form $\pair{\ ,\ }$ is both $G$-invariant and $\theta$-invariant. When we want to emphasize the index $m$, we write $\FG_m,\FH_m,\theta_m,\fs_m$. Notice that the case $m=n$ and $D=F$ is just the case denoted by $(\FG',\FH')$ in \S\ref{section. intro}.
We will consider the action of $H\times H$ on $G$ by left and right translation, and the adjoint action of $H$ on $S$ or $\fs(F)$. These actions are related by $$s\left((h_1,h_2)\cdot g\right)=h_1\cdot s(g),$$ for $(h_1,h_2)\in H\times H$ and $g\in G$.
Now we recall some notions attached to a general symmetric pair $(\FG,\FH,\theta)$. We refer the reader to \cite{ag1} and \cite{ag2} for more details. Define $Q(\fs)=\fs/\fs^\FH$. Since $\FH$ is reductive, there exists a unique $\FH$-equivariant splitting $Q(\fs)\hookrightarrow\fs$. Denote by $\phi:\fs\ra\fs/\FH$ the standard projection. Let $\CN$ be the set of elements of $\fs$ that belong to the null-cone of $\fg$. We call $\CN$ the null-cone of $\fs$, since $\CN=\phi^{-1}(\phi(0))$ by \cite[Lemma 7.3.8]{ag1}. Note that $\CN\subset Q(\fs)$. Let $R(\fs)=Q(\fs)-\CN$. We call an element $g\in G$ \emph{admissible} if (i) $\Ad(g)$ commutes with $\theta$
and (ii) $\Ad(g)|_\fs$ is $\FH$-admissible. Notice that, in our case, $Q(\fs)=\fs$.
A symmetric pair $(\FG,\FH,\theta)$ is called \emph{good}, if for any closed $H\times H$ orbit $O$ in $G$, $\iota(O)=O$.
A symmetric pair $(\FG,\FH,\theta)$ is called \emph{regular} if for any admissible $g\in G$ such that $\CD(R(\fs)(F))^H\subset\CD(R(\fs)(F))^{\Ad(g)}$ we have $$\CD(Q(\fs)(F))^H\subset\CD(Q(\fs)(F))^{\Ad(g)}.$$
For each nilpotent element $X\in\fs(F)$, there exists a group homomorphism $\phi:\SL_2(F)\ra G$ such that $X={\mathrm{d}} \phi\left(\begin{pmatrix}0&1\\0&0\end{pmatrix}\right)$, $Y:={\mathrm{d}} \phi\left(\begin{pmatrix}0&0\{\mathbbold{1}}&0\end{pmatrix}\right)$ belongs to $\CN$, and ${\bf d}(X):={\mathrm{d}} \phi\left(\begin{pmatrix}1&0\\0&-1\end{pmatrix}\right)$ belongs to $\fh(F)$ and is semisimple (cf. \cite[Lemma 7.1.11]{ag1}). We call $(X,{\bf d}(X),Y)$ an $\fs\fl_2$-triple. Such a triple depends on the choice of $\phi$. However, the choice does not really matter (cf. \cite[Notation 7.1.12]{ag1}).
We say that a symmetric pair $(\FG,\FH,\theta)$ is of \emph{negative defect} if for any nilpotent $X\in\fs(F)$ we have
$$\RTr(\ad({\bf d}(X))|_{\fh_X})<\dim Q(\fs),$$ where $\fh_X$ is the centralizer of $X$ in $\fh$. By \cite[Proposition 7.3.5, Proposition 7.3.7, Remark 7.4.3]{ag1}, we see that if a symmetric pair is of negative defect it is regular.
Let $g\in G$ be $H\times H$-semisimple and $x=s(g)$. Then $x$ is semisimple both as an element of $G$ and with respect to the $H$-action (cf. \cite[Proposition 7.2.1]{ag1}). The triple
$(\FG_x,\FH_x,\theta|_{\FG_x})$ is still a symmetric pair (clearly $\theta$ preserves $\FG_x$ for $x\in\FG^\iota(F)$). A symmetric pair obtained this way is called a descendant of $(\FG,\FH,\theta)$. Notice that the ``Lie algebra'' of $\FG_x/\FH_x$ can be identified with $\fs_x$, where $\fs_x$ is the set of elements in $\fs$ commuting with $x$. It was shown in \cite[Theorem 7.4.5]{ag1} that if $(\FG,\FH,\theta)$ is a good symmetric pair such that all its descendants are regular then it is a GK-pair. Here, the statement that $(\FG,\FH)$ is a GK-pair means: $$\CD(G)^{H\times H}\subset\CD(G)^\iota.$$
\paragraph{Descendants} Now we return to the specific symmetric pair that concerns us in the paper. To study the property of multiplicity one, as we have explained, it is important to know all descendants of $(\FG,\FH,\theta)$. The following proposition gives a list of all possible descendants. \begin{prop}\label{prop. descendant 0} All descendants of $(\FG,\FH,\theta)$ are products of symmetric pairs of the following types \begin{enumerate} \item $\left(\R_{L/F}\left(\GL_r(D')\times\GL_r(D')\right), \Delta(\R_{L/F}\GL_r(D')),\delta\right)$ for some field extension $L/F$ and some central division algebra $D'$ over $L$, \item $\left(\R_{L'/F}\GL_r(D'\otimes_LL'),\R_{L/F}\GL_r(D'),\gamma\right)$ for some field extension $L/F$, a quadratic extension $L'/L$ and some central division algebra $D'$ over $L$, \item $(\GL_{2r}(D),\GL_r(D)\times\GL_r(D),\theta)$. \end{enumerate} \end{prop}
\begin{remark} Here we use $\Delta$ to denote the diagonal embedding and use $\R_{L/F}$ to denote the Weil restriction with respect to the field extension $L/F$. The involution $\delta$ in (1) of the above proposition is $(x,y)\mapsto(y,x)$, $\gamma$ in (2) is induced by the nontrivial element of $\Gal(L'/L)$, and $\theta$ in (3) is the one introduced before. \end{remark}
\begin{proof} In the case $D=F$, this proposition was first proved in \cite[Proposition 4.3]{jr} and reproved in \cite[Theorem 7.7.3]{ag1}. Our proof is similar to that of \cite[Theorem 7.7.3]{ag1}.
Let $x\in\FG^\iota(F)$ be a semisimple element. Put $V=D^{2m}$ and view $x$ as an element of $\GL_D(V)$. Let $m(t)=\prod_{i=1}^sp_i(t)$ be the minimal polynomial of $x$, where $p_i(t)\in F[t]$ is a monic irreducible polynomial and $p_i\neq p_j$ if $i\neq j$. Set $L_i:=F[t]/(p_i(t))$, which can be viewed as a field extension of $F$. Then $F[x]\simeq\prod_{i=1}^s L_i$. $V$ is an $(F[x],D)$-bimodule, and has a decomposition $V=\bigoplus_{i=1}^s V_i$ where $V_i$ is a $D$-submodule and $L_i$ acts faithfully on it. Thus $V_i$ is a $D\otimes_FL_i$-module. Since $D\otimes_FL_i$ is a central simple algebra over $L_i$, $D\otimes_F L_i=\M_{c_i\times c_i}(D_i)$ for some central division algebra $D_i$ over $L_i$. Set $V_i=W_i^{\oplus t_i}$ where $W_i\simeq D_i^{\oplus c_i}$. The above discussion on linear algebra over $D$ can be found in \cite[\S3]{yu}.
Therefore, $\FG_x\simeq\prod_{i=1}^s\R_{L_i/F}\GL_{t_i}(D_i)$. The rest of the proof is the same as that of \cite[Theorem 7.7.3]{ag1}. We remark that only the condition $x\in\FG^\iota(F)$ is used in the proof of \cite[Theorem 7.7.3]{ag1}. By this condition, we can only deduce a weaker result, namely that all descendants of $(\FG,\FH,\theta)$ are products of symmetric pairs of types 1, 2 or 3, where a type 3 symmetric pair is of the form $$(\GL_{q+r}(D),\GL_{q}(D)\times\GL_r(D),\theta_{q,r}),$$ where $\theta_{q,r}(g)=\epsilon_{q,r}g\epsilon_{q,r}$ with $\epsilon_{q,r}=\begin{pmatrix}{\bf 1}_q&0\\0&-{\bf 1}_r\end{pmatrix}$ and $q$ may not equal $r$. Since $x$ lies in $S\subset\FG^\iota(F)$, the list of possibilities for $\fs_x$ computed in Proposition \ref{prop. descendant 1 group} below lets us eliminate factors of the form $(\GL_{q+r}(D),\GL_q(D)\times\GL_r(D),\theta_{q,r})$ with $q\neq r$. \end{proof}
\paragraph{Multiplicity one} From the above classification of the descendants, we can see that, for any $H\times H$-semisimple $g\in G$, $H^1(F,\FH_{s(g)})$ is trivial. By \cite[Corollary 7.1.5]{ag1}, this implies that the symmetric pair $(\FG,\FH)$ is good.
This classification also implies that all the descendants of $(\FG,\FH)$ are regular. The reason is that, after base change to some extension field $F'$, they are of negative defect over $F'$ (proved in \cite[Theorem 7.6.5]{ag1} for symmetric pairs of types 1 and 2 and in \cite[Lemma 7.7.5]{ag1} for symmetric pairs of type 3), and hence they are of negative defect over $F$ by \cite[Lemma 4.2.7]{ag2}.
Therefore $(\FG,\FH)$ is a GK-pair. In particular, by \cite[Corollary 8.1.6]{ag1}, it satisfies the following property, which is a weaker variant of the property defining Gelfand pairs.
\begin{prop}\label{prop. multiplicity} For any irreducible admissible representation $\pi$ of $G$ we have $$\dim\Hom_H(\pi,\BC)\cdot\dim\Hom_H(\wt{\pi},\BC)\leq 1,$$ where $\wt{\pi}$ is the contragredient of $\pi$. \end{prop}
\begin{cor}\label{cor. multiplicity one for unitary repn} If $\pi$ is an irreducible unitary admissible representation of $G$, we have $$\dim\Hom_H(\pi,\BC)\leq1.$$ \end{cor}
\begin{proof} The following ``well known'' argument was pointed out by Prasad to the author. If $\pi$ is unitary, then $\bar{\pi}=\wt{\pi}$, where $\bar{\pi}$ denotes the complex conjugate of $\pi$. Observe that if $\pi$ has an $H$-invariant form $\ell$, taking the complex conjugate of the form, one can obtain an $H$-invariant form $\bar{\ell}$ on $\bar{\pi}\simeq\wt{\pi}$.
\end{proof}
\begin{remark} In general, we expect that $(\FG,\FH)$ is a Gelfand pair. When $D=F$, this is the main theorem of \cite{jr}. For $m=1$ and general $D$, it was proved by Prasad in \cite[\S7]{pr}. For general $m$ and $D$, we do not know how to prove it, because we do not know a Gelfand-Kazhdan type realization for the contragredient representation of an irreducible admissible representation of $G$. However, if $D$ is a quaternion algebra, there is an anti-automorphism $\tau$ of $G$ such that $\tau^2\in\Ad(G(F))$, $\tau$ preserves any closed conjugacy class in $G(F)$, and $\tau(H)=H$ (cf. \cite[Theorem 3.1]{ra}). Thus, by \cite[Proposition 2.1.6]{ag2}, we have the following result. \end{remark}
\begin{cor} If $D$ is a quaternion algebra, for any irreducible admissible representation $\pi$ of $G$ we have $$\dim\Hom_H(\pi,\BC)\leq1.$$ \end{cor}
\begin{remark}\label{rem. archimedean} The results of this section also hold when $F=\BR$, and they can be proved by the same arguments. All the notions we have introduced and all the propositions and theorems we have quoted hold in the archimedean case. \end{remark}
\section{Smooth transfer}\label{section. smooth transfer} We keep the notations as before, and continue to let $F$ be a $p$-adic field. In this section, we will show the existence of the smooth transfer with respect to the relative trace formulae concerned in this paper. Our strategy here follows the somewhat standard procedure, that was also used to study smooth transfer in other cases (cf. \cite{zh1} or \cite{zh} for more details). This strategy arises from the work of Waldspurger on endoscopic transfer (cf. \cite{wa97}). After several reduction steps, we reduce the question to proving the property that ``the Fourier transform commutes with smooth transfer''. We refer the reader to Theorem \ref{thm. fourier} for the exact statement. This property will be proved in Section \ref{section. local orbital integrals}.
The local orbital integrals that we consider are \begin{equation}\label{equation. local orbital 1}O(g,f)=\int_{H_g\bs(H\times H)}f(h_1^{-1}gh_2)\ {\mathrm{d}} h_1\ {\mathrm{d}} h_2,\end{equation} where $g\in G$ is $H\times H$-regular semisimple, and $f\in\CC_c^\infty(G)$. The quotient map $q:G\ra G/H=S$ gives rise to a surjection $\wt{q}:\CC_c^\infty(G)\ra\CC_c^\infty(S)$ defined by $$(\wt{q}f)(\bar{y})=\int_Hf(yh)\ {\mathrm{d}} h$$(cf. \cite[Lemma in Section 1.21]{bz}). Let $\wt{f}=\wt{q}(f)$. We identify $S$ with its image in $\FG^\iota(F)$ under the symmetrization map $s$, and view $\wt{f}$ as a $\CC_c^\infty$-function on the image of $S$. Thus, by definition, $\wt{f}(x)=\wt{f}(\bar{g})$ if $x=s(g)$. Now let $g\in G$ be $H\times H$-regular semisimple. Then $x=s(g)$ is $H$-regular semisimple (cf. \cite[Proposition 7.2.17]{ag1}) and we have the relations $$H_g\bs(H\times H)\twoheadrightarrow H_g(\{1\}\times H)\bs(H\times H)\stackrel{\pr_1}{\simeq}H_x\bs H,$$ and $H_g\cap(\{1\}\times H)=\{1\}\times\{1\}$. Therefore we have $$\begin{aligned}O(g,f) &=\int_{H_g(\{1\}\times H)\bs(H\times H)}\int_{\{1\}\times H}f(h_1^{-1}xh_2)\ {\mathrm{d}} h_2\ {\mathrm{d}} h_1\\ &=\int_{H_x\bs H}\wt{f}(h^{-1} x h)\ {\mathrm{d}} h. \end{aligned}$$ Henceforth it suffices to consider the orbital integrals for $\CC_c^\infty(S)$ with respect to the $H$-action. Eventually, we also have to consider the orbital integrals for $\CC_c^\infty(\fs(F))$ with respect to the $H$-action.
\paragraph{Orbits} First, we classify all $H$-semisimple orbits of $S$ and $\fs(F)$. We remark that the results here on the orbits also hold when we replace $F$ by a number field $k$. For each semisimple element $x$ in $S$ or $\fs(F)$, it is also important to determine the couple $(\FH_x,\fs_x)$ which is also called the descendant of $(\FH,\fs)$ at $x$. The reason for considering a semisimple element $x$ of $S$ (resp. $\fs(F)$) and the descendant of $(\FH,\fs)$ at $x$ is the following. We can reduce the study of the orbital integrals of an element $f$ in $\CC_c^\infty(S)$ (resp. $\CC_c^\infty(\fs(F))$) at regular semsimple elements near $x$ to the study of orbital integrals for the $H_x$-action of an appropriate element $f^\#_x$ in $\CC_c^\infty(\fs_x(F))$ at regular semisimple elements close to 0 in $\fs_x(F)$. Here $f^\#_x$ is obtained from $f$ by a process called semisimple descent. We refer the reader to \cite[Proposition 3.11]{zh1} for more details.
\begin{prop}\label{prop. descendant 1 group} \begin{enumerate} \item Each semisimple element $x$ of $S$ is $H$-conjugate to an element of the form $$x(A,m_1,m_2):=\begin{pmatrix}A&0&0&{\bf1}_r&0&0\\ 0&{\bf1}_{m_1}&0&0&0&0\\ 0&0&-{\bf1}_{m_2}&0&0&0\\ A^2-{\bf1}_r&0&0&A&0&0\\ 0&0&0&0&{\bf1}_{m_1}&0\\ 0&0&0&0&0&-{\bf1}_{m_2}\end{pmatrix},$$ with $m=r+m_1+m_2$, $A\in \fg\fl_r(D)$ being semisimple without eigenvalues $\pm1$ and unique up to conjugation. Moreover, $x(A,m_1,m_2)$ is regular if and only if $m_1=m_2=0$ and $A$ is regular in $\fg\fl_m(D)$. \item Let $x=x(A,m_1,m_2)$ in $S$ be semisimple. Then the descendant $(\FH_x,\fs_x)$ is isomorphic to the product $$\left(\GL_r(D)_A,\fg\fl_r(D)_A\right)\times(\FH_{m_1},\fs_{m_1}) \times(\FH_{m_2},\fs_{m_2}).$$ Here $\GL_r(D)_A$ and $\fg\fl_r(D)_A$ are the centralizers of $A$ in $\GL_r(D)$ and $\fg\fl_r(D)$ respectively, and $\GL_r(D)_A$ acts on $\fg\fl_r(D)_A$ by the adjoint action. \end{enumerate} \end{prop}
\begin{proof}
The first assertion was proved in \cite[Proposition 4.1]{jr} in the case $D=F$. The reader can check that the same proof goes through for general $D$ without difficulty. We only provide some steps in the proof.
Let $x=\begin{pmatrix}A&B\{\mathrm{P}}&Q\end{pmatrix}$ be a semisimple element of $S$ inside $\FG^\iota(F)$, with $A,B,P,Q$ in $\fg\fl_m(D)$. We claim that $A,Q,BP,PB$ and $\begin{pmatrix}0&B\{\mathrm{P}}&0\end{pmatrix}$ are semisimple matrices. This is the case when $D=F$ by \cite[Lemma 4.2]{jr}. Actually, in the proof of \cite[Lemma 4.2]{jr}, one can assume that $F$ is algebraically closed, since the condition of being semisimple does not depend on the ground field. Hence the claim for any general $D$ follows. Since $x\in\FG^\iota(F)$, we have the relations \begin{equation}\label{equation. matrix condition} A^2={\bf 1}_m+BP,\quad Q^2={\bf 1}_m+PB,\quad AB=BQ,\quad QP=PA. \end{equation} Replacing $x$ by a conjugate under $H$, we may assume $B=\begin{pmatrix}{\bf 1}_r&0\\0&0\end{pmatrix}$. Since $\begin{pmatrix}0&B\{\mathrm{P}}&0\end{pmatrix}$ is semisimple, by \cite[Proposition 2.1]{jr} (which is valid here), $x$ is $H$-conjugate to an element of the form $\begin{pmatrix}A&\begin{pmatrix}{\bf 1}_r&0\\0&0\end{pmatrix}\\ \begin{pmatrix}C_r&0\\0&0\end{pmatrix}&D\end{pmatrix}$ where $C_r\in\GL_r(D)$ is semisimple. The relations (\ref{equation. matrix condition}) force such an $H$-conjugate of $x$ to be of the form $\begin{pmatrix}A&0&{\bf1}_r&0\\ 0&A'&0&0\\ A^2-{\bf1}_r&0&A&0\\ 0&0&0&Q'\end{pmatrix}$, where $A\in\fg\fl_r(D)$ is semisimple without eigenvalues $\pm1$, $A'$ and $Q'$ are elements of order 2. By the same discussion as in the proof of \cite[Proposition 4.1]{jr}, we see that $x$ is $H$-conjugate to some $x(A,m_1,m_2)$.
Conversely, by similar discussions as in the proofs of \cite[Lemma 4.3, Proposition 4.1]{jr}, we see that each $x(A,m_1,m_2)$ is semisimple and lies in $S$. The remaining assertions such as the uniqueness of $A$ up to conjugation are straightforward.
For the second assertion, since descendants $(\FH_x,\fs_x)$ can be obtained by computing the centralizers of $x$ in $\FH$ and $\fs$, we leave the details to the reader.
\end{proof}
\begin{prop}\label{prop. descendant 1 lie} \begin{enumerate} \item Each semisimple element $X$ of $\fs(F)$ is $H$-conjugate to an element of the form $$X(A)=\begin{pmatrix}0&0&{\bf1}_r&0\\0&0&0&0\\A&0&0&0\\0&0&0&0\end{pmatrix}$$ with $A\in\GL_r(D)$ being semisimple and unique up to conjugation. Moreover, $X(A)$ is regular if and only if $r=m$ and $A\in\GL_m(D)$ is regular. \item Let $X=X(A)$ in $\fs(F)$ be semisimple. Then the descendant $(\FH_X,\fs_X)$ is isomorphic to the product $$\left(\GL_r(D)_A,\fg\fl_r(D)_A\right)\times(\FH_{m-r},\fs_{m-r}).$$ \end{enumerate} \end{prop}
\begin{proof} This proposition was proved in \cite[Proposition 2.1, Proposition 2.2]{jr} in the case $D=F$. The proof is simpler than that of Proposition \ref{prop. descendant 1 group} and can be applied to our more general situation directly. We leave the details to the reader. \end{proof}
\paragraph{Matching between the orbits} Now we give a description for the matching between $H$-semisimple orbits in $S$ or $\fs(F)$ and $H'$-semisimple orbits in $S'$ or $\fs'(F)$. Let $L$ be a field extension of $F$ with degree $d$ contained in $D$. Since $D$ is a $d$-dimensional $L$-vector space, we can obtain an embedding $D\hookrightarrow\fg\fl_d(L)$. This induces embeddings $(G,H)\hookrightarrow(\FG'(L),\FH'(L))$, $S\hookrightarrow \FG'(L)/\FH'(L)$ and $\fs(F)\hookrightarrow \fs'(L)$. We identify $\FG'(L)/\FH'(L)$ with its image in $\FG'^\iota(L)$.
\begin{prop}\label{prop. match orbits} \begin{enumerate} \item For each semisimple element $x$ of $S$, there exists $h\in\FH'(L)$ such that $hxh^{-1}$ belongs to $S'$. This establishes an injection from the set of $H$-semisimple orbits in $S$ into the set of $H'$-semisimple orbits in $S'$. This injection carries the orbit of $x(A,m_1,m_2)$ in $S$ to the orbit of $x(B,m_1d,m_2d)$ in $S'$ such that $A\in\fg\fl_{m-m_1-m_2}(D)$ and $B\in\fg\fl_{(m-m_1-m_2)d}(F)$ have the same characteristic polynomial. \item For each semisimple element $X$ of $\fs(F)$, there exists $h\in\FH'(L)$ such that $hXh^{-1}$ belongs to $\fs'(F)$. This establishes an injection from the set of $H$-semisimple orbits in $\fs(F)$ into the set of $H'$-semisimple orbits in $\fs'(F)$. This injection carries the orbit of $X(A)$ in $\fs(F)$ to the orbit of $X(B)$ in $\fs'(F)$ such that $A\in\GL_{r}(D)$ and $B\in\GL_{rd}(F)$ have the same characteristic polynomial. \end{enumerate} \end{prop}
\begin{proof} We only prove the matching between the orbits in $\fs(F)$ and $\fs'(F)$. The proof of the matching between the orbits in $S$ and $S'$ is similar. It is harmless to assume that $x$ is of the form $X(A)$ with $A\in\GL_r(D)$. We view $A$ as an element in $\GL_{rd}(L)$. Since the coefficients of the characteristic polynomial of $A$ are in $F$, there exists $h_0\in\GL_{rd}(L)$ such that $B=h_0^{-1}Ah_0$ is in $\GL_{rd}(F)$. Let $h=\begin{pmatrix}h_0&&&\\&{\bf1}_{(m-r)d}&&\\&&h_0&\\&&&{\bf 1}_{(m-r)d}\end{pmatrix}$. Then $h^{-1}\cdot X(A)\cdot h=X(B)$. \end{proof}
\begin{defn} We say that $x\in S_{\mathrm{ss}}$ (resp. $X\in\fs_{\mathrm{ss}}(F)$) matches $y\in S'_{\mathrm{ss}}$ (resp. $Y\in\fs'_{\mathrm{ss}}(F)$), and write $x\leftrightarrow y$ (resp. $X\leftrightarrow Y$), if the above map sends the orbit of $x$ (resp. $X$) to the orbit of $y$ (resp. $Y$). \end{defn}
Now we discuss some properties of the above matching. These properties will be used in Section \ref{section. local orbital integrals}.
\begin{remark}\label{rem. matching orbits} For a regular semsimple element $Y=\begin{pmatrix}0&A\\B&0\end{pmatrix}\in\fs'_{\mathrm{rs}}(F)$, suppose we wish to know whether there exists $X\in\fs_{\mathrm{rs}}(F)$ such that $X\leftrightarrow Y$. Note that $Y$ is $H'$-conjugate to $X(AB)$. It is well known that there exists $C\in\GL_m(D)$ such that $C$ and $AB$ have the same characteristic polynomial if and only if the degree of every irreducible factor (over $F$) of the characteristic polynomial of $AB$ can be divided by $d$. Hence there exists $X\in\fs_{\mathrm{rs}}(F)$ such that $X\leftrightarrow Y$ if and only if the degree of every irreducible factor (over $F$) of the characteristic polynomial of $AB$ is divisible by $d$. \end{remark}
\begin{remark}\label{rem. matching descendants} Suppose that $x\in S_{\mathrm{ss}}$ and $y\in S'_{\mathrm{ss}}$ match. We want to compare $(\FH_x,\fs_x)$ with $(\FH'_y,\fs'_y)$. It is harmless to assume that $x=x(A,m_1,m_2)$ and $y=x(B,m_1d,m_2d)$. Then, according to Proposition \ref{prop. descendant 1 group} $$(\FH_x,\fs_x)\simeq\left(\GL_r(D)_A,\fg\fl_r(D)_A\right)\times(\FH_{m_1},\fs_{m_1}) \times(\FH_{m_2},\fs_{m_2}),$$ and $$(\FH'_y,\fs'_y)\simeq\left(\GL_{rd,B},\fg\fl_{rd,B}\right)\times(\FH'_{m_1d},\fs'_{m_1d}) \times(\FH'_{m_2d},\fs'_{m_2d}).$$ Note that $\left(\GL_r(D)_A,\fg\fl_r(D)_A\right)$ is an inner form of $\left(\GL_{rd,B},\fg\fl_{rd,B}\right)$. Also note that the other factors are related in a similar manner as $(\FH,\fs)$ and $(\FH',\fs')$ are. For $X\in\fs_{\mathrm{ss}}(F)$ and $Y\in\fs'_{\mathrm{ss}}(F)$ such that $X\leftrightarrow Y$, according to Proposition \ref{prop. descendant 1 lie}, the descendants $(\FH_X,\fs_X)$ and $(\FH'_Y,\fs'_Y)$ satisfy a similar relation as above. \end{remark}
\begin{remark}\label{matching stabilizer} Suppose that $x$ in $S_{\mathrm{rs}}$ (resp. $\fs_{\mathrm{rs}}(F)$) and $y$ in $S'_{\mathrm{rs}}$ (resp. $\fs'_{\mathrm{rs}}(F)$) match. By Propositions \ref{prop. descendant 1 group}, \ref{prop. descendant 1 lie} and \ref{prop. match orbits}, we have $$\FH_x\simeq\FH'_y.$$ \end{remark}
\begin{remark}\label{rem. matching cartan} Recall that a Cartan subspace of $\fs$ is by definition a maximal abelian (with respect to the Lie bracket on $\fg$) subspace consisting of semisimple elements. An element $X\in\fs$ is regular semisimple if the centralizer of $X$ in $\fs$ is a Cartan subspace (cf. \cite[page 471]{Vi}). For a Cartan subspace $\fc$, we denote by $\fc_\reg(F)$ the subset of regular elements in $\fc(F)$. For a Cartan subspace $\fc$ of $\fs$ and a Cartan subspace $\fc'$ of $\fs'$, we say that $\fc$ and $\fc'$ match and write $\fc\leftrightarrow\fc'$ if there exist $X\in\fc_\reg(F)$ and $Y\in\fc'_\reg(F)$ such that $X\leftrightarrow Y$. Note that if $\fc\leftrightarrow\fc'$, there is an isomorphism $\varphi_\fc:\fc\ra\fc'$ defined over $F$ such that $X\leftrightarrow\varphi_\fc(X)$ for any $X\in\fc_\reg(F)$. To see this, suppose that $X\in\fc_\reg(F)$ and $Y\in\fc'_\reg(F)$ match. We may assume that $X=X(A)$ and $Y=Y(B)$. Then we have $$\fc(F)=\fs_X(F)=\left\{\begin{pmatrix}0&C\\AC&0\end{pmatrix};C\in \fg\fl_m(D),\ AC=CA\right\}\simeq{\mathfrak{gl}}_{m}(D)_A.$$ We also have $$\fc'(F)=\fs'_Y(F)=\left\{\begin{pmatrix}0&D\\BD&0\end{pmatrix};D\in \fg\fl_n,\ BD=DB\right\}\simeq\fg\fl_{n,B}.$$ Since $A$ and $B$ are regular semisimple and have the same characteristic polynomial, there is an isomorphism $\varphi:\fg\fl_m(D)_A\ra\fg\fl_{n,B}$ over $F$ such that $\varphi(A)=B$. The isomorphism $\varphi_c$ can be obtained from $\varphi$. \end{remark}
\paragraph{Smooth transfer} Now we can introduce the notion of smooth transfer and state the main theorem of the paper. We first fix Haar measures on $H$ and $H'$, and fix a Haar measure on $H'_y$ for each $y$ in $S'_{\mathrm{rs}}$ or $\fs'_{\mathrm{rs}}(F)$. We may and do assume that for $y_1,y_2$ in $S'_{\mathrm{rs}}$ or $\fs'_{\mathrm{rs}}(F)$ that lie in the same $H'$-orbit, the Haar measures on $H'_{y_1}$ and $H'_{y_2}$ are compatible in the obvious sense. For any $x$ in $S_{\mathrm{rs}}$ or $\fs_{\mathrm{rs}}(F)$, choose any $y$ in $S'_{\mathrm{rs}}$ or $\fs'_{\mathrm{rs}}(F)$ respectively so that $x\leftrightarrow y$. Then $H_x\simeq H'_y$. We choose the Haar measure on $H_x$ compatible with that on $H'_y$.
\begin{defn} For $x\in S_{\mathrm{rs}}$ (resp. $x\in\fs_{\mathrm{rs}}(F)$), and $f\in\CC_c^\infty(S)$ (resp. $f\in\CC_c^\infty(\fs(F))$), we define the orbital integral of $f$ at $x$ to be $$O(x,f)=\int_{H_x\bs H}f(h^{-1}xh)\ {\mathrm{d}} h,$$ which is convergent since any semisimple orbit is closed. \end{defn}
\begin{defn}\label{defn. transfer} For $f\in\CC_c^\infty(S)$ (resp. $\CC_c^\infty(\fs(F))$), we say that $f'\in\CC_c^\infty(S')$ (resp. $\CC_c^\infty(\fs'(F))$) is a smooth transfer of $f$ if for each $y\in S'_{\mathrm{rs}}$ (resp. $y\in\fs'_{\mathrm{rs}}(F)$) \begin{equation}\label{equation. condition of orbital integrals} \begin{array}{lll}O(y,f')=\left\{\begin{array}{ll} \begin{aligned}O(x,f),&\quad \textrm{if there exists } x\in S_{\mathrm{rs}} \textrm{ (resp. $x\in\fs_{\mathrm{rs}}(F)$) such that }x\leftrightarrow y,\\ 0,&\quad \textrm{otherwise}. \end{aligned} \end{array}\right. \end{array}\end{equation} Sometimes we will write transfer instead of smooth transfer for short. If $f'$ is a transfer of $f$, we write $f\leftrightarrow f'$ for simplicity. \end{defn}
\begin{remark}\label{rem. converse smooth transfer} Conversely, for $f'\in\CC_c^\infty(S')$ (resp. $f'\in\CC_c^\infty(\fs'(F))$) satisfying the following condition \begin{equation}\label{equation. condition} O(y,f')=0\ \textrm{if there does not exist $x$ in $S_{\mathrm{rs}}$ (resp. $\fs_{\mathrm{rs}}(F)$) such that $x\leftrightarrow y$}, \end{equation} we say that $f\in\CC_c^\infty(S)$ (resp. $f\in\CC_c^\infty(\fs(F))$) is a smooth transfer of $f'$ if for each $x\in S_{\mathrm{rs}}$ (resp. $x\in\fs_{\mathrm{rs}}(F)$) $$O(x,f)=O(y,f'),$$ where $y$ in $S'_{\mathrm{rs}}$ (resp. $\fs'_{\mathrm{rs}}(F)$) is such that $x\leftrightarrow y$. \end{remark}
\begin{remark}\label{rem. transfer on descendant} For semisimple $x\in S$ (resp. $x\in\fs(F)$) and semisimple $y\in S'$ (resp. $y\in\fs'(F)$) such that $x\leftrightarrow y$, we can also define smooth transfer from $\CC_c^\infty(\fs_x(F))$ to $\CC_c^\infty(\fs'_y(F))$ determined by the orbital integrals with respect to the action of $H_x$ on $\fs_x(F)$ and the action of $H'_y$ on $\fs'_y(F)$. According to Remark \ref{rem. matching descendants}, there are two types of smooth transfer to consider. The first type is associated to $(\FH_{m'},\fs_{m'})$ and $(\FH'_{m'd},\fs'_{m'd})$ with $m'\leq m$. The second type is associated to $(\GL_r(D)_A,\fg\fl_r(D)_A)$ and its inner form $(\GL_{rd,B},\fg\fl_{rd,B})$. In this case, the orbital integrals are with respect to the adjoint action and the existence of smooth transfer is known (cf. \cite{wa97}). \end{remark}
Our main theorem on the smooth transfer is the following.
\begin{thm}\label{thm. main1} For each $f\in\CC_c^\infty(S)$, there exists $f'\in\CC_c^\infty(S')$ that is a smooth transfer of $f$. \end{thm}
Showing the existence of smooth transfer essentially is a local issue. Via the Luna Slice Theorem and descent of orbital integrals, we can reduce to proving the existence of smooth transfer between the descendants $(\FH_x,\fs_x(F))$ and $(\FH'_y,\fs'_y(F))$ for each semisimple $x\in S$ and $y\in S'$ such that $x\leftrightarrow y$. According to Remark \ref{rem. transfer on descendant}, we reduce to proving the following Lie algebra version of smooth transfer. We refer the reader to \cite[\S3]{zh1} or \cite[\S5.3]{zh} for more details of such reduction steps. The arguments there can be applied for our situation without modification.
\begin{thm}\label{thm. main2} For each $f\in\CC_c^\infty(\fs(F))$, there exists $f'\in\CC_c^\infty(\fs'(F))$ that is a smooth transfer of $f$. \end{thm}
To prove Theorem \ref{thm. main2}, the following theorem, which roughly says that the Fourier transform commutes with smooth transfer, is the key input.
\begin{thm}\label{thm. fourier} There exists a nonzero constant $c\in\BC$ satisfying that: if $f'\in\CC_c^\infty(\fs'(F))$ is a transfer of $f\in\CC_c^\infty(\fs(F))$, then $\widehat{f'}$ is a transfer of $c\widehat{f}$. \end{thm}
\begin{remark} We now briefly recall why Theorem \ref{thm. fourier} implies Theorem \ref{thm. main2}. We use induction and assume that Theorem \ref{thm. main2} holds for functions in $\CC_c^\infty(\fs_{m'}(F))$ for every $m'<m$. Via the Luna Slice Theorem and descent of orbital integrals again, we can reduce to proving the existence of smooth transfers on the descendants for each semisimple $X\in\fs_{\mathrm{ss}}(F)$ and $Y\in\fs'_{\mathrm{ss}}(F)$ such that $X\leftrightarrow Y$. If $X$ is nonzero, the factor $(\FH_{m'},\fs_{m'})$ in the descendant $(\FH_X,\fs_X)$ satisfies that $m'<m$. Thus, by Remark \ref{rem. transfer on descendant} and the inductive hypothesis, smooth transfers exist for functions whose supports are contained in a neighborhood of $X$. Moreover, this shows the existence of smooth transfer for $f\in\CC_c^\infty(\fs(F)-\CN)$, where $\CN$ is the null-cone of $\fs(F)$. We have explained that the symmetric pair $(\FG,\FH)$ is of negative defect, which implies that $(\FG,\FH)$ is special (cf. \cite[Proposition 7.3.7]{ag1}). The speciality means the following statement. If $T$ is an $H$-invariant distribution on $\fs(F)$ such that $\Supp(T)\subset\CN$ and $\Supp(\widehat{T})\subset\CN$, then $T$ must be zero. This fact has the following direct consequence. Let $\CC_0=\bigcap\limits_{T}\ker(T)$ where $T$ runs over all $H$-invariant distributions on $\fs(F)$. Then each $f\in\CC_c^\infty(\fs(F))$ can be written as $f=f_0+f_1+\widehat{f_2}$ with $f_0\in\CC_0$ and $f_i\in\CC_c^\infty(\fs(F)-\CN)$ for $i=1,2$. Therefore, it remains to prove the existence of the smooth transfer for $\widehat{f}$ with $f$ belonging to the space $\CC_c^\infty(\fs(F)-\CN)$, which is exactly what Theorem \ref{thm. fourier} shows. \end{remark}
\begin{remark} To prove the existence of smooth transfer in the converse direction, in the sense of Remark \ref{rem. converse smooth transfer}, it suffices to prove that each $f\in\CC_c^\infty(\fs'(F))$ satisfying condition (\ref{equation. condition}) can be written as $f=f_0+f_1+\widehat{f_2}$. Here $f_0$ is in $\CC'_0$ which is defined similarly as $\CC_0$ for $\fs$, and $f_i\in\CC_c^\infty(\fs'(F)-\CN')$, for $i=1,2$, is also required to satisfy condition (\ref{equation. condition}) (here $\CN'$ is the null-cone of $\fs'(F)$). However we do not know how to prove such a decomposition. \end{remark}
\section{Local orbital integrals}\label{section. local orbital integrals}
Let $F$ be a $p$-adic field as before. This section is devoted to proving Theorem \ref{thm. fourier}. We employ several techniques used by Waldspurger on endoscopic smooth transfer (cf. \cite{wa95} and \cite{wa97}). These techniques also involve some more classical results of Harish-Chandra on harmonic analysis for $p$-adic groups (cf. \cite{hc1} and \cite{hc}). We also establish various analogous results for the $p$-adic symmetric spaces considered here. A large part of this section can be viewed as a generalization of the results in \cite{zh}.
\subsection{Preparations} \paragraph{Inequalities} Fix a nonzero $X_0$ in the null-cone $\CN$ of $\fs(F)$. Let $(X_0,{\bf d},Y_0)$ be an $\fs\fl_2$-triple with ${\bf d}\in\fh(F)$
and $Y_0\in\CN$. We set ${\bf r}=\dim_F\fs_{Y_0}$ and ${\bf m}=\frac{1}{2}\RTr\left(\ad(-{\bf d})|_{\fs_{Y_0}}\right)$. The inequalities below are used to bound the orbital integrals for elements of $\CC_c^\infty(\fs(F))$ along a Cartan subspace of $\fs$.
\begin{prop}\label{prop. inequality for nilpotent} We have the relations \begin{enumerate} \item ${\bf r}\geq n$, \item ${\bf r}+{\bf m}> n^2+\frac{n}{2}$. \end{enumerate} \end{prop}
\begin{proof} Let $L$ be an extension field of $F$ with degree $d$ contained in $D$. Then $$(\FG\times_FL,\FH\times_FL )\simeq(\FG'\times_FL,\FH'\times_FL)=:(\FG'',\FH'').$$ Denote by $\fs''$ the ``Lie algebra'' associated to $\FG''/\FH''$. We can, in a canonical way, view $X_0$ and $Y_0$ as elements of $\fs''(L)$ and also ${\bf d}$ as an element of $\fh''(L)$
($\fh'':=\Lie(\FH'')$). Let ${\bf r}'=\dim_L\fs''_{Y_0}$ and ${\bf m}'=\frac{1}{2}\RTr\left(\ad(-{\bf d})|_{\fs''_{Y_0}}\right)$. Since $\fs''_{Y_0}\simeq\fs_{Y_0}\otimes_FL$, it is not hard to see that ${\bf r}={\bf r}'$ and ${\bf m}={\bf m}'$. Therefore the required inequalities follow immediately by \cite[Proposition 4.4]{zh}. \end{proof}
\paragraph{Representability} With the aid of Proposition \ref{prop. inequality for nilpotent}, we can generalize all the results in \cite[\S5, \S6, \S7]{zh} when $d=1$ to the more general case at hand. We will only state the results and omit the proofs since they are obtained as almost verbatim reproductions of those in \cite{zh}.
Let $X\in\fs_{\mathrm{rs}}(F)$ lie in a Cartan subspace $\fc$ of $\fs$. Then the centralizer $\FT$ of $\fc$ in $\FH$ equals $\FH_X$. Thus $T$ is a torus by Proposition \ref{prop. descendant 1 lie}. Write
$\ft=\Lie(\FT)$. We define the normalizing factor $|D^\fs(X)|_F$ to be
$$|\det(\ad(X);\fh/\ft\oplus\fs/\fc)|_F^{\frac{1}{2}},$$ which is also equal to $|\det(\ad(X);\fg/\fg_X)|_F^{\frac{1}{2}}$. We consider the normalized orbital integral:
$$I(X,f)=|D^\fs(X)|_F^{\frac{1}{2}}O(X,f),\ \textrm{ for }f\in\CC_c^\infty(\fs(F)),$$ which is a distribution on $\fs(F)$. We also consider its Fourier transform: $$\widehat{I}(X,f):=I(X,\widehat{f}),\quad\textrm{ for }f\in\CC_c^\infty(\fs(F)).$$
If $X\in\fs_{\mathrm{rs}}(F)$ and $Y\in\fs'_{\mathrm{rs}}(F)$ are such that $X\leftrightarrow Y$, viewed as elements of $\M_{2m\times 2m}(D)$ and $\M_{2n\times 2n}(F)$ respectively, they have the same characteristic polynomial (cf. Proposition \ref{prop. match orbits}). Since the normalizing factor is determined by characteristic polynomial, we see that
$$|D^\fs(X)|_F=|D^{\fs'}(Y)|_F.$$ Hence it does not matter if we consider the smooth transfer with respect to the normalized orbital integrals.
The following theorem is a generalization of \cite[Theorem 6.1]{zh}. Its proof can be copied word for word from that of \cite[Theorem 6.1]{zh}. The ingredients of its proof are the analogues of parabolic induction and Howe's finiteness theorem for our symmetric spaces, together with bounds for normalized orbital integrals along Cartan subspaces of $\fs(F)$.
\begin{thm}\label{thm. representability} For each $X\in\fs_{\mathrm{rs}}(F)$, there exists a locally constant $H$-invariant function $\widehat{i}_X$ defined on $\fs_{\mathrm{rs}}(F)$ which is locally integrable on $\fs(F)$, such that for any $f\in\CC_c^\infty(\fs(F))$ we have $$\widehat{I}(X,f)=\int_{\fs_{\mathrm{rs}}(F)}\widehat{i}_X(Y)f(Y)
|D^\fs(Y)|_F^{-1/2}\ {\mathrm{d}} Y.$$ \end{thm}
We will need a proposition that shows up in the course of the proof. Recall that an element $X\in\fs_{\mathrm{rs}}(F)$ is called elliptic if its centralizer $\FH_X$ is an elliptic torus. Denote by $\fs_\el(F)$ the set of elliptic elements in $\fs_{\mathrm{rs}}(F)$. Suppose that $X\in\fs_{\mathrm{rs}}(F)$ is of the form $\begin{pmatrix}0&{\bf1}_m\\A&0\end{pmatrix}$. Also suppose that $X$ is not elliptic. Then $A\in\GL_m(D)$ is not elliptic in the usual sense. Then there exists a proper Levi subgroup ${\mathbf{M}}_0$ of $\GL_m(D)$ such that $A\in M_0:={\mathbf{M}}_0(F)$. Set $\fm_0:=\Lie({\mathbf{M}}_0)$. Identify $\fs^+$ (resp. $\fs^-$) with $\fg\fl_m(D)$, and let $\fr^+\subset\fs^+$ (resp. $\fr^-\subset\fs^-$) be the subspace that corresponds to $\fm_0$ under this identification. Finally, set $\fr=\fr^+\oplus\fr^-$. Then $X$ lies in $\fr(F)$ and is regular semisimple with respect to the adjoint action of $M=M_0\times M_0$
on $\fr(F)$. Choosing a Haar measure on $M$, we also consider the orbital integral with respect to the action of $M$ on $\fr(F)$. Note that $\ft$ is contained in $\fm$ and $\fc$ is contained in $\fr$. The normalizing factor $|D^\fr(X)|_F$ is defined to be
$$|\det\left(\ad(X);\fm/\ft\oplus\fr/\fc\right)|_F^{\frac{1}{2}}.$$ The normalized orbital integral $I^\fr(X,f')$, for $f'\in\CC_c^\infty(\fr(F))$, is defined to be
$$|D^\fr(X)|_F^{\frac{1}{2}}\int_{H_X\bs M}f'(m^{-1}Xm)\ {\mathrm{d}} m,$$ which is convergent since $X$ is semisimple with respect to the action of $M$. $I^\fr(X,\cdot)$ is a distribution on $\fr(F)$. We also consider its Fourier transform $$\widehat{I}^\fr(X,f'):=I^\fr(X,\widehat{f'}).$$ Then, with suitable choices of Haar measures, there is a relation between the orbital integrals $I(X,\cdot)$ and $I^\fr(X,\cdot)$, the so-called parabolic descent of orbital integrals, $$I(X,f)=I^\fr(X,f^{(\fr)}),\ \textrm{ for all }f\in\CC_c^\infty(\fs(F)).$$ Here $f^{(\fr)}\in\CC_c^\infty(\fr(F))$ is a sort of ``constant term'' of $f$. We refer the reader to \cite[\S6.1]{zh} for the precise definition. The exact same formula for $f^{(\fr)}$ as there still works in our situation.
Applying Theorem \ref{thm. representability} to lower rank situations, we see that there exists a locally constant $M$-invariant function $\widehat{i}^\fr_X$ defined on $\fr_{\mathrm{rs}}(F)$ which is locally integrable on $\fr(F)$, such that for any $f'\in\CC_c^\infty(\fr(F))$ we have $$\widehat{I}^\fr(X,f')=\int_{\fr_{\mathrm{rs}}(F)}\widehat{i}_X^\fr(Y) f'(Y)
|D^\fr(Y)|_F^{-\frac{1}{2}}\ {\mathrm{d}} Y.$$ Not surprisingly, there is a relation between $\widehat{i}_X$ and $\widehat{i}^\fr_X$. The following formula for $\widehat{i}_X$ in terms of $\widehat{i}^\fr_X$ will be needed.
\begin{prop}\label{prop. i(X,Y) parabolic} Keep the notations and assumptions above. We have $$\widehat{i}_X(Y)=\sum_{Y'}\widehat{i}_X^\fr(Y'),\quad Y\in\fs_{\mathrm{rs}}(F),$$ where $Y'$ runs over a set of representatives for the finitely many $M$-conjugacy classes of elements in $\fr(F)$ which are $H$-conjugate to $Y$. In particular, if there is no element in $\fr(F)$ which is $H$-conjugate to $Y$, we have $$\widehat{i}_X(Y)=0.$$ \end{prop}
\paragraph{Limit formula} We also write $\widehat{i}(X,Y)$ for $\widehat{i}_X(Y)$. There is a limit formula for $\widehat{i}(X,Y)$ shown in \cite[Proposition 7.1]{zh}, which takes care of a situation where $d=1$ (and where there is an additional quadratic character present to deal with the more general, twisted, periods considered there). A similar formula still holds for the case at hand and will be stated below. Notice that changing the Haar measures on $H$ and $H_X$ multiplies $\widehat{i}(X,Y)$ by a nonzero scalar. We do not specify the Haar measures, and instead refer the reader to \cite{zh} for more details. Results that follow this limit formula (Proposition \ref{prop. i(X,Y)} below) do not depend on the choices of the measures.
Let $\fc$ be a Cartan subspace of $\fs$, $\FT$ the centralizer of $\fc$ in $\FH$, and $\ft$ the Lie algebra of $\FT$. For $X,Y\in\fc_\reg(F)$, we define a bilinear form $q_{X,Y}$ on $\fh(F)/\ft(F)$ by $$q_{X,Y}(Z,Z')=\pair{[Z,X],[Y,Z']},$$ where the pairing $\pair{\cdot,\cdot}$ is the one introduced before. One can check that $q_{X,Y}$ is nondegenerate and symmetric. One can also verify that $q_{X,Y}=q_{Y,X}$. We will write $\gamma_\psi(X,Y)=\gamma_\psi(q_{X,Y})$ for simplicity.
\begin{prop}\label{prop. i(X,Y)} Let $X\in\fs_{\mathrm{rs}}(F)$ and $Y\in\fc_\reg(F)$. Then there exists $N\in\BN$ such that if $\mu\in F^\times$ satisfies $v_F(\mu)<-N$, we have the equality $$\widehat{i}(\mu X,Y)=\sum_{h\in T\bs H,\ h\cdot X\in\fc}\gamma_\psi \left(\mu h\cdot X,Y\right)\psi\left(\pair{\mu h\cdot X,Y}\right).$$ \end{prop}
\begin{proof} One can make an obvious modification of the proof of \cite[Proposition 7.1]{zh} to apply it here. \end{proof}
\paragraph{Construction of test functions} For $X,Y\in\fc_\reg(F)$, there is a formula for $\gamma_\psi(X,Y)$, which is exhibited in \cite[Proposition 7.3]{zh} when $d=1$. The formula for general $d$ has the same form. We will not state it here, since it involves much more notation. The following lemma is used to construct certain test functions required in Proposition \ref{prop. local prop} below.
\begin{prop}\label{prop. compare lemma} Let $\fc$ be a Cartan subspace of $\fs$. Fix a Cartan subspace $\fc'$ of $\fs'$ such that $\fc\leftrightarrow\fc'$. Then for any $X,X'\in\fc_\reg(F)$ we have the equality $$\gamma_\psi\left(X,X'\right)=\gamma_\psi(\fh(F)) \gamma_\psi(\fh'(F))^{-1}\gamma_\psi(\varphi_\fc(X),\varphi_\fc(X')).$$ Here $\varphi_\fc$ is an isomorphism from $\fc$ to $\fc'$ as in Remark \ref{rem. matching cartan}. \end{prop}
The following proposition is an analogue of \cite[Proposition 7.6]{zh}, and its proof involves Propositions \ref{prop. i(X,Y)} and \ref{prop. compare lemma}. It plays an important role in proving the existence smooth transfer using the global method that we are following here.
\begin{prop}\label{prop. local prop} Let $X_0\in\fc_\reg(F)$ and $Y_0\in\fc'_\reg(F)$ be such that $X_0\leftrightarrow Y_0$. Then there exist functions $f\in\CC_c^\infty(\fs(F))$ and $f'\in\CC_c^\infty(\fs'(F))$ satisfying the following conditions. \begin{enumerate} \item If $X\in\Supp(f)$, $X$ is $H$-conjugate to an element in $\fc_\reg(F)$. If $Y\in\Supp(f')$, there exists $X'\in\fc_\reg(F)$ such that $X'\leftrightarrow Y$. \item $f'$ is a transfer of $f$. \item There is an equality $$\widehat{I}(X_0,f)=c\widehat{I}(Y_0,f')\neq0,$$ where $c=\gamma_\psi(\fh(F))\gamma_\psi(\fh'(F))^{-1}$. \end{enumerate} \end{prop}
\begin{proof} The same proof as that of \cite[Proposition 7.6]{zh} applies. \end{proof}
\subsection{Proof of Theorem \ref{thm. fourier}}
In this subsection, we fix two $\CC_c^\infty$-functions $f'\in\CC_c^\infty(\fs'(F))$ and $f\in\CC_c^\infty(\fs(F))$ such that $f\leftrightarrow f'$. The proof of Theorem \ref{thm. fourier} can be divided into two parts: \begin{enumerate} \item the first part is to prove that $\widehat{I}(Y,f')=0$ for any $Y\in\fs'_{\mathrm{rs}}(F)$ such that there exists no element in $\fs_{\mathrm{rs}}(F)$ matching $Y$; \item the second part is to search for a nonzero constant $c\in\BC$, independent of $f$ and $f'$, such that $$\widehat{I}(Y,f')=c\widehat{I}(X,f)$$ for any $X\in\fs_{\mathrm{rs}}(F),Y\in\fs'_{\mathrm{rs}}(F)$ such that $X\leftrightarrow Y$. \end{enumerate}
\paragraph{First part of the proof} Now we fix a $Y_0\in\fs'_{\mathrm{rs}}(F)$ such that there exists no element in $\fs_{\mathrm{rs}}(F)$ matching $Y_0$. Suppose that $Y_0$ belongs to a Cartan subspace $\fc'_0$ of $\fs'$. By Theorem \ref{thm. representability} (in the case where $d=1$) and the Weyl integration formula, we have \begin{equation}\label{equation. weyl}\begin{aligned} \widehat{I}(Y_0,f')&=\int_{\fs'_{\mathrm{rs}}(F)}\widehat{i}_{Y_0}(Z)f'(Z)
|D^{\fs'}(Z)|_F^{-\frac{1}{2}}\ {\mathrm{d}} Z\\ &=\sum_{\fc'} \frac{1}{w_{\fc'}}\int_{\fc'_\reg(F)}\widehat{i}_{Y_0}(Z) I(Z,f')\ {\mathrm{d}} Z, \end{aligned}\end{equation} where $\fc'$ runs over a set of representatives for the finitely many $H'$-conjugacy classes of Cartan subspaces in $\fs'$ and $w_{\fc'}$ is the cardinality of the relative Weyl group associated to $\fc'$. For the Weyl integration formula in the setting of symmetric spaces, we refer the reader to \cite[page 106]{rr}.
We denote by $\sC^D$ the set of Cartan subspaces $\fc'$ of $\fs'$ such that there exists a Cartan subspace $\fc$ of $\fs$ with $\fc\leftrightarrow\fc'$. By the condition on $Y_0$, we see that $\fc'_0\notin\sC^D$.
For any $\fc'\notin\sC^D$, we automatically have $I(Z,f')=0$ for each $Z\in\fc'_\reg(F)$ by the condition on $f'$. If $\fc'\in\sC^D$, we claim that $\widehat{i}_{Y_0}(Z)=0$ for any $Z\in\fc'_\reg(F)$. We can assume that $Y_0$ is of the form $\begin{pmatrix}0&{\bf 1}_n\\ A&0 \end{pmatrix}$ with $A\in\GL_n(F)_{\mathrm{rs}}$. By the condition on $Y_0$, there exists an irreducible factor (over $F$) of the characteristic polynomial of $A$ with degree $r$ such that $d\nmid r$. Then there exists a subspace $\fr$ of $\fs$ of the form $\left(\fg\fl_{r}\oplus\fg\fl_{n-r}\right)\bigoplus \left(\fg\fl_{r}\oplus\fg\fl_{n-r}\right)$ such that $Y_0\in\fr(F)$ (see Proposition \ref{prop. i(X,Y) parabolic} for the notation). Since $\fc'\in\sC^D$, there exists no element in $\fr(F)$ which is $H'$-conjugate to any $Z\in\fc'_\reg(F)$. Thus the claim follows from Proposition \ref{prop. i(X,Y) parabolic}. Therefore, in any case, we have showed that the terms appearing in the sum of (\ref{equation. weyl}) are zero, thus obtaining that $\widehat{I}(Y_0,f')=0$.
\paragraph{Second part of the proof} The arguments in this part are almost the same as those in \cite[\S8]{zh}. We shall explain them briefly.
Now, we fix $f\in\CC_c^\infty(\fs(F))$, and $f'\in\CC_c^\infty(\fs'(F))$ which is a transfer of $f$, and fix $X_0\in\fs_{\mathrm{rs}}(F),Y_0\in\fs'_{\mathrm{rs}}(F)$ such that $X_0\leftrightarrow Y_0$. Next, we choose some global data as follows.
\s{$\bullet$ \emph{Fields}}. We choose a number field $k$ and a central division algebra $\BD$ over $k$ so that: \begin{enumerate} \item $k$ is totally imaginary; \item there exists a finite place $w$ of $k$ such that $k_w\simeq F$ and $\BD(k_w)\simeq D$; \item there exists another finite place $u$ of $k$ such that $\BD$ does not split over $k_u$. By conditions 1 and 2, such a finite place $u$ exists. \end{enumerate} Such a number field $k$ and a division algebra $\BD$ do exist. See \cite[Proposition in \S 11.1]{wa97}. From now on, we identify $k_w$ with $F$. We denote by $\CO_k$ the ring of integers of $k$, and by $\BA$ the ring of adeles of $k$. We fix a maximal order $\CO_\BD$ of $\BD$ containing $\CO_k$. We fix a continuous character on $\BA/k$ whose local component at $w$ is $\psi$, and henceforth use the letter $\psi$ to denote this new (global) character.
\s{$\bullet$ \emph{Groups}}. We define a global symmetric pair $(\BG,\BH)$ over $k$ with respect to $\BD$, so that the base change of $(\BG,\BH)$ to $k_w$ is isomorphic to $(\FG,\FH)$. Thus if the index of $\BD$ is $d'$, let $(\BG,\BH)=(\GL_{2m'}(\BD),\GL_{m'}(\BD)\times\GL_{m'}(\BD))$ where $m'd'=n$. Define the symmetric pair $(\BG',\BH')=(\GL_{2n},\GL_n\times\GL_n)$ over $k$ as usual. We now use $\fh$ (resp. $\fh'$) to denote the Lie algebra of $\BH$ (resp. $\BH'$), and $\fs$ (resp. $\fs'$) to denote the ``Lie algebra'' of $\BG/\BH$ (resp. $\BG'/\BH'$). Hence $X_0\in\fs_{\mathrm{rs}}(k_w)$ and $Y_0\in\fs'_{\mathrm{rs}}(k_w)$.
\s{$\bullet$ \emph{Places}}. Denote by $V$ (resp. $V_\infty,\ V_{\mathrm{f}}$) the set of all (resp. archimedean, non-archimedean) places of $k$. Fix two $\CO_k$-lattices: $\FL=\fg\fl_{m'}(\CO_\BD)\oplus\fg\fl_{m'}(\CO_\BD)$ in $\fs(k)$ and $\FL'=\fg\fl_n(\CO_{k})\oplus\fg\fl_n(\CO_k)$ in $\fs'(k)$. For each $v\in V_{\mathrm{f}}$, set $\FL_v=\FL\otimes_{\CO_k}\CO_{k,v}$ and $\FL'_v=\FL'\otimes_{\CO_k}\CO_{k,v}$. We fix a finite set $S\subset V$ such that: \begin{enumerate} \item $S$ contains $u,w$ and $V_\infty$; \item for each $v\in V-S$, everything is unramified, i.e. $\BG$ and $\BG'$ are unramified over $k_v$, and $\FL_v$ and $\FL'_v$ are self-dual with respect to $\psi_v$ and $\pair{\ ,\ }$. \end{enumerate} We denote by $S'$ the subset $S-V_\infty-\{w\}$ of $S$.
\s{$\bullet$ \emph{Orbits}}. For each $v\in V_{\mathrm{f}}$, we choose an open compact subset $\Omega_v$ of $\fs(k_v)$ such that: \begin{enumerate} \item if $v=w$, we require that: $X_0\in\Omega_w$ and $\Omega_w\subset\fs_{\mathrm{rs}}(k_w)$, $\widehat{I}(\cdot,f)$ is constant on $\Omega_w$, and $\widehat{I}(\cdot,f')$ is constant and hence equal to $\widehat{I}(Y_0,f')$ on the set of $Y\in\fs'_{\mathrm{rs}}(k_w)$ which matches an element $X$ in $\Omega_w$; \item if $v=u$, we require $\Omega_u\subset\fs_\el(k_u)$; \item if $v\in S$ but $v\neq w,u$, choose $\Omega_v$ to be any open compact subset of $\fs(k_v)$; \item if $v\in V_{\mathrm{f}}-S$, let $\Omega_v=\FL_v$. \end{enumerate} Recall that a semisimple regular element $X\in\fs(k)$ is called elliptic if its centralizer $\BH_X$ is an elliptic torus. Denote by $\fs_\el(k)$ (resp. $\fs'_\el(k)$) the set of elliptic regular semisimple elements in $\fs(k)$ (resp. $\fs'(k)$). Then by the strong approximation theorem, there exists $X^0\in\fs(k)\subset\fs(\BA)$ such that for each $v\in V_{\mathrm{f}}$ we have $X^0\in\Omega_v$. Furthermore, by the condition (2) on the $\Omega_v$'s, $X^0\in\fs_\el(k)$. Take an element $Y^0\in\fs'_\el(k)$ such that $X^0\leftrightarrow Y^0$.
\s{$\bullet$ \emph{Functions}}. For each $v\in V$, we choose Bruhat-Schwartz functions $\phi_v\in\CS(\fs(k_v))$ and $\phi'_v\in\CS(\fs'(k_v))$ in the following way: \begin{enumerate} \item if $v=w$, let $\phi_v=f$ and $\phi'_v=f'$; \item if $v\in S'$, by Proposition \ref{prop. local prop}, we can require that: \begin{itemize} \item if $X_v\in\Supp(\phi_v)$, there exists $X'_v\in\fc_{X^0}(k_v)$ such that $X_v$ and $X'_v$ are $\BH(k_v)$-conjugate, where $\fc_{X^0}$ is the Cartan subspace of $\fs$ containing $X^0$; \item if $Y_v\in\Supp(\phi'_v)$, there exists $X_v\in\fc_{X^0}(k_v)$ such that $X_v\leftrightarrow Y_v$; \item $\phi'_v$ is a transfer of $\phi_v$; \item $\widehat{I}(X^0,\phi_v)=c_v\widehat{I}(Y^0,\phi'_v)$, where $c_v=\gamma_\psi(\fh(k_v))\gamma_\psi(\fh'(k_v))^{-1}$; \end{itemize} \item for $v\in V-S$, since we required $\BG$ to be unramified over $k_v$, that is to say, $\BD$ to be split over $k_v$, we can make suitable identifications $\BG(k_v)=\BG'(k_v)$, $\FL_v=\FL'_v$, and set $\phi_v=\phi_v'={\bf1}_{\FL_v}$; moreover, since $\FL_v$ is self-dual with respect to $\psi_v$ and $\pair{\ ,\ }$, $\phi_v=\widehat{\phi}_v$; \item for $v_0\in V_\infty$, identifying $(\BH(k_{v_0}),\fs(k_{v_0}))$ with $(\BH'(k_{v_0}),\fs'(k_{v_0}))$, we can choose $\phi_{v_0}=\phi'_{v_0}\in\CS(\fs(k_{v_0}))$ such that: \begin{itemize} \item $\widehat{I}(X^0,\phi_{v_0})=\widehat{I}(Y^0,\phi'_{v_0})\neq0$; \item if $X\in\fs(k)$ is $\BH(k_v)$-conjugate to an element in the support of $\widehat{\phi_v}$ at each place $v\in V$, then $X$ is $\BH(k)$-conjugate to $X^0$; \item if $Y\in\fs'(k)$ is $\BH'(k_v)$-conjugate to an element in the support of $\widehat{\phi'_v}$ at each place $v\in V$, then $Y$ is $\BH'(k)$-conjugate to $Y^0$. \end{itemize} \end{enumerate} The condition 4 can be satisfied, and was discussed in \cite[Lemma in \S10.7]{wa97} in the endoscopic case. The key point is that we have a morphism $\fs/\BH\ra\FA_k^\ell$ where $\FA_k^\ell=\Spec\left(\CO(\fs)^\BH\right)$ is an affine space. Then the discussion is the same as in \cite[Lemma in \S10.7]{wa97}.
Now we set $\phi\in\CS(\fs(\BA))$ and $\phi'\in\CS(\fs'(\BA))$ to be: $$\phi=\prod_{v\in V}\phi_v,\quad \phi'=\prod_{v\in V}\phi'_v.$$
\s{$\bullet$ \emph{The end of the proof}}. As shown in \cite[Theorem 8.2]{zh} the following integrals $I(\phi)$ and $I(\phi')$ are absolutely convergent: $$I(\phi)=\int_{\BH(k)\bs\BH(\BA)^1}\sum_{X\in\fs_\el(k)}\phi(X^h)\ {\mathrm{d}} h,\quad I(\phi')=\int_{\BH'(k)\bs\BH'(\BA)^1}\sum_{Y\in\fs'_\el(k)}\phi'(Y^h)\ {\mathrm{d}} h,$$ where
$$\BH(\BA)^1=\bigcap_{\chi\in\Hom_k(\BH,\BG_m)}\ker|\chi|,
\quad \BH'(\BA)^1=\bigcap_{\chi\in\Hom_k(\BH',\BG_m)}\ker|\chi|.$$ Here $\abs{\chi}$ is the function on $\BH(\BA)$ or $\BH'(\BA)$ defined in the usual way. Actually \cite[Theorem 8.2]{zh} only treats the case of $(\BG',\BH')$, but the arguments also work for $(\BG,\BH)$. It is obvious that $$I(\phi)=\sum_{X\in[\fs_\el(k)]}\tau(X)\prod_v I(X,\phi_v),$$ and $$I(\phi')=\sum_{Y\in[\fs'_\el(k)]}\tau(Y)\prod_v I(Y,\phi'_v),$$ where $[\fs_\el(k)]$ denotes the set of $\BH(k)$-orbits in $\fs_\el(k)$, $$\tau(X)=\vol(\BH_X(k)\bs (\BH_X(\BA)\cap\BH(\BA)^1)),$$ and the definitions of $[\fs'_\el(k)]$ and $\tau(Y)$ are similar. If $X\in\fs_\el(k)$ and $Y\in\fs'_\el(k)$ are such that $X\leftrightarrow Y$, then $\BH_X\simeq\BH'_Y$ (the justification is the same as in the local field case). We choose Haar measures on $\BH_X(\BA)$ and $\BH'_Y(\BA)$ so that they are compatible. In particular, $$\tau(X)=\tau(Y).$$
According to the conditions on $\phi_u$ (resp. $\phi'_u$), we know that if $X\in\fs(k)$ (resp. $Y\in\fs'(k)$) is such that $X\in\Supp(\phi)^{\BH(\BA)}$ (resp. $Y\in\Supp(\phi')^{\BH'(\BA)}$), then $X\in\fs_\el(k)$ (resp. $Y\in\fs'_\el(k)$). Here we use $\Supp(\phi)^{\BH(\BA)}$ to denote the union of all $\BH(\BA)$-orbits intersecting $\Supp(\phi)$. We have a similar defined set $\Supp(\phi')^{\BH'(\BA)}$. By the conditions on $\phi'_v$ at each place $v$, we know that, if $Y\in\fs'_\el(k)$ is such that $I(Y,\phi'_v)\neq0$ for each $v\in V$, then there exists $X_v\in\fs_{\mathrm{rs}}(k_v)$ such that $X_v\leftrightarrow Y$ at each place $v\in V$ and hence there exists $X\in\fs_\el(k)$ such that $X\leftrightarrow Y$. Therefore we have $$I(\phi)=I(\phi'),$$ since $\phi_v$ is a transfer of $\phi'_v$ at each place $v\in V$ by the requirements we have imposed.
On the other hand, according to the conditions on $\widehat{\phi_v}$ and $\widehat{\phi'_v}$, we know that if $X\in\fs(k)$ (resp. $Y\in\fs'(k)$) is such that $X\in\Supp(\widehat{\phi})^{\BH(\BA)}$ (resp. $Y\in\Supp(\widehat{\phi'})^{\BH'(\BA)}$), $X$ is $\BH(k)$-conjugate to $X^0$ (resp. $Y$ is $\BH'(k)$-conjugate to $Y^0$).
By the Poisson summation formula, we have $$\sum_{X\in\fs(k)}\phi(X^h)=\sum_{X\in\fs(k)}\widehat{\phi}(X^h),\quad \forall\ h\in\BH(\BA),$$ and $$\sum_{Y\in\fs'(k)}\phi'(Y^h)=\sum_{Y\in\fs'(k)}\widehat{\phi'}(Y^h),\quad \forall\ h\in\BH'(\BA).$$ Actually, by the conditions on $\phi$ and $\phi'$, we can replace $\fs(k)$ (resp. $\fs'(k)$) by $\fs_\el(k)$ (resp. $\fs'_\el(k)$) on both sides of the above two equations. Thus, we have $$I(\phi)=I(\widehat{\phi}),\quad I(\phi')=I(\widehat{\phi'}).$$ Hence we have $$I(\widehat{\phi})=I(\widehat{\phi'}),$$ which amounts to saying, $$\tau(X^0)\prod_{v\in V}\widehat{I}(X^0,\phi_v) =\tau(Y^0)\prod_{v\in V}\widehat{I}(Y^0,\phi'_v).$$ For $v\in V-S$ or $v\in V_\infty$, we have $$\widehat{I}(X^0,\phi_v)=\widehat{I}(Y^0,\phi'_v)\neq0.$$ For $v\in S'$ we have $$\widehat{I}(X^0,\phi_v)=c_v\widehat{I}(Y^0,\phi'_v)\neq0.$$ Therefore $$c\widehat{I}(X^0,f)=\widehat{I}(Y^0,f'),$$ where $$c=\prod_{v\in S'}c_v=\prod_{v\in S'}\gamma_\psi(\fh(k_v)) \gamma_\psi(\fh'(k_v))^{-1}.$$ Notice that if $v\in V_\infty$ or $v\in V-S$, $$\gamma_\psi(\fh(k_v))=\gamma_\psi(\fh'(k_v))=1.$$ Also notice that $$\prod_{v\in V}\gamma_\psi(\fh(k_v))=\prod_{v\in V}\gamma_\psi(\fh'(k_v))=1.$$ Therefore $$c=\gamma_\psi(\fh(k_w))^{-1}\gamma_\psi(\fh'(k_w)).$$ Since $$\widehat{I}(X_0,f)=\widehat{I}(X^0,f), \quad \widehat{I}(Y_0,f')=\widehat{I}(Y^0,f'),$$ we complete the proof of the theorem.
\paragraph{Acknowledgements} This work was supported by the National Key Basic Research Program of China (No. 2013CB834202). The author would like to thank Dipendra Prasad for his valuable comments, and Wen-Wei Li for his long list of useful comments and suggestions. He also thanks Dihua Jiang and Binyong Sun for helpful discussions. He expresses gratitude to Ye Tian and Linsheng Yin for their constant encouragement and support. The anonymous referee pointed out a gap and numerous mathematical and grammatical inaccuracies, made a lot of useful comments, and helped the author to greatly improve the exposition. The author is grateful to him or her.
\s{\small Chong Zhang\\ School of Mathematical Sciences, Beijing Normal University,\\ Beijing 100875, P. R. China.\\ E-mail address: \texttt{[email protected]}}
\end{document} | arXiv |
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\begin{document}
\title{Entanglement and Symmetry: A Case Study in \\ Superselection Rules, Reference Frames, and Beyond}
\author{S. J. Jones}
\affiliation{Centre for Quantum Computer Technology, Centre for
Quantum Dynamics, School of Science, Griffith University, Brisbane,
4111 Australia} \author{H. M. Wiseman} \email{[email protected]} \affiliation{Centre for Quantum Computer Technology, Centre for
Quantum Dynamics, School of Science, Griffith University, Brisbane,
4111 Australia} \author{S. D. Bartlett} \affiliation{School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia}
\author{J. A. Vaccaro}
\affiliation{Centre for Quantum Computer Technology, Centre for
Quantum Dynamics, School of Science, Griffith University, Brisbane,
4111 Australia}
\author{D. T. Pope}
\affiliation{Centre for Quantum Computer Technology, Centre for
Quantum Dynamics, School of Science, Griffith University, Brisbane,
4111 Australia}
\date{October 20, 2006}
\begin{abstract} In recent years it has become apparent that constraints on possible quantum operations, such as those constraints imposed by superselection rules (SSRs), have a profound effect on quantum information theoretic concepts like bipartite entanglement. This paper concentrates on a particular example: the constraint that applies when the parties (Alice and Bob) cannot distinguish among certain quantum objects they have. This arises naturally in the context of ensemble quantum information processing such as in liquid NMR. We discuss how a SSR for the symmetric group can be applied, and show how the extractable entanglement can be calculated analytically in certain cases, with a maximum bipartite entanglement in an ensemble of $N$ Bell-state pairs scaling as $\log(N)$ as $N \to \infty$. We discuss the apparent disparity with the asymptotic ($N \to \infty)$ recovery of unconstrained entanglement for other sorts of superselection rules, and show that the disparity disappears when the correct notion of applying the symmetric group SSR to multiple copies is used. Next we discuss reference frames in the context of this SSR, showing the relation to the work of von Korff and Kempe [Phys. Rev. Lett. {\bf 93}, 260502 (2004)]. The action of a reference frame can be regarded as the analog of activation in mixed-state entanglement. We also discuss the analog of distillation: there exist states such that one copy can act as an imperfect reference frame for another copy. Finally we present an example of a stronger operational constraint, that operations must be non-collective as well as symmetric. Even under this stronger constraint we nevertheless show that Bell-nonlocality (and hence entanglement) can be demonstrated for an ensemble of $N$ Bell-state pairs no matter how large $N$ is. This last work is a generalization of that of Mermin [Phys. Rev. D {\bf 22}, 356 (1980)].
\end{abstract}
\pacs{03.67.-a, 03.67.Mn, 03.65.Ud, 03.65.Ta}
\maketitle
\section{Introduction}\label{Intro}
The entanglement of disjoint (typically spatially separate) quantum systems is at the heart of quantum information processing ~\cite{NieChu00}. For bipartite pure states under LOCC (local operations and classical communication) the quantification and transformation of entanglement is now well understood. However, it is also now well understood that the non-ideal situation of mixed states, which pertains in practice, is far more complicated (or richer, to put a different spin on it) \cite{Hor01}. In recent years it has also become apparent that a situation,
in which only certain operations can be performed, also leads to an interesting theory of entanglement, even if the states are pure. One approach, leading to a generalized notion of entanglement, dismisses altogether with the bipartite setting \cite{Barnum03,Barnum04}. A less radical, and more obviously applicable, idea is to restrict the local operations to those that are invariant under a superselection rule (SSR) \cite{Ver03,WisVac03,BarWis03,KMP04,SchVerCir04,BarDohSpeWis06,VacAnsWisJac06}. At the same time, the nature of quantum reference frames in the bipartite setting has also been hotly debated (see for example Refs.~\cite{RudSan01,EnkFuc02a,SanBarRudKni03,Wis04}).
Much of the work in this area \cite{Ver03,WisVac03,SchVerCir04,BarDohSpeWis06,RudSan01, EnkFuc02a,SanBarRudKni03,Wis04} has concentrated upon the case of a U(1)-SSR. This is the SSR that can be motivated by considering the conservation of a locally additive scalar quantity with a discrete spectrum \cite{Wis04}. It can also be applied to quantum optics experiments which lack an optical phase reference (that is, which lack a shared clock of sufficient precision) \cite{SanBarRudKni03}. Many simplifications arise from this SSR because U(1) is Abelian (there is only one generator, corresponding to the local operator of the conserved quantity). Non-Abelian Lie-group SSRs (with non-commuting generators) have also been considered \cite{KMP04} but relatively little attention has been paid to SSRs arising from discrete groups. An example with obvious application to ensemble quantum information processing is the symmetric group $S_N$ (the group of permutations of $N$ objects) \cite{BarWis03}.
This paper explores issues in entanglement under operations constrained by symmetry. We use the $S_N$-SSR formalism of Ref.~\cite{BarWis03}, but also go beyond that work. This work is important for a number of reasons. First, as noted above, the symmetric group has been relatively neglected in studies of entanglement constrained by a SSR. For the U(1)-SSR concepts like bound entanglement (of two distinct types), activation, and distillation have been shown to apply, in analogy to these concepts in mixed-state entanglement. Although not immediately obvious, we construct specific examples to show how these concepts apply to the $S_N$-SSR. Second, we clarify the notion of a reference frame for the $S_N$ group, linking in with the work of von Korff and Kempe \cite{vKK04}. Finally, we give an example where it is not obvious that the symmetry constraints on the system can be formulated as a SSR.
Nevertheless we show that, even under such constraints, it is possible to exhibit Bell-nonlocality \cite{Bel64} for an ensemble of identically prepared singlets.
\section{Entanglement and SSRs}
\subsection{Concepts of Entanglement} \label{sec:concepts}
The term entanglement was coined by Schr\"odinger\ \cite{Sch35} as the property that bipartite pure states have when they are not product states. Schr\"odinger\ showed that for such an entangled state, one party (say Bob) could, via a measurement on his system, collapse Alice's system with some probability to {\em any} state vector (except those in the null space of Alice's reduced state matrix). Schr\"odinger\ thought this nonlocality was unreasonable enough to be called a ``paradox'' \cite{Sch36}. A generation later, Bell \cite{Bel64} discovered that such states had an even stronger form of nonlocality: for certain measurement schemes, the correlations between the results of Alice and Bob cannot be explained by any locally causal theory. This property, which we will call Bell-nonlocality, we regard as the strongest operational notion of entanglement.
\subsubsection{Separability and Local Preparability} When correlations in mixed states were first studied in earnest \cite{Wer89}, it became clear that the question as to whether a state was entangled was no longer straightforward. In particular, Werner showed that there were nonseparable states such that the measurement correlations of Alice and Bob could nevertheless be explained by a local theory involving hidden variables. Nonseparable states are states that cannot be written in the form \begin{eqnarray} \rho &=&
\sum_k \wp_k |\psi_k\rangle\langle \psi_k| \otimes |\phi_k\rangle\langle\phi_k|\nonumber\\ &\equiv& \biguplus_k \sqrt{\wp_k} \ro{ \ket{\psi_k}\otimes \ket{\phi_k} } \nonumber\\ &\equiv& \uplus \sqrt{\wp_1} \ro{ \ket{\psi_1}\otimes \ket{\phi_1} } \uplus \sqrt{\wp_2} \ro{ \ket{\psi_2}\otimes \ket{\phi_2} } \uplus \cdots \nl{} \end{eqnarray} Here, following Ref.~\cite{VacAnsWisJac06}, we have defined a notation that we will use throughout this paper, that for an arbitrary {\em ray} $\ket{r}$, we have $\uplus \ket{r} \equiv + \ket{r}\bra{r}$. Werner called nonseparable states ``EPR correlated states''. They are sometimes identified with ``entangled'' states but we will call them non-locally-preparable states. This name captures the physical significance of such states: they cannot be prepared by LOCC from a product state.
\subsubsection{$n$-Distillability and Bound Entanglement}\label{Distillable} Since Werner, the richness of the entanglement of mixed states has been further developed, involving concepts such as bound entanglement, distillation, $n$-distillability, and activation \cite{Hor01}. Here, following Ref.~\cite{BarDohSpeWis06}, we concentrate upon those properties of mixed state entanglement for which there are obvious analogs in pure state entanglement constrained by SSRs. First, as noted above, it is useful to define the class of \emph{locally preparable} states, which are those states that are preparable from a product state using LOCC. Another useful class is the class of states that are \emph{distillable}~\cite{Ben96}. States in the distillable class are such that $n$ copies can be converted into $nr$ pure maximally entangled states via LOCC for some $r>0$ in the limit $n\to\infty$. A \emph{pure} state is either locally preparable or distillable, depending on whether it is a product state or not. On the other hand, there are mixed states that are neither locally preparable nor distillable. These are the so-called \emph{bound entangled} states~\cite{Hor98}.
For mixed states, deciding whether a state is locally preparable is known to be an NP-hard problem computationally \cite{Gur02}, but algorithms to do so exist \cite{DohParSpe02}. It is not known if it is even possible to determine whether a state is distillable. For this reason, a related, but simpler to characterize, class has been defined: the states that are \emph{1-distillable}~\cite{Div00,Dur00}. A state $\rho$ is 1-distillable if by LOCC Alice and Bob can, with some probability, create from it a non-separable two-qubit state. (Note that for two qubits, there are no bound entangled states~\cite{Hor97}.) By extension, a state $\rho$ is \emph{$n$-distillable} if $\rho^{\otimes n}$ is 1-distillable. (If a state is $n$-distillable for some $n$ then it is distillable.) Thus, the set of distillable states includes the 1-distillable states,
and in fact it has recently been shown that the $n$-distillable states are a subset of the distillable states $\forall\ n$~\cite{Wat04}. Since the 1-distillable states are a subset of distillable states, there are clearly mixed states that are neither locally preparable nor 1-distillable. We shall refer to these states as being \emph{1-bound}.
Note that although a nonseparable two-qubit state is always distillable, this does not mean that undistilled copies can be used to demonstrate Bell-nonlocality, as Werner showed \cite{Wer89}. However, in our work pertaining to SSRs, when we demonstrate that a state is 1-distillable, we do this by showing that it is possible for Alice and Bob by LOCC to create with some probability a {\em pure} entangled state, as this is strictly stronger than the requirements for being 1-distillable. Thus, for these purposes, a state that is 1-distillable allows Alice and Bob to demonstrate Bell-nonlocality, which, as noted above, we regard as the strongest notion of entanglement.
\subsubsection{Closing the gap: PPT-Channels}\label{Become}
Returning to the 1-bound states in general, this class can be divided into two by considering what would happen if we were to give Alice and Bob a PPT-channel. That is, a channel that can distribute only bipartite states for which the partial transpose is positive.
This allows Alice and Bob to perform PPT-operations as well as LOCC. A PPT operation is one that preserves the positivity of the partial (with respect to Alice or Bob) transpose of states~\cite{Rai01}. With this addition, Alice and Bob can locally prepare all states with a positive partial transpose, which includes some states which are 1-bound \cite{Hor98}. We will call these the bound states that \emph{become locally preparable}. Conversely, the rest of the 1-bound states, those that are not PPT, become 1-distillable under LOCC plus all PPT operations \cite{Egg01}. Hence we call this class (which is also non-empty \cite{Wat04}) \emph{become 1-distillable}.
\subsubsection{Activation}
Physically, a PPT-channel is equivalent to
supplying Alice and Bob with an infinite number of copies of every state in the become locally preparable class. Access to these states automatically makes them locally preparable. However, it is not necessary to use all of the states to make 1-distillable a state in become 1-distillable. Rather, for every $\rho$ in become 1-distillable there exists a state $\sigma$ in become locally preparable such that $\rho \otimes \sigma$ is 1-distillable. This is known as {\em activation}~\cite{Hor99}. Note the distinction from {\em distillation}, in which for some $\rho$ which is in become 1-distillable, there exists an $n$ such that $\rho^{\otimes n}$ is 1-distillable \cite{Wat04}. Note that it is trivially the case that any state $\rho$ which can become locally preparable does so given a suitable state $\sigma$ which can become locally preparable: one simply chooses $\sigma = \rho$.
\subsubsection{Measures of Entanglement}
Finally for this section, we define some {\em measures} of entanglement. The entanglement of formation, $E_F$, of a mixed state $\rho$ is the minimum ratio, in the asymptotic limit, of the number of singlets used to the number of copies of $\rho$ created thereby, using LOCC \cite{HillWoot97}. Similarly, the distillable entanglement, $E_D$, is the asymptotic yield of arbitrarily pure singlets that can be prepared by LOCC from copies of $\rho$ \cite{BennetDiVSmolWoot96}. By definition, both of these measures are {\em partially additive}. That is, $n$ copies of a state $\rho$ contains $n$ times the entanglement of a single copy; $E(\rho^{\otimes n})=nE(\rho)$. Also by definition \cite{Hor99}, and the fact that LOCC cannot increase entanglement, $E_F$ is an upper bound on $E_D$. In general it is a strict upper bound, which is obvious from the existence of bound entangled states where $E_F \neq 0$ when $E_D = 0$. However for pure states $E_F = E_D$. Since we will be concerned with states that can be made pure by LOCC, there is no need to distinguish between $E_F$ and $E_D$. For a bipartite pure state $\ket{\Psi}$ the entanglement (measured in e-bits \cite{BennetDiVSmolWoot96}) is defined as the von Neumann entropy of either subsystem's reduced density matrix, \begin{equation} E(\ket{\Psi})=-\tr{\rho_A\log_2\rho_A}=-\tr{\rho_B\log_2\rho_B}, \end{equation} where the reduced density matrices for Alice and Bob are defined as $\rho_A={\rm Tr_{B}}[\ket{\Psi}\bra{\Psi}]$ and $\rho_B={\rm Tr_{A}}[\ket{\Psi}\bra{\Psi}]$ respectively. ${\rm Tr_{A,B}}$ signifies the partial trace operation with respect to Alice or Bob.
\subsection{Superselection Rules}\label{sec:SSRs}
\subsubsection{SSRs as an Operational Restriction}
Originally \cite{Wic52}, SSRs were regarded as restrictions on the states that a system can be in. This could be restated operationally, as a restriction on the means of preparing a system. Since any operation could be part of a system-preparation procedure, it is only sensible to say that a SSR is a restriction on the operations that can be performed on a system. For an SSR for charge (the first such SSR ever proposed) \cite{Wic52}, this restriction would amount to saying that it is not possible to create superpositions of different charge eigenstates. Alternatively, all operations on the system must commute with charge-preserving operations such as measurement of charge. Charge-preserving operations can be built up from transformations in the Lie group U(1) generated by the charge operator. This formulation allows the concept of SSRs to be generalized to arbitrary compact Lie groups, or finite groups \cite{BarWis03,KMP04}, as we now explain.
The SSR for a group $G$ of physical transformations can be defined operationally as follows. Consider for the moment a single party, Alice, who possesses a quantum system, described by a Hilbert space $\mathbb{H}_A$. Let the physical transformation corresponding to an element $g$ of $G$ be denoted $\hat T_A(g)$. Then the $G$-SSR is the rule that all operations must be $G$-invariant. That is, if ${\cal O}$ is the completely positive map $\rho \to {\cal O}{\rho}$ representing the operation, then \begin{equation} \forall \rho \textrm{ and } \forall g \in G\,, \;{\cal O}[\hat T_A(g) \rho \hat T^\dagger_A(g)] = \hat T_A(g)[{\cal O}\rho]\hat T^\dagger_A(g). \end{equation} Note that ``operations" includes unitaries, where ${\cal O}\rho = \hat{U}\rho\hat{U}^\dagger$, and also measurements, where for example ${\cal O}_r\rho = \hat{M}_r\rho\hat{M}_r^\dagger$ and $\sum_r \hat{M}_r^\dagger \hat{M}_r = \hat{1}$.
According to this definition, we would say that a SSR for charge $\hat{Q}_A$, for example, would be a SSR for the group U(1) generated by $\hat{Q}_A$. Such a SSR can be motivated from consideration of a conservation law for global charge $\hat{Q}_A$. Note however that we do not assume that the operational restriction described by a general SSR must be derivable from a conservation law. For the purposes of this paper, it is more fruitful to regard a $G$-SSR as being due to the lack (by Alice) of an appropriate {\em reference frame} \cite{AhaSus67,BarRudSpe05,BarDohSpeWis06}.
This idea will be explored later in the particular context of the $S_N$-SSR.
\subsubsection{SSRs and Mixing} \label{SSRmixing}
All quantum information processing ultimately ends in measurement. If a $G$-SSR is in force over the entire process, then no outcomes will be changed if the state matrix for the quantum system $\rho$ is replaced by the state matrix $\hat T_A(g)\rho \hat T^\dagger_A(g)$ for any $g \in G$. That is, under the $G$-SSR the state of the quantum system is represented by an equivalence class of state matrices. The
{\em most mixed} state matrix to which $\rho$ is physically equivalent is \begin{equation} \mathcal{G}_A\rho \equiv |G|^{-1} \sum_{g \in G}
\hat T_A(g) \rho \hat T^\dagger_A(g) \end{equation} for finite groups, where $|G|$ is the group order, and \begin{equation} \mathcal{G}_A\rho \equiv \int_{{}_{G}} \text{d}\mu(g)\, \hat T_A(g) \rho \hat T^\dagger_A(g) \end{equation} for compact Lie groups, where $\text{d}\mu(g)$ is the Haar measure. We call this the $G$-invariant state, as \begin{equation} \label{AlocalGinv} \forall g \in G, \; \hat T_A(g) [\mathcal{G}_A\rho] \hat T^\dagger_A(g) = \mathcal{G}_A\rho. \end{equation} For traditional SSRs, i.e. groups with a single generator $\hat{Q}_A=\sum_q q\, \hat\Pi_q $, the $G$-invariant state is simply the block-diagonal state ${\cal G}_A \rho = \sum_q \, \hat\Pi_q \, \rho \, \hat\Pi_q.$
This maximum-entropy member of the equivalence class is the one containing no irrelevant information, and hence it is the natural representation of the state of the system as a state matrix. This state can also be given an operational interpretation \cite{VacAnsWisJac06}. Given the $D$-dimensional quantum system with state $\rho$ and a heat bath at temperature $T$, work can be extracted by allowing the system to come to thermal equilibrium. The maximum amount of extractable work is $k_{\rm B}T[\log D - S(\rho)]$, where $S$ is the von Neumann entropy \cite{OppHor02}. Under the constraint of a $G$-SSR, the amount of extractable work is reduced by (the positive quantity) $k_{\rm B}T \Delta_G(\rho)$, where $\Delta_G(\rho) = S({\cal G}_A\rho)-S(\rho)$ is precisely the amount of ``irrelevant information'' in $\rho$.
It is very important to note that if Alice has two systems with states $\rho_1$ and $\rho_2$, such that ${\cal G}_A(\rho_1\otimes
\rho_2)$ equals \begin{equation} |G|^{-1}
\sum_{g \in G}
[\hat T_1(g)\otimes \hat T_2(g) ] \rho [\hat T_1^\dagger(g)\otimes \hat T_2^\dagger(g)
], \end{equation} then this state is not the same as ${\cal G}_A\rho_1 \otimes
{\cal G}_A \rho_2$, which equals \begin{equation} |G|^{-2} \sum_{g,g' \in G}
[\hat T_1(g)\otimes \hat T_2(g') ] \rho [\hat T_1^\dagger(g)\otimes \hat T_2^\dagger(g') ] . \end{equation} That is why in the above we have referred to {\em the} quantum system, not {\em a} quantum system. If we are considering the whole quantum system (or at least all parts to which the SSR applies), then the state $\rho$ of the system can be replaced by ${\cal G}_A\rho$. But if there are other quantum systems that may enter into the quantum information processing at a later time, then it is {\em not} true in general that ${\cal G}_A\rho$ contains all of the relevant information about that system.
\subsubsection{Bipartite SSRs} In this paper we are concerned with the impact of SSRs on entanglement, rather than extractable work (although the latter is, in the bipartite setting, also related to entanglement \cite{VacAnsWisJac06}). In this context we have to define the concept of {\em local} SSRs. That is, the local operations of Alice and Bob (say) must respect local SSRs, rather than a global SSR. This is obviously applicable in the case when a SSR is motivated by a conservation law for a locally additive quantity. It is also applicable more generally if Alice and Bob each lack a reference frame. It turns out that for the purpose of non-local quantum information processing, what is important is that Alice and Bob have a shared reference frame. Furthermore, such a reference frame need
only be correlated between the two parties. This point will be clarified by later examples.
For the concept of a local SSR or local reference frame to make sense, the physical transformation on the joint Hilbert space $\mathbb{H}_A\otimes\mathbb{H}_B$ corresponding to an element $g$ of the group $G$ must have the following form: \begin{equation} \hat T(g) = \hat T_A(g)\otimes \hat T_B(g). \end{equation} Now if Alice and Bob lack reference frames, then the effective state for the bipartite system is the locally $G$-invariant state \cite{BarWis03} \begin{equation}
({\cal G}_A \otimes {\cal G}_B) \rho,
\end{equation} where ${\cal G}_A$ is defined as above, and ${\cal G}_B$ similarly, and these act locally according to the tensor-product structure of the joint system. Note that this state is in general very different from the globally $G$-invariant state
\begin{equation}
{\cal G}\rho = \sum |G|^{-1}
\sum_{g \in G}
[\hat T_A(g) \otimes \hat T_B(g)] \rho [\hat T_A^\dagger (g) \otimes \hat T_B^\dagger
(g)].
\end{equation} Just as in the case of a single party, it is important to remember that $\rho$ can be replaced by $({\cal G}_A \otimes {\cal G}_B) \rho$ only if it is the state of the entire quantum system shared by Alice and Bob (or at least all parts to which the SSR applies).
\subsubsection{SSRs and Hilbert Space (Technicalities)}\label{Technicalities}
To determine the effect of SSRs on entanglement it is necessary to understand how a SSR induces a structure on Hilbert space. A local $G$-SSR for Alice splits $\mathbb{H}^A$ into ``charge sectors'' labeled by $y$: \begin{equation}
\mathbb{H}^A = \bigoplus_y \mathbb{H}^A_y , \end{equation} where each $\mathbb{H}^A_y$ carries inequivalent representations $\hat T^A_y$ of $G$. The sectors are then further decomposed into tensor products: \begin{equation}
\mathbb{H}^A_y = \mathbb{M}^A_{y} \otimes \mathbb{Q}^A_y . \end{equation} This is technically known as dividing the system into subsystems. The subsystem $\mathbb{M}^A_y$ carries an irreducible representation (irrep) $\hat t^A_y(g)$ and the subsystem $\mathbb{Q}^A_y$ carries a trivial representation of $G$. That is to say, \begin{equation} \hat T^A_y(g) = \hat t^A_y(g) \otimes \hat{I}^A_y. \end{equation} For an Abelian SSR such as charge, the subsystems $\mathbb{M}^A_y$ are one-dimensional, and so the additional tensor product structure within the irreps is not required. However, for a non-Abelian SSR such as we will consider later, they are nontrivial.
The subsystems $\mathbb{Q}^A_y$ are clearly $G$-invariant. They have been called noiseless subsystems, or decoherence-free subsystems, relative to the decoherence map $\mathcal{G}_A$~\cite{Kni00}. By contrast, the subsystems $\mathbb{M}^A_y$ become completely mixed under the action of $\mathcal{G}_A$, because $\hat t^A_y(g)$ is irreducible. Thus the action of $\mathcal{G}_A$ on an arbitrary state matrix $\rho$ is, in terms of this decomposition, \begin{equation}
\mathcal{G}_A\rho = \sum_y \mathcal{D}^A_{y} \otimes
\mathcal{I}^A_{y}(\hat\Pi^A_y \rho \hat\Pi^A_y ). \end{equation} Here $\hat{\Pi}^A_y$ is the projection onto the charge sector $y$, $\mathcal{D}^A_{y}$ is the trace-preserving map that takes every operator for the subsystem $\mathbb{M}^A_y$ to a maximally mixed operator (i.e. proportional to the identity operator on that space), and $\mathcal{I}^A_{y}$ is the identity map over operators for the subsystem $\mathbb{Q}^A_y$. The effect of the local superselection rule, then, is to remove the coherence between different local charge sectors (as in the Abelian case) {\em and} to make the subsystems $\mathbb{M}^A_y$ completely mixed. The same structure arises for $\mathbb{H}^B$ and provides an analogous decomposition of $\mathcal{G}_B$. For further details, see~\cite{BarWis03,KMP04}.
\subsection{Concepts of Entanglement Constrained by SSRs}\label{sec:EntSSR} In this section we summarize the results of Ref.~\cite{BarDohSpeWis06}, showing the analogies between mixed-state entanglement and pure-state entanglement constrained by a SSR. The various concepts of entanglement explored in Sec.~\ref{sec:concepts} arise from considering two parties able to perform LOCC. Adding the constraint of a local $G$-SSR (that is, that the local operations must be $G$-invariant) we say that the two parties can perform $G$-LOCC.
\subsubsection{Local Preparability}
The class of pure bipartite states that are locally preparable under $G$-LOCC will call \emph{${G\text{-SSR}}$ locally preparable}. Just as preparable under LOCC means preparable from states that are local (separable), so preparable under $G$-LOCC means preparable from states that respect the $G$-SSR (i.e. that are locally $G$-invariant). It is trivial to see that a pure bipartite state $\ket{\psi}$ is ${G\text{-SSR}}$ locally preparable iff (i) the state is a product state, and (ii) it is locally $G$-invariant. Note that not all pure product states are ${G\text{-SSR}}$ locally preparable; it is a {\em smaller} class than the locally preparable states.
\subsubsection{$n$-Distillability and Bound Entanglement}
The class of pure states that are 1-distillable under $G$-LOCC, which we call \emph{${G\text{-SSR}}$ 1-distillable},
is defined as those states $\ket{\psi}$ for which the following is true: The two parties can, by local measurements, project $\ket{\psi}$ onto a $2{\times}2$-dimensional subspace with nonzero probability, such that the projected state is (i) locally $G$-invariant and (ii) non-separable. The significance of the first condition is that the SSR is now irrelevant, so that the usual condition (nonseparability)
is all that is required for 1-distillability. It is not difficult to see \cite{BarWis03} that $|\psi\rangle$ is ${G\text{-SSR}}$ 1-distillable iff $\mathcal{G}_A \otimes \mathcal{G}_B[|\psi\rangle\langle\psi |]$ is 1-distillable under LOCC.
Both the class of ${G\text{-SSR}}$ locally preparable and ${G\text{-SSR}}$ 1-distillable states are non-empty in general (i.e. for a general SSR). Moreover, as with mixed-state entanglement, there is a proper gap between these two classes. The class of states in the gap contains both product and non-product pure states, and is analogous to the class of 1-bound states in mixed-state entanglement.
The concepts of $n$-distillability with the SSR constraint (and the corresponding classes of pure states, \emph{${G\text{-SSR}}\ n$-distillable}) can be defined analogously to the case of unconstrained entanglement. It is not difficult to illustrate the phenomenon of distillation; that is, to find examples of states that are ${G\text{-SSR}}$ distillable but not
${G\text{-SSR}}$ 1-distillable \cite{KMP04}. Here ${G\text{-SSR}}\ {\rm distillable}= {G\text{-SSR}}\ \infty$-distillable is the class of distillable pure states under this constraint.
\subsubsection{Closing the gap}
Just as in mixed-state entanglement adding a PPT channel removes the 1-bound class, so it is possible to augment $G$-LOCC in such a way that any pure state in the gap between ${G\text{-SSR}}$ locally preparable and ${G\text{-SSR}}$ 1-distillable becomes either locally preparable or 1-distillable. In this case the augmentation is very simple: one simply lifts the restriction of the local SSR by providing Alice and Bob with a shared reference frame.
With this additional resource, Alice and Bob can now implement any operation in LOCC.
Augmenting $G$-LOCC to LOCC divides the proper gap of pure states between ${G\text{-SSR}}$ locally preparable and ${G\text{-SSR}}$ 1-distillable into two classes, both of which are non-empty. All product states that are not locally $G$-invariant (i.e., product states not in ${G\text{-SSR}}$ locally preparable) \emph{become} locally preparable with $G$-LOCC plus the shared reference frame for $G$. We call this class \emph{${G\text{-SSR}}$ become locally preparable}. This result follows directly from the fact that all pure product states are locally preparable with unrestricted LOCC. Similarly all non-product pure states which are not in ${G\text{-SSR}}$ 1-distillable \emph{become} 1-distillable under $G$-LOCC plus the shared reference frame for $G$. We thus call this class \emph{${G\text{-SSR}}$ become 1-distillable}. This result follows directly from the fact that all pure non-product states are 1-distillable with unrestricted LOCC.
\subsubsection{Activation}
Again, just as in the mixed-state case, it is not necessary to completely lift the SSR constraint in order to make any particular state $\ket{\psi}$ either ${G\text{-SSR}}$ locally preparable or ${G\text{-SSR}}$ 1-distillable. Rather, all that is needed is some other pure state $\ket{\phi}$ which is ${G\text{-SSR}}$ become locally preparable. Again, this is trivial if $\ket{\psi}$ is ${G\text{-SSR}}$ become locally preparable; one simply chooses $\ket{\phi}=\ket{\psi}$. But the result is nontrivial when $\ket{\psi}$ is ${G\text{-SSR}}$ locally preparable, and says that a state $\ket{\phi}$ which is ${G\text{-SSR}}$ become locally preparable exists such that $\ket{\phi}\otimes\ket{\psi} \in\ {G\text{-SSR}}$ 1-distillable. This is analogous to activation and is an example of a partial reference frame.
\subsubsection{Measures of Entanglement} As discussed in Sec.~\ref{SSRmixing}, in the unipartite setting a SSR in general reduces the maximum work that can be extracted from a system, and that is quantified by the $G$-invariant state. Similarly, in the bipartite setting the amount of entanglement that can be extracted from a system under $G$-LOCC is less than under LOCC, and the locally $G$-invariant state again quantifies this reduction. The extractable entanglement \footnote{In Ref. \cite{BarWis03} they actually term this the \emph{entanglement constrained by a superselection rule}.}
from a single copy is given by \cite{BarWis03}\begin{equation} E_{G\textrm{-SSR}}(\rho) = E_D[({\cal G}_A \otimes {\cal G}_B) \rho]. \end{equation} As noted earlier, there is no way known to compute the distillable entanglement for a general mixed state. Thus we will restrict our attention to cases \cite{VacAnsWisJac06} where it is identical to the entanglement of formation. Also note that if a state (mixed or otherwise) can be used to demonstrate Bell-nonlocality then it necessarily has nonzero extractable entanglement.
\section{The Symmetric Group SSR}
\subsection{The constraint of symmetry}
The importance of symmetry as a constraint becomes apparent when dealing with many identical systems, that is, ensembles. By the term \emph{ensemble quantum information processing} we mean: (i) there are $N$ (typically $\gg 1$) identical ``molecules'' each consisting of $M$ ``atoms'' (typically qubits); (ii) all operations are {\em symmetric} (i.e. affect each molecule identically).
For example, in a nuclear magnetic resonance (NMR) experiment \cite{NMRQIP} each molecule contains $M$ atoms typically having a spin-$\frac12$ nucleus. Operations may be implemented using radio frequency (RF) magnetic pulses and an antenna. For the case of $M=4$ the qubits could be the spin-$\frac12$ nuclei of ${}^1$H, ${}^{17}$O, ${}^{13}$C, ${}^{19}$F. Another example occurs in spin squeezing experiments \cite{Ueda}. In this case each molecule is a single two-level, or multi-level, atom ($M=1$). Operations are implemented using uniform laser fields (and detectors for them), and thus affect all molecules identically.
In NMR quantum information processing it is also the case that the molecules are typically prepared in highly mixed states, and the detection efficiency is very small. These are practical constraints that apply to current experimental techniques rather than fundamental constraints such as those previously studied as SSRs. The consequences of such practical constraints will be discussed later (the first of these can be overcome at least for small molecules \cite{Anw04}).
There are $N!$ possible permutations of $N$ molecules. The set of these permutations $p$ (under the permutation operation) form the {\em symmetric group} $S_N$. The fact that symmetric operations must affect the identical molecules in the same way leads to what is known as the $S_N$-SSR. Another way of stating this is to say that only symmetric operations can be performed on ensemble quantum information processing systems.
Using the SSR formalism of Ref. \cite{BarWis03}, the restriction on operations ${\cal O}$ for ensemble quantum information processing systems can be stated as \begin{equation} \mathcal{O}[\hat T(p) \rho \hat T^\dag(p)] = \hat T(p)[ \mathcal{O} \rho]\hat T^\dag(p),\forall p\in S_N,\end{equation} where $p$ is a permutation of the $N$ molecules and $\hat T(p)$ is the unitary operator that implements that permutation. The $N$ molecules can each be thought of as subsystems of $M$ atoms (e.g. for $M=4$, the $N$ subsystems could be made up of a $\ {}^{1}$H atom, $\ {}^{17}$O atom, $\ {}^{13}$C atoms, and $\ {}^{19}$F atom). Each of the atoms within a subsystems is acted on by the same $\hat T(p)$, because they are attached to the same molecule.
When the $S_N$-SSR is in effect the allowable operations on the system are restricted to being symmetric. Under such operations the state $\rho$ is indistinguishable from the states $\hat T(p) \rho \hat T^\dag(p)$ for any $p\in S_N$. Thus we define the most mixed state with which $\rho$ is equivalent (the $S_N$-invariant or randomly permuted state) as \begin{equation} \mathcal{P} \rho \equiv \frac{1}{N!} \sum_{p \in S_N} \hat T(p) \rho \hat T^\dag(p). \end{equation} Under the $S_N$-SSR it is operationally appropriate to use $\mathcal{P}\rho$ to describe the state $\rho$.
\subsection{Local $S_N$-SSR}
NMR quantum information processing with pure states may allow the possibility of scalable quantum computing. In this paper we are not concerned with this question, but rather a question of principle: even with pure states, is there entanglement between different subsystems comprising atoms of the same species? Say we can create molecules such that there is entanglement between two species of atom (call them $A$ and $B$) on each molecule, as in Ref. \cite{Anw04}. Then if we could isolate an individual molecule, and give one of the relevant atoms to Alice and the other to Bob, then Alice and Bob would share \color{nblack} entanglement. We could even ``give'' one atom ($A$) to Alice and one ($B$) to Bob without splitting the molecule, merely by saying that Alice can control an applied magnetic field and antenna resonant with the frequency of $A$'s nucleus, and Bob similarly with $B$'s nucleus. However, the symmetry constraint means that Alice and Bob cannot isolate a single molecule. So the question then becomes: what is the nature of the entanglement between Alice's ensemble of $A$ atoms and Bob's ensemble of $B$ atoms?
Both Alice and Bob are restricted from individually addressing the $N$ molecules in their possession,
so we must apply the $S_N$-SSR locally. That is to say, the effective quantum state is $\left({\cal P}_A \otimes {\cal P}_B\right) \rho$. To understand this, it is helpful to consider a simple example; say $M=3$ (nuclei $A$, $A'$ and $B$, per molecule) and $N=2$ (there are two molecules, 1 and 2). We consider that the $A$s and $A'$s belong to Alice and the $B$s to Bob. The typical situation in NMR is to assume that the two molecules are prepared identically. However, for illustrative purposes it will be useful to consider the following state, where the molecules are not prepared identically: \begin{equation} \ket{\psi} = \ket{\uparrow_A^1 \uparrow_{A'}^1 \uparrow_B^1}\ket{\downarrow_A^2 \downarrow_{A'}^2\downarrow_B^2}.\end{equation} This is so that we can allow for (and see the effect of the local $S_N$-SSR on) correlations between Alice's atoms and Bob's atom without considering entangled states or mixed states. Here the states $\ket{\uparrow}$ and $\ket{\downarrow}$ are orthogonal states of the nucleus (spin up and spin down).
Now if Alice's local operations (acting only on $A$s and $A'$s) cannot distinguish molecules 1 and 2, then this state is { equivalent} to \begin{equation} { \hat{T}_A(p_1)\ket{\psi} = \ket{\downarrow_A^1 \downarrow_{A'}^1\uparrow_B^1}\ket{\uparrow_A^2 \uparrow_{A'}^2 \downarrow_B^2},} \end{equation} where $p_1$ is the swap permutation. Thus under the action of ${\cal P}_A$ (or ${\cal P}_B$, or ${\cal P}_A\otimes {\cal P}_B$), $\ket{\psi}$ goes to an { equal mixture}: \begin{eqnarray} \ket{\psi} &\stackrel{{\cal P}_A \otimes {\cal P}_B}{\longrightarrow} & {\cal P}_A \otimes {\cal P}_B[\ket{\psi}\bra{\psi}] \nonumber\\ &=& \uplus \smallfrac{1}{\sqrt{2}} \ket{\uparrow_A^1 \uparrow_{A'}^1 \uparrow_B^1}\ket{\downarrow_A^2 \downarrow_{A'}^2\downarrow_B^2} \nl{\uplus} \smallfrac{1}{\sqrt{2}} \ket{\downarrow_A^1 \downarrow_{A'}^1\uparrow_B^1}\ket{\uparrow_A^2 \uparrow_{A'}^2 \downarrow_B^2}\label{Mixture} \end{eqnarray} \color{nblack} Recall the notation $\uplus$ defined in Sec.~II as a shorthand for describing a projector. The two terms in the mixture are due to the two elements in the $S_2$ group. Thus, under the $S_N$-SSR Alice knows that both her atoms' spins are aligned. However, she loses knowledge of their orientation with respect to Bob's atom. Similarly, applying the $S_N$-SSR locally for Bob causes him to lose information about the orientation of his spin with respect to Alice's atoms.
\subsection{General Action of ${\cal P}$}\label{HilSpace}
Consider the general action of ${\cal P}$ on $N$ copies of a $d$-dimensional system. For our purposes $d$ is the total Hilbert space dimension of a single molecule in the ensemble. For example, if the molecules are made up of $M$ qubits, then $d=2^M$. The general action of ${\cal P}$ can be understood by analyzing the structure that it induces on the Hilbert space of the total system, $(\mathbb{C}_d)^{\otimes N}$. When the $S_N$-SSR applies to the system, as is the case for an ensemble of identical particles or subsystems, this Hilbert space carries a reducible representation $\hat{T}$ of $S_N$. Recall from Sec. \ref{Technicalities} that this splits the Hilbert space into `charge sectors': \begin{equation}\label{SNDecomposition}
(\mathbb{C}_d)^{\otimes N} = \bigoplus_{y \in Y} \mathbb{C}_y
\,. \end{equation} The sectors are further decomposed into irreps of $S_N$: \begin{equation}\label{SNIrepDecomposition}
\mathbb{C}_y = \mathbb{M}_y \otimes \mathbb{Q}_y
\,, \end{equation} where $\mathbb{M}_y$ carries an irrep $\hat{T}_y$ of $S_N$, $\mathbb{Q}_y$ carries the trivial irrep and has dimension given by the multiplicity of $\hat{T}_y$ in $\hat{T}$. The label $y$ can now be interpreted as a Young frame corresponding to an irrep of $S_N$. The set of Young frames $Y$, viewed as Young diagrams, are those consisting of $N$ boxes in up to $d$ rows of non-increasing length. We define $D_y \equiv \text{dim}(\mathbb{M}_y)$. For further details on the representations of $S_N$, see~\cite{Ful91}.
\subsubsection{Spin-1/2 particles}\label{spinhalf} There are two cases where the structure of the Hilbert space induced by ${\cal P}$ is particularly straightforward. The first is when the subsystems are identical spin-$\half$ particles. This means that the ensemble is composed of $d=2$ dimensional systems and the possible Young diagrams are those consisting of $N$ boxes in no more than $d=2$ rows. This limits the set of possible Young frames $Y$ to having $\lfloor N/2\rfloor+1$ elements, where $\lfloor N/2\rfloor$ is the largest integer less than or equal to $N/2$. Thus we are able to label each element by a single number. In this case, since we are dealing with spin systems, it is sensible to set the label $y$ for the Young frames equal to $j$, the ``total angular momentum" of the ensemble.
Consider the one-party case of $N=2J$ spin-$\half$ particles (i.e. $M=1$ qubit per molecule). The Hilbert space for each of the particles is given by the $2$-dimensional complex vector space, $\mathbb{C}_2$. Using Eqs. (\ref{SNDecomposition}) and (\ref{SNIrepDecomposition}) along with the fact that there are $\lfloor J\rfloor+1$ Young frames labelled by $j$, the total Hilbert space can be decomposed into \begin{equation} \left(\mathbb{C}_2\right)^{\otimes 2J} = \bigoplus_{j=J-\lfloor J\rfloor}^{J}\mathbb{M}_{j}
\otimes \mathbb{Q}_{j}.\end{equation} $\mathbb{M}_{j}$ and $\mathbb{Q}_{j}$ correspond to permutation and angular momentum subspaces respectively. Thus permutations of the spins $\hat{T}(p)$ act only upon $\mathbb{M}_{j}$ and joint operations such as rotations act only upon $\mathbb{Q}_{j}$. The dimensions of the subspaces are \begin{equation}{\rm dim}(\mathbb{M}_{j})=d_j \equiv \binom{2J}{J-j}\frac{2j+1}{J+j+1},\end{equation} for the permutation subspace and\begin{equation}
{\rm dim}(\mathbb{Q}_{j})=2j+1,\end{equation} for the angular momentum subspace.
Thus the basis for $\mathbb{C}_2^{\otimes 2J}$ in terms of these subspaces can be written as: $\cu{\ket{j,n}\otimes\ket{j,m}{\,:\,}_{j=J-\lfloor J\rfloor}^{J}{\,;\,}_{m=-j}^{j}{\,;\,}_{n=1}^{d_j}}$, where $n$ is a permutation label and $m$ is the magnetic quantum number. Now consider the action of the permutation operator ${\cal P}$. Physically, this operator destroys coherence between the `charge sectors' and also acts to randomly permute the particles. Mathematically this corresponds to $\mathcal{P}$ having the following effect on a state matrix $\rho$ for an ensemble of $N=2J$ qubits, \begin{equation}\mathcal{P}\rho=\sum_{j=J-\lfloor J\rfloor}^{J}\mathcal{D}_j \otimes \mathcal{I}_j(\hat\Pi_j \rho \hat\Pi_j ). \label{EGspinhalf}\end{equation} Here $\hat{\Pi}_j$ is the projection onto the charge sector $j$, and $\mathcal{D}_j$ is the trace-preserving map that acts on the permutation subspace to completely mix over the $\ket{j,n}$ basis states. $\mathcal{I}_j$ is the identity map over operators for the angular momentum subspace, $\mathbb{Q}_j$.
\subsubsection{Ensemble of two molecules}\label{N2} The second instance where it is straightforward to study the Hilbert space structure is when there are only two molecules in the ensemble (that is, $N=2$, so the $S_2$ group applies). In general the molecules are $d$-dimensional systems, so the Hilbert space for each molecule is given by $\mathbb{C}_d$. In this case there are only two possible Young frames, corresponding to the symmetric and antisymmetric representations of $S_2$. These are both $1$-dimensional representations meaning that the total Hilbert space can be decomposed as, \begin{eqnarray} \left(\mathbb{C}_d\right)^{\otimes 2}& =& \bigoplus_{y=s,a} \mathbb{M}_1 \otimes \mathbb{Q}_y \nonumber\\ & =& \bigoplus_{y=s,a} \mathbb{Q}_y,\end{eqnarray} since $D_1={\rm dim}(\mathbb{M}_1)=1$. The components of the angular momentum subspace, $\mathbb{Q}_s$ and $\mathbb{Q}_a$, correspond to symmetric and antisymmetric subspaces respectively. Their dimensions are given by dim$(\mathbb{Q}_s)=(d^2+d)/2$ and dim$(\mathbb{Q}_a)=(d^2-d)/2$.
This structure can be simply understood from the fact that there are only two permutations in the $S_2$ group, which can be represented by $\hat{T}(p_0)=\hat{I}$, and the operator $\hat{T}(p_1)=\hat{T}$ which swaps the two molecules. The group structure of $S_2$ ensures that $\hat{T}^2=\hat{I}$, which means that $\hat{T}$ can be written as $\hat{T}=\hat{\Pi}_s-\hat{\Pi}_a$, where the operators $\hat{\Pi}_s$ and $\hat{\Pi}_a$ project onto the symmetric and antisymmetric subspaces respectively. Also note that the identity operator can be represented as $\hat{I}=\hat{\Pi}_s+\hat{\Pi}_a$. Hence the action of ${\cal P}$ on the density matrix $\rho$ for an $N=2$ ensemble state is given by \begin{equation} \mathcal{P}\rho=\frac{1}{2}\left(\hat{I}\rho\hat{I}+\hat{T}\rho\hat{T}^\dagger\right). \label{N2ensemble}\end{equation} Using the expressions for $\hat{I}$ and $\hat{T}$ in terms of projection operators gives, \begin{eqnarray} \mathcal{P}\rho&=&\frac{1}{2}[(\hat{\Pi}_s+\hat{\Pi}_a)\rho(\hat{\Pi}_s+\hat{\Pi}_a)+(\hat{\Pi}_s-\hat{\Pi}_a)\rho(\hat{\Pi}_s-\hat{\Pi}_a)]\nonumber\\ &=& \hat{\Pi}_s\rho\hat{\Pi}_s+\hat{\Pi}_a\rho\hat{\Pi}_a. \label{N2ensembleProj}\end{eqnarray} This illustrates the fact that $\mathcal{P}$ destroys coherence between the angular momentum `charge sectors', which in this case means destroying coherence between the symmetric and antisymmetric subspaces.
\subsection{Multiple Copies under the $S_N$-SSR}\label{sec:SSR:mcopies}
We have seen in Sec. \ref{sec:EntSSR} that pure states subject to a SSR show remarkable similarities to mixed states. In order to obtain entanglement from mixed states we often consider preparing many copies of the state and performing distillation protocols to recover maximally entangled states. Similarly for pure states subject to a SSR, it is possible to use many copies of the state to obtain extractable entanglement.
However, care must be taken when applying the notion of multiple copies to ensemble states which are subject to the $S_N$-SSR. If one were simply to double the number of molecules in the ensemble, there would be more possible ways of permuting them and the system would in fact be constrained by a different SSR (i.e. $S_{2N}$-SSR rather than $S_N$-SSR). Applying the notion of multiple copies under the $S_N$-SSR means duplicating an ensemble of $N$ molecules, each with $M$ atoms, by creating an ensemble of $N$ molecules, each with $2M$ atoms. This way, each molecule now contains two copies of the original state, and Alice and Bob possess two copies of the original ensemble. In general they can obtain $C$ copies of the original ensemble by increasing the number of atoms in each of the $N$ molecules to $M'=CM$. If the original ensemble of $N=2$ molecules had $M=2$ atoms (with Alice and Bob each `owning' one atom from each molecule in the original ensemble), two copies of the ensemble is given by an ensemble of $N=2$ molecules with $M'=4$ atoms. This is illustrated in Fig. \ref{Multicopies}. This concept will be discussed further in the context of recovering entanglement ostensibly lost due to the SSR. \begin{figure}
\caption{\color{nblack} Creating multiple copies of an ensemble described by $\ket{\psi}$. $N=2$ and $M'=4$, which means that Alice and Bob share two copies of the $N=2$, $M=2$ ensemble.}
\label{Multicopies}
\end{figure}
\section{Applications of $S_N$-SSR}
\subsection{ Asymptotic Loss of Entanglement}\label{Loss}
Typically with the $S_N$-SSR applied to many identical copies of an entangled state the amount of extractable entanglement will be less than in the unconstrained case. This was actually done first by Eisert {\it et al.~}\ \cite{Eis00}. Consider an ensemble of $N=2J$ identically prepared molecules each consisting of two nuclei in the following state: \begin{equation} \ket{\psi} = \alpha\ket{\downarrow_A \downarrow_B} + \beta\ket{\uparrow_A \uparrow_B},\label{spinstate} \end{equation} where $\alpha$ and $\beta$ can be taken to be real, so that $\alpha^2+\beta^2=1$ is the normalization condition. Using the Hilbert space decomposition into permutation and angular momentum subspaces from Sec. \ref{spinhalf}, the total state of the ensemble can be written as, \begin{eqnarray} \ket{\psi}^{\otimes N} &=& \sum_{j=J-\lfloor J\rfloor}^{J}\sum_{n=1}^{d_j} \sum_{m=-j}^{j} \alpha^{J-m}\beta^{J+m} \nonumber \\ && \times \ket{j,n}_A \ket{j,m}_A\, \otimes\, \ket{j,n}_B\ket{j,m}_B, \label{BasisState}\end{eqnarray} where the condition $\alpha^2+\beta^2=1$ also indicates that $\ket{\psi}^{\otimes N}$ is normalized. From the spin representation [Eq. (\ref{spinstate})] it is easy to see that the entanglement for the ensemble is $E(\ket{\psi}^{\otimes N}) = N (-\alpha^2\log\alpha^2 - \beta^2
\log\beta^2)= N E(|\psi\rangle)$.
We now consider the amount of extractable entanglement under the {$S_N$-SSR}. To do so, we must take into account the effect of the SSR on the ensemble state. The permutation operator ${\cal P}$ results in a completely mixed state for both Alice and Bob in the permutation subspace. That is, \begin{eqnarray} \ket{\psi}^{\otimes N}\!\!\! &&\stackrel{{\cal P}_A \otimes {\cal P}_B}{\longrightarrow} \ \ \sum_{j=J-\lfloor J\rfloor}^{J}\; \sum_{n=1}^{d_j}\ \frac{I_A^j}{d_j}\otimes\frac{I_B^j}{d_j}\nonumber \\ &\otimes&\!\!\left(\uplus\sum_{m=-j}^{j} \alpha^{J-m}\beta^{J+m}\ket{j,m}_A \ket{j,m}_B\right)\!\!, \label{43}\end{eqnarray} where implementation of the $S_N$-SSR has also destroyed coherence between different $j$ terms. Note that $I_A^j$ is the identity operator on the Hilbert space $\mathbb{M}_j$ for Alice, and similarly $I_B^j$ for Bob. Eq. (\ref{43}) can be simplified by defining the (normalized) angular momentum part of the state as $\ket{\phi_j} =(1/\sqrt{d_j \wp_j}\ ) \sum_{m=-j}^{j} \alpha^{J-m}\beta^{J+m} \ket{j,m}_A \otimes \ket{j,m}_B$. The term $\wp_j = \sum_{m=-j}^j \alpha^{2(J-m)}\beta^{2(J+m)}/d_j$ is the probability of obtaining the $j$th angular momentum value and a particular irrep, indexed by $n_A$ and $n_B$. Since there are actually $d_j^2$ irreps for each $j$ value, the probability of obtaining a particular $j$ is $d_j^2\wp_j$. It can be verified that $\sum_{j=J-\lfloor J\rfloor}^J d_j^2\wp_j=1$ as required by conservation of probability. Using these definitions allows \erf{43} to be rearranged as, \begin{eqnarray} {\cal P}_A\!&\otimes&\!{\cal P}_B\ \left(\ket{\psi}\bra{\psi}\right)^{\otimes N} \nonumber \\ &=&\sum_{j=J-\lfloor J\rfloor}^{J}\; \!\!\!\!\!d_j \left(\frac{I_A}{d_j}\otimes\frac{I_B}{d_j}\right) \otimes\ d_j\wp_j\left(\frac{}{}\!\!\uplus\ket{\phi_j}\right)\!. \end{eqnarray}
For convenience we will omit writing the completely mixed states on the permutation subspace, although when we write the $S_N$-invariant state they are assumed to be there. Using this convention, the $S_N$-invariant state can be written compactly as \begin{equation} {\cal P}_A\otimes{\cal P}_B\left(\ket{\psi}\bra{\psi}\right)^{\otimes N} =\sum_{j=J-\lfloor J\rfloor}^{J}\;d_j^2\wp_j\ \left(\!\!\frac{}{} \uplus\ket{\phi_j}\right)\!.\label{SNcompact} \end{equation}
Since no observed quantities can be changed by replacing $\ket{\psi}^{\otimes N}$ with the $S_N$-invariant state, calculating the constrained entanglement of $\ket{\psi}^{\otimes N}$ is equivalent to \begin{equation} E_{S_N\textrm{-SSR}}(\ket{\psi}^{\otimes N}) = E_D \ro{ {\cal P}_A\otimes {\cal P}_B \left(\ket{\psi}\bra{\psi}\right)^{\otimes N} }.\label{ConstrainedE}\end{equation} If the state of interest is composed of states that are are locally distinguishable (for both Alice and Bob) then it is known as a \emph{biorthogonal mixture} (see Ref. \cite{Herbut03} for more details). The expected entanglement of such a state is simply a weighted sum of the entanglement present in each of the the possible states. That is, \begin{equation} E(\rho)=\wp_1E(\rho_1)+\wp_2E(\rho_2)+\ldots,\label{ExpectedE}\end{equation} where $E(\rho_{1,2,\ldots})$ is the entanglement of the locally distinguishable states making up the mixture, and $\wp_{1,2,\ldots}$ are the corresponding probabilities of each state occurring. The $S_N$-invariant state is of this form, with $\uplus\ket{\phi_j}$ the possible states and $d_j^2\wp_j$ the corresponding probabilities. Therefore, Eq. (\ref{ConstrainedE}) can be rewritten as \begin{equation} E_{S_N\textrm{-SSR}}(\ket{\psi}^{\otimes N}) = \sum_{j=J-\lfloor J\rfloor}^{J} d_j^2 \wp_j E(\ket{\phi_j}), \end{equation} where $E(\ket{\phi_j})$ is the entanglement of the angular momentum state $\ket{\phi_j}$. We expect the total amount of constrained entanglement to be less than the $E(\ket{\psi}^{\otimes N}) = N (-\alpha^2\log\alpha^2 - \beta^2 \log\beta^2)$ ebits calculated for the unconstrained system.
To demonstrate this, consider the particular case of Bell states, where $\alpha=\beta=\smallfrac{1}{\sqrt{2}}$. This gives $ E(\ket{\psi}^{\otimes N}) = N$, but, as shown by Bartlett and Wiseman \cite{BarWis03}, \begin{eqnarray}
E_{S_N\textrm{-SSR}} ( \ket{\psi}^{\otimes N} ) &=&\!\!\!\! \sum_{j=J-\lfloor J\rfloor}^{J}\!\! d_j^2 \wp_j\log_2 (2j+1).\label{Eloss}
\end{eqnarray} This expression can be simplified significantly in the asymptotic limit (i.e. $J=N/2\rightarrow \infty$) because the probability distribution $ d_j^2 \wp_j$ becomes sharply peaked at a single $j$ value. Thus, a single term in the sum essentially determines the value of the entanglement. It can be shown that for large ensembles ($N\gg1$) the significant term in the sum is specified by $j\approx \sqrt{J}$. This means that in the asymptotic limit Eq. (\ref{Eloss}) reduces to approximately $(1/2) \log_2 N$. Since this is the maximum total entanglement, the entanglement per molecule must always $\to 0$ as $N \to \infty$. Hence, under the $S_N$-SSR for an ensemble of maximally entangled pure states we asymptotically lose the ability to access the entanglement.
\subsection{Asymptotic Recovery of Entanglement}\label{Recovery}
We have just shown that under the $S_N$-SSR we apparently `lose' much of the entanglement in the ensemble. This might seem contrary to the intuition obtained from the U(1) case, for example, where in the limit of a large number of particles, the entanglement per particle is \emph{recovered} asymptotically approaching the unconstrained entanglement \cite{WisVac03}. This discrepancy arises from taking an inappropriate form of the asymptotic limit for the $S_N$-SSR. As explained in Sec. \ref{sec:SSR:mcopies}, having multiple copies under an $S_N$-SSR does not mean changing $N$. The asymptotic limit for the number of copies thus should be considered with $N$ fixed.
We begin by considering an ensemble of $N=2$ molecules. As discussed in Sec. \ref{N2} this is a special case that considerably simplifies the action of $\mathcal{P}$. To relate to Sec. \ref{Loss}, imagine that Alice and Bob share an ensemble of two molecules each of which is a Bell state. The difference here is that we allow each molecule to be larger and to contain $C$ copies of a Bell state. That is, we allow Alice and Bob to share $C$ copies of the original $N=2$ ensemble.
For convenience we define the density matrix for $C=1$ copy of the ensemble of $N=2$ Bell states as \begin{equation} \rho_{AB}=\Big[\ket{\psi^-}\bra{\psi^-}\Big]^{\otimes 2},\end{equation} where the Bell singlet state \footnote{For simplicity with our formalism we make use of the singlet Bell state, however, our results also hold for the triplet Bell states.} is defined as $\ket{\psi^-}=\frac{1}{\sqrt{2}}(\ket{\uparrow_A\downarrow_B}-\ket{\downarrow_A\uparrow_B})$. The state $\rho_{AB}$ can also be expressed as \begin{eqnarray} \rho_{AB}&=&\uplus\frac{1}{2}\Big[\ket{A}+\sqrt{3}\ket{S}\Big], \end{eqnarray} where we define normalised states in the antisymmetric and symmetric subspaces in terms of the $\ket{j,m}$ basis (recall Sec. \ref{spinhalf}) as $\ket{A}=\ket{j=0,m=0}_A\ket{j=0,m=0}_B$ and $\ket{S}=(1/\sqrt{3})\sum_{m=-1}^1\ket{j=1,m}_A\ket{j=1,-m}_B$ respectively.
Using this representation for the state, it becomes apparent that $\mathcal{P}$ simply destroys coherence between the symmetric and antisymmetric subspaces which can be represented as \begin{equation}\mathcal{P}\rho_{AB}=\uplus\sqrt{\frac{1}{4}} \ket{A}\uplus\sqrt{\frac{3}{4}}\ket{S}. \label{Erecover}\end{equation}
Since Alice and Bob share a biorthogonal mixture, they can each make local measurements to distinguish between the symmetric and antisymmetric subspaces. This is equivalent to the situation considered by Eisert \emph{et al.} \cite{Eis00}. With probability $1/4$ they find that they have the locally antisymmetric state and they retain no entanglement (as this is a separable state). However, with probability $3/4$ they obtain the locally symmetric state, which is equivalent to a maximally entangled qutrit state. In that case they retain $\frac{3}{4}\log_2(3)\approx 1.19$ ebits of entanglement. Without the $S_2$-SSR constraining their two Bell states, Alice and Bob would possess 2 ebits of entanglement.
One might expect that by using the concept of multiple copies it would be possible to ameliorate the effect of the SSR. This is indeed the case, as we now show. For the $S_2$-SSR to apply, Alice and Bob must share entanglement contained in 2 molecules. In the simplest case, each molecule is simply a Bell singlet state and the combined state is $\rho_{AB}$, as discussed above. To apply the concept of multiple copies, Alice and Bob must share $C$ copies of $\rho_{AB}$ (see Fig. \ref{BellStates}). With no restrictions in place Alice and Bob would share $2C$ ebits of entanglement. \begin{figure}
\caption{\color{nblack} Two copies ($C=2$) of $\rho_{AB}$ (which is composed of two Bell states $\ket{\psi^-}$). Each molecule can be extended to include more Bell states to increase the number of copies $C$ of $\rho_{AB}$.}
\label{BellStates}
\end{figure}
The calculation of how much entanglement is retained using multiple copies can be significantly simplified by noting that in this case, each of the molecules (containing $C$ Bell pairs) can be considered as a maximally entangled \emph{qudit} pair. This is possible due to the global symmetry of the ensemble state chosen. In this case, each molecule can be described as a maximally entangled pair of qudits, with the qudits dimension given by
$d=2^C$. This simplifies calculations, as the maximum entanglement of a pair of entangled qudits is readily calculated to be $E_{\rm max}=\log_2 d$. Thus, without considering the $S_2$-SSR constraint, the total entanglement for the two maximally entangled qudit pairs is $E=2C$ ebits, as already derived.
We can express the state of $C$ copies of $\rho_{AB}$ under the $S_2$-SSR explicitly as a biorthogonal mixture of a locally symmetric and a locally antisymmetric state, \begin{equation}\mathcal{P}\left(\rho_{AB}\right)^{\otimes C}=\wp_s\rho_s+\wp_a\rho_a,\label{biomix}\end{equation} where the weightings $\wp_s$ and $\wp_a$ are the probabilities of both Alice and Bob obtaining a locally symmetric or locally antisymmetric state respectively. These probabilities depend upon the dimension of the subspace that each of the local states occupy: $\wp_s={\rm dim}\left(\mathbb{Q}_s\right)/d^2$ and $\wp_a={\rm dim}\left(\mathbb{Q}_a\right)/d^2$ (recall the expressions for the subspace dimensions defined in Sec. \ref{N2}).
The structure of Eq. (\ref{biomix}) means that it is quite straightforward to calculate the extractable entanglement of $\left(\rho_{AB}\right)^{\otimes\ C}$. It is simply a weighted average of the entanglement in the two subspaces: \begin{equation}E=\frac{d^2-d}{2d^2}\log_2\!\!\left(\frac{d^2-d}{2}\right)+\frac{d^2+d}{2d^2}\log_2\!\!\left(\frac{d^2+d}{2}\right).\label{Ent}\end{equation} For a large number of copies ($C \gg 1$ ) the dimension $d$ is large and Eq. (\ref{Ent}) reduces to approximately $E = 2C-1$. Thus in the asymptotic limit, nearly all of the entanglement has been recovered (only a single ebit has been lost).
Another way to consider this problem is that Alice and Bob share many copies of the state $\rho_{AB}$ via a channel (see Fig. \ref{Channel}). The channel is deterministic and either does nothing or performs a swap of the molecules. If Alice and Bob were unable to make collective measurements on their entire collection of qubits then they could still make use of their copies of $\rho_{AB}$ to asymptotically retain much of their entanglement. A non-optimal procedure that they could implement would be to use up a small number of copies to find out what map the channel performs (either identity or swapping). Once they know what the channel does they can then safely use the 1 ebit of entanglement in each of their remaining Bell pairs. This method is non-optimal because Alice and Bob lose at least a few ebits of entanglement in characterizing the channel (and asymptotically with collective measurements they need lose only 1 ebit). \begin{figure}
\caption{\color{nblack} Alice and Bob share $N$ copies of $\rho_{AB}$ via a channel. In case (a) the channel distributes the states in order. In (b) the channel swaps the ordering within each pair.}
\label{Channel}
\end{figure}
In general for the case of the $S_N$-SSR with $N>2$ it is difficult to optimally calculate the exact asymptotic amount of entanglement recovered. However, considering the non-optimal procedure just discussed it is intuitive that Alice and Bob could recover most of their entanglement (in the asymptotic limit) simply by using up some copies of the state to characterize the `channel'. They would then retain the entanglement in the remaining copies. As the size of the ensemble ($N$) increases, more copies of the state will be required to satisfactorily characterize the `channel' and thus more entanglement will be lost.
\section{Reference Frames}
In general, a reference frame for a SSR is something that removes its effect. For example, a perfect reference frame completely removes the effect of, or `lifts', the SSR. This is the ideal case, although in practice it is possible to have partial reference frames which only partially remove the effect of the SSR.
Usually a reference frame is an extra system added to the system of interest which allows access to degrees of freedom otherwise unaccessible due to the SSR. Thus, for an ensemble of molecules, for which the $S_N$-SSR applies, one might naively expect to add an extra ensemble of molecules to act as a reference frame. However, as discussed in Sec. \ref{sec:SSR:mcopies}, due to the nature of the $S_N$ group, adding molecules would in fact alter the SSR for the system. That is, the reference molecules would actually be permuted with the system molecules, making it more, not less, difficult to gain information about the system.
Instead, the type of reference frame needed for an ensemble system is analogous to a labelling. Classically, one would think of physically writing a label (say a number) on each object, to serve as a reference ordering. Physically this corresponds not to adding molecules to the ensemble, but adding an extra nucleus (or group of nuclei) to each molecule in the ensemble.
To illustrate this, consider a simple example,
with $N=3$ molecules. In this instance a pure state [where the three molecules happen to be uncorrelated, see Fig. (\ref{ReferenceFrames})] with a reference frame is \begin{equation} \ket{\Psi} = \ket{\psi^1,1}\ket{\psi^2,2}\ket{\psi^3,3} = \ket{\psi^1,\psi^2,\psi^3} \otimes { \ket{1,2,3}} \end{equation} Here $\ket{\psi^k}$ is the state of the $M$ nuclei in the $k$th molecule (not including the reference frame) which we have assumed to factorize. In regards to the tensor product structure it is important to remember that the second system is not in the same state as the first (it need not even have the same Hilbert space dimension).
\begin{figure}
\caption{\color{nblack} Classical versus quantum reference frames. The classical reference frames on the left are represented by boxes and are uncorrelated. On the right the reference frames are quantum systems and we allow for correlations between the label systems.}
\label{ReferenceFrames}
\end{figure}
\subsection{Quantum reference frames}
In the classical example above, we placed each $N$-dimensional attached label system (nucleus or group of nuclei) in a unique product state. An obvious question is whether or not it is possible to use label systems of smaller dimension if we allow entanglement between the states of the $N$ label systems. As demonstrated by von Korff and Kempe~\cite{vKK04}, it is indeed possible to reduce the dimension of the label systems by a constant factor in the limit $N\rightarrow \infty$.
Recalling the structure of the Hilbert space from Sec.
\ref{HilSpace}, a state of the $N$ label systems $|0\rangle \in (\mathbb{C}_d)^{\otimes N}$ that works as a \emph{perfect} quantum reference frame would satisfy the property that the $N!$ states \begin{equation}\label{OrthogonalStates}
|p_n\rangle = \hat{T}(p_n)|p_0\rangle \,, \end{equation}
for all $p_n \in S_N$ satisfy $|\langle p_n| p_{n'} \rangle|^2 = \delta_{n,n'}$. This property ensures that every different ordering is classically distinguishable (i.e., is associated with an orthogonal quantum state). So the problem reduces to the following: What is the minimum $d$ such that such a set of orthogonal states exists?
First, we note that the space $\mathbb{H}_R$ spanned by $\{
|p_n\rangle,\, p_n\in S_N\}$ is $N!$-dimensional and that the representation $\hat{T}$ when restricted to this space is isomorphic to the (left) \emph{regular representation}. (The regular representation $R$ of a group $G$ has $G$ as a carrier space, and acts as $R(g)g' = gg'$.) It is well-known~(\cite{Ful91}, p.~17) that the regular representation of $S_N$ contains every irrep $\hat{T}_y$ of $S_N$, each with a multiplicity equal to $D_y$, the dimension of $\hat{T}_y$. Thus, for $\hat{T}$ to contain the regular representation, it must contain every irrep $\hat{T}_y$ of $S_N$ with a multiplicity of at least $D_y$. In particular, this must hold true for the fully-antisymmetric representation of $S_N$ (the irrep labeled by a Young diagram consisting of a single column of $N$ boxes), and $\hat{T}$ only contains the fully-antisymmetric representation if $d\geq N$. Thus, if we demand that the label systems act as a \emph{perfect} reference frame for $S_N$, then each label system must be at least $N$-dimensional.
However, von Korff and Kempe~\cite{vKK04} have shown that it is possible to use label systems with any dimension $d > \lfloor N/e \rfloor$ if the requirement of a perfect reference frame is relaxed to the less-stringent demand that, for $p_n\neq p_{n'}$,
$|\langle p_n| p_{n'}\rangle|^2 \rightarrow 0$ as $N \rightarrow \infty$. (That is, that the reference frame states are distinguishable only in the asymptotic limit.) The basic idea is that if $d > \lfloor N/e \rfloor$ then, although $\hat{T}$ does not contain \emph{all} irreps of $S_N$ with the required multiplicity, the set that are missing has measure approaching zero as $N \rightarrow \infty$. We refer the reader to~\cite{vKK04} for details.
We now explicitly construct states of the form of Eq.~(\ref{OrthogonalStates}), using the general construction of~\cite{KMP04} that was subsequently applied specifically to the $S_N$ group in~\cite{vKK04}. Let $\overline{Y}$ be the set of irreps that are contained in $\hat{T}$ and have sufficient multiplicity, i.e., that satisfy \color{nblack} $\text{dim}\mathbb{Q}_y \geq D_y$. For each $y \in \overline{Y}$, choose an arbitrary subspace $\mathbb{Q}'_y \subset \mathbb{Q}_y$ of dimension $D_y$. Let $\{
|y,i,j\rangle, i,j=1,\ldots,D_y \}$ be a basis for $\mathbb{M}_y \otimes \mathbb{Q}'_y$, where $i$ labels a basis for $\mathbb{M}_y$ and $j$ labels a basis for $\mathbb{Q}'_y$. Define $D = \sum_y D_y^2$. Then the state \begin{equation}\label{PermutationRFstate}
|p_0\rangle = \sum_{y \in \overline{Y}} \sum_{i=1}^{D_y}
\sqrt{\frac{D_y}{D}}|y,i,i\rangle \,, \end{equation}
can be used to define a set of states $\{|p_n\rangle =
\hat{T}(p_n)|p_0\rangle \}$ for $p_n\in S_N$ as in Eq.~(\ref{OrthogonalStates}). As demonstrated in~\cite{vKK04},
$\lim_{N\to\infty} D = N!$ and $\lim_{N\to\infty}|\langle p_n| p_{n'} \rangle|^2 = \delta_{n,n'}$ provided that $d > \lfloor N/e \rfloor$.
\subsection{Shared reference frames} The simplest \emph{shared} reference frame is for Alice and Bob each to have a reference frame. In general, if Alice and Bob share $N$ tensor product states and both have a reference frame for each state, then the total system can be described as \begin{eqnarray}\ket{\Psi} &=& \bigotimes_{i=1}^N\ket{\psi^i_{AB},i_A,i_B}.\end{eqnarray} \color{nblack} For example, this can be written out explicitly for the case when two product states are shared, \begin{eqnarray}\ket{\Psi} &=& \ket{\psi^1_{AB},1_A,1_B}\ket{\psi^2_{AB},2_A,2_B}\nonumber\\ &=&\ket{\Psi_{AB}}\ket{p_0}_A\ket{p_0}_B,\end{eqnarray} \color{nblack} where in the second line we have written the shared states first, followed by Alice and Bob's reference frames. Note that we have rewritten Alice's reference state $\ket{\cdots,1_A,\cdots}\ket{\cdots,2_A,\cdots}$ as the fiducial reference state $\ket{\cdots}\ket{p_0}_A\ket{\cdots}_B$, and similarly for Bob's.
Although these states are separable, they cannot be prepared {\em locally} by { ${\cal P}_A\otimes {\cal P}_B$-invariant} operations from a { ${\cal P}_A\otimes {\cal P}_B$-invariant} state. Hence they are bound entangled states which may become locally preparable. (Recall the definitions in Sec. \ref{sec:EntSSR}.) Note that such states are not globally ${\cal P}$-invariant. However, using the final reference frame basis above we can write a separable ${\cal P}$-invariant reference frame: \begin{equation}{ \biguplus_{p_n\in S_N} \frac{1}{\sqrt{N!}}\ket{p_n}_A\ket{p_n}_B}.\end{equation} This reference frame is an incoherent mixture of reference states which is an example of a \emph{shared} reference frame. The key point is that the same permutation is applied to both Alice and Bob's reference states resulting in perfect correlation between each of Alice and Bob's labels. That is, this reference frames gives no indication of labels for individual states, but indicates that Alice and Bob's particles are in the same order. States of this form are mixed (separable) and hence not part of the classification scheme of Sec. \ref{sec:EntSSR}.
Alternatively, a pure globally ${\cal P}$-invariant reference frame can be constructed by considering non-separable states: \begin{equation}{ \sum_{p_n\in S_N} \frac{1}{\sqrt{N!}}\ket{p_n}_A\ket{p_n}_B}.\end{equation} This state is a coherent superposition of reference states which are perfectly correlated between Alice and Bob. Once again for an explicit example we consider a reference frame for the $S_2$ group \begin{eqnarray}\ket{\Psi}_{\rm RF}&=&\frac{1}{\sqrt{2}}\sum_{p_n\in S_2}\ket{p_n}_A\ket{p_n}_B\nonumber\\&=&\frac{1}{\sqrt{2}} \left[\ket{p_0}_A\ket{p_0}_B+\ket{p_1}_A\ket{p_1}_B\right],\label{ReferenceFrameStateS2} \end{eqnarray} where $p_0$ is the identity permutation and $p_1$ is the swap permutation. In this case it can be shown that the partial transpose of the state matrix $\rho_{\rm RF}=\ket{\Psi}_{\rm RF}\bra{\Psi}_{\rm RF}$ is actually equal to $\rho_{\rm RF}$. Thus it is a valid state matrix which means that $\rho_{\rm RF}$ has a positive partial transpose \cite{Peres96}. This shows that for the $S_2$ group, which is actually an Abelian group, a shared reference state of the form of Eq. (\ref{ReferenceFrameStateS2}) is become 1-distillable (this is because it contains no entanglement under the $S_2$-SSR but becomes 1-distillable if the SSR is lifted).
\section{Analogies with mixed-state entanglement}
\subsection{Activation}
Recall from section \ref{Distillable} that a general state $\rho$ is called 1-distillable if by LOCC Alice and Bob can, with some probability, create from it a nonseparable two-qubit state. Also recall that there are bound entangled states that become 1-distillable when the two parties have their LOCC supplemented by a shared PPT-channel. These states, as we have mentioned in section \ref{Become}, are called become 1-distillable states.
Since $S_N$ is a finite group, reference frames for the $S_N$ group can be finite (this is quite different to the case for Lie group SSRs such as the U(1)-SSR). Moreover, the $S_N$ reference frames can be used without being disturbed because they form an orthonormal set. Thus under the $S_N$-SSR there is no distinction between activation of a bound entangled state (by a bound entangled state which becomes locally preparable) and lifting the $S_N$-SSR to make become 1-distillable states 1-distillable.
Activation of a bound entangled state can be seen in the following example. If $N=2$ and $M=2$, (i.e Alice and Bob own one nucleus per molecule), then the state \begin{equation}{ \sqrt{2}\ket{\psi} = \ket{+}_A\ket{-}_B + \ket{-}_A\ket{+}_B},\end{equation} is bound entangled that can become 1-distillable. Here $\ket{+} = \ket{j=1,m=0}$ and $\ket{-}=\ket{j=0,m=0}$, so ${\hat{T}(p_1)\ket{\pm}=\pm \ket{\pm}}$. From this it is easy to see that $\ket{\psi}$ is globally symmetric, but under the local SSR, \begin{equation}{ \sqrt{2}\ket{\psi} \stackrel{{\cal P}_A \otimes {\cal P}_B}{\longrightarrow} \;\uplus\; \ket{+}_A\ket{-}_B \;{ \uplus}\; \ket{-}_A\ket{+}_B},\end{equation} which is clearly separable. Hence, with the SSR the state has no distillable entanglement.
It is possible to completely lift the SSR and regain 1-ebit of entanglement from this state. This is achieved by adding an extra shared state $\ket{\phi}$ to \emph{activate} the bound entanglement in $\ket{\psi}$. This is shown in Fig. \ref{Activate}. For instance, the simplest perfect reference frame $\ket{\phi}$ would label each of Alice and Bob's nuclei, for example, $\ket{\phi}=\ket{1_A,2_A}\ket{1_B,2_B}=\ket{p_0}_A\ket{p_0}_B$. Then it becomes possible for Alice to find out which of her nuclei is correlated with which of Bob's simply through measurement of the shared reference state. Thus by use of a reference frame (that is, activating the bound entanglement), it is possible to access 1-ebit of entanglement from the become 1-distillable state.
\begin{figure}
\caption{\color{nblack} Using an extra state $\ket{\phi}$ to activate the bound entanglement in $\ket{\psi}$. In this case $\ket{\phi}$ acts as a perfect reference frame and all the entanglement in $\ket{\psi}$ is recovered.}
\label{Activate}
\end{figure}
\subsection{Distillation}
We now illustrate the phenomenon of distillation using the same example state $\ket{\psi}$. That is, although without a reference frame the state ${ \sqrt{2}\ket{\psi} = \ket{+}_A\ket{-}_B + \ket{-}_A\ket{+}_B}$ has $E_{S_2\textrm{-SSR}}=0$, with two copies some entanglement can be obtained.
Recall from Section \ref{sec:SSR:mcopies} that two copies does {\em not} mean four molecules. Since $S_2$ is fixed, we still have $N=2$ molecules, but instead of $M=2$ we now have $M'=4$, that is, Alice and Bob each have \emph{two} nuclei per molecule. This is demonstrated in Fig. \ref{Duplication}. \begin{figure}
\caption{\color{nblack} With two copies of the state $\ket{\psi}$ the second can act as a reference frame for the first allowing one ebit of entanglement to be accessed. This is considered an imperfect reference frame for the system as we would expect two copies of $\ket{\psi}$ to contain two ebits of entanglement. }
\label{Duplication}
\end{figure} The state for the two copies can be written as, \begin{eqnarray} (\sqrt{2}\ket{\psi} )^{\otimes 2} &=& \ket{++}_A\ket{--}_B + \ket{-+}_A\ket{+-}_B \nl{+} \ket{+-}_A\ket{-+}_B + \ket{--}_A\ket{++}_B, \end{eqnarray} which with a perfect reference frame contains two ebits of entanglement. The effect of the SSR is to create a mixture of the unchanged state with the state formed by applying the swap $\hat{T}(p_1)$ to Alice's particles (or Bob's). Thus under the $S_2$-SSR, the state becomes,
\begin{eqnarray} (\sqrt{2}\ket{\psi} )^{\otimes 2} &\stackrel{{\cal P}_A \otimes {\cal P}_B}{\longrightarrow}& \uplus \left( \ket{++}_A\ket{--}_B + \ket{--}_A\ket{++}_B\right) \nl{\uplus} \left(\ket{-+}_A\ket{+-}_B + \ket{+-}_A\ket{-+}_B\right). \nl{} \end{eqnarray} Now Alice and Bob share a mixture of two superpositions. By Alice and Bob each measuring a suitable observable (such as $\hat{O}=\uplus\ket{++}\uplus\ket{--}$ for example), they can perform a local measurement to discriminate the two superposition states superpositions they actually
(without destroying the superposition). Thus they have access to 1 ebit of constrained entanglement. In this case we started with two copies of $\ket{\psi}$, therefore with a perfect reference frame we would expect to be able to recover two ebits of entanglement. However, even without an external reference frame it is possible to access entanglement from two copies of the state. This is because one of the states acts a reference for the other, activating its entanglement. Alternatively, one could consider that each of the entangled states acts a as partial reference frame for the other, allowing half of its entanglement to be accessed. This is an example of a case where no entanglement could be distilled from a single copy of the state (with no reference frame), but two copies of the state allows entanglement to be distilled. Hence the state $\ket{\psi}$ is not 1-distillable, but it is 2-distillable. That is to say that this state demonstrates the fact that the 1-distillable states are a subset of the 2-distillable states for the $S_2$-SSR.
\section{Beyond the $S_N$-SSR}
\subsection{Adding a stronger constraint} So far we have considered the problem of describing ensemble quantum information processing using the formalism for SSRs associated with some group. The $S_N$-SSR says that all elements (molecules) are subject to identical operations. This constraint has a demonstrable effect on the properties of the system, which can however be removed through use of additional resources such as reference frames. We now wish to consider the case where a stronger constraint than a SSR may apply to a system.
First we point out a difference between NMR experiments and spin-squeezing experiments, for which the $S_N$-SSR also applies. In the latter, it is possible to perform symmetric operations which entangle the elements (atoms), such as spin-squeezing unitaries \cite{Ueda} or quantum non-demolition measurements of $\hat{J}_z$ \cite{Kuz00}. By contrast, in NMR it is not possible to induce correlations between different molecules. The reasons for this difference are subtle, and relate to practical constraints due to decoherence during the read-out. This constraint also manifests itself in very low measurement efficiencies, but here we ignore that issue.
Consider the $M=1$ case for simplicity. Then all that can be done in practice in NMR experiments is \begin{itemize} \item \color{nblack} Rotations $\exp(-i\underline{\theta}\cdot\underline{\hat J}) = \exp(-i\underline{\theta}\cdot\sum_{k=1}^{N} \underline{\hat\sigma}^k/2)$. \item Destructive measurement of $\hat J_z = \sum_{k=1}^{N} \hat\sigma_z^k /2$. \end{itemize} Here $\underline{\hat\sigma}^k$ denotes $I\otimes\ldots I\otimes\underline{\hat\sigma}\otimes I\ldots\otimes I$ with $\underline{\hat\sigma}$ in the $k$th position. When making a measurement of this type (e.g. measuring $\hat J_z$) we actually get out an overall signal which is proportional to the sum of the spin ($\hat{\sigma}_z$) for each particle. Moreover, the final state of the ensemble is unrelated to the measurement result, due to thermal decoherence. Thus in general, the only operations possible in NMR are to make destructive measurements of symmetric observables that are additive over the ensemble: \begin{equation}\hat{O}_{\rm total}=\sum_{k}\hat{O}^k, \label{NMROperations}\end{equation} where $\hat{O}^k $ is the operator for the $k$th particle as above.
We call such operations {\em non-collective}. This terminology is
appropriate because the result of the measurement could be
obtained by individually measuring each element of the ensemble
and summing the results.
We can contrast such non-collective operations with a collective operation like measuring (destructively or otherwise) $\hat{J}^2$ to find out the value of the total angular momentum $j$ for the ensemble. This could not be done by measuring each particle and summing the results. Previous work using the $S_N$-SSR assumed that such collective measurements are possible. We will now consider the case where operations need not only be \emph{symmetric} but \emph{also non-collective}, as a stronger constraint on the system.
We suspect that we cannot completely characterize these constraints by any $G$-SSR. Instead we must supplement the $S_N$-SSR with the extra constraint that the operations also be non-collective. This complicates matters, as we are now unable to write down an equivalent state which is invariant under all the allowable operations. Despite this, we wish to determine if any
entanglement survives under this stronger constraint.
Since we are unable to determine an operationally equivalent state matrix for the constrained state we cannot calculate the extractable entanglement directly. However, if a Bell inequality violation can be demonstrated then this proves that entanglement is present in some form. So the question becomes, using the $S_N$-invariant state as a description for the system, is it possible to demonstrate Bell nonlocality using non-collective operations?
\subsection{Bell inequality for ensembles} For specificity, we consider the problem of demonstrating Bell nonlocality under symmetric, non-collective measurements on an ensemble of $N=2J$ Bell singlets, $\ket{\psi}=\ket{\psi^-}^{\otimes N}$. As discussed in Sec. \ref{Loss} the interesting part of this state can be written for simplicity as \begin{equation}
\mathcal{P}\left[\left(\ket{\psi}\bra{\psi}\right)^{\otimes N}\right]=\sum_{j=J-\lfloor J\rfloor}^{J}{d_j}^2\wp_j|\phi_j\rangle\langle\phi_j|,\label{Ensemble} \end{equation} which is an incoherent mixture of different spin ($j$) states. The added constraint means that we are unable to measure $\hat{J}^2$ directly, but can only measure components of spin (such as $\hat{J}_z$). Thus we must be derive a Bell inequality that allows for particles of different spin (i.e. different $j$ values).
Mermin \cite{Mer**} developed a Bell inequality for spin-$j$
particles by considering a generalization of the Bohm-EPR experiment. The only assumption that needs to be satisfied for this inequality to be applicable is that the desired state exhibit perfect anticorrelation in the spins of the two particles. The inequality can be written as, \begin{eqnarray} \left\langle\left|
m_A(\hat{a})- m_B(\hat{b})\right| \right\rangle &\geq& \frac{1}{J}\Big(\langle m_A(\hat{a})m_B(\hat{c})\rangle \Big.\nonumber\\ &&+\left.\langle m_A(\hat{b})m_B(\hat{c})\rangle\right),\label{MInequality}\end{eqnarray} where $m_i(\hat{a})$ represents the spin component of the $i$th particle in the $\hat{a}$ direction and $J$ is an upper bound on the $m_i(\hat{a})$. For Mermin's case one can (and Mermin does) choose $J=j$. However, we require that the parameter $J$ because we cannot distinguish between different $j$-values. Inequality (\ref{MInequality}) will be satisfied by any theory obeying local causality. For ease of analysis we define a quantity
\begin{eqnarray}M_J(\theta)&=& \left\langle\left|
m_A(\hat{a})-m_B(\hat{b})\right| \right\rangle\ \nonumber \\ && -\frac{1}{J}\left(\langle m_A(\hat{a})m_B(\hat{c})\rangle+\langle m_A(\hat{b})m_B(\hat{c})\rangle\right).\label{MJ}\end{eqnarray} The condition for local causality to be satisfied can thus be expressed as $M_J(\theta)\geq 0$.
Consider a Stern-Gerlach experiment such that the spin can be measured along one of three axes defined by coplanar vectors $\hat{a}$, $\hat{b}$, and $\hat{c}$. Mermin defined these axes such that the vectors $\hat{a}$ and $\hat{b}$ make the same angle $\pi/2+\theta$ with $\hat{c}$, and the angle $\pi-2\theta$ with each other. Using this set up for two perfectly anticorrelated spin-$j$ particles, quantum mechanics predicts that Eq. (\ref{MJ}) can be expressed as \begin{equation} M_J^{\rm spin-j}(\theta) =
f_j(\theta)-\frac{1}{J}\frac{2j}{3}\left(j+1\right)\sin\theta,\label{Mj}\end{equation} where the functions $f_j(\theta)$ are defined as \begin{equation} f_j(\theta)=\frac{1}{2j+1}\sum_{m,m'}\left| m-m'\right| \left|
\bra{m}e^{-2i\theta \hat{S}_y}\ket{m'}\right|^2, \label{fj}\end{equation}
and $\hat{S}_y$ is a spin matrix.
Now an ensemble of Bell singlet states is perfectly anticorrelated in spin and thus \erf{Ensemble} satisfies the necessary assumption for inequality (\ref{MInequality}) to be applicable. Also, when Mermin evaluated \erf{Mj} he assumed measurements of spin components, that is, non-collective measurements. Thus it is possible to use the same method as Mermin to evaluate the Bell inequality for an NMR ensemble, as all the relevant constraints are accounted for. The ensemble state simply behaves like an incoherent mixture of different spin-$j$ states.
Thus, for an ensemble of Bell singlet states, quantum mechanics predicts Eq. (\ref{MJ}) can be written as \begin{equation} M_J^{\rm Ensemble}(\theta)= \sum_{j=J-\lfloor J\rfloor}^Jd_j^2\wp_j M_J^{\rm spin-j}(\theta),\label{EnsembleMJ} \end{equation} where $M_J^{\rm Ensemble}(\theta)<0$ demonstrates Bell-nonlocality.
\subsection{Demonstrating Bell nonlocality}
We are now in a position to show that Bell-nonlocality survives under stronger constraints than those imposed by a SSR alone. To do this we must evaluate $M_J^{\rm Ensemble}(\theta)$ and show that it can become negative. To simplify this task it is instructive to recall the form of Eq. (\ref{Mj}). When Mermin evaluated these terms, he found to a good approximation (particularly for large $J$) that he was able to use a quadratic form to simplify their evaluation. Using the same approximation allows $M_J^{\rm Ensemble}(\theta)$ to be simplified to the expression \begin{equation}M_J^{\rm Ensemble'}(\theta)=\!\!\!\!\! \sum_{j=J-\lfloor J\rfloor}^J\!\!\!\!\!d_j^2\wp_j\left[\frac{2}{3} j\left(j+1\right)\sin\theta\left(2\sin\theta-\frac{1}{J}\right)\right]\!\!,\label{EnsembleApprox} \end{equation} where the prime indicates an approximation.
Now, the probability terms $d_j^2\wp_j$ in Eq. (\ref{EnsembleApprox}) are always positive, so the question becomes, can the remaining factor be negative? If this factor is negative for all terms in the sum, then $M_J^{\rm Ensemble'}(\theta)$ is negative and the \color{nblack} state exhibits Bell nonlocality. Examining the terms in the sum more closely reveals that there is always a linear (in $\sin\theta$) term subtracted from a quadratic (in $\sin\theta$) term. Hence, if $\theta$ (and thus $\sin\theta$) is small enough, then the linear term will always be dominant, resulting in a negative contribution to the sum. It is possible to choose $\theta$ to be small enough that every term in the sum will be negative, thus $M_J^{\rm Ensemble'}(\theta)<0$ and the ensemble state exhibits Bell nonlocality.
To put it explicitly (by solving for $\theta$ in terms of $J$) the ensemble state exhibits Bell-nonlocality despite the constraints when the detectors can be arranged to make measurements defined by $\theta$ where \begin{equation} 0 < \sin\theta <1/2J.\label{Range} \end{equation} This is actually a lower bound on the range of $\sin\theta$ for which a violation is possible. For small values of $J$ ($\leq3$), Eq. (\ref{EnsembleMJ}) can be explicitly calculated (without resorting to approximations). Even for these small values of $J$ the exact numerical results agree quite well \footnote{We expect the approximation to work well for large $J$, however, even for $J=1$ there is only 20\% difference between using the exact evaluation of Eq. (\ref{EnsembleMJ}) and the approx. given by Eq. (\ref{Range}). This difference drops to less than 8\% for $J=3$. As $J\rightarrow\infty$ the difference between Eq.s (\ref{EnsembleMJ}) and (\ref{Range}) vanishes.} with the range of angles specified by Eq. (\ref{Range}) and the agreement improves with larger $J$. This lends confidence that for large $J$ the approximation leading to Eq. (\ref{Range}) is a valid one.
Somewhat surprisingly, Eq. (\ref{Range}) gives exactly the same angular range for which Mermin demonstrated a pair of (unconstrained) entangled spin-$J$ particles exhibit Bell nonlocality. One may then ask which of the two systems, an ensemble of Bell states or a pair of spin-$J$ particles, violates the inequality more strongly. A way to measure this is to consider the depth of the violation, that is, how negative $M_J(\theta)$ becomes. For a pair of perfectly anticorrelated spin-$J$ particles, the minimum value of $M_J(\theta)$ converges to a constant value of $-1/12$ for large $J$. In contrast, for an ensemble of $N=2J$ Bell states, the minimum of $M_J^{\rm Ensemble'}(\theta)$ scales as $-1/J$. That is, the violation depth tends to zero for large ensembles. Thus, a pair of spin-$J$ particles violates this Bell inequality more strongly than an ensemble of $2J$ Bell states under our stronger constraint.
\section{Summary} In this paper we have classified groups of states based on their mixed state entanglement properties and related these states to the well known concepts of activation and distillation. We have also reviewed the analogy between mixed state entanglement and that of pure state entanglement constrained by a SSR. In particular we have focused on the symmetric group SSR. We have demonstrated that the $S_N$-SSR limits the amount of entanglement that can be accessed from an ensemble of entangled states. In comparison with U(1)-SSRs such as the particle number SSR we show how to apply the correct notion of multiple copies of an ensemble state to asymptotically recover the entanglement lost due to the SSR. We have also discussed the concepts of reference frames and given examples to illustrate the similarities between concepts of activation, distillation and use of reference frames (or multiple copies of states) to recover entanglement. For the $S_2$-SSR we showed that by using multiple copies of the ensemble, it is possible to only lose 1 ebit of entanglement (asymptotically).
Finally we gave an example where it does not seem possible to formulate the constraints on a system as a SSR. This situation arises naturally in the context of a liquid NMR ensemble. The lack of individual addressability requires that the $S_N$-SSR be considered. However, other technical constraints arise due to the large amount of thermal noise present in NMR ensembles. This noise manifests itself in two ways: low measurement efficiency and the fact that only non-collective measurements are possible. We addressed the latter manifestation and went on to show that despite this stronger constraint it is still possible in principle to demonstrate Bell nonlocality. It may prove interesting to attempt also to include the effect of the low efficiency constraint.
Further studies of physical constraints which cannot be formalized as SSRs may prove a fruitful area of research, not only for explaining experiments but also for understanding the properties of entanglement in general.
\begin{acknowledgments} This work was supported by the Australian Research Council and the State of Queensland. We thank P. Turner and A. C. Doherty for useful discussions.
\end{acknowledgments}
\end{document} | arXiv |
Izv. RAN. Ser. Mat.:
Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 4, Pages 75–82 (Mi izv4080)
Steiner symmetrization and the initial coefficients of univalent functions
V. N. Dubinin
Institute of Applied Mathematics, Far-Eastern Branch of the Russian Academy of Sciences
Abstract: We establish the inequality $|a_1|^2-\operatorname{Re}a_1a_{-1}\ge |a_1^*|^2-\operatorname{Re}a_1^*a_{-1}^*$ for the initial coefficients of any function $f(z)=a_1z+a_0+{a_{-1}}/z+\dotsb$ meromorphic and univalent in the domain $D=ż\colon |z|>1\}$, where $a_1^*$ and $a_{-1}^*$ are the corresponding coefficients in the expansion of the function $f^*(z)$ that maps the domain $D$ conformally and univalently onto the exterior of the result of the Steiner symmetrization with respect to the real axis of the complement of the set $f(D)$. The Pólya–Szegő inequality $|a_1|\ge |a_1^*|$ is already known. We describe some applications of our inequality to functions of class $\Sigma$.
Keywords: Steiner symmetrization, capacity of a set, univalent function, covering theorem.
DOI: https://doi.org/10.4213/im4080
Izvestiya: Mathematics, 2010, 74:4, 735–742
UDC: 517.54
MSC: Primary 30C50; Secondary 30C85
Citation: V. N. Dubinin, "Steiner symmetrization and the initial coefficients of univalent functions", Izv. RAN. Ser. Mat., 74:4 (2010), 75–82; Izv. Math., 74:4 (2010), 735–742
\Bibitem{Dub10}
\by V.~N.~Dubinin
\paper Steiner symmetrization and the initial coefficients of univalent functions
\jour Izv. RAN. Ser. Mat.
\mathnet{http://mi.mathnet.ru/izv4080}
\crossref{https://doi.org/10.4213/im4080}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2010IzMat..74..735D}
\jour Izv. Math.
\crossref{https://doi.org/10.1070/IM2010v074n04ABEH002505}
http://mi.mathnet.ru/eng/izv4080
https://doi.org/10.4213/im4080
http://mi.mathnet.ru/eng/izv/v74/i4/p75
V. N. Dubinin, "Lower bounds for the half-plane capacity of compact sets and symmetrization", Sb. Math., 201:11 (2010), 1635–1646
V. N. Dubinin, "Asymptotic Behavior of the Capacity of a Condenser as Some of Its Plates Contract to Points", Math. Notes, 96:2 (2014), 187–198
V. N. Dubinin, "Geometric estimates for the Schwarzian derivative", Russian Math. Surveys, 72:3 (2017), 479–511
Dubinin V.N., "Some Unsolved Problems About Condenser Capacities on the Plane", Complex Analysis and Dynamical Systems: New Trends and Open Problems, Trends in Mathematics, ed. Agranovsky M. Golberg A. Jacobzon F. Shoikhet D. Zalcman L., Birkhauser Verlag Ag, 2018, 81–92
First page: 15 | CommonCrawl |
Journal of Infrastructure Preservation and Resilience
Enclosing contour tracking of highway construction equipment based on orientation-aware bounding box using UAV
Yapeng Guo1,
Yang Xu2,
Zhonglong Li1,
Hui Li2 &
Shunlong Li1
Journal of Infrastructure Preservation and Resilience volume 4, Article number: 4 (2023) Cite this article
Construction equipment tracking of highway construction site can obtain the spatiotemporal location in real time and provide data basis for construction risk control. The complete 2D moving of construction equipment in surveillance videos could be spatially represented by the translation, rotation and size change of corresponding images. To describe the temporal relationships of these variables, this study proposes a construction equipment enclosing contour tracking method based on orientation-aware bounding box (OABB), where UAV surveillance videos are employed to alleviate the occlusion problem. The method balances the rotation insensitivity of horizontal bounding box and the complexity of pixel-level segmented contour, which has three modules. The first module integrates OABB into a deep learning detector to provide detected contours. The second module updates OABBs with Kalman prediction to output tracked contours. The third module manages IDs of multiple tracked contours for construction equipment motions. Five in-situ UAV videos including 4325 frames were employed as the evaluation dataset. The tracking performance achieved 2.657 degrees in angle error, 97.523% in MOTA and 83.243% in MOTP.
The motion of construction equipment in the 2D plane based on computer vision can be defined by translation and rotation. Considering that the distance from the photography plane to construction equipment might change, pixel size of corresponding equipment image also needs to be included. These constitute a complete 2D spatial description of the plane moving pattern of construction equipment, which is represented by the enclosing contour in this study. Precise spatial–temporal information of construction equipment is one of the most important datatypes in construction sites [1,2,3], which can be used to provide location feedback for equipment engaged in hazardous operations and early warning for construction personnel around the equipment. Furthermore, such information can provide the basis for the organization and guidance of traffic flow at key nodes of construction sites and for the analysis of working productivity efficiency [4, 5]. Enclosing contour tracking of construction equipment, used for relatively precise spatial–temporal information acquisition, has become critical to improve efficiency and ensure safety in construction sites.
Kinematic-based construction equipment tracking methods using installed devices (e.g., radio frequency identification, global positioning systems, ultra-wideband, Bluetooth low-energy, accelerator) [2,3,4,5,6,7,8,9,10,11,12] have been validated with good accuracy and real-time processing speed for moving trajectories extraction. In addition to those approaches, vision-based sensing methods have become promising due to non-contact, low cost and abundant data. Many methods have been conducted treating equipment as a point, i.e., trajectory identification), including 2D trajectory [13,14,15,16] and 3D trajectories [17, 18]. These methods concentrate on the translations of construction equipment, but when the construction equipment is close to each other or close to the workers, its volume cannot be ignored. Therefore, the identification of more accurate information of construction equipment has attracted the attention of researchers, i.e., treating equipment as an enclosing contour.
Using horizontal bounding box (HBB) to represent the construction equipment enclosing contour and track the size (width and height) in addition to the translation (centre point coordinates) can alleviate the above limitations. HBB-based construction equipment enclosing contour tracking methods can detect rough equipment regions [19,20,21,22,23,24]. However, HBB has no rotation sensitivity, and its region contains a large number of non-equipment parts. Pixel-level segmented contour tracking is an appropriate way to accurately represent the construction equipment spatial–temporal information [1]. But robust segmented contour tracking based on deep learning needs complex manual labelling and temporal contour association, which would be superfluous for the 2D spatio-temporal description.
Thus, to balance the rotation insensitivity of the HBB and the high calculation complexity of pixel-level segmented contour, this study proposes an enclosing contour tracking method for construction equipment based on OABB using UAV surveillance videos. This study is arranged as follows: Sect. " Literature review" presents a literature review on vision-based tracking for construction equipment and arbitrary-oriented object detection; Sect. " Methodology" illustrates the methodology of the proposed approach; Sect. " Evaluation and implementation details" describes the dataset used to evaluate the algorithm, the evaluation metrics and the implementation details; Sect. " Results and discussions" shows the tracking results both qualitatively and quantitatively, with a discussion of the key update factor; Sect. " Conclusions and future works" concludes the research.
In this section, tracking methods on vision-based for construction equipment will be reviewed. Because this research integrates OABB into the tracking method, research work in the field of arbitrary-oriented object detection will also be reviewed comprehensively.
Vision-based tracking methods for construction equipment
Many studies on construction equipment tracking based on computer vision techniques have been conducted. Some of them focus on the translation (moving trajectory) identification which treat the construction equipment as one point. Kim et al. [13] presented a mobile construction equipment 2D trajectory extraction method based on deep learning detector and image rectification technique using UAV videos. Tang et al. [14] took 2D tracks of construction equipment and predicted their locations using long short-term memory network and mixture density network. Zhao et al. [15] proposed a construction equipment tracking for 2D trajectory extraction using deep learning. Zhu et al. [16] proposed a particle filter-based construction equipment tracking method to acquire 2D trajectories. To calculate more accurate spatial locations of construction equipment, they [18] also developed a novel Kalman filter-based tracking method to estimate 3D positions using stereo vision. Jog. et al. [17] developed a multiple equipment position monitoring method using 3D coordinates. These studies can timely and accuratel + y track construction equipment and obtain their trajectories. However, when construction equipment are close to each other or workers, only treating the construction equipment as a point will lead to the loss of information, which cannot be effectively described its spatial–temporal information.
The enclosing contour of the construction equipment using HBB can provide more information than the aforementioned point-represented construction equipment methods, in addition to the trajectory there are time-varying width and height. Zhu et al. [24] presented an automatic construction equipment detection and tracking method using HBBs for better precision and recall. Kim and Chi [20] adapted a 2D long-term construction equipment tracking method integrated with real-time online learning-based detector and tracker. Kim and Chi [21] also conducted researches on excavator and truck tracking method based on cross-camera matching techniques. Chen et al. [19] proposed a detection and tracking method for construction equipment to recognize their activities. Xiao and Kang [22] developed a construction equipment tacker using deep learning detector integrated technique. They [23] also proposed a robust night-time construction equipment tracker using deep learning illumination enhancement. These HBB-based tracking methods can reflect the size changes of the construction equipment. But when the aspect ratio of the construction equipment is much greater than 1 or the spatial distribution is dense, the HBB-based enclosing contour would contain a lot of non-target information. Wang et al. and Bang et al. [1, 25] employed instance segmentation method to extract the pixel-level segmented contours of construction equipment. This is an appropriate way for the construction equipment representation. But robust segmented contour tracking based on deep learning needs complex manual labelling and temporal contour association, which would be superfluous for the moving pattern recognition and tracking.
Arbitrary-oriented object detection methods
OABB is a rotatable rectangle with one more parameter rotating angle than HBB, which is the basis of arbitrary-oriented object representation. Because the perspective of the overhead-view images can better reflect the moving patterns of targets, the basic five parameters can be extracted from images intuitively and accurately, so OABB is more used to detect the enclosing contour of targets in overhead-view images [26, 27].
In recent years, many researchers have devoted their efforts on five-parameter detection based on OABBs. In overhead-view images, targets are distributed with random orientations, which makes detecting targets in this field challenging. Chen et al. [28] designed a OABB-based detection model consisted of two CNN networks, in which one CNN was for arbitrary-oriented regions with the orientation information and the other was for object recognition with multi-level feature extraction. Ma et al. [29] proposed a two-stage multi-oriented detector based on CNN in optical remote sensing images using for OABB prediction. Guo et al. [26] developed a single-stage orientation-aware construction vehicle precise detection approach using CNN with feature fusion technique.
Research challenges and objectives
As mentioned before, vision-based enclosing contour tracking of construction equipment is an important mean to obtain spatial–temporal information in large construction sites. The current vision-based construction equipment tracking methods needed to be strengthened in two aspects: in addition to the translation and size change information obtained by the point-represented or HBB-represented tracking methods, the rotation information should be included; considering the complex manual labelling and temporal association in the pixel-level segmented contour, the concise tracking methodology balancing the accuracy and complexity should be considered.
The objective of this study is to develop an enclosing contour tracking method of construction equipment to acquire not only moving trajectories but also temporal sizes and rotating angles. OABB instead of HBB was employed to establish the robust and accurate tracking model for construction equipment using UAV surveillance videos.
In this section, the three modules of the OABB based tracking method of construction equipment, including enclosing contour detection, enclosing contour update, and tracking ID managing, are described in detail. Firstly, the enclosing contour is parameterized using five variables of OABB, a CNN-based contour detection model with multi-level features is built and the loss function is defined; secondly, the video frames are input to the model to get detected contours, and the motion model of the construction equipment is built to get predicted contours, tracked contours are updated from predicted contours using the detected contours; finally, the intersection over union (IOU) of OABBs is used to add, keep or delete multiple construction equipment IDs to obtain the tracking status of each equipment.
Enclosing contour detection
The CNN-based detection module describes the construction equipment in images by OABB enclosing contours. Figure 1 shows the difference between HBB and OABB. HBB is defined by four parameters: centre point coordinate (x, y), width (w), and height (h), while OABB is defined by five parameters: x, y, w, h and rotating angle (r). Figure 1(b) compares the effects of equipment representations with two kinds of bounding box. The enclosing contour detection model, which aims to generate and regress OABBs, is modified from the CenterNet [30]. The model consists of two parts: backbone and detection head, as shown in Fig. 2.
Difference between HBB and OABB: (a) description parameters, (b) equipment representations
Detailed architecture of the anchor-free equipment OABB detector
Backbone provides multi-level features of construction equipment. A modified ResNet-18 base network (mResNet-18) is employed with four residual blocks, each comprising four convolutional layers with two shortcut connections. The residual network has a better fitting ability for extracting more accurate features, and it can also solve the problem of optimisation training when the number of layers increases. Four deconvolution layers are added to recover the spatial information. To speed up the detection efficiency, the output size of the mResNet18 is M / 4 × N / 4 (the size of the input image is M × N).
There are four regression parts in the detection head based on the OABB: centre point regression (x, y), offset regression (offx, offy), width and height regression (w, h), and angle regression (r). The four regression parts aim to learn the integers of the centre point coordinates, decimals of the centre point coordinate, width and height, and rotating angle of the OABBs with feature maps processed by (3 × 3 × 64, 1 × 1 × 2), (3 × 3 × 64, 1 × 1 × 2), (3 × 3 × 64, 1 × 1 × 2), and (3 × 3 × 64, 1 × 1 × 1) convolutional kernels, respectively. In the network inference stage, the heat maps from the centre point regression are processed based on 3 × 3 max-pooling, which functions as non-maximum suppression.
To decrease the difficulty of training and increase the efficiency of inference, a Gaussian heat map generated from the ground truth centre point coordinates (x0, y0) is employed in this research. wxy and \(\hat{w}_{xy}\) are the actual and predicted weights in the Gaussian heat map, respectively. The Gaussian heat map weight at coordinate (x, y) is calculated based on a Gaussian kernel with six parameters: the Gaussian mean (μ1, μ2), Gaussian variance (σ1, σ1), and window size (r1, r2), using Eq. (1), as follows:
$$\begin{gathered} w_{x,y} = \left\{ {\begin{array}{*{20}c} {\exp \left\{ { - \frac{1}{2}\left[ {\frac{{(x - \mu_{1} )^{2} }}{{\sigma_{{1}}^{{2}} }} + \frac{{(y - \mu_{2} )^{2} }}{{\sigma_{{2}}^{{2}} }}} \right]} \right\},x_{0} - \frac{{r_{1} }}{2} < x < x_{0} + \frac{{r_{1} }}{2},y_{0} - \frac{{r_{2} }}{2} < y < y_{0} + \frac{{r_{2} }}{2}} \\ {0,others} \\ \end{array} } \right. \\ \mu_{1} = x_{0} ,\mu_{2} = y_{0} ,\sigma_{1} = \lambda w,\sigma_{2} = \lambda h,r_{1} = 2\sigma_{1} + 1,r_{2} = 2\sigma_{2} + 1 \\ \end{gathered}$$
The final Gaussian heat map weights at the coordinates (xg, yg) are modified based on the rotating angle of the construction equipment as shown in Eq. (2).
$$w_{{x_{g} ,y_{g} }} = \left\{ {\begin{array}{*{20}c} {w_{x,y} ,if\left[ {\begin{array}{*{20}c} {x_{g} = (x - x_{0} )\cos ag - (y - y_{0} )\sin ag + x_{0} } \\ {y_{g} = (x - x_{0} )\sin ag + (y - y_{0} )\cos ag + y_{0} } \\ \end{array} } \right]} \\ {0,others} \\ \end{array} } \right.$$
The training loss of the enclosing contour detector (Ldet, defined by Eq. (3)) is divided into four components, designed based on the detection head: the centre loss (Lc), offset loss (Lo), width and height loss (Lwh), and angle loss (Lag). λc, λo, λwh, and λag are the corresponding weights, respectively. The centre loss employs focal loss for better training convergence, as controlled by Eq. (4), where α and β are adjustment parameters, and N is the number of heat map points, and the other three employ the L1 loss to regress the corresponding parameters.
The enclosing contour detection model is pretrained by construction equipment in MOCS proposed by An et al. [31]. For better generalization, the trained network is then fine-tuned by the collected overhead-view construction equipment dataset. The images of this dataset are captured by drone-borne cameras at different heights and angle, containing 600 images and 1570 equipment.
$$L_{\det } = \lambda_{c} L_{c} + \lambda_{o} L_{o} + \lambda_{wh} L_{wh} + \lambda_{ag} L_{ag}$$
$$L_{c} = \frac{1}{N}\sum\limits_{xy} {\left\{ {\begin{array}{*{20}c} {(1 - \hat{w}_{xy} )^{\alpha } \log (\hat{w}_{xy} ){\text{, if }}w_{xy} = 1} \\ {(1 - w_{xy} )^{\beta } (\hat{w}_{xy} )^{\alpha } \log (1 - \hat{w}_{xy} ){\text{, otherwise}}} \\ \end{array} } \right.}$$
Enclosing contour update
The detection module could generate high-confidence enclosing contour of construction equipment at each frame without considering the temporal context information, resulting in an inability to match construction equipment between different frames. Inspired by Bewley et al. [32], this module employs a Kalman filter to model the frame-by-frame enclosing contours from detection module in the time domain. The Kalman filter predicts the enclosing contours based on the previous contours, and weights the predicted contours with the detected contours for much more accuracy. The state variables of OABB-based construction equipment motion (translation, size change and rotation) can be described as shown in Eq. (5):
$${\mathbf{x}} = \left[ {c_{x} ,c_{y} ,w,h,r,c^{\prime}_{x} ,c^{\prime}_{y} ,w^{\prime},h^{\prime},r^{\prime}} \right]^{{\text{T}}}$$
where \(c^{\prime}_{x}\),\(c^{\prime}_{y}\),\(w^{\prime}\),\(h^{\prime}\) and \(r^{\prime}\) are the first derivatives of the corresponding OABB parameters. Assuming that the construction equipment is moving at a relatively low speed (reasonable for equipment at construction sites), the size and orientation of the equipment will change uniformly over a short time Δt. The state function describing OABB-based construction equipment motion could be expressed as Eq. (6):
$${\hat{\mathbf{x}}}_{k|k - 1} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & {\Delta t} & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & {\Delta t} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & {\Delta t} & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & {\Delta t} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & {\Delta t} \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]\left\{ \begin{gathered} c_{x,k - 1} \hfill \\ c_{y,k - 1} \hfill \\ w_{k - 1} \hfill \\ h_{k - 1} \hfill \\ r_{k - 1} \hfill \\ c^{\prime}_{x,k - 1} \hfill \\ c^{\prime}_{y,k - 1} \hfill \\ w^{\prime}_{k - 1} \hfill \\ h^{\prime}_{k - 1} \hfill \\ r^{\prime}_{k - 1} \hfill \\ \end{gathered} \right\} = {\mathbf{Fx}}_{k - 1} { + }{\mathbf{w}}_{k - 1}$$
where \({\mathbf{x}}_{k - 1}\) represents the construction equipment state at the (k-1)th frame and \({\hat{\mathbf{x}}}_{k|k - 1}\) is calculated state estimation at the kth frame using \({\mathbf{x}}_{k - 1}\) and state function; \(\Delta t\) is the time interval of per frame, and F is the state transition matrix; \({\mathbf{w}}_{k - 1}\) indicates process noise of the investigated equipment motion model, assumed to be white noise with 0 mean and \({\mathbf{Q}}_{k - 1} = E\left( {{\mathbf{w}}_{k - 1} {\mathbf{w}}_{k - 1}^{{\text{T}}} } \right)\) covariance. The covariance estimation of the state variables, described by the state covariance matrix P, can be obtained by linearization of the equipment motion model from Eq. (7):
$${\hat{\mathbf{P}}}_{k|k - 1} = {\mathbf{FP}}_{k - 1} {\mathbf{F}}^{{\text{T}}} + {\mathbf{Q}}_{k - 1}$$
where \({\hat{\mathbf{P}}}_{k|k - 1}\) illustrates the predicted state covariance matrix using optimal estimation \({\mathbf{P}}_{k - 1}\) and the investigated equipment motion model.
In Kalman prediction stage, the predicted contours have certain difference with actual situations. Therefore, at this stage, the contour information of the detected construction equipment would be used as the measured value(s) for the Kalman update. The state transition from the state vector to the measurements is shown in Eq. (8), where zk is the measurement of the kth frame, and H is the measurement matrix. Only the former five parameters can be acquired from the actual detected contours; thus, the size of H is 5 × 10.
$${\mathbf{z}}_{k} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ \end{array} }\,\,\right]\left[ {\begin{array}{*{20}c} {c_{x,k} } \\ {c_{y,k} } \\ {w_{k} } \\ {h_{k} } \\ {r_{k} } \\ {c^{\prime}_{x,k} } \\ {c^{\prime}_{y,k} } \\ {w^{\prime}_{k} } \\ {h^{\prime}_{k} } \\ {r^{\prime}_{k} } \\ \end{array} } \right] = {\mathbf{Hx}}_{k} + {\mathbf{v}}_{k}$$
where \({\mathbf{v}}_{k}\) represents the measurement noise, assumed to be white noise with 0 mean and \({\mathbf{R}}_{k} = E\left( {{\mathbf{v}}_{k} {\mathbf{v}}_{k}^{{\text{T}}} } \right)\) covariance. The Kalman gain (\({{\varvec{\upkappa}}}\)), calculated using Eq. (9), is the core matrix in the Kalman filter, considering both the prediction and the measurements to update
$${{\varvec{\upkappa}}}_{k} {\mathbf{ = \hat{P}}}_{k|k - 1} {\mathbf{H}}^{{\text{T}}} \left( {{\mathbf{H\hat{P}}}_{k|k - 1} {\mathbf{H}}^{{\text{T}}} {\mathbf{ + R}}_{k} } \right)^{ - 1}$$
Using the Kalman gain, the state vectors and state variances of the construction equipment from the Kalman prediction can be updated using Eqs. (10) and (11). And the updated OABB information of the construction equipment considering temporal detection information can be set as the final tracked enclosing contour of the kth frame.
$${\mathbf{x}}_{k} {\mathbf{ = \hat{x}}}_{{k{|}k - 1}} {\mathbf{ + \kappa }}_{k} {\mathbf{(z}}_{k} {\mathbf{ - H\hat{x}}}_{k} {\mathbf{)}}$$
$${\mathbf{P}}_{k} {\mathbf{ = }}\left( {{\mathbf{I - \kappa }}_{k} {\mathbf{H}}} \right){\hat{\mathbf{P}}}_{k|k - 1}$$
Tracking ID managing
The allocation of construction equipment IDs is a core issue in multiple construction equipment tracking. Most HBB-based tracking methods lead to the overlapping of boxes for multiple objects, resulting in a high complexity in the data associations between frames. For the OABB represented construction equipment, there is hardly no overlap between the OABBs. Therefore, this research employs the IOU of the OABB as the indicator for the ID managing part (calculated by Eq. (12)).
$$I(U\left(a,b\right)=\frac{OABB_a\bigcap oABB_b}{OABB_a\bigcup AAB_b}$$
The ID allocation of construction equipment can be divided into three states: add, keep, and delete. The result of the detected contours and that of the predicted contours are used to calculate the IOU. When the ratio is greater than the pre-setting threshold (IOUt), the situation is denoted as 'matched'; otherwise, it is denoted as 'unmatched'. When there is an unmatched detected contour and the situation lasts for three consecutive frames, a new equipment ID should be added. When there is an unmatched predicted contour and the situation lasts for three consecutive frames, the corresponding equipment ID should be deleted. The matched detected OABB is used as the measurement for participating in the Kalman update to generate the final tracked contour, and the corresponding equipment ID is maintained.
Evaluation and implementation details
Dataset description
This dataset contains five video clips in various construction environments, captured by cameras mounted on UAVs. All videos were captured in 1080 × 1080 pixels and filmed at 30 frames per second (FPS) at different heights and view angles. The dataset includes single and multiple equipment, static and moving equipment, hovering and fast-moving cameras, with a total length of 4325 frames, 18 equipment, and 8174 contours, typical frames of evaluation videos are shown in Fig. 3. A detailed description is provided in Table 1. For convenience, annotation was performed every 10 frames. The labelling format is as follows: frame number, equipment ID, centre point coordinates, width and height, angle, and category (confidence score).
Example frames of evaluation videos
Table 1 Description of evaluation dataset for overhead-view construction equipment tracking
Evaluation metrics
The multiple object tracking (MOT) challenge [33] is a multiple object tracking benchmark, and is widely used to evaluate tracker performance. The evaluation metrics employed in this research are modified from the MOT challenge.
Multiple object tracking accuracy (MOTA) and multiple object tracking precision (MOTP) are core evaluation indexes used to jointly measure a tracker's ability to continuously track objects (i.e. accurately determining the number of objects in consecutive frames, and accurately delineating their positions, so as to achieve uninterrupted continuous tracking). MOTA mainly considers the accumulation of object-matching errors in tracking, and mainly includes FP, FN, and IDs (described as Eq. (13)).
$$MOTA = 1 - \frac{{\sum {(FN + FP + IDs)} }}{{\sum {GT} }} \in ( - \infty ,1)$$
FP and FN represent the wrongly tracked equipment and unmatched ground truth equipment in the unmatched status, respectively. IDs denotes the number of ID switches assigned to ground truth equipment, and GT is the total number of ground truth equipment. MOTA measures the performance of trackers in detecting objects and tracking, and is not affected by the detector performance. MOTP reflects the accuracy of determining the object position and size, and is highly affected by detector performance. The MOTP is calculated using Eq. (14).
$$MOTP = \frac{{\sum\nolimits_{b,a} {IOU(a,b)} }}{{\sum\nolimits_{a} {c_{a} } }} \in (0,1)$$
where a is the frame number, b is the equipment number, ca is the number of trackers in the matched status, and IOU(a,b) is the IOU value of the matched equipment OABBs.
AR represents the mean square error of tracking rotating angles in degrees. MT represents the number of trajectories matching the ground truth successfully in over 80% of the total frames, respectively. RC and PR are the recall and precision, and represent the ratio of TP OABBs to ground truth OABBs and ratio of TP OABBs to all detected OABBs, respectively. Hz is the processing speed of the algorithms, including the detector in this research; which is different from that used in the MOT challenge.
In the enclosing contour detection module, the excavator, truck, loader, roller and concrete mixer truck categories from the MOCS dataset [31] were selected for pretraining with 1000 epochs. The proposed dataset was processed using augmentation techniques, and then was re-trained or fine-tuned using the weights from pretraining. The total re-training epoch was 350, with an initial learning rate of 1.25 × 10–4, and a 0.1-fold decay was performed at epochs 200 and 300. The loss weights in Eq. (3), i.e. λc, λo, λwh, and λag were set to 1.0, 1.0, 0.5, and 1.0, respectively. An Adam optimiser was employed in this training with default hyperparameters to achieve better convergence.
In the enclosing contour update module, as shown in Eq. (15), the state covariance matrix P0 was initiated, and the measurement covariance matrix Rk was set as the identity matrix. To find the proper parameter of the process covariance matrix Qk, λ was used to represent the relationship between Rk and Qk, and is set as 5.0. IOUt was set as 0.8.
$${\mathbf{P}}_{0} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {10} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {10} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {10} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {10} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {10} \\ \end{array} } \right],{\mathbf{R}}_{k} = {\mathbf{I}},{\mathbf{Q}}_{k} = \lambda {\mathbf{I}}$$
In the experiments, a modified tracking method from SORT (mSORT) [32] was chosen as the baseline method to compare with the proposed method in this study to test the tracking results of evaluation videos. Because SORT method is one of the state-of-the-art methods in the field of multiple object tracking, characterized by a flexible framework and fast tracking speed. In addition, the mSORT used for comparison with the proposed method employed the same detector backbone, based on HBB generation and regression to detect construction equipment, and was trained using the same dataset. It also used Kalman filtering for HBB prediction of construction equipment and used more complex linear assignment and IOU of HBBs for ID management.
The hardware platform employed mainly includes an Intel Xeon(R) E5-2620 v4 CPU, a Nvidia GTX 1080Ti GPU, and 32 GB of memory.
Results and discussions
The experimental results using the proposed method and the baseline method are shown in Table 2. To better compare the differences between the two methods, Fig. 4 shows the tracking results of five video example frames, where the solid line box represents results from the proposed method and the dashed line box from mSORT. The tracking performance of the five video clips from the evaluation dataset was averaged. The proposed method achieved the recall of 99.381%, precision of 98.165%, MOTA of 97.523%, MOTP of 83.243%. Meanwhile, MT = 18 indicates that the proposed method successfully tracked all 18 trajectories of construction equipment. From the tracking results, it can be seen that the proposed method can accurately and robustly track construction equipment from the overhead-view videos. Specifically, the proposed method improves 25.387% over mSORT on precision and 24.549% on MOTP. It is worth noting that the proposed method achieves 97.523% MOTA, which proves high robustness. The MOTP metric can also be improved by improving the backbone with higher feature extraction efficiency and increasing the amount of training data. The overall AR achieved an averaged 2.657 degrees, which validates the effectiveness of the rotation tracking. There is no significant difference between the tracking speed of the proposed method and mSORT, both up to about 30 frames per second, which can be called real-time processing algorithms. If the speed of the algorithm needs to be further increased, it can be done by improving the hardware capability or by using techniques such as parallel coding.
Table 2 Quantitative evaluation tracking results for the evaluation dataset
Tracking results comparison between the proposed method and mSORT
In the evaluation results, CVT-01 contains only one moving construction equipment, and the proposed method achieved 88.025% of MOTP, which improved 24.801% comparing to mSORT. That proves the effectiveness of the proposed OABB for single equipment representation. The two parked construction equipment filmed with a fast-rotating camera are continuously assigned two IDs in CVT-02, with a MOTP of 81.804%. The proposed method improved 27.406% of MOTP than mSORT. CVT-03 contains dense multiple construction equipment and has a construction equipment moving out of view and another equipment moving into view, and the proposed method successfully deleted the ID of the former when it disappeared, and allocated a new ID for the latter with a MOTP of 84.790%. There are eight successive different construction equipment entering in CVT-04 with a MOTP of 76.59%, and the proposed method correctly handles the complex destruction and creation of equipment IDs with accurate detections. The AR achieved 4.374 degrees, which is the highest among the five videos. That validates the difficulty of small equipment rotation identification. CVT-05 contains two construction equipment in cooperative operation; one of them moves out of view, and then moves into view again. The proposed method achieved 84.366% of MOTP. The equipment was allocated to different IDs, because the proposed method could not re-recognise the same equipment which re-entered the view. In conclusion, the tracking results illustrate that the proposed method can accurately detect construction equipment and stably track different equipment, and has a significant improvement on tracking accuracy comparing to mSORT.
Influences of OABB update parameter
The enclosing contour update is conducted by the fusion of detected OABB and predicted OABB. The measurement covariance matrix R represents the detection noise in the equipment OABB generation and regression, which is validated as a high-confidence detector. Thus, R is set to a small value (the identity matrix in this research). The process covariance matrix Q reflects the process noise of the assumed dynamic motion model, and is abstracted from the complex actual situation. λ controls the ratio of Q to R, and Table 3 shows the quantitative evaluation results for different λs.
Table 3 Quantitative evaluation tracking results with different λs
Table 3 indicates that when λ is greater than or equal to 5.0, that is, the measurement error is relatively small, there is an increase in the MOTA, but there are no evident changes in the other indicators. Therefore, in this study, λ is set to 5.0. This experiment also proves that the proposed tracking method is robust to the assumptions of the construction equipment motion model.
Conclusions and future works
This study proposes a fully automated vision-based enclosing contour tracking method for construction equipment of highway construction sites to obtain the spatial–temporal information of equipment motion. The conclusions could be drawn as follows:
(1) The proposed method integrated OABB to CNN enclosing contour detection of construction equipment; presented a ten-parameter motion model of construction equipment for enclosing contour prediction and updating using Kalman filtering; and finally employed IOU metric instead of complex data association process for ID management of multiple construction equipment.
(2) The proposed method was tested using five evaluation videos, obtaining 2.657 degrees in angle error, 97.523% of MOTA and 83.269% of MOTP, a satisfactory level in multiple object tracking field. And the proposed method could track all 18 trajectories of construction equipment. The experimental results show the advantage of arbitrary-oriented object tracking compared to the widely-used mSORT method.
In this study, the proposed method is suitable for accurate tracking of construction equipment within the field of view. The limitation of this paper is that when the tracked construction equipment gradually moves out of the field of view and then enters the field of view again, the proposed method will renumber the equipment as a new construction equipment, that is, the proposed method does not have the ability to re-identify the equipment. The future work will focus on improving the re-identification capability to track construction equipment in re-entering view. Another future direction is to lightweight the contour detection network which is expected to be deployed on mobile devices.
The data and code are available upon request.
OABB:
Orientation-aware bounding box
UAV:
HBB:
Horizontal bounding box
CNN:
mResNet-18:
The modified ResNet-18 base network proposed in this paper
IOU:
Intersection over union
MOT:
Multiple object tracking
MOTA:
Multiple object tracking accuracy
MOTP:
Multiple object tracking precision
FP:
False Positive
FN:
TP:
True Positive
GT:
Ground Truth
The mean square error of tracking rotating angles in degrees
The number of trajectories matching the ground truth successfully in over 80% of the total frames
RC:
MOCS:
A dataset named Moving objects in construction sites
mSORT:
The modified tracking method from SORT employed in this paper
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The authors appreciate the National Natural Science Foundation of China, Heilongjiang Natural Science Foundation and Fundamental Research Funds for Central Universities for support of this research.
The financial support for this study was provided by the NSFC [Grant Nos. 51922034 and 52278299], the Heilongjiang Natural Science Foundation for Excellent Young Scholars [Grant No. YQ2019E025] and Fundamental Research Funds for Central Universities (Grant No. FRFCU5710051018).
School of Transportation Science and Engineering, Harbin Institute of Technology, Harbin, 150090, China
Yapeng Guo, Zhonglong Li & Shunlong Li
School of Civil Engineering, Harbin Institute of Technology, Harbin, 150090, China
Yang Xu & Hui Li
Yapeng Guo
Yang Xu
Zhonglong Li
Hui Li
Shunlong Li
Yapeng Guo: conduct literature review, build models and analyse, draft the manuscript; Yang Xu: provide assistance on building models; Zhonglong Li, provide assistance on the dataset; Hui Li, refine the manuscript; Shunlong Li, envision the study. The authors read and approved the final manuscript.
Correspondence to Shunlong Li.
Guo, Y., Xu, Y., Li, Z. et al. Enclosing contour tracking of highway construction equipment based on orientation-aware bounding box using UAV. J Infrastruct Preserv Resil 4, 4 (2023). https://doi.org/10.1186/s43065-023-00071-y
Construction equipment tracking
UAV surveillance videos
Highway construction site
Rotating angle | CommonCrawl |
Approximate identity
In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element.
Definition
A right approximate identity in a Banach algebra A is a net $\{e_{\lambda }:\lambda \in \Lambda \}$ such that for every element a of A, $\lim _{\lambda \in \Lambda }\lVert ae_{\lambda }-a\rVert =0.$ Similarly, a left approximate identity in a Banach algebra A is a net $\{e_{\lambda }:\lambda \in \Lambda \}$ such that for every element a of A, $\lim _{\lambda \in \Lambda }\lVert e_{\lambda }a-a\rVert =0.$ An approximate identity is a net which is both a right approximate identity and a left approximate identity.
C*-algebras
For C*-algebras, a right (or left) approximate identity consisting of self-adjoint elements is the same as an approximate identity. The net of all positive elements in A of norm ≤ 1 with its natural order is an approximate identity for any C*-algebra. This is called the canonical approximate identity of a C*-algebra. Approximate identities are not unique. For example, for compact operators acting on a Hilbert space, the net consisting of finite rank projections would be another approximate identity.
If an approximate identity is a sequence, we call it a sequential approximate identity and a C*-algebra with a sequential approximate identity is called σ-unital. Every separable C*-algebra is σ-unital, though the converse is false. A commutative C*-algebra is σ-unital if and only if its spectrum is σ-compact. In general, a C*-algebra A is σ-unital if and only if A contains a strictly positive element, i.e. there exists h in A+ such that the hereditary C*-subalgebra generated by h is A.
One sometimes considers approximate identities consisting of specific types of elements. For example, a C*-algebra has real rank zero if and only if every hereditary C*-subalgebra has an approximate identity consisting of projections. This was known as property (HP) in earlier literature.
Convolution algebras
An approximate identity in a convolution algebra plays the same role as a sequence of function approximations to the Dirac delta function (which is the identity element for convolution). For example, the Fejér kernels of Fourier series theory give rise to an approximate identity.
Rings
In ring theory, an approximate identity is defined in a similar way, except that the ring is given the discrete topology so that a = aeλ for some λ.
A module over a ring with approximate identity is called non-degenerate if for every m in the module there is some λ with m = meλ.
See also
• Mollifier
• Nascent delta function
• Summability kernel
Spectral theory and *-algebras
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• Involution/*-algebra
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• B*-algebra
• C*-algebra
• Noncommutative topology
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• Spectral radius
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Spectrum
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Banach spaces are:
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Function space Topologies
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Theorems
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Types of sets
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Subsets / set operations
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Examples
• Absolute continuity AC
• $ba(\Sigma )$
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• Hardy Hp
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• Morrey–Campanato $L^{\lambda ,p}(\Omega )$
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Applications
• Differential operator
• Finite element method
• Mathematical formulation of quantum mechanics
• Ordinary Differential Equations (ODEs)
• Validated numerics
Authority control: National
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| Wikipedia |
In the diagram, the area of triangle $ABC$ is 27 square units. What is the area of triangle $BCD$?
[asy]
draw((0,0)--(32,0)--(9,15)--(0,0));
dot((0,0));
label("$A$",(0,0),SW);
label("6",(3,0),S);
dot((6,0));
label("$C$",(6,0),S);
label("26",(19,0),S);
dot((32,0));
label("$D$",(32,0),SE);
dot((9,15));
label("$B$",(9,15),N);
draw((6,0)--(9,15));
[/asy]
Let $h$ be the distance from $B$ to side $AD$. The area of $ABC$ is 27, so $\frac{1}{2}\cdot6\cdot h = 27$, which implies $h=9$. The area of $BCD$ is $\frac{1}{2}\cdot26\cdot9=\boxed{117}$ square units. | Math Dataset |
han chinese physical characteristics
$$ \mathrm{MCC}=\frac{TP\times TN- FP\times FN}{\sqrt{\left( TP+ FP\right)\left( TP+ FN\right)\left( TN+ FP\right)\left( TN+ FN\right)}} $$, $$ \widehat{N}{um}_i/\widehat{D}{en}_i $$, $$ \widehat{N}{um}_i=\left(w-x\right)\left(y-z\right) $$, $$ \widehat{D}{en}_i=\left(w+x-2 wx\right)\left(y+z-2 yz\right) $$, $$ {F}_2\left(A,B\right)=E\left[{\left({a}^{\hbox{'}}-{b}^{\hbox{'}}\right)}^2\right] $$, $$ {F}_3\left(C:A,B\right)=E\left[\left({c}^{\hbox{'}}-{a}^{\hbox{'}}\right)\left({c}^{\hbox{'}}-{b}^{\hbox{'}}\right)\right] $$, $$ {F}_4\left(A,B:C,D\right)=E\left[\left({a}^{\hbox{'}}-{b}^{\hbox{'}}\right)\left({c}^{\hbox{'}}-{d}^{\hbox{'}}\right)\right] $$, $$ {F}_4\left(B,O;C,A\right)=\left(1-\alpha \right)m $$, http://creativecommons.org/licenses/by/4.0/, http://creativecommons.org/publicdomain/zero/1.0/, https://doi.org/10.1186/s41065-018-0057-5. [43] In the late 20th century, the sorrowful "Western style" of pansori overtook the vigorous "Eastern style" of pansori, and pansori began being called the "sound of han". East Asia is one of the world's most populated places, consisting of about 38% of the Asian population or about 22% of the world-wide population. Each vertical bar represents an individual and each color stands for a genetic component (generated by R 2.15.2). [13] Traditional Korean stories almost always have happy endings. [4] All surviving pansori epics end happily, but contemporary pansori focuses on the trials and tribulations of the characters, commonly without reaching the happy ending because of the contemporary popularity of excerpt performances. volume 155, Article number: 19 (2018) [6] According to Kim Yol-kyu, "똥구멍이 찢어지게 가난하다" (as poor as a church mouse)[61] is about han. A heat map of pair-wise FST. According to the Translation Journal, "Han is frequently translated as sorrow, spite, rancor, regret, resentment or grief, among many other attempts to explain a concept that has no English equivalent. (A) The NJ tree was constructed according pair-wise genotyping difference; (B) Individual NJ Tree of Han Chinese, Japanese, Korean and Mongolian individuals (generated by MEGA version 4.0). We screened Ancestry Informative Markers (AIMs) [8] to distinguish Han Chinese, Japanese and Korean. Now, Chinese as a language is still only a broad category. Individual level Neighbor-Joining Tree. These estimations and findings facilitate to understanding population history and mechanism of human genetic diversity in East Asia, and have implications for both evolutionary and medical studies. Physical attributes Typical faces of Chinese, Japanese and Korean people. [51] Peterson also disagrees with the Japanese colonial view of Korea as stagnant, inefficient, and corrupt. Japanese and Korean each has each independent cluster, while two Han Chinese populations and two Mongolian both have some overlaps, and some Mongolian individuals are located toward CHB. Genetic structure, divergence and admixture of Han Chinese, Japanese and Korean populations. To examine fine-scale population structure and assess the genetic make-up of East Asian groups, we applied a model-based method, STRUCTURE [10], to analyze the genome-wide data with that of worldwide populations. Models used in gene flow study. We'll see if that's true or not, but just in case, here's a breakdown. All the East Asian populations share a clade, and Mongolians are much closer to European than any other East Asian groups.
Han is a very small and minor part of classical Korean literature. [12][17][18][19][20] Following the March First Movement, an independence movement that ended with the death of about 7,000 Koreans at the hands of the Japanese police and military,[21] the Japanese art critic Yanagi Sōetsu wrote articles in 1919 and 1920, expressing sympathy for the Korean people and appreciation for Korean art.
Genetics. 5. In fact, China is made up of 56 distinct ethnic groups.
First, we asked what the genetic make-up of the three populations is and how well they are differentiated. More detailed information about the populations can be found in Additional file 3: Table S1. For the purposes of this study, only SNPs with Reference Sequence (RS) numbers and vendor-specified strands were used in combining data.
We could also observe considerable TC contribution in CHB and JPRK (5.2 and 2.6%, respectively) (Additional file 9: Figure S7, K = 7, in purple). We further applied D test for detecting gene flow between Japanese/Korean and south/north Han Chinese, the D values were not significantly different from 0, indicating the gene flow between north Han Chinese and Japanese/Korean are almost equal to that between south Han Chinese and Japanese/Korean.
© 2020 BioMed Central Ltd unless otherwise stated. You can test out of the Our results showed that Mongolian populations (BMON and QHM) were admixed by European and some East Asian populations (for example, Han Chinese or Japanese, or their common ancestry, Additional file 12: Table S3), consistent with their known history. [59], Frost can fall even in May and June, if a woman harbors a grudge [han]. Each vertical bar represents an individual and each color stands for a genetic component (generated by R 2.15.2).
Log in here for access. [25] To justify the colonization of Korea, the Japanese propagated an image of Koreans as an inferior, uncivilized people, who were incapable of being independent and prone to being invaded and oppressed. Geographic Distribution. 2006;2(12):e190. Am J Hum Genet. PubMed Common ancestor of Han Chinese, Japanese and Koreans dated to 3000 – 3600 years ago. Early depictions of some form of han as an individual expression are found in examples of traditional Korean stories, poems, and songs. SEAC (in green) constituted the majority (92.7%) of the CDX genome; Mongolian and Tibetan showed distinct composition but both were influenced by NEAC significantly (5.7%~ 15.0%) (Additional file 10: Figure S8, K = 4, in red). In this project, we conducted a genome-wide study of three East Asian populations based on genome-wide high-density SNP data. (PDF 152 kb), Figure S9. [36][48][49] Yanagi's interpretation of Korean history and art has been disputed. As a result, we could use only 89 SNPs (CHB/KOR), 46 SNPs (CHB/JPT), 44 SNPs (CHS/KOR), 26 SNPs (CHS/JPT) and 73 SNPs (JPT/KOR) respectively to perfectly distinguish each population pairs. Article A central aspect of han today is loss of identity. The Europeans described its manifestations as terrifying. At K = 5, Mongolian had different composition from Tibetan but showed considerable contribution from the Tibetan component (TC) (3.1~ 14.5%), especially in QHM, who migrated to Qinghai-Tibetan Plateau about 500 years ago and were reported to have adapted to local environment [6]. Divergence time (TF) can be estimated by 2NeFST (4) [9]. China is home to 56 ethnic groups, but the largest by far is the Han Chinese. Lipson M, Loh PR, Levin A, Reich D, Patterson N, Berger B. Yes, admittedly it can be challenging to differentiate the three, even for fellow Asians. We further explored whether the three populations could be well distinguished with a small number of ancestry informative markers (AIMs, see Methods). However, the genetic history of the northern Han Chinese is still … Moreover, East Asian people use similar languages and words, for example, Chinese characters are shared in Japanese, and also existed in Korean until their recent abolition in the 1940s [1]. All rights reserved. Many genetic studies have shown that Han Chinese can be divided into two distinct groups: northern Han Chinese and southern Han Chinese. EJHG: European journal of human genetics; 2013. Mapping human genetic diversity in Asia. Felsenstein J. Maximum-likelihood estimation of evolutionary trees from continuous characters. (A) Outgroup is YRI or CEU, donate populations are TIB and CDX, target population is Han Chinese or KOR; (B) Outgroup is YRI or CEU, donate populations are CDX and JPRK, target population is Han Chinese or JPT or KOR; (C) Outgroup is YRI or CEU, donate populations are TIB and JPRK, target population is JPT or KOR (generated by Microsoft Excel 2010). Integrated genotype calling and association analysis of SNPs, common copy number polymorphisms and rare CNVs. However, we observed considerable presence of European ancestry in Mongolian population (15.5~ 16.1%) and some other East Asian populations (up to 6.4%). And the marked number are bootstrap value; (B) The top two PCs of individuals representing six East Asian populations, mapped to their corresponding geographic locations (generated by R 2.15.2 and Microsoft Excel 2010). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. CAS All rLD2 are adjust as (rLD2–1/n), where n is sample size (number of chromosomes) [9]. We confirmed the ancestry inference results with another commonly used method, ADMIXTURE [11] (Fig. Result of K = 5 was shown and each vertical bar represents genome makeup of an sample individual (calculated by ADMIXTURE 1.23 [1] and R 2.15.2).
[46] Choe criticized Yanagi's view, that Korean art has a "beauty of sorrow" because Korea has long suffered at the hands of foreign countries, as being in accordance with Japanese colonial policies, which, he said, were intended to instill a sense of defeat and shame in Koreans about Korean history. Results of K = 3 and K = 4 are shown (generated by R 2.15.2). A maximum likelihood (ML) tree reconstructed based on pairwise allele frequency difference provides a better visualization of the genetic relationship of populations (Fig. YJ contributed partly to sample collection and genotype data. The Han Chinese are one of the largest ethnic groups in the world. 2013;9(7):e1003634. [64], "Unique Korean Cultural Concepts in Interpersonal Relations", "Yanagi exhibit navigates critic's controversies", Interactions between the emotional and executive brain systems, https://en.wikipedia.org/w/index.php?title=Han_(cultural)&oldid=979155225, Articles containing Japanese-language text, Creative Commons Attribution-ShareAlike License, Sandra So Hee Chi Kim argues that the current usage of the word, According to Joshua D. Pilzer, the idea of, This page was last edited on 19 September 2020, at 03:45. In D test, D value indicates the difference of the two allele patterns ABBA and BABA (Additional file 13: Figure S11A). These results suggested that although similar in appearance, Han Chinese, Japanese and Korean are different in terms of genetic make-up, and the difference among the three groups are much larger than that between northern and southern Han Chinese. According to coordinate of PC2 which explained 0.7% of the total variance, Mongolian and Tibetan are closely located in one side, while the island populations (Japanese and Ryukyuan) other side. By using this website, you agree to our [38], Sandra So Hee Chi Kim's article on han says that "the term han itself emerged as a significant ideological concept during the 1970s" and "[s]ome contend that it was during the Park Chung Hee regime that the idea of han transformed from a personal sense of sorrow and resentment to a broader, national experience of unrelenting suffering and injustice". An example of han as a collective expression was observed by Westerners in 1907, but a national culture of han did not exist in Korea. Do Community College Classes Transfer to CSU? (C) PCA result of groups within East Asia populations, excluding JPRK individuals for a higher marker density.
Inference of population structure using multilocus genotype data: linked loci and correlated allele frequencies.
Trends Immunol. We conducted a genome-wide study and evaluated the population structure of 182 Han Chinese, 90 Japanese and 100 Korean …
The Dong-a Ilbo wrote: "The anger, bitterness, and sorrow built up inside us have become mixed together, and it could be said that the passing of the Yunghui Emperor [Sunjong] has touched the hearts of the Joseon people and released their pent-up sadness." Many people also belong to various Christian denominations due to the influence of western culture. In fact, some linguists have suggested that the Chinese language will be the international standard in the future.
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Who gains the most from improving working conditions? Health-related absenteeism and presenteeism due to stress at work
Beatrice Brunner ORCID: orcid.org/0000-0002-6010-51841,
Ivana Igic2,
Anita C. Keller3 &
Simon Wieser1
The European Journal of Health Economics volume 20, pages 1165–1180 (2019)Cite this article
Work stress-related productivity losses represent a substantial economic burden. In this study, we estimate the effects of social and task-related stressors and resources at work on health-related productivity losses caused by absenteeism and presenteeism. We also explore the interaction effects between job stressors, job resources and personal resources and estimate the costs of work stress. Work stress is defined as exposure to an unfavorable combination of high job stressors and low job resources. The study is based on a repeated survey assessing work productivity and workplace characteristics among Swiss employees. We use a representative cross-sectional data set and a longitudinal data set and apply both OLS and fixed effects models. We find that an increase in task-related and social job stressors increases health-related productivity losses, whereas an increase in social job resources and personal resources (measured by occupational self-efficacy) reduces these losses. Moreover, we find that job stressors have a stronger effect on health-related productivity losses for employees lacking personal and job resources, and that employees with high levels of job stressors and low personal resources will profit the most from an increase in job resources. Productivity losses due to absenteeism and presenteeism attributable to work stress are estimated at 195 Swiss francs per person and month. Our study has implications for interventions aiming to reduce health absenteeism and presenteeism.
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A loss of work productivity can be a result of health impairments and arise from absenteeism (being away from work due to illness or disability) and presenteeism (being present at work but constrained in certain aspects of job performance by health problems) [1]. Maintaining a healthy and productive workforce is increasingly challenging due to the continuing structural changes in the working environment, an aging workforce and an increasing number of employees affected by stress at work [2]. Gaining better knowledge of the stress-related causes of absenteeism and presenteeism is therefore of high social and economic importance. A detailed analysis of the drivers of work stress-related productivity losses may be particularly useful to understand which employees are most at risk of incurring stress-related productivity losses and to identify those who might profit the most from interventions that improve work conditions.
Productivity losses are determined by multiple factors [3], but work-related factors have often been proposed as especially important [4]. According to the models developed in occupational health psychology, such as the Job-Demands Control model (JDC) [5, 6] and the Job-Demands Resources model (JDR) [7], unfavorable job conditions are associated with high levels of job stressors and a lack of job resources. Exposure to such job conditions can lead to stress among employees, resulting in decreased performance and motivation and, over time, in serious health problems [8]. However, only a handful of empirical studies have analyzed these propositions in relation to productivity losses caused by absenteeism and presenteeism (e.g., [4, 9, 10]). For example, a lack of job control, which is a well-established work resource [11] defined as the ability to determine when and where work is done, has been shown to increase the risk of presenteeism [12]. Another study found a similar relation with sickness absence but only for women [13]. Additionally, high time demands and physical demands at work have been shown to be associated with presenteeism and absenteeism [14, 15].
In this study, we estimate the effects of work stressors and resources on health-related productivity losses caused by absenteeism and presenteeism, and add to the current literature in three ways.
First, we estimate the effects of task-related and social job stressors and resources on health-related productivity losses, whereas the current literature mainly focused on task-related factors. Empirical evidence suggests that social stressors may be especially harmful to employee health and well-being, even more so than other job stressors [16]. Furthermore, the recent "Stress-as-Offense-to-Self" model underlines the relevance of social job resources, such as appreciation at work, and highlights its absence as particularly stressful for employees [17]. Absenteeism and presenteeism can also be explained by the social exchange perspective [18]. According to this approach, the employee–organization relationship is a trade of effort and loyalty for benefits such as pay, social support and recognition [19]. When employees are satisfied with this mutual exchange, they will be engaged in their jobs. However, when employees perceive the benefits received as too low compared to their contribution, they may withdraw from the relationship. Absenteeism and presenteeism can thus be seen as a method of restoring equity in the employee–organization relationship [20]. Some previous studies support these assumptions, although evidence is still scarce. Injustice at work [20], low organizational support [18] and low workgroup cohesiveness [21] have, for example, been shown to increase the risk of absenteeism. Similarly, negative relationships with colleagues [22], role ambiguity [23], and workplace bullying [24] have been shown to increase the risk of presenteeism. A few studies also provide evidence of the relevance of positive social aspects at work. Employees working under a supportive supervisor [25] who demonstrated strong integrity [26] showed less presenteeism and absenteeism.
Second, in addition to social and task-related stressors and resources at work, we consider personal resources. According to the JDR model, job and personal resources can affect health and organizational outcomes both directly and indirectly; personal resources might enable employees not only to deal with job demands in a resilient way but also to make better use of available job resources [7]. Previous studies have shown that personal resources are related to absenteeism; however, studies on presenteeism rarely consider personal factors [27]. We included occupational self-efficacy as a relevant personal resource for individuals in organizations [28, 29], expecting self-efficacy to act as a buffer for the negative effects of job stressors [7]. Occupational self-efficacy is defined as the belief or confidence in one's ability to successfully fulfill a task or cope with difficult tasks or problems [30]. Previous research has shown direct beneficial effects of occupational self-efficacy on productivity [28, 29], work-related behavior [31] and job attitudes [32] and demonstrated its moderating effects in the stressors–strain relationship [33, 34]. However, to date, no studies have explored its effects on health-related productivity losses. Based on the conservation of resources theory (COR, [35]), according to which individuals who lack resources are more vulnerable to resource loss and less capable of resource gain (negative spiral), we expected that employees lacking both job and personal resources are most at risk of experiencing health-related productivity losses due to absenteeism and presenteeism when job stressors increase. Furthermore, in line with the "gain paradox principle" [35] according to which resource gains become more important when resources are loss is high, we expect that an increase in job resources is especially important for employees with low self-efficacy and high stressors.
Third, we contribute to the literature on the economic burden of work stress. Although work stress and its consequences for employees and employers are high on the political agendas of European institutions and policy-makers [36], evidence of the economic burden of work stress is scant, especially regarding stress-related presenteeism [37]. The few available studies suggest that the costs of work stress are substantial [38]. We add to previous studies on the productivity losses caused by work stress by estimating the cost of employees' health-related productivity loss due to presenteeism and absenteeism of being exposed to an imbalance between job stressors and job resources. Such an imbalance, according to occupational stress models (e.g., JDC, JDCR [39]), results in work stress and has a high probability of leading to serious health problems. We calculated the total health-related productivity loss due working under unfavorable job conditions per employee and month, considering both absenteeism and presenteeism.
The aim of this study was threefold. First, we estimated the effects of task-related and social stressors and resources on health-related productivity losses due to absenteeism and presenteeism. We assessed stressors and resources at work based on six indices measuring (1) task-related work stressors (time pressure, task uncertainty, performance constraints, and mental and qualitative overload), (2) social work stressors (social stressors from supervisor and co-workers), (3) task-related work resources (job control and task significance), (4) social work resources (social support from supervisor, appreciation at work), as well as (5) overall work stressors and (6) overall work resources. We controlled for a wide range of confounding factors, such as socio-economic characteristics, job characteristics, private demands, and personal characteristics (self-efficacy). Second, we explored the interaction effects between job stressors, job resources, and personal resources. We aimed to understand which employees are most at risk if job stress increases and which employees would benefit the most from interventions improving the balance between job stressors and resources. Third, we built an economic model estimating the productivity losses caused by employee exposure to an imbalance between job stressors and resources.
We used data from a Swiss workforce survey carried out in two measurement waves. The survey consisted of two datasets: a representative cross-sectional dataset based on the first wave and a longitudinal dataset based on both waves. We used both datasets because they have different strengths and weaknesses. The cross-sectional wave 1 dataset is representative of the Swiss workforce regarding gender, age, region and industry branch. Moreover, it contains information on occupational self-efficacy, allowing us to explore the interaction effects between job stressors, job resources and occupational self-efficacy. However, due to its cross-sectional nature, it could not be used to identify causal effects. The longitudinal wave 1–2 dataset allowed us to overcome this weakness, as it permits the application of methodologically superior panel data estimation methods. However, wave 1–2 suffers from considerable attrition in the second wave and does not allow us to explore interaction effects because occupational self-efficacy was not assessed in the second wave.
Wave 1 The first wave was conducted in February 2014. The recruitment of participants was based on a large Swiss Internet panel including full- and part-time employees. The sample was stratified by gender, age, region and industry branch. Participants were recruited randomly from the sample by phone and e-mail to complete the online questionnaire [40]. A total of 3758 employees completed the questionnaire. Of these, 59 were excluded because of timing and response patterns and 318 because of missing or implausible information. The final cross-sectional sample consisted of 3381 employees who are representative of the Swiss workforce regarding gender, age, region and industry branch.
Wave 1–2 The second wave was conducted in February 2015. Of the 3381 participants of wave 1, 352 had left the panel and 196 were no longer economically active and were therefore excluded. Hence, 2833 individuals were re-contacted, of whom 2125 (75%) participated and 1759 (62%) completed the questionnaire. We excluded 93 individuals because of timing and response patterns and 153 because of missing information on industry branch or work productivity. Plausibility checks on income and hours worked led to the exclusion of an additional 14 individuals. Our final longitudinal sample included N = 1513 individuals who had participated in both waves. The longitudinal wave 1–2 data set was used to test the robustness of the cross-sectional estimations. We accounted for selective attrition by estimating and applying inverse-probability-of-attrition weights.
Our dependent variable was individual health-related productivity loss, corresponding to the sum of the percentage of absenteeism (percentage of work time missed due to health) and percentage of presenteeism (percentage of work time affected by productivity impairment due to health problems while working). These data were collected with the Work Productivity and Activity Impairment-General Health (WPAI-GH) questionnaire. The WPAI-GH is a psychometrically tested instrument measuring absenteeism, presenteeism and overall health-related work productivity losses (corresponding to the sum of absenteeism and presenteeism) with good reliability, validity, generalizability and practicability [41,42,43].
The WPAI-GH questionnaire is composed of five questions: Q1 = currently employed; Q2 = hours missed due to health problems; Q3 = hours missed due to other reasons (e.g., vacation); Q4 = hours actually worked; Q5 = degree to which health affected productivity while working (using a 0–10 Visual Analogue Scale). Following the coding and scoring rules of the WPAI developers, we obtained the percentage of health-related work productivity losses [Q2/(Q2 + Q4) + ((1 − Q2/(Q2 + Q4)) × Q5/10)], which constitutes our dependent variable.
The main advantages of the WPAI over other productivity questionnaires [e.g., health and work questionnaire (HWQ) or health and work performance questionnaire (HPQ)] are the possibility of transforming outcomes into monetary values, as outcomes are expressed as impairment percentages. Furthermore, it uses a 1-week rather than a 4-week recall period, which significantly reduces recall bias [42].
Main explanatory variables
Our main explanatory variables are six indices measuring task-related and social stressors and resources at work. The indices were constructed based on several task-related and social work conditions proposed by the theoretical and empirical literature to be relevant regarding health and key organizational variables such as productivity and motivation (e.g., JDC, JDCR [7, 8, 11]). We included four well-established task-related stressors (time pressure, task uncertainty, performance constraints, and mental and qualitative overload) and two resources (job control and task significance). In addition to the task-related factors, we included two social resources (social support from supervisor, appreciation at work) [25, 44, 45] and two social stressors (social stressors from supervisor and co-workers) [46, 47] as suggested by theory [8] and empirical research [48, 49]. We also included occupational self-efficacy, a personal resource, to test proposed interaction effects [33, 34].
Job stressors
We assessed four task-related stressors with items from the Instrument for Stress-Oriented Task Analysis (ISTA; [50]), including time pressure (e.g., "How often must you finish work later because of having too much to do?"), task uncertainty (e.g., "How often do you receive contradictory instructions from different supervisors?"), performance constraints (e.g., having to work with inadequate devices or obsolete information) [50], and mental and qualitative overload at work (three items, e.g., having to perform tasks that exceed one's skills) [51]. With the exception of the last scale, which has three items, each of the scales contains four items. We assessed social stressors with the social stressors scale by Frese and Zapf, which includes two scales each with five items. One scale focuses on conflicts or animosities and negative group climate among co-workers (e.g., "With some colleagues there is often conflict"), and the other focuses on conflicts with supervisors (e.g., "I often quarrel with my boss") [52].
We assessed job control using the ISTA [50]. The five items measured job control by evaluating respondents' freedom to choose the time (e.g., "To what degree are you able to decide on the amount of time you will be working on a certain task?") and method (e.g., "Can you decide yourself which way to carry out your work?") for accomplishing tasks at work. The second task-related resource was task significance ("In my job, one can produce something or carry out an assignment from A to Z"), which was measured with one item from the Salutogenetische Subjektive Arbeitsanalyse (SALSA; [51]) instrument. Social job resources were measured by four items evaluating supportive behavior from supervisors (e.g., a line manager lets a worker know how well a job was done), which was also measured with items from the SALSA [51], and appreciation at work, which was assessed with a single item based on the Appreciation at Work Scale ("I feel generally appreciated in my job") [53]. With the exception of appreciation, all items were answered on a 5-point Likert scale, with responses ranging from 1 (very little/not at all) to 5 (very much). Appreciation, originally answered on a 7-point Likert scale, was transformed into a 5-point scale.
We assessed occupational self-efficacy with a four-item scale from Rigotti, Schyns, and Mohr [54]. Work-related self-efficacy measures the belief in one's ability to cope with difficult tasks and problems at work (e.g., "I can remain calm when facing difficulties in my job because I can rely on my abilities") and was assessed only in the first wave of the survey (wave 1).
We created six indices measuring the level of job demands and job resources. First, to test the overall effects of job stressors and resources on employees' health-related productivity losses, we built an overall job stressors and an overall job resources measure. These measures were constructed by averaging over all six stressors and four resources described above, representing demands and resources from the JDC model. This procedure has been previously used [55]. Second, to test the distinct productivity effects of task-related and social stressors and resources, we constructed four additional indices measuring (1) task-related stressors, (2) social stressors, (3) task-related resources and (4) social resources. The four measures were constructed similarly, by averaging over the single task-related and social job stressors and resources. Table A.1 presents the Cronbach alpha values for the single stressors and resources as well as for the indices. For the analysis, we used the standardized values of the six job stressor and resource measures.
We considered a variety of potential confounders that, based on previous evidence, were expected to be associated with work productivity as well as job stressors and resources. First, we controlled for several demographic and socio-economic characteristics (gender, age, number of children, marital status, educational level, and whether the respondent had Swiss citizenship) [56, 57]. Second, we controlled for labor market and job characteristics such as industry branch, occupation, company size, job tenure, average number of working hours, shiftwork, part-time employment, and managerial function, as has been done in previous studies [56, 58, 59]. Third, we controlled for chronic physical health conditions such as asthma, allergies, cancer, chronic bronchitis or emphysema, diabetes, kidney disease, osteoarthritis or rheumatoid arthritis, osteoporosis, and permanent injury after an accident, as the negative relationship between health problems and work productivity is well established [60, 61]. We did not, however, consider diseases with often psychosomatic causes, such as migraines or depression, as they may be a part of the outcome [11, 62] and therefore represent bad control variables for our research question [63]. Finally, we controlled for family-to-work conflict [64] to account for the potential productivity effects of mood spillovers, which have been identified in previous studies [65].
Econometric framework
Since our analysis was based on a cross-sectional dataset (wave 1) and on a longitudinal dataset (wave 1–2), we applied both cross-sectional and panel-data estimation methods. Cross-sectional methods were used to explore the association of health-related productivity loss with job stressors and job resources as well as to explore the interaction between job stressors, job resources, and personal resources in wave 1. The wave 1 dataset is representative of the Swiss workforce and holds information on occupational self-efficacy, which the second wave does not. However, cross-sectional estimation methods require the key regressors to be strictly exogenous conditional on covariates in order to have a causal interpretation. Panel data methods allow for relaxing this strong assumption by controlling for unobserved time-invariant heterogeneity. We used the wave 1–2 panel data set to test the robustness of the cross-sectional estimation results estimating fixed effects models while accounting for selective attrition using inverse-probability-of-attrition weights.
Cross-sectional estimation
The associations between health-related productivity losses and job stressors and resources were examined based on five model specifications with hierarchical adjustment and estimated by ordinary least squared (OLS). The fully specified model takes the following form:
$$\begin{aligned} Y_{i} & = \alpha + \beta_{1} \dot{R}_{i}^{j} + \beta_{2} \dot{S}_{i}^{j} + \gamma {\mathbf{X}}'_{i} + \delta {\mathbf{J}}'_{i}\\ & \quad + \varphi_{\text{r}} + \mu \dot{S}_{i}^{p} + \theta {\mathbf{H}}'_{i} + \vartheta \dot{R}_{i}^{p} + \varepsilon_{i} , \\ & \quad {\text{with}}\quad \dot{R}_{i}^{j} \in \left( {\dot{R}_{i}^{j} ,(\dot{R}_{i}^{{j,{\text{social}}}} ,\dot{R}_{i}^{{j,{\text{task}}}} ) } \right)\\ & \quad\quad {\text{and}}\quad \dot{S}_{i}^{j} \in \left( {\dot{S}_{i}^{j} , (\dot{S}_{i}^{{j,{\text{social}}}} ,\dot{S}_{i}^{{j,{\text{task}}}} )} \right). \\ \end{aligned} ,$$
\(Y_{i}\) denotes the percentage productivity losses of individual i due to sickness absenteeism and presenteeism. \(\dot{R}_{i}^{j}\) and \(\dot{S}_{i}^{j}\) represent the level of resources and stressors at individual i's current job (the dots representing standardized values). The fully specified model distinguishes between task-related and social job stressors and resources. The hierarchical adjustment involves the following five model specifications. We started by estimating simple correlations with Model 1 (CS-1), including only job resources \((\dot{R}_{i}^{j} )\) and job stressors \((\dot{S}_{i}^{j} )\). Model 2 (CS-2) additionally included known confounding variables related to socio-economic and job characteristics (\({\mathbf{X^{\prime}}}\) and \({\mathbf{J^{\prime}}}\), see covariates section for more details). This model also included regional fixed effects, denoted by \(\varphi_{\text{r}}\) with r indexing the canton of residence of individual i. Model 3 (CS-4) additionally included family-related stressors (\(\dot{S}_{i}^{p}\)), as previous literature suggests that mood disturbances can spill over from the family domain to the work domain [65]. Model 4 (CS-4) added a set of nine dummy variables indicating chronic health conditions (\({\mathbf{H^{\prime}}}\), see "Covariates" section) to account for the relationship between chronic conditions, such as asthma and diabetes, and work productivity [60, 61]. Finally, our fully specified Model 5 (CS-5) included occupational self-efficacy (\(\dot{R}_{i}^{p}\)). Under the assumption of strict exogeneity, \(\beta_{1}\) and \(\beta_{2}\) represent the percentage-point change in health-related productivity losses due to a one-standard-deviation change in job resources and job stressors. Note that there is a potential issue of reverse causality. However, this problem is likely to be mitigated as the dependent variable referred to the week before the interview, while the key regressors referred to the current work situation in general and may therefore be considered predetermined.
Panel data estimation and inverse-probability-of-attrition weighting
We assessed the robustness of the fully specified cross-sectional model (CS-5) by estimating a fixed effects model based on the longitudinal wave 1–2 dataset while accounting for selective attrition using inverse-probability-of-attrition weighting. The fixed effects model differed from the fully specified cross-sectional model in Eq. (1) in three ways. First, it included individual fixed effects. This allowed us to relax the assumption of strict exogeneity as the model controlled for unobserved time-invariant heterogeneity. Second, it excluded occupational self-efficacy because it was not observed in the second wave. While the time-invariant component of occupational self-efficacy was captured by the individual fixed effect, we could not control for its time-variant component, i.e., potential productivity effects resulting from changes in occupational self-efficacy. However, self-efficacy is considered to be stable over time [66]. Third, for the fixed effects model to provide unbiased estimates of \(\beta_{1}\) and \(\beta_{2}\), it is essential to avoid attrition bias; thus, the fixed effects model weights observations by inverse-probability-of-attrition weights.
Inverse-probability-of-attrition weighting involved two steps. First, for each period with potential selective attrition (in our case, wave 2), a dummy variable indicating second-wave participation was regressed on a series of covariates in wave 1, and probabilities \(\hat{P}_{i2}\) were fitted using logistic regression. The covariates also included variables on attitudes, character traits, mental health and well-being, and many other variables not used in Eq. (1) (see Appendix A.2 for specification details). In the second step, the objective function was weighted by the inverse probability weights 1/\(\hat{P}_{i2}\). The intuition behind these weights was that respondents with characteristics similar to those of individuals missing due to attrition are up-weighted in the analysis and vice versa. The method of inverse-probability-of-attrition weighting corrects for selection bias under the assumption that conditional on observables in the first wave, second-wave participation is independent of health-related productivity and job stressors and resources in the second wave [67].
We estimated the costs of job stress based on the representative wave 1 dataset. Using the results of Eq. (1), we proceeded in four steps: first, job stress was defined as a binary variable taking the value 1 if job stressors exceeded job resources, which was the case if the net effect of workplace conditions on productivity losses was positive, and 0 otherwise (\({\text{job}}\;{\text{stress}}_{i} = 1[ \beta_{1} \dot{R}_{i}^{j} + \beta_{2} \dot{S}_{i}^{j} > 0 ]\)). Second, we converted the individual percentage productivity losses into monetary values by multiplying them with monthly earnings. This yielded the observed monthly production loss in Swiss francs (CHF) caused by health problems for an average employee in February 2014. The third step involved a counterfactual prediction. We predicted the health-related production loss that would have been observed if each employee experiencing job stress had a net workplace condition effect of zero, i.e., would not have been exposed to job stress. This yielded the predicted monthly production loss caused by health problems for an average employee in the absence of job stress. The fourth and final step consisted of taking the difference between the observed and the predicted production losses, which yielded the part of the health-related production loss attributable to job stress.
Descriptive statistics of wave 1
Of the 3381 wave 1 participants, 54% were female, and the average age was 42.3 years (Table 1). Almost two-thirds were employed full-time (64%), and approximately one-fifth performed shift work (20%). In terms of job category, approximately 18% were self-employed, were firm owners or worked in independent professions, 31% were executive employees, 37% were non-executive employees, 17% were skilled workers, and 2.3% were unskilled manual workers.
Table 1 Descriptive statistics of the cross-sectional data (wave 1)
The average health-related productivity losses amounted to 14.3% of the working time, corresponding to 6 h per week for a full-time employee. At 10.9%, presenteeism had a more important role than absenteeism (3.4%). These findings are in line with those of other studies using the WPAI (e.g., [68]). Moreover, Fig. 1a shows that 65% of the participants reported health-related productivity losses of zero. Of those with a non-zero loss, the majority reported a loss between 10 and 20%, corresponding to 4–8 h per week. On average, job resources (M = 3.85, SD = 0.66) were higher than job stressors (M = 2.03, SD = 0.51), and this difference was more pronounced for social than for task-related job stressors and resources. Furthermore, both job stressors and job resources exhibited a distinctive asymmetrical distribution with opposite skewness, with the majority of employees reporting above-average resources and below-average stressors (Fig. 1b).
Distribution of key variables
Correcting for selective attrition in wave 2
Table 2 reports the scale of selective attrition in the wave 1–2 subsample and shows how well the inverse-probability-of-attrition weights performed in adjusting for it. Comparing the characteristics between the wave 1 sample (column 1) and the unweighted wave 1–2 subsample (column 2) suggests that attrition was non-random as participation in the second wave was significantly related to age, office size, working in the art sector, and absenteeism. In particular, participants in the second wave were on average younger (43 vs. 42.3 years), worked in smaller offices (9.7 vs. 3.3 co-workers), showed higher absenteeism (2.5% vs. 3.4%) and were more likely to work in the art sector (2.6% vs. 6%) than participants in the first wave.
Table 2 Attrition and the results of inverse-probability-of-attrition weighting
The comparison of the characteristics between the wave 1 sample (column 1) and the weighted wave 1–2 subsample (column 3) illustrates that inverse-probability-of-attrition weighting is capable of reducing the differences between the two samples considerably. Small differences only remain with respect to office size and the probability of working in the art sector.
Productivity effect of job stressors and job resources
Table 3 presents the main regression results using cross-sectional wave 1 data, with the first five models building up from simple correlations (CS-1) to the fully specified model presented in column five (CS-5). Column six (FE-5) shows the results of the fully specified fixed effects model based on longitudinal data (waves 1–2). Finally, columns seven and eight (CS-6 and FE-6) present the effects of task-related and social job stressors and resources on productivity separately.
Table 3 Effects of job stressors and resources on health-related productivity losses
The results for the cross-sectional data show that (CS-1) a one-standard-deviation increase in job stressors is associated with an increase in health-related productivity losses of 4.4 percentage points (95% CI 3.2–5.7), and a one-standard-deviation increase in job resources is associated with a decrease in health-related productivity losses of 1.7 percentage points (95% CI − 0.5 to − 2.9). Adding socio-economic and job characteristics (CS-2) increases the point estimate of job stressors to 4.8 (95% CI 3.6–6.1) but decreases the point estimate of job resources to − 1.5 (95% CI − 0.3 to − 2.8). As expected, including family conflicts (CS-3) drives down the point estimate of job stressors to 4.1 (95% CI 2.7–5.4), while the point estimate of job resources remains unchanged. Adjusting for chronic diseases (CS-4) leaves both coefficients nearly unchanged, implying that chronic conditions are related neither to job stressors nor to resources. Finally, adding occupational self-efficacy (CS-5) reduces the point estimate of job resources to − 1.3 (95% CI − 0.1 to − 2.5) and, to a lesser extent, the point estimate of job stressors to 4 (95% CI 2.7–5.3). This implies a positive correlation between occupational self-efficacy and both job stressors and resources. One explanation might be that individuals with a high level of occupational self-efficacy may also seek job tasks (or jobs) that are more challenging and demanding, as they feel capable of mastering them, and such jobs are typically combined with high job resources (e.g., [69]). Also note that occupational self-efficacy significantly reduces productivity losses due to absenteeism and presenteeism.
The point estimates of the fixed effects regression based on wave 1–2 (FE-5) appear statistically equivalent to those of the fully specified model using wave 1 (CS-5). A one-standard-deviation increase in job stressors leads to an increase in productivity losses of 3.8 percentage points (95% CI 1.4–6.2), whereas a one-standard-deviation increase in job resources leads to a decrease in productivity losses of − 1.2 percentage points (95% CI 1.3 to − 3.7). While the point estimates are statistically identical, the standard errors have nearly doubled due to the smaller sample size, which renders the coefficient of job resources insignificant.
Converted into elasticities, our results suggest an elasticity of health-related productivity losses of 1.09 with respect to job stressors and an elasticity of − 0.53 with respect to job resources (as shown in the last two rows).
The results of distinguishing between social and task-related job stressors and job resources are presented in the last two columns of Table 3 with CS-6, which present OLS and FE-6 fixed effects regression results. The coefficients on both task-related and social job stressors are positive and statistically significant in both models. Moreover, although productivity losses seem to be slightly more affected by task-related than by social job stressors, the hypothesis of equal effects cannot be rejected (see Table 3, notes). A similar pattern emerges for job resources, as we cannot reject the hypothesis that social and task-related job resources affect productivity equally. However, in the fixed effects model (Table 3, FE-6), neither social nor task-related resources are statistically significant. The elasticity of lost productivity with respect to social job stressors ranged between 0.30 and 0.48, and that with respect to task-related stressors ranged between 0.72 and 0.8.
In terms of the comparability between the wave 1 and wave 1–2 results (OLS vs. fixed effects regression), it should be kept in mind that although the weights reduced sample differences considerably (see Table 2), we cannot rule out the possibility that a certain attrition bias is still present. Nonetheless, the fact that both models yield statistically equivalent results can be interpreted as strong evidence that omitted time-invariant variables in the cross-sectional data hardly bias the results.
Robustness checks
We performed a set of robustness checks. The first robustness check was related to the fact that the dependent variable is strictly non-negative and contains a large mass of zeros, which might lead to biased estimates when estimated by OLS. We therefore transformed the dependent variable into a count variable corresponding to the weekly number of hours lost due to health problems, and we tested the robustness of the baseline results with respect to the use of a negative binomial model (NBM), which is especially suited to account for zero-inflated over-dispersed count data [70]. The results are shown in the first two columns of Table 4. Column 1 (C1) replicates our baseline estimates (Table 3, CS-5) using the transformed dependent variable. C2 presents the NBM results based on the same covariate specification. The comparison shows that the models lead to very similar marginal effects. The OLS point estimates lie between the marginal effects estimated by the NBM at mean characteristics and the average marginal effects. This is strong evidence that our results are not driven by the choice of the model. The next robustness check refers to the specification of both the covariates and the main explanatory variables. The specification in C3 differs from the baseline model, as it allows for interaction effects between gender, age, education and industry sector and includes the gender-age-education-sector distribution with sixteen sectors and five education categories. Again, the resulting point estimates are virtually identical to our baseline estimates. The last two robustness checks refer to the functional form of the relationship between the dependent variable and job stressors and resources. Model C4 estimates different effects for below- and above-average job resources and stressors, and C5 tests for a quadratic form of the relationships providing no evidence for a significant non-linear relationship.
Table 4 Robustness checks
Interaction effects
In our baseline model (Eq. 1), we assumed that job stressors affect health-related productivity losses independently of the level of job resources and vice versa and that both the impact of job stressors and the impact of job resources do not depend on the level of occupational self-efficacy. We relaxed these assumptions and estimated two additional models. In Model 1 (CS-7), we added an interaction term between job stressors and resources to the baseline model (CS-5). In Model 2 (CS-8), we additionally included interaction terms between job stressors, job resources, and occupational self-efficacy (Table 5).
Table 5 Heterogeneous effects
Comparing CS-7 (Table 5; Fig. 2) with our baseline results in CS-5 (Table 3) shows that the impact of job resources on productivity—when estimated at average stress levels—is similar, although somewhat smaller than the constant resource effect estimated by the baseline model. The same applies to the effect of job stressors. However, the coefficient of the interaction term turns out negative (and only closely misses the 10% significance level), indicating a decreasing marginal effect of job stressors on health-related productivity losses with increasing levels of job resources. Furthermore, the productivity effect of job stressors is significant at all levels of job resources and approximately twice as large at the minimum level than at the maximum level of job resources (Fig. 2a). By contrast, job resources affect work productivity only at higher levels of job stressors (> fourth decile). In summary, the productivity effect of a change in job stressors is larger for individuals with low, rather than high, job resources. In contrast, the effect of a change in job resources is larger for individuals with high compared to low stressors.
Marginal effects of job stressors and resources allowing for interaction effects. Notes: a The marginal effects of job stressors on lost productivity depending on job resources. b The marginal effects of job resources on lost productivity depending on job stressors. The estimates and the 90% CI are based on the results shown in column 1 of Table 5
The results of Model 2 (CS-8) are presented in Table 5 and, for easier interpretation, in Fig. 3. Graph (a) shows the marginal effects of job stressors on lost productivity depending on job resources at low (first decile), medium (mean) and high (ninth decile) levels of occupational self-efficacy. Graph (a) shows that at low levels of occupational self-efficacy, the marginal effects of job stressors heavily depend on the level of job resources: the lower the job resources, the larger the negative productivity effect of job stressors. With increasing levels of occupational self-efficacy, however, this relationship becomes weaker until it disappears at about the sixth decile of occupational self-efficacy. Graph (b) shows the marginal effects of job resources. In contrast to job stressors, job resources do not affect every individual's productivity loss. Positive effects of job resources are found for individuals who have above-average job stressors and below-average occupational self-efficacy. The effects for individuals with low occupational self-efficacy who face high job stressors are largest. In sum, individuals with low occupational self-efficacy are the most vulnerable in the sense that negative and positive changes in job stressors and resources have the biggest impact on work productivity in the expected direction.
Marginal effects of job stressors and job resources depending on occupational self-efficacy. a The marginal effects of job stressors on health-related productivity losses depending on job resources and at low (1st decile), average and high (9th decile) values of occupational self-efficacy. b The marginal effects of job resources on health-related productivity losses depending on job stressors at low (1st decile), average and high (9th decile) values of occupational self-efficacy. The estimates and the 90% CI are based on the results shown in column 2 of Table 5
Costs of work stress
Table 6 presents our estimates on the costs of job stress due to absenteeism and presenteeism. Our results suggest that job stress accounts for 23.8% of the total health-related production losses, which, in monetary terms, corresponds to CHF 195 per person and month. This corresponds to 3.2% of the average monthly earnings in Switzerland.
Table 6 Average monthly per capita costs of job stress
We estimated the impact of job stressors and job resources on productivity losses due to sickness absenteeism and presenteeism based on a representative survey of Swiss employees conducted in 2014 and 2015. First, we found that health-related productivity losses increase with an increase in job stressors and decrease with an increase in job resources, with social and task-related stressors and resources being equally important determinants. Second, the analysis of heterogeneous effects revealed that an increase in job stressors is especially harmful if job resources are low. These effects are even more pronounced if occupational self-efficacy is low as well. On the other hand, an increase in job resources is most effective in reducing health-related productivity losses if job stressors are high and occupational self-efficacy is low. Third, the results of a counterfactual analysis suggest that job stress (defined as job stressors exceeding job resources) accounts for 23% of the total health-related productivity losses due to absenteeism and presenteeism. This corresponds to CHF 195 per person and month.
Our findings contribute to studies on the effects of positive and negative social aspects of work on presenteeism and absenteeism. In line with research showing that social aspects of work may be especially relevant to employee health and organizational behavior [25, 46], we found that social stressors and resources at work are important determinants of health-related productivity losses due to absenteeism and presenteeism in addition to task-related job stressors and resources. Moreover, we found that social and task-related stressors have direct and equal effects on health-related productivity losses, and while social resources remain a significant predictor, task-related resources do not. If employees work under unfavorable work conditions characterized by high levels of job demands, do not feel appreciated or respectfully treated at work, or lack social support, health-related productivity losses due to absenteeism and presenteeism might increase. This behavior can be seen as a method of employees restoring equity in the employee-organization relationship, as proposed by the social exchange perspective [18]. These results have scientific and practical implications. Our findings suggest that both social and task-related factors should be considered in future studies and in planning interventions aiming to reduce health-related productivity losses by improving workplace conditions.
As expected, our results confirm that job resources buffer the negative effects of job stressors on productivity losses. These findings are in line with the buffering hypothesis of the JDC model as well as with the postulation that high-strain jobs, characterized by a combination of high job demands and low resources, should see the most harmful effects, while the combination of high demands and high resources is considered to be the most beneficial (active job) [8]. Moreover, our results show that an increase of 1% in job stressors results in a larger effect on health-related productivity losses than a decrease of 1% in job resources. These results are in line with those of previous studies showing that negative conditions and events typically have stronger effects than good conditions [71]. This implies that an increase in demands at work should always be accompanied by an even larger increase in job resources in order to prevent the negative consequences regarding health-related productivity impairments.
Our results also show that not only job resources but also occupational self-efficacy buffer the negative effects of job stressors on health-related productivity losses. Furthermore, we find that employees with a simultaneous lack of personal and job resources are the most vulnerable with respect to an increase in job stressors. This finding is in line with the vicious cycle postulated by the COR model: individuals who lack resources are more vulnerable to resource loss and less capable of resource gain. We also find that employees with low personal resources facing high job stressors are the ones who would profit the most from an increase in job resources. This is in line with the "gain paradox principle" of the COR model [35], stating that resources are even more important when resource losses are high.
We do not find a significant productivity effect of an increase in job resources for employees with high personal resources and low level of job stressors. Therefore, an increase in job resources without a reduction in job stressors may not always be sufficient to reduce health-related productivity losses.
We add to the economic literature by estimating the total health-related productivity loss due to unfavorable job conditions. Our estimated productivity loss of CHF 195 per person and month may seem modest at first. However, extrapolation indicates that job stress may have cost Swiss companies up to CHF 10 billion in 2014, corresponding to 1.7% of the gross domestic product. This emphasizes the economic importance of interventions aiming to improve work conditions in general and the balance between work demand and resources.
Our study has several methodological and theoretical strengths. First, the cross-sectional data were representative of Swiss employees with respect to gender, age, region, and industry branch. Second, we tested the robustness of our cross-sectional results using longitudinal data, as this allowed the application of methodologically superior panel-data estimation methods. Third, we included several task-related and social work conditions and explored the relevance of positive and negative social aspects at work beyond the task-related aspect. Fourth, in addition to job resources, we considered personal resources—occupational self-efficacy—and explored interaction effects with job stressors and job resources.
Several limitations need to be taken into account. First, self-reported measures such as the WPAI-GH may suffer from social desirability and recall bias. While a recall bias is unlikely, given the 1-week recall period of WPAI-GH, a social desirability bias is likely to be present. Studies comparing self-reported with company-registered absenteeism show that employees tend to underreport absenteeism [22]. If this were due to social desirability, we would also expect employees to underreport presenteeism. While this would lead to an underestimate of the magnitude of health-related productivity losses, it would not necessarily bias the validity of the associations between workplace conditions and health-related productivity losses. A second shortcoming related to the WPAI-GH is that it has not (yet) been validated against objective work productivity data. We thus do not know whether an employee-reported productivity impairment of 10% translates into a 10% loss of an employee's value to the employer. A study comparing self-reported measures from the Work Limitations Questionnaire (WLQ) with objective productivity outcomes found that a self-reported 10% health-related limitation at work translated into a 4–5% reduction in work output. However, the generalizability of these results is unclear because the study was carried out in a single work setting and did not consider quality of work [14]. If this overestimation in self-reporting of productivity losses applied to our data, it would imply an overestimation in our job stress-induced productivity losses. There is a clear need for more research on the extent to which employee-reported productivity measures translate into production losses for employers. A third limitation relates to the high dropout rate in the second wave of the survey. Although we show that the inverse-probability-of-attrition weights are capable of correcting for selective attrition to a large extent, we cannot rule out the possibility that our panel data estimations are still biased.
Our results suggest that improvements in work conditions could help organizations to reduce previously undetected productivity losses by implementing programs targeting an improved balance between job stressors and job resources. We also show that an increase in job demands affects employees to different degrees depending on their levels of job and personal resources and that not everyone benefits from increased job resources. This finding highlights the need for organizations to take a tailored approach by providing additional attention to the most vulnerable employees. Moreover, our data suggest that job stressors and resources as well as health-related productivity losses vary greatly across occupations. Our sample size prevents the estimation of occupation specific effects though, offering an opportunity for future research.
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Winterthur Institute of Health Economics, Zurich University of Applied Sciences, Gertrudstrasse 15, 8401, Winterthur, Switzerland
Beatrice Brunner & Simon Wieser
Department of Work and Organizational Psychology, University of Bern, Fabrikstrasse 8, 3012, Bern, Switzerland
Ivana Igic
Department of Organizational Psychology, University of Groningen, Grote Kruisstraat 2/1, 9712 TS, Groningen, The Netherlands
Anita C. Keller
Beatrice Brunner
Simon Wieser
Correspondence to Beatrice Brunner.
Below is the link to the electronic supplementary material.
Supplementary material 1 (DOCX 23 kb)
Brunner, B., Igic, I., Keller, A.C. et al. Who gains the most from improving working conditions? Health-related absenteeism and presenteeism due to stress at work. Eur J Health Econ 20, 1165–1180 (2019). https://doi.org/10.1007/s10198-019-01084-9
Issue Date: November 2019
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Analysis of the development of the wind power industry in China — from the perspective of the financial support
Lin Wu1 &
Han Li2
Energy, Sustainability and Society volume 7, Article number: 37 (2017) Cite this article
The wind energy industry is an important part of the renewable energy industry. Helpful financial support plays an essential role in the process of its development. This study analyzes the financial support efficiency not only from the aspect of capital raise, but also from the aspect of the allocation of up-, middle- and down-stream of the Chinese wind power industry chain (which includes the fan component manufacturing enterprises, the fan production enterprises, the wind farm generation and operation enterprises).
Based on a data envelopment analysis (DEA) model, this study selected 30 representative public companies which sampled and extracted their financial data in a panel analysis from 2010 to 2015, in which the financial support efficiency for the Chinese wind power industry was investigated from the aspect of capital raise and allocation.
In terms of capital raise, the comprehensive efficiencies of these three streams all reached their peak in 2011, and then indicated a slight decline, whereas the fan component manufacturing enterprises had both the largest pure technical efficiency value in 2011 and a relatively high scale efficiency value during the sampling period. In terms of capital allocation, the fan component manufacturing enterprises and the wind farm generation and operation enterprises indicated both the highest comprehensive efficiency in 2011, which is merely the expansion period of the new energy industry. The wind farm generation and operation enterprises showed the lowest pure technical value in 2012, which is also the depression period of the new energy industry. The scale efficiency of the fan production enterprises as well as the wind energy generation and operation enterprises had a relatively high scale efficiency value from 2010 to 2015.
The overall efficiency of financial support in the Chinese wind power industry has a close relationship to the macro-economic environment and capital raise while the allocation efficiency of up-, middle-and down-stream show different characteristics. Moreover, the lag of the core technology is the biggest barrier to the financial support efficiency of the wind power industry.
China is vast in territory of wind energy resources. Wind energy reserves that can be exploited and used are at the forefront of the world. Supporting the fan component manufacturing enterprises vigorously and promoting wind energy integration has been a priority for the development of a new energy strategy in China. After a tireless struggle, within the last 10 years the Chinese wind energy industry has jumped to the top of the world. By the end of 2012, fan manufacturing, wind energy installed capacity and integrated wind energy as a whole in China was the largest in the world [1].
Although China's wind energy industry development has gained gratifying achievements under the haze of financial crisis, the European debt crisis and the global economic slowdown, there were signs of a sharp drop in energy demand at home and abroad, as the wind energy industry was not spared this fate and fell into a long period of adjustment. Since the "Abandon wind" problem is as yet still not understood, growth in the wind energy industry has slowed [2]. In the past 2 years, business performance of wind energy enterprises has been generally poor and has entered an era of meager profit.
In order to improve this situation, the ministries and commissions of China have introduced a lot of new energy industry stimulus policies, regulations, standards and guidance catalogs both in the period of "the eleventh five-year plan" and "the twelfth five-year plan". With respect to the wind energy industry, it mainly includes taxation policies such as investment subsidies and tax breaks, industry policies such as rationing and forced integration, etc. But as the biggest development bottleneck, the financing problem has not yet been resolved effectively. Thus based on the correlation mechanisms of the finance as well as industry analyses and the industries financial support efficiency, we can determine the efficiency of financing channels and realize the key development bottlenecks of the industry to guide resources, capital, technology and the demand for optimization and agglomeration [3].
Given the background of policy support, this paper uses a data envelopment analysis (DEA) model to study the financial support efficiency for the Chinese wind energy industry from the aspects of capital raise and allocation. In addition, the up-, middle- and down-stream of the wind energy industry chain are analyzed individually. It can be useful to understand the financial support level of each link by comparison in order to achieve more precise conclusions.
The rest of the paper is structured as follows. Methods contains a review of the literature relevant to this topic. Results introduces the DEA model, whereas the methodology and data sources are provided in Conclusions. The results of the empirical application in Chinese wind energy industry are given in Results and discussion, and the last section concludes the paper.
There are many investigations dealing with the capital raise of the wind energy industry. "Aeolus power" (2007) thought that the largest investment in the process of the wind energy project development should be used to purchase wind power units. In order to raise these funds, wind power enterprises should employ the following financing methods: cooperative bank loans, carbon funds, community shares, energy saving trust funds etc. Cory et al. [4] pointed out that wind power enterprises in the USA should create a financing model for the commercial development of wind power projects in 2008.
Following the financial crisis in 2008, existing financing methods for wind power enterprises became imbalanced, and new financing approaches had to be developed. Under this macroscopic background, David [5] suspected that the equipment financing lease had a certain competitive, effectiveness and profitability relative to business borrowing by comparing their residual value structure, complexity and accounting methods; Lee [6] suspected that the American renewable energy development projects could rely on the following three kinds of financing approaches: renewable energy project financing, clean and renewable energy bonds and providing loan guarantees to existing capital markets from the energy departments; Frølunde et al. [7] studied the Danish evaluation method of wind energy enterprises in the project financing process and the conclusions showed that the real option valuation method could provide valuable decision making strategies for managers in wind energy enterprises. Wu [8] suggested that China's wind energy enterprises should gradually achieve transformation from a single financing model to a diversified financing mode; Zhong et al. [9] supposed that the financing methods in wind projects would force wind energy companies to face a higher asset-liability ratio and larger funding pressures, whereas the author put forward the "joint tenant mode of the wind power project financing lease" model, as this model can be implemented to save costs, on the one hand, and does not affect an enterprises preferential Value added tax (VAT) benefit based on commercial loan, on the other.
Apart from capital raise, most of the literature focusses on capital allocation of the wind energy industry. Nikos [10] analyzed the life cycle cost of a wind park and set up a life cycle cost model; Laura et al. [11] put forward a method to evaluate the cost breakdown structure for offshore wind farms.
Some literature studied the financial support of the wind energy industry from a macro perspective. Campoccia et al. [12] presented a comparison of the main support strategies for Wind Farms (Feed-in tariffs and Green Tags), taking into account the situation in some European Countries; Bolinger et al. [13] discussed the limitations of incentives in supporting farmer- or community-owned wind projects and described four ownership structures that potentially overcome such limitations, and finally conducted a comparative financial analysis of those four structures by using a hypothetical 1.5 MW farmer-owned project located in the state of Oregon as an example. Ozkan et al. [14] considered the financial viability of an offshore wind project as dependent on many interrelated factors.
The application of the data envelopment analysis (DEA) for measuring the efficiency of the industrial financial support has widely been used. Emerging strategic industries, such as biotechnology, new energy and advanced equipment manufacturing has often been the research object. For instance, Zhou et al. [15] assessed the performance of China's renewable resource industry from two aspects—the overall industrial development and the listed companies based on the DEA Model. Moreover, Xiong et al. [16] analyzed the financial support efficiency in the new energy, new materials, energy conservation and environmental protection industries as the research object was based on the DEA model. Some studies applied the Information Technology industry as the research object, such as Sueyoshi et al. [17], who discussed the use of DEA-DA to assess the corporate value of IT firms. There are plenty of industries which have been studied. For example, Suo et al. [18] utilized the DEA to analyze the importance of financial support in agricultural development; and Tong [19] applied the DEA to assess business performance in the car industry. Pang et al. [20] used the DEA to analyze the efficiency of financial support in a commercial bank.
According to the above literature review, there are plenty of articles which analyze the capital raise and allocation of the wind energy industry. Even though the DEA model has always been widely used, only a few studies refined the industries financial segments to include the up-, middle- and down-stream of the wind energy industry chain. Furthermore, there are only a few studies investigating the financial support efficiency from the aspect of capital raise and allocation. These are therefore the two innovation points claimed in this paper.
The DEA method was first put forward by some famous operational research experts. According to Charnes et al. [21], DEA is a non-parametric method for efficiency. The main principle of DEA is to keep the input or output of the decision-making units (DMUs) as the same, and determine not only the relative effective production frontier but also the statistical data by a mathematical programming method. Then, each decision-making unit is projected onto the production frontier, and their relative effectiveness is evaluated by comparing the degree of deviation from the relative effective production frontier. The DEA method does not need to estimate parameters or the hypothesis of the index weight in advance, and is therefore appropriate for border production functions of multi-input and multi-output, and can avoid any deviation caused by the subjectivity which is not eligible for the data dimension. Thus, the DEA model is broadly applicable and has become a very important and effective analysis tool in management science. The basic idea of DEA is shown in Fig. 1.
The basic idea of DEA
In Fig. 1, A, B, C, D and E represent the five DMUs. In production activities, each DMU uses the two inputs; XI and X2, whereas Y denotes the output. From Fig. 1 it is evident that D is the ineffective DMU and the other DMUs are all effective, as they are on the production frontier.
In general, a new efficient DMU can be built through the effective linear combination of the DMU and the production frontier. Taking the DMU D as an example, the intersection point of the DO line and the production frontier is apparently E, which can be described as a linear combination of A and B. Under the same outputs, the inputs of E are less than D. This means that the inputs of D are higher than E, i.e. they are invalid. At this moment, the efficiency of E can be described by EO/DO. If EO/DO < 1, D is an invalid DMU and if EO/DO = 1, D is a valid DMU. By this way, based on the linear programming model, the DEA can evaluate the relative efficiency of the DMUs.
The DEA model allows for calculating both the allocative efficiency and the technical efficiency, whereas the technical efficiency can be decomposed into scale efficiency and pure technical efficiency. Each model has two forms: an input- and output-oriented one. An output-oriented DEA model is appropriate for calculating the largest output value for a given amount of inputs and the input-oriented DEA model is appropriate for minimizing the cost on a given level of output. The DEA model can be used for a constant return to the scale (CRS) and a variable return to the scale (VRS). The relative efficiency value of decision-making units (DMUs) for the distribution in (0, 1) and at the effective forefront value is calculated to be 1.
The CCR model is the first model of the DEA method named after A. Charnes, W.W. Cooper and E. Rhodes, and is also known as CRS model (constant return to the scale). A CCR model presumes that there are n DMUs whereas each DMU has m types of input and s types of output, whereas vector xj and yj are used to represent the j-th DMU: the input vector xj = (x1j, x2j, …xmj)Twhile the output vector yj = (y1j, y2j, …, ysj)T, (i = 1,2,3,…n.). x represents the m × n dimensional input matrix and y represents the s × n dimensional output matrix. It should measure the proportion of all outputs and inputs for each of the DMUs, namely u ′ y i /v ′ x i , where u is the s × 1 dimensional output weight vector and v is the m × 1 dimensional input weight vector. When constant returns to the scale are presumed, the optimal weight can be obtained using the following formula:
$$ {\displaystyle \begin{array}{lll}& \max u,\kern0.5em v\left({u}^{\prime }{y}_i/{v}^{\prime }{x}_i\right)& \\ {}\mathrm{s},\mathrm{t}& {u}^{\prime }{y}_i/{v}^{\prime }{x}_i\le 1& \mathrm{j}=1,2,3,\kern0.5em \dots \mathrm{n}\\ {}& u,v\ge 0& \end{array}} $$
In order to avoid infinite multiple solutions, the constraint conditions v ′ x i = 1 are increased, so that formula 1 can be changed to the follow model:
$$ {\displaystyle \begin{array}{lll}& \max \kern0.5em u,\kern0.5em v\left({u}^{\prime }{y}_i/{v}^{\prime }{x}_i\right)& \\ {}&\ {v}^{\prime }{x}_i=1& \\ {}\mathrm{s},\mathrm{t}&\ {u}^{\prime }{y}_i/{v}^{\prime }{x}_i\le 1& \mathrm{j}=1,2,3,\dots \mathrm{n}\\ {}& u,v\ge 0& \end{array}} $$
Therefore, the efficiency of each DMU can be obtained by using the following model:
$$ {\displaystyle \begin{array}{lll}& \min e-\upvarepsilon \left({\sum}_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{s}}_{\mathrm{i}}^{-}+{\sum}_{\mathrm{r}=1}^{\mathrm{s}}{\mathrm{s}}_{\mathrm{r}}^{+}\right)& \\ {}\mathrm{s}.\mathrm{t}.& {\sum}_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{x}}_{\mathrm{i}\mathrm{j}}{\uplambda}_{\mathrm{j}}+{\mathrm{s}}_{\mathrm{i}}^{-}={\uptheta \mathrm{x}}_{\mathrm{i}0}& \mathrm{i}=1,2,\dots, m;\\ {}& {\sum}_{j=1}^n{y}_{rj}{\lambda}_j-{s}_i^{+}={y}_{r0}& \mathrm{r}=1,2,\dots, s;\\ {}& {\lambda}_j,{s}_i^{-},{s}_r^{+}\ge 0\forall i,j,r& \end{array}} $$
where the \( {\mathrm{s}}_{\mathrm{i}}^{-} \) and \( {\mathrm{s}}_{\mathrm{r}}^{+} \) are slack variables, m and s denote the input and output indexes, respectively. λ denotes the mix proportion of the n decision unit in a reformulated and effective DMU, which is relative to DMU0. Θ denotes the effective utilization degree of the input, (which is relative to the output in the DMU0), which is the efficiency value, indicating whether the financial support is effective in the sample enterprises and whether the output is maximized. 1 − θ represents the proportion of an extra investment in the DMU0, namely the maximum investment proportion which can be reduced.
Input-output index selection
This paper investigates the financial support efficiency for the Chinese wind energy industry from the two aspects of capital raise and allocation. The capital raise efficiency represents the capability of an enterprise to raise funds at the lowest cost through various financing channels in order to support the enterprise operation. Thus, the input-oriented model is adopted to study the capital raise. The capital allocation efficiency is the ability to maximize the output by allocating the given funds to the production and business operation activities of the enterprise. Thus an output-oriented model is adopted to study the capital allocation. The input and output indexes can be found in Table 1.
Table 1 The input and output indexes
Data sources and processing
This paper selects 30 public companies as samples which are from the fan component manufacturing enterprises, the fan production enterprises and the wind farm generation and operation enterprises, respectively, and extracts their financial data as a panel analysis from 2010 to 2015. The data was primarily sourced from the Shanghai stock exchange and www.hexun.com.
As in the DEA analysis model, the data of input and output cannot be negative, we employed the following formula to normalize the data, which leads to a decline in the value within the scope of [0.1, 1]: in the formula (5), where Zmnis the n-th indicators'value of the m-th enterprise.
$$ {\mathrm{Z}}_{\mathrm{mn}}=0.1+\left({\mathrm{Z}}_{\mathrm{mn}}-{\mathrm{minZ}}_{\mathrm{mn}}\right)/\left({\mathrm{maxZ}}_{\mathrm{mn}}-{\mathrm{minZ}}_{\mathrm{mn}}\right)\times 0.9 $$
The DEA analysis for the efficiency of capital raise
This paper used the data envelopment analysis program version 2.1(DEAP version 2.1) software to analyze 30 public companies' comprehensive efficiency, i.e. the pure technical efficiency and the scale efficiency of capital raise and allocation in 2010–2015. The comprehensive efficiency also reflected comprehensive measures and an evaluation of the ability of the DMUs in resource allocation and use. Thus, in this paper, the comprehensive efficiency referred to a comprehensive measure for the financial support efficiency of the wind power industry. The pure technical efficiency referred to the production efficiency of the inputs, provided the DMU was on a certain scale. In this paper, the pure technical efficiency referred to a change in the financial support efficiency, which was affected by the management and technology factors of the wind power enterprises. The scale efficiency did reflect the gap between the actual scale and the optimal production scale, whereas in this paper, the scale efficiency referred to the change in the financial support efficiency due to the scale factor of the wind power enterprises.
If the value of the comprehensive efficiency amounted to 1, it showed that the input and output of this DMU are comprehensive and effective, namely, the technology and the scale were both valid. If the value of the pure technical efficiency amounted to 1, it showed that the use of its resources was efficient at the present technical level, and the root cause of a value that failed to be comprehensively valid was based on its invalid scale, so that its reform should be focused on how to better represent its scale benefit. If the value of the scale efficiency was 1, it showed that its scale was valid, however, its technology was invalid and ineffective, and more attention should be given to its technical innovation.
Analysis for the comprehensive efficiency of capital raise
The comprehensive efficiency referred to the product of pure technical efficiency and scale efficiency. Table 2 presents the comprehensive efficiency of capital raise for the sample enterprises from 2010 to 2015. The data analysis is as follows:
Table 2 The sample enterprises' comprehensive efficiency of capital raise in 2010–2015
1. The comprehensive efficiency of these three streams all reached their peak in 2011, and then had a slight decline. The fan component manufacturing enterprises and the fan production enterprises are both appearing at their lowest value in 2015, which was especially true for the fan production enterprises, where the lowest value amounted to only 0.497. The comprehensive efficiency had however rebounded for the wind farm generation and operation enterprises in 2015, which demonstrated that the capital preferred to enter the construction of the wind farm rather than the production of the fan in these hard times for the development of the wind energy industry.
2. The fan component manufacturing enterprises have the highest comprehensive efficiency value in 2013, where 8 enterprises achieved a comprehensive efficiency of higher than the average value; 2011 was the best year for fan production enterprises, in which the comprehensive efficiency for 8 enterprises reached 1, i.e. the effective comprehensive efficiency. The wind farm generation and operation enterprises included 7 enterprises in 2010, for which the comprehensive efficiency was higher than the average value 0.
Analysis for the pure technical efficiency of capital raise
The pure technical efficiency is employed to represent the use efficiency of the inputs, i.e. the higher the value of the pure technical efficiency, the more effective the investment of the project and the higher the efficiency of the company will be. Table 3 presents the pure technical efficiency of capital raise for the sample enterprises from 2010 to 2015. The data analysis is as follows:
Table 3 The sample enterprises' pure technical efficiency of capital raise in 2010–2015
1.The fan component manufacturing enterprises had the largest efficiency value in 2011, where the value was 0.938. however, since then, the pure technical efficiency value began to drop, leading to a value of only 0.715 in 2015; Relative to the fan component manufacturing enterprises and the wind farm generation and operation enterprises, the pure technical efficiency of the fan production enterprises had a relatively low level during the sample period, whereas this value further dropped to 0.694 in 2015. This indicated that the technology investment efficiency of the fan production enterprises is not high and will affect its capital raise efficiency; the wind farm generation and operation enterprises showed the lowest value in the depression period of the new energy industry, which leads us to the conclusion that it has a close relationship to macro-environment.
2. The fan component manufacturing enterprises have 8 enterprises below the pure technical efficiency from 2010 to 2013 and the fan production enterprises both have 8 enterprises where the effective pure technical efficiency value obtained was 1 in 2011 and 2014. There were only 7 farm generation and operation enterprises which had reached the effective pure technical efficiency in the best year. This suggests that the majority of technology investments by the wind farm generation and operation enterprises did not bring any improvement to the capital raise efficiency.
Analysis of the technical scale efficiency of capital raise
The scale efficiency presented the appropriateness of the proportion of the aspect of input and output in a period of time. The higher the scale efficiency value is, the more appropriate the scale of the company will be. In Table 4, the scale efficiency of capital raise is listed for the sample enterprises from 2010 to 2015. The data analysis is as follows:
The fan component manufacturing enterprises had a relatively high scale efficiency value during the sampling period, whereas the average value was above 0.8, indicating that the investment scale of the fan component manufacturing enterprises had a positive impact on the capital raise efficiency. The scale efficiency of the fan production enterprises were fluctuant in 2010–2015, indicating that its investment scale did not attract stable capital. The wind farm generation and operation enterprises reached the largest scale efficiency value in 2011.
Among the fan component manufacturing enterprises were 8 enterprises reached the effective pure technical efficiency in 2013. The fan production enterprises included the most (8) companies which had reached the effective scale technical efficiency in 2011. There were only 6 wind farm generation and operation enterprises which had reached the effective scale technical efficiency in the best year. This showed that the investment scale of the majority of wind farm generation and operation enterprises did not bring any improvement in the capital raise efficiency.
Table 4 The sample enterprises' scale efficiency of capital raise in 2010–2015
DEA analysis for the efficiency of capital allocation
This paper used the DEAP Version 2.1 software to analyze the comprehensive efficiency, the pure technical efficiency and the scale efficiency of capital allocation in 2010–2015.
Analysis of the comprehensive efficiency of capital allocation
Table 5 illustrates the comprehensive efficiency of capital allocation for the sample enterprises from 2010 to 2015. The data analysis is as follows:
The fan component manufacturing enterprises include most enterprises that reached the comprehensive effective efficiency in 2011 as well as the wind farm generation and operation enterprises with the highest comprehensive efficiency value in 2011, which comprises the expansion period of the new energy industry, indicating that a good macro-environment can improve the efficiency of capital allocation. The comprehensive efficiency of the fan production enterprises were fluctuant in 2010–2015; The wind farm generation and operation enterprises showed a higher comprehensive efficiency in 2011 than in other years, where the comprehensive efficiencies of 7 enterprises were higher, i.e. showed a higher level than the average value.
Most of the fan component manufacturing enterprises were not below the effective comprehensive efficiency during the sampling period, with only 3 in 2015. The fan production enterprises included 7 companies that had reached the effective comprehensive efficiency in 2012, i.e. there were more than 5 wind farm generation and operation enterprises below the effective comprehensive efficiency from 2010 to 2015.
Table 5 The sample enterprises' comprehensive efficiency of capital allocation in 2010–2015
Analysis for the pure technical efficiency of capital allocation
In Table 6, the pure technical efficiency of capital allocation for the sample enterprises from 2010 to 2015 is presented. The data analysis is as follows:
The fan component manufacturing enterprises had the largest pure technical efficiency value in 2010, where it reached 0.965. Subsequently, the pure technical efficiency value began to drop, and the lowest value appeared in 2012, and afterwards the efficiency value began to rebound. This shows that the fan component manufacturing enterprises are trying to improve the efficiency of capital allocation through the investment of technology following an industrial recession. The pure technical efficiency of the fan production enterprises remain in an upward trend from 2010 to 2014, however, a sharp fall was observed in 2015, where the wind farm generation and operation enterprises reached the lowest value in the depression period of the new energy industry, which indicated a close relationship to the macro-environment.
The fan component manufacturing enterprises included 7 enterprises below the effective pure technical efficiency in 2010 and 2015, and among the fan production enterprises were the most numbers of enterprises with a pure technical efficiency value of 1. In 2014, this were 8 enterprises. There are, in general, more wind farm generation and operation enterprises, which had reached the effective pure technical efficiency compared to the other two types of enterprises, where 38 enterprises were below the effective pure technical efficiency from 2010 to 215. This indicated that the technology investment of the majority of wind farm generation and operation enterprises was able to optimize the configuration of funds.
Table 6 The sample enterprises' pure technical efficiency of capital allocation in 2010–2015
Analysis for the scale efficiency of capital allocation
In Table 7, the scale efficiency of capital allocation is listed for sample enterprises from 2010 to 2015. The data analysis is as follows:
The fan component manufacturing enterprises had a relatively low scale efficiency value during the sampling period, in which low values were found in 2010 and 2015. 2010 is the year in which the wind energy industry expanded rapidly, whereas 2015 is the year in which the wind energy industry produced excess capacity. This shows that its investment scale has a great influence on the efficiency of capital allocation. The scale efficiency of the fan production enterprises and the wind energy generation and operation enterprises both had a relatively high scale efficiency value from 2010 to 2015, which indicated that their investment scale was appropriate for capital allocation efficiency.
Most of the fan component manufacturing enterprises did not reach the effective pure technical efficiency from 2010 to 2015, which was reached by only 6 in the best year. In this period of time, the fan production enterprises included the highest number of enterprises with a pure technical efficiency value of 1, which were 7 in 2012. The best year for the wind farm generation and operation enterprises was 2011, where only 7 enterprises were below the effective pure technical efficiency.
Table 7 The sample enterprises' scale efficiency of capital allocation in 2010–2015
Comparative analysis of the financial support efficiency
This paper compares the comprehensive efficiency, the pure technical efficiency and the scale efficiency of capital raise and allocation of 30 public companies from 2010 to 2015.
Comparative analysis of capital raise efficiency
Figure 2 shows the comparative analysis with regard to the capital raise efficiency. The analysis of the results is as follows:
For the fan component manufacturing enterprises, the comprehensive efficiency value showed a slight change when the scale efficiency value showed a sharp rise; the pure technical efficiency was less than the scale efficiency within each year. This means that the capital raise efficiency of the fan component manufacturing enterprises is highly correlated with technology investment.
For the fan production enterprises, the comprehensive efficiency value indicated the same change trend as the pure technical efficiency value and the scale efficiency value, whereas the pure technical efficiency was always less than the scale efficiency. This showed that the fan production enterprises should focus on technology investment in order to improve the capital raise efficiency.
For the wind farm generation and operation enterprises, the comprehensive efficiency value indicated the same change trend as the pure technical efficiency value and the scale efficiency value, whereas the scale efficiency is consistently less than the pure technical efficiency. These results tell us that the investment scale is the main factor, which significantly affects the wind farm generation and operation enterprises.
Comparative analysis of the efficiency on capital raise
Comparative analysis of the capital allocation efficiency
Figure 3 presents a comparative analysis with regard to the capital allocation efficiency. The analysis of the results is as follows:
For the fan component manufacturing enterprises, the scale efficiency remained "almost flat" during the sampling period, as the change of the comprehensive efficiency is mainly based upon a change in the pure technical efficiency, whereas the pure technical efficiency is below the scale efficiency most of the time. This means that the capital allocation efficiency of the fan component manufacturing enterprises is highly correlating with technology investment.
For the fan production enterprises, the comprehensive efficiency value indicated the same change trend as the pure technical efficiency value, whereas the pure technical efficiency was, for the most part, below the scale efficiency. This showed that the fan production enterprises should focus on technology investment in order to improve the capital raise efficiency.
For the wind farm generation and operation enterprises, the scale efficiency remained flat from 2012 to 2015, whereas a change of the comprehensive efficiency was mainly based upon a change in the pure technical efficiency. These results tell us that the wind farm generation and operation enterprises should spend more money on technology investment.
Comparative analysis of the efficiency on capital allocation
Comparative analysis of an industrial chain of the financial support efficiency of wind energy
Figure 4 shows the comparative analysis for capital raise efficiency. The analysis results are as follows:
For the comprehensive efficiency, excluding 2010 and 2015, the fan component manufacturing enterprises showed a higher comprehensive efficiency value than the other two types of enterprises, which indicated that the expansion of the wind energy industry had attracted a lot of funds, but this advantage disappeared with the realization of the financing risk. Thus the capital raise efficiency of the fan component manufacturing enterprises was mainly influenced by macro-factors; The fan production enterprises had the lowest average and that average was volatile, which demonstrated that the fund raise efficiency of the fan production enterprises was also volatile. The wind farm generation and operation enterprises maintained a relatively stable and gradually rising status which meant that the capital raise efficiency of the wind farm generation and operation enterprises were stable and not sensitive external factors.
For the pure technical efficiency, excluding 2014, the fan production enterprises had a lower pure technical efficiency than the other two types of enterprises, demonstrating that the fan production enterprises should pay more attention to technology investment than the other two types of enterprises in order to improve the capital raise efficiency.
With regard to the scale efficiency, the wind farm generation and operation enterprises indicated the lowest scale efficiency from 2010 to 2014, so that the scale investment would be the greatest concern for the wind farm generation and operation enterprises.
Comparative analysis of the wind energy industrial chain' capital raise efficiency
Comparative analysis of the industrial chain capital allocation efficiency of wind energy
Figure 5 shows the comparative analysis for capital allocation efficiency. The results of the analysis are as follows:
For the comprehensive efficiency, the fan production enterprises indicated a higher capital allocation efficiency than the other two types of enterprises most of the time. On the contrary, the fan component manufacturing enterprises indicated the lowest fund allocation efficiency; the capital raise efficiency of the wind farm generation and operation enterprises fluctuated slightly during the sampling period.
For the pure technical efficiency, excluding 2011 and 2015, the wind farm generation and operation enterprises indicated a lower capital allocation efficiency than the other two types of enterprises. Thus, the wind farm generation and operation enterprises should spend more money on technology investment in order to improve the capital allocation efficiency.
For the scale efficiency, excluding 2011, the fan component manufacturing enterprises indicated a lower capital allocation efficiency than the other two types of enterprises. Thus, technology investment should be the in the focus of the fan component manufacturing enterprises.
Comparative analysis of the wind energy industrial chain' capital allocation efficiency
The development of renewable energy is one way to solve the energy crisis. The wind energy industry is an important part of the renewable energy industry and has been a priority for the development of new energy strategies in the world. However, the growth of the wind energy industry has stopped and fallen to a long period of adjustments under the haze of the financial crisis in China. One of the development bottlenecks is the financing problem. Under this macroscopic background, it is very necessary to analyze the financial support efficiency of the wind energy industry in China.
The analysis of financial support efficiency of the wind energy industry is carried out in a very detailed way in this paper. First, this study thoroughly investigated the wind energy industry in the up-, middle- and down-stream chain (including not only the fan component, the manufacturing enterprises and the fan production enterprises, but also the wind farm generation and operation enterprises). Furthermore, it studied the financial support efficiency from the two aspects of capital raise and capital allocation.
In this paper, the use of the DEA model represented the very basis for carrying out a further index and regression analysis for determining the more specific factors, which would affect the financial support efficiency of the wind power industry in China and provide more targeted suggestions for promoting its benign development.
The following conclusions could be drawn based on the analysis above.
On the whole, the overall efficiency of the financial support of the Chinese wind energy industry had a close relationship to the macroeconomic environment, also, the capital raise and the allocation efficiency of the up-, middle- and down-stream chain showed different characters. Moreover, the lag of the core technology represented the biggest barrier to the financial support efficiency of the wind energy industry.
For the fan component manufacturing enterprises, its expansion of the production scale could attract more money, but with the advent of the financial crisis, this advantage reduced rapidly, as its capital raise efficiency had a close relationship to the macro-economic environment. Therefore, the fan component manufacturing enterprises should pay more attention to technology investment in order to improve its capital raise efficiency. Regardless of a change in the macroeconomic environment, the technical level was a major factor for improving its capital allocation efficiency. Thus, the fan component manufacturing enterprises should concentrate on their capacity of technology innovation, digestion and absorption.
The fund raise efficiency of the fan production enterprises was volatile, as the capital preferred to enter the construction of the wind farm rather than the production of the fans during the financial crisis of the wind energy industry. The scale expansion of the fan production enterprises cannot attract stable capital. Instead, its investment scale was appropriate for capital allocation efficiency. Therefore, fan production enterprises should strengthen their research and development of new technology and products.
The capital raise efficiency of the wind farm generation and operation enterprises was always higher than those of the other two types of enterprises, even during the financial crisis. Moreover, its scale expansion could attract more investments. The wind farm generation and operation enterprises should therefore increase investments in technology if they wish to improve their capital allocation efficiency.
Apart from improving the wind power industries own technology and scale, for enhancing the efficiency of the financial support for the wind power industry, a corresponding policy support is needed. FIT (Feed-in-Tariff), for example, are a kind of new energy subsidies, where their purpose is to encourage investors to invest in the new energy field, as the cost of new energy technology and production is higher than that of the traditional energy sources. Governments should adopt FIT, as the new energy would be able to reduce the greenhouse effect and carbon emissions and has important environmental and social effects.
Using the FIT policy, a government can improve not only the price of wind power and eventually improve the wind product prices in various fields of the power industry, but also improve their profit margins and encourage the development of the wind power industry. On the other hand, another characteristic of FIT are the subsidies which should be provided less and less as time goes on. The government has to encourage investors to invest in this new energy field, and also intend to reduce the production costs of the industry step by step. In addition, the government should encourage enterprises to carry out technology research and development, improve processes and reduce the production costs by cutting the amount of subsidies year by year in order to reach a final level which competes with traditional energy costs.
DEA:
data envelopment analysis
DEAP version 2.1:
data envelopment analysis program version 2.1
DMUs:
decision-making units
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The authors are grateful for the support provided by the Technical and Economic Research Institute of Business School in Hohai University.
Business School, Nanjing Xiaozhuang University, Hongjing Avenue 3601, Nanjing, Jiangsu, 211171, People's Republic of China
Lin Wu
Business School, Hohai University, Xikang Road 1, Nanjing, Jiangsu, 210098, People's Republic of China
Han Li
LW and HL designed and performed the experiments; LW collected the data; all authors wrote the paper. Both authors read and approved the final manuscript.
Correspondence to Han Li.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Wu, L., Li, H. Analysis of the development of the wind power industry in China — from the perspective of the financial support. Energ Sustain Soc 7, 37 (2017). https://doi.org/10.1186/s13705-017-0140-1
Wind power industry
DEA model | CommonCrawl |
Godunov's theorem
In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high-resolution schemes for the numerical solution of partial differential equations.
The theorem states that:
Linear numerical schemes for solving partial differential equations (PDE's), having the property of not generating new extrema (monotone scheme), can be at most first-order accurate.
Professor Sergei Godunov originally proved the theorem as a Ph.D. student at Moscow State University. It is his most influential work in the area of applied and numerical mathematics and has had a major impact on science and engineering, particularly in the development of methods used in computational fluid dynamics (CFD) and other computational fields. One of his major contributions was to prove the theorem (Godunov, 1954; Godunov, 1959), that bears his name.
The theorem
We generally follow Wesseling (2001).
Aside
Assume a continuum problem described by a PDE is to be computed using a numerical scheme based upon a uniform computational grid and a one-step, constant step-size, M grid point, integration algorithm, either implicit or explicit. Then if $x_{j}=j\,\Delta x$ and $t^{n}=n\,\Delta t$, such a scheme can be described by
$\sum _{m=1}^{M}{\beta _{m}}\varphi _{j+m}^{n+1}=\sum _{m=1}^{M}{\alpha _{m}\varphi _{j+m}^{n}}.$
(1)
In other words, the solution $\varphi _{j}^{n+1}$ at time $n+1$ and location $j$ is a linear function of the solution at the previous time step $n$. We assume that $\beta _{m}$ determines $\varphi _{j}^{n+1}$ uniquely. Now, since the above equation represents a linear relationship between $\varphi _{j}^{n}$ and $\varphi _{j}^{n+1}$ we can perform a linear transformation to obtain the following equivalent form,
$\varphi _{j}^{n+1}=\sum \limits _{m}^{M}{\gamma _{m}\varphi _{j+m}^{n}}.$
(2)
Theorem 1: Monotonicity preserving
The above scheme of equation (2) is monotonicity preserving if and only if
$\gamma _{m}\geq 0,\quad \forall m.$
(3)
Proof - Godunov (1959)
Case 1: (sufficient condition)
Assume (3) applies and that $\varphi _{j}^{n}$ is monotonically increasing with $j$.
Then, because $\varphi _{j}^{n}\leq \varphi _{j+1}^{n}\leq \cdots \leq \varphi _{j+m}^{n}$ it therefore follows that $\varphi _{j}^{n+1}\leq \varphi _{j+1}^{n+1}\leq \cdots \leq \varphi _{j+m}^{n+1}$ because
$\varphi _{j}^{n+1}-\varphi _{j-1}^{n+1}=\sum \limits _{m}^{M}{\gamma _{m}\left({\varphi _{j+m}^{n}-\varphi _{j+m-1}^{n}}\right)}\geq 0.$
(4)
This means that monotonicity is preserved for this case.
Case 2: (necessary condition)
We prove the necessary condition by contradiction. Assume that $\gamma _{p}^{}<0$ for some $p$ and choose the following monotonically increasing $\varphi _{j}^{n}\,$,
$\varphi _{i}^{n}=0,\quad i<k;\quad \varphi _{i}^{n}=1,\quad i\geq k.$
(5)
Then from equation (2) we get
$\varphi _{j}^{n+1}-\varphi _{j-1}^{n+1}=\sum \limits _{m}^{M}{\gamma _{m}}\left({\varphi _{j+m}^{n}-\varphi _{j+m-1}^{n}}\right)={\begin{cases}0,&j+m\neq k\\\gamma _{m},&j+m=k\\\end{cases}}$
(6)
Now choose $j=k-p$, to give
$\varphi _{k-p}^{n+1}-\varphi _{k-p-1}^{n+1}={\gamma _{p}\left({\varphi _{k}^{n}-\varphi _{k-1}^{n}}\right)}<0,$
(7)
which implies that $\varphi _{j}^{n+1}$ is NOT increasing, and we have a contradiction. Thus, monotonicity is NOT preserved for $\gamma _{p}<0$, which completes the proof.
Theorem 2: Godunov’s Order Barrier Theorem
Linear one-step second-order accurate numerical schemes for the convection equation
${{\partial \varphi } \over {\partial t}}+c{{\partial \varphi } \over {\partial x}}=0,\quad t>0,\quad x\in \mathbb {R} $
(10)
cannot be monotonicity preserving unless
$\sigma =\left|c\right|{{\Delta t} \over {\Delta x}}\in \mathbb {N} ,$
(11)
where $\sigma $ is the signed Courant–Friedrichs–Lewy condition (CFL) number.
Proof - Godunov (1959)
Assume a numerical scheme of the form described by equation (2) and choose
$\varphi \left({0,x}\right)=\left({{x \over {\Delta x}}-{1 \over 2}}\right)^{2}-{1 \over 4},\quad \varphi _{j}^{0}=\left({j-{1 \over 2}}\right)^{2}-{1 \over 4}.$
(12)
The exact solution is
$\varphi \left({t,x}\right)=\left({{{x-ct} \over {\Delta x}}-{1 \over 2}}\right)^{2}-{1 \over 4}.$
(13)
If we assume the scheme to be at least second-order accurate, it should produce the following solution exactly
$\varphi _{j}^{1}=\left({j-\sigma -{1 \over 2}}\right)^{2}-{1 \over 4},\quad \varphi _{j}^{0}=\left({j-{1 \over 2}}\right)^{2}-{1 \over 4}.$
(14)
Substituting into equation (2) gives:
$\left({j-\sigma -{1 \over 2}}\right)^{2}-{1 \over 4}=\sum \limits _{m}^{M}{\gamma _{m}\left\{{\left({j+m-{1 \over 2}}\right)^{2}-{1 \over 4}}\right\}}.$
(15)
Suppose that the scheme IS monotonicity preserving, then according to the theorem 1 above, $\gamma _{m}\geq 0$.
Now, it is clear from equation (15) that
$\left({j-\sigma -{1 \over 2}}\right)^{2}-{1 \over 4}\geq 0,\quad \forall j.$
(16)
Assume $\sigma >0,\quad \sigma \notin \mathbb {N} $ and choose $j$ such that $j>\sigma >\left(j-1\right)$. This implies that $\left({j-\sigma }\right)>0$ and $\left({j-\sigma -1}\right)<0$.
It therefore follows that,
$\left({j-\sigma -{1 \over 2}}\right)^{2}-{1 \over 4}=\left(j-\sigma \right)\left(j-\sigma -1\right)<0,$
(17)
which contradicts equation (16) and completes the proof.
The exceptional situation whereby $\sigma =\left|c\right|{{\Delta t} \over {\Delta x}}\in \mathbb {N} $ is only of theoretical interest, since this cannot be realised with variable coefficients. Also, integer CFL numbers greater than unity would not be feasible for practical problems.
See also
• Finite volume method
• Flux limiter
• Total variation diminishing
References
• Godunov, Sergei K. (1954), Ph.D. Dissertation: Different Methods for Shock Waves, Moscow State University.
• Godunov, Sergei K. (1959), A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations, Mat. Sbornik, 47, 271-306, translated US Joint Publ. Res. Service, JPRS 7226, 1969.
• Wesseling, Pieter (2001). Principles of Computational Fluid Dynamics. Berlin: Springer-Verlag. ISBN 9783540678533. OCLC 44972030.
Further reading
• Hirsch, Ch (1990). Numerical Computation of Internal and External Flows. Vol. 2. Chichester [England]: Wiley. ISBN 0-471-91762-1. OCLC 16523972.
• Laney, Culbert B. (1998). Computational Gasdynamics. Cambridge: Cambridge University Press. ISBN 978-0-511-77720-2. OCLC 664017316.
• Toro, Elewterio F. (2009). Riemann Solvers and Numerical Methods for Fluid Dynamics a Practical Introduction (3rd ed.). Berlin. ISBN 978-3-540-25202-3. OCLC 391057413.{{cite book}}: CS1 maint: location missing publisher (link)
• Anderson, Dale A.; Tannehill, John C.; Pletcher, Richard H.; Munipalli, Ramakanth; Shankar, Vijaya (2020). Computational Fluid Mechanics and Heat Transfer (Fourth ed.). Boca Raton, FL: Taylor & Francis. ISBN 978-1-351-12400-3. OCLC 1237821271.
| Wikipedia |
\begin{document}
\title {On the radical of multigraded modules}
\author {Viviana Ene}
\address{Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527 Constanta, Romania
\newline \indent Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700, Buchaest, Romania}
\email{[email protected]}
\author{Ryota Okazaki}
\thanks{The first author was partially supported by the JSPS Invitation Fellowship Programs for Research in Japan and by the grant UEFISCDI, PN-II-ID-PCE- 2011-3-1023. The second author is partially supported by JST, CREST and also by KAKENHI (no. 20624109)}
\address{Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan}
\email{[email protected]}
\begin{abstract} We define a functor ${\frk r}^\ast$ from the category of positively determined modules to the category of squarefree modules which plays the role of passing from a monomial ideal to its radical. By using this functor, we generalize several results on properties that are shared by a monomial ideal and its radical. Moreover, we study the connection of ${\frk r}^\ast$ to the Alexander duality and Auslander-Reiten translate functor.
\end{abstract}
\keywords{Multigraded modules, squarefree modules, Cohen-Macaulay modules, Alexander duality, Auslander-Reiten translate}
\maketitle
\section*{Introduction}
Our original motivation for this work was to generalize some of the results obtained by Herzog, Takayama, and Terai in \cite{HTT}. They proved that
many properties of a monomial ideal pass to its radical. It is well known that every monomial ideal $I$ in the polynomial ring $S=K[x_1,\ldots,x_n]$
over a field $K$ is a positively ${\bold t}$-determined $S$-module for an appropriate ${\bold t}\in {\NZQ N}^n$ as it was defined by Miller in \cite{Mill}. Thus, a
natural way to generalize the results of \cite{HTT} is to consider positively determined modules instead of monomial ideals. We show that one may
define a functor ${\frk r}^\ast$ from the category $\Modtb$ of positively ${\bold t}$-determined modules to the category $\Modone$ of squarefree $S$-modules
which plays the role of passing from a monomial ideal to its radical. As it was shown in \cite{BF}, to any ordering preserving map ${\bold q}:{\NZQ N}^n\to
{\NZQ N}^n$, one may associate a functor ${\bold q}^\ast$ from the category of ${\NZQ N}^n$-graded $S$-modules to itself which is defined as follows. For any $M\in
{\bf Mod}_{S}^{\mathbb{N}^n}$ and ${\bold a}\in {\NZQ N}^n,$ $({\bold q}^\ast M)_{{\bold a}}=M_{q({\bold a})}$ and the $S$-module structure of ${\bold q}^\ast M$ is given by the multiplication $(q^\ast
M)_{{\bold b}}\stackrel{\cdot{\bold x}^{\bold a}}{\rightarrow}(q^\ast M)_{{\bold b}+{\bold a}}$ that maps every homogeneous element $y\in (q^\ast M)_{{\bold b}}$ to ${\bold x}^{q({\bold a}+{\bold b})-q({\bold b})}y.$ If $f:M\to N$ is a graded morphism of ${\NZQ N}^n$-graded $S$-modules, then for every ${\bold a}\in {\NZQ N}^n,$ the ${\bold a}$-degree component of $q^\ast f: q^\ast M \to q^\ast N$ is $f_{q({\bold a})}.$ In \cite{BF} it is shown that $q^\ast$ is an exact functor. In Section~\ref{radicalsection}, we have considered the following map. For ${\bold t}\in {\NZQ N}^n$ with ${\bold t}\geq {\bf 1}$ where ${\bf 1}=(1,\ldots,1), $ we define ${\frk r}: {\NZQ N}^n \to {\NZQ N}^n$ by ${\frk r}({\bold a})=(r_i(a_i))_{1\leq i\leq n}$ where \[ r_i(a_i)=\left\{ \begin{array}{ll}
t_i, & \text{ if } a_i>0,\\
0, & \text{ otherwise. } \end{array}\right. \] This is an ordering preserving map which induces a functor ${\frk r}^\ast$ which depends on ${\bold t}$ from the category ${\bf Mod}_{S}^{\mathbb{N}^n}$ to itself. We showed in Section~\ref{radicalsection} that this functor transports the category $\Modtb$ into the category of squrefree modules and, moreover, for any monomial ideal $I\subset S,$ we have ${\frk r}^\ast I\cong \sqrt{I}$ as $S$-modules.
As it was explained in \cite{HTT}, the Betti numbers do not increase when one passes from a monomial ideal to its radical. We show in Theorem~\ref{bettiradical} that passing from a positively ${\bold t}$-determined module to its ''radical'' module has a similar behavior. In particular, one obtains $\depth M\leq \depth{\frk r}^\ast M$ for any $M\in \Modtb$.
Unlike the monomial case, for a positively ${\bold t}$-determined module $M,$ we show in Corollary~\ref{dim}, that one has only the inequality $\dim {\frk r}^\ast M\leq \dim M.$ Easy examples show that the inequality may be strict. By using the inequalities between depth and Krull dimension, we show in Theorem~\ref{CM} that the (sequentailly) Cohen-Macaulay property of $M$ passes to the ''radical'' of $M$ for any postively ${\bold t}$-determined module $M$with ${\frk r}^\ast M\neq 0.$
In Section~\ref{extsection} we study the connection between the functor ${\frk r}^\ast$ and $\Ext$. The main result of the section is Theorem~\ref{ext} which states that for every module $M\in \Modtb$ there exists a natural isomorphism $ \grExt^p_S(M, \omega_S)_{\geq \Zero} \cong \grExt^p_S({\frk r}^\ast(M),\omega_S)$ for all $p,$ where $\omega_S$ is the canonical module of $S.$ In particular, for any Cohen-Macaulay ideal $I\subset S$ such that $S/I$ is a positively ${\bold t}$-determined module, it follows that the canonical module of $S/\sqrt{I}$ is isomorphic to the positive part of $\omega_{S/I}.$ From Theorem~\ref{ext}, under an additional condition, it follows that if $M\in \Modtb$ is a generalized Cohen-Macaulay (Buchsbaum) module, then ${\frk r}^\ast M$ shares the same property.
Finally, in the last two sections we show how our ''radical'' functor is connected to the Alexander duality (Proposition~\ref{prop:r_and_A}) and Auslander-Reiten translate functor (Proposition~\ref{ART}).
\section*{Acknowledgment}
A part of this research was done when the second author visited the Faculty of Mathematics and Informatics of Ovidius University. He is deeply grateful for warm hospitality.
\section{The ${\frk r}^\ast$ functor and first applications}
\label{radicalsection}
In this section we define the ${\frk r}^\ast$ functor on the category ${\bf Mod}_{S}^{\mathbb{N}^n}$ of the ${\NZQ N}^n$-graded $S$-modules where $S=K[x_1,\ldots,x_n]$ is the polynomial ring in $n$ variables over a field $K.$ We first recall the basic notions and set the notation. Let ${\NZQ N}$ be the set of non-negative integers. For ${\bold a}=(a_1,\ldots,a_n)\in {\NZQ N}^n$ we set ${\bold x}^{\bold a}=x_1^{a_1}\cdots x_n^{a_n}$ and call ${\bold a}$ the degree of the monomial ${\bold x}^{\bold a}$. $\nu_i(u)$ denotes the exponent of variable $x_i$ in the monomial $u\in S$. Let
$\leq$ be the partial order on ${\NZQ Z}^n$ which is defined as follows. If ${\bold a}=(a_1,\ldots,a_n),{\bold b}=(b_1,\ldots,b_n)\in{\NZQ Z}^n$, then ${\bold a}\leq {\bold b}$ if $a_i\leq b_i$ for $1\leq i\leq n.$ Of course, this order induces a partial order on ${\NZQ N}^n.$
Let ${\bold t}\in {\NZQ N}^n$ with ${\bold t}\geq {\bf 1},$ where ${\bf 1}=(1,\ldots,1).$ According to \cite{Mill}, a ${\NZQ Z}^n$-graded $S$-module $M$ is called {\em positively ${\bold t}$-determined} if it is finitely generated, ${\NZQ N}^n$-graded, and if the multiplication map $M_{{\bold a}}\stackrel{x_i}{\rightarrow}{M_{{\bold a}+{\bold e}_i}}$ is an isomorphism of $K$-vector spaces whenever $a_i \ge t_i$. Here, ${\bold e}_i$ is the vector of ${\NZQ Z}^n$ with its $i$-th component equal to $1$ and all the others equal to $0.$ A monomial ideal $I$ is positively ${\bold t}$-determined if and only if it is generated by some elements $x^{\bold a}$ with $\Zero \le {\bold a} \le {\bold t}$. Every finitely generated ${\NZQ N}^n$-graded $S$-module is positively ${\bold t}$-determined for some ${\bold t} \gg {\bf 1}$. In particular, for any 2 monomial ideals $I,J$ with $J \supseteq I$, $J/I$ is positively ${\bold t}$-determined for some ${\bold t} \gg {\bf 1}$.
Let $\Modtb$ be the full subcategory of ${\bf Mod}_{S}^{\mathbb{Z}^n}$ consisting of positively ${\bold t}$-determined $S$-modules.
According to \cite{BF}, with any order preserving map $q:{\NZQ Z}^n\to {\NZQ Z}^n,$ one may associate a functor $q^{\ast}: {\bf Mod}_{S}^{\mathbb{Z}^n} \to {\bf Mod}_{S}^{\mathbb{Z}^n}$. Since we are concerned only with ${\NZQ N}^n$-graded modules, that is, ${\NZQ Z}^n$-graded modules whose components of degree ${\bold a}\in {\NZQ Z}^n\setminus {\NZQ N}^n$ are all zero, we may consider the map $q:{\NZQ N}^n\to {\NZQ N}^n.$
$q^\ast$ acts on modules and morphisms as follows. For a ${\NZQ Z}^n$-graded $S$-module $M,$ the ${\bold a}$-degree component of $q^\ast M$ is $(q^\ast M)_{{\bold a}}=M_{q({\bold a})}.$ The multiplication which gives the $S$-module structure of $q^\ast M$ is the following. For a monomial ${\bold x}^{\bold a}\in S,$ the map $(q^\ast M)_{{\bold b}}\stackrel{\cdot{\bold x}^{\bold a}}{\rightarrow}(q^\ast M)_{{\bold b}+{\bold a}}$ maps every homogeneous element $y\in (q^\ast M)_{{\bold b}}$ to ${\bold x}^{q({\bold a}+{\bold b})-q({\bold b})}y.$
We describe now the action of $q^\ast$ on the morphisms of the category ${\bf Mod}_{S}^{\mathbb{Z}^n}$ following \cite{BF}. If $f:M\to N$ is a graded morphism of ${\NZQ Z}^n$-graded $S$-modules, then for every ${\bold a}\in {\NZQ Z}^n,$ the ${\bold a}$-degree component of $q^\ast f: q^\ast M \to q^\ast N$ is $f_{q({\bold a})}.$ In \cite{BF} it is shown that $q^\ast$ is an exact functor.
We consider now the following order preserving map. Let ${\bold t} \in {\NZQ N}^n, {\bold t}\geq {\bf 1},$ and ${\frk r}: {\NZQ N}^n \to {\NZQ N}^n$ given by ${\frk r}({\bold a})=(r_i(a_i))_{1\leq i\leq n}$ where \[ r_i(a_i)=\left\{ \begin{array}{ll}
t_i, & \text{ if } a_i>0,\\
0, & \text{ otherwise. } \end{array}\right. \] It is easily seen that ${\frk r}$ is an order preserving map, hence we may consider the functor ${\frk r}^\ast: {\bf Mod}_{S}^{\mathbb{N}^n}\to {\bf Mod}_{S}^{\mathbb{N}^n}$ associated with ${\frk r}.$
\begin{Proposition}\label{sqfree} Let $M$ be a positively ${\bold t}$-determined ${\NZQ N}^n$-graded $S$-module. Then ${\frk r}^\ast M$ is a positively ${\bf 1}$-determined module, that is, ${\frk r}^\ast M$ is a squarefree $S$-module. \end{Proposition}
\begin{proof} It is enough to show that, for any ${\bold a}\in {\NZQ N}^n$ and $i\in \supp({\bold a}),$ the multiplication map \[ ({\frk r}^\ast M)_{{\bold a}} \stackrel{\cdot x_i}{\rightarrow}({\frk r}^\ast M)_{{\bold a}+{\bold e}_i} \] is an isomorphism of $K$-vector spaces. But this is almost obvious, since ${\bold x}^{e_i}\cdot u={\bold x}^{{\frk r}({\bold a}+{\bold e}_i)-{\frk r}({\bold a})}u=u.$ The last equality is true since $\supp({\bold x}^{{\frk r}({\bold a}+{\bold e}_i)-{\frk r}({\bold a})})=\emptyset$ for any $i\in \supp({\bold a})$. Therefore, the multiplication by $x_i$ is the identity map, hence it is an isomorphism of vector spaces. \end{proof}
\begin{Proposition}\label{classicradical} Let ${\bold t}\in {\NZQ N}^n$ with ${\bold t}\geq {\bf 1},$ and let $I\subset S$ be a monomial ideal which is positively ${\bold t}$-determined, that is, for every $u\in G(I),$ $\deg(u)\leq {\bold t}.$ Then ${\frk r}^\ast I\cong \sqrt{I}.$ \end{Proposition}
\begin{proof} Firstly, we claim that ${\bold x}^{\bold a}\in \sqrt{I}$ if and only if ${\bold x}^{{\frk r}({\bold a})}\in I.$ Let us prove this claim. If ${\bold x}^{{\bold a}}\in \sqrt{I},$ then there exists $k\geq 1$ such that ${\bold x}^{k{\bold a}}\in I.$ Obviously, we may choose $k$ such that $ka_i\geq t_i$ for all
$a_i>0.$ Then there exists ${\bold x}^{\bold b}\in G(I)$ such that ${\bold x}^{\bold b}\ |\ {\bold x}^{k{\bold a}},$ which implies that ${\bold b}\leq k{\bold a}.$ Since $I$ is positively
${\bold t}$-determined, we also have ${\bold b}\leq {\bold t}.$ It then follows that ${\bold b}\leq {\frk r}(k{\bold a})={\frk r}({\bold a})$ which implies that ${\bold x}^{\bold b}\ |\ {\bold x}^{{\frk r}({\bold a})}$ and, therefore, ${\bold x}^{{\frk r}({\bold a})}\in I.$
Conversely, let ${\bold x}^{{\frk r}({\bold a})}\in I.$ We obviously may find $k\geq 1$ such that $k{\bold a}\geq {\frk r}({\bold a}),$ hence ${\bold x}^{k{\bold a}}\in I$ and, therefore, ${\bold x}^{{\bold a}}\in \sqrt{I},$ which ends the proof of our claim.
Let $f:\sqrt{I}\to {\frk r}^\ast I$ be the map given by $f=\oplus_{{\bold a}}f_{{\bold a}}$ where $f_{{\bold a}}:(\sqrt{I})_{\bold a}\to ({\frk r}^\ast I)_{{\bold a}}$ is defined by $f_{{\bold a}}({\bold x}^{\bold a})={\bold x}^{{\frk r}({\bold a})}.$ The map $f$ is obviously a graded isomorphism of $K$-vector spaces. We show that $f$ is an $S$-module isomorphism. Indeed, for any ${\bold a}, {\bold b},$ \[ f({\bold x}^{\bold b} \cdot {\bold x}^{\bold a})={\bold x}^{{\frk r}({\bold a}+{\bold b})}= {\bold x}^{{\frk r}({\bold a}+{\bold b})-{\frk r}({\bold a})}{\bold x}^{{\frk r}({\bold a})}= {\bold x}^{\bold b}\cdot f({\bold x}^{\bold a}). \] \end{proof}
In order to state the first main result, we need a preparatory lemma. Before stating it, let us set some more notation. For ${\bold a}\in {\NZQ N}^n$, let $S(-{\bold a})$ be the graded free $S$-module whose all graded components are obtained from those of $S$ by shifting with the vector ${\bold a},$ and let $\sqrt{{\bold a}}$ be the following vector of ${\NZQ N}^n:$ \[ (\sqrt{{\bold a}})_i=\left\{ \begin{array}{ll}
1, & \text{ if } a_i>0,\\
0, & \text{ if } a_i=0. \end{array}\right. \]
\begin{Lemma}\label{shift} Let ${\bold t}\in {\NZQ N}^n$ with $ {\bold t}\geq {\bf 1}.$ Then $r^\ast (S(-{\bold a}))\cong S(-\sqrt{{\bold a}})$ for every ${\bold a}\in {\NZQ N}^n$ with ${\bold a}\leq {\bold t}.$ \end{Lemma}
\begin{proof} We obviously have the following isomorphisms: \[ {\frk r}^\ast(S(-{\bold a}))\cong {\frk r}^\ast({\bold x}^{\bold a})\cong ({\bold x}^{\sqrt{{\bold a}}})\cong S(-\sqrt{{\bold a}}). \] \end{proof}
In the sequel we will always assume that ${\frk r}^\ast M\neq 0$. Note that ${\frk r}^\ast M=0$ if and only if $M_{{\bold a}}=0$ for all ${\bold a}\in {\NZQ N}^n$ such that $a_i\in\{0,t_i\}$ for $1\leq i\leq n.$
The following theorem shows that the graded Betti numbers go down when passing from the module to its radical. In particular, we may derive inequalities for the corresponding depths.
\begin{Theorem}\label{bettiradical} Let ${\bold t}\in {\NZQ N}^n,$ ${\bold t}\geq {\bf 1}$, and let $M$ be a positively ${\bold t}$-determined ${\NZQ N}^n$-graded $S$-module. Then \[ \beta_{i,{\bold a}}(M)\geq \beta_{i,\sqrt{{\bold a}}}({\frk r}^\ast M) \] for all $i$ and ${\bold a}.$ In particular, the following inequality holds: \[ \depth M\leq \depth {\frk r}^\ast M. \] \end{Theorem}
\begin{proof} Let \[ {\NZQ F}_{\bullet}: \hspace{0.8cm} 0\to \bigoplus_{{\bold a}}S(-{\bold a})^{\beta_{p,{\bold a}}}\to \cdots \to \bigoplus_{{\bold a}}S(-{\bold a})^{\beta_{1,{\bold a}}} \to \bigoplus_{{\bold a}}S(-{\bold a})^{\beta_{0,{\bold a}}}\to M\to 0 \] be a minimal free resolution of $M$ over $S.$ Since $M$ is positively ${\bold t}$-determined, by \cite[Proposition 2.5]{Mill}, it follows that all the shifts in the above resolution are $\leq {\bold t}.$ We apply ${\frk r}^\ast$ to ${\NZQ F}_{\bullet}$. By the exactness of ${\frk r}^\ast,$ we get a free $S$-resolution of ${\frk r}^\ast M,$ possibly non-minimal. Therefore, we get the inequalities between the graded Betti numbers of $M$ and, respectively, ${\frk r}^\ast M.$ These inequalities imply that $\projdim_S M\geq \projdim_S( {\frk r}^\ast M)$ and, by using Auslander-Buchsbaum formula, we get the inequalities between depths. \end{proof}
\begin{Remark}\label{resol}{\em By the above proof it follows that if \[ {\NZQ F}_{\bullet}: 0\to \bigoplus_{{\bold a}}S(-{\bold a})^{\beta_{p,{\bold a}}}\to \cdots \to \bigoplus_{{\bold a}}S(-{\bold a})^{\beta_{1,{\bold a}}} \to \bigoplus_{{\bold a}}S(-{\bold a})^{\beta_{0,{\bold a}}}\to M\to 0 \] is a minimal ${\NZQ Z}^n$-graded free resolution of $M,$ then \[ {\frk r}^\ast{\NZQ F}_{\bullet}: 0\to \bigoplus_{{\bold a}}S(-\sqrt{{\bold a}})^{\beta_{p,{\bold a}}}\to \cdots \to \bigoplus_{{\bold a}}S(-\sqrt{{\bold a}})^{\beta_{1,{\bold a}}} \to \bigoplus_{{\bold a}}S(-\sqrt{{\bold a}})^{\beta_{0,{\bold a}}}\to {\frk r}^\ast M\to 0 \] is a free resolution of ${\frk r}^\ast M$. Moreover, if the map $\partial_i: \bigoplus_{{\bold a}}S(-{\bold a})^{\beta_{i,{\bold a}}} \to \bigoplus_{{\bold a}}S(-{\bold a})^{\beta_{i-1,{\bold a}}}$ is given by the matrix $({\bold x}^{{\bold a}_j-{\bold b}_k})_{j,k}$, then ${\frk r}^\ast \partial_i: \bigoplus_{{\bold a}}S(-\sqrt{{\bold a}})^{\beta_{i,{\bold a}}} \to \bigoplus_{{\bold a}}S(-\sqrt{{\bold a}})^{\beta_{i-1,{\bold a}}}$ is given by $({\bold x}^{\sqrt{{\bold a}_j}-\sqrt{{\bold b}_k}})_{j,k}$. } \end{Remark}
In the following corollary we derive some consequences of Theorem~\ref{bettiradical}. To begin with, let $I\subset S$ be a monomial ideal. Then $I$ is a positively
${\bold t}$-determined ${\NZQ N}^n$-graded $S$-module if we choose, for instance, ${\bold t}=(t_1,\ldots,t_n)$ where $t_i=\max\{\nu_i(u)\ |\ u\in G(I)\}$. Here $G(I)$ denotes the minimal system of monomial generators of the ideal $I.$ Proposition~\ref{classicradical} says that ${\frk r}^\ast I$ is actually $\sqrt{I}.$ Therefore, from Theorem~\ref{bettiradical}, we obtain as a consequence the following corollary which extends results of \cite{HTT}.
\begin{Corollary}\label{extensionHHT} Let $I\subset J\subset S$ be monomial ideals with $\sqrt{I}\neq \sqrt{J}$. Then $$\beta^S_{i,{\bold a}}(J/I)\geq \beta^S_{i,\sqrt{{\bold a}}}(\sqrt{J}/\sqrt{I})$$ for all $i\geq 0$ and ${\bold a}\in {\NZQ N}^n.$ Consequently, $$\depth_S(\sqrt{J}/\sqrt{I})\geq \depth_S(J/I).$$ \end{Corollary}
In the following we study the relationship between the Krull dimension of $M$ and ${\frk r}^\ast M.$ We first introduce the following notation. For ${\bold a}\in {\NZQ N}^n,$ $\supp({\bold a})=\{i: a_i>0\}$ and $\supp^{\bold t}({\bold a})=\{i: a_i\geq t_i\}.$ We use the following convention. For ${\bold a}, {\bold b} \in {\NZQ Z}^n$, let ${\bold a} \cdot {\bold b}$ denote the vector whose $i$-th component is $a_ib_i$.
\begin{Proposition}\label{Ass} Let $M$ be a positively ${\bold t}$-determined $S$-module where ${\bold t}\in {\NZQ N}^n, {\bold t}\geq {\bf 1}.$ Then $\Ass({\frk r}^\ast M)\subset \Ass(M).$ \end{Proposition}
\begin{proof} Let $F\subset [n]$ and $P=P_F:=(x_i : i\notin F)$ be an associated prime of ${\frk r}^\ast(M)$. Then, by \cite[Lemma 2.2]{Y}, there exists $0\neq u\in ({\frk r}^\ast M)_{{\bold e}_F}$ such that $x_i u=0$ for all $i\notin F,$ where ${\bold e}_F:=\sum_{i\in F}{\bold e}_i.$ This means that there exists $0\neq u\in M_{{\bold t}\cdot {\bold e}_F}$ such that \[ {\bold x}^{{\frk r}({\bold e}_{F\cup\{i\}})-{\frk r}({\bold e}_i)} u={\bold x}^{{\bold t}\cdot {\bold e}_i}u=x_i^{t_i}u=0 \] for all $i\notin F.$ Then we may choose a maximal monomial (with respect to divisibility) $w\in K[\{x_i : i\notin F\}]$ such that $wu\neq 0.$ We claim that $P_F=\ann(wu)$, which will end the proof.
If $i\notin F,$ then $x_i(wu)=0$ by the choice of $w,$ hence $P_F\subset\ann(wu).$ Let now $v$ be a monomial in $\ann(wu)$, that is, $v(wu)=0$. Clearly, for every monomial $w^\prime\in K[\{x_i: i\in F\}]$, we have $w^\prime wu\neq 0$ since $\supp(w^\prime)\subset \supp^{\bold t}(wu)$ and $M$ is positively ${\bold t}$-determined. This implies that there exists $i\notin F$ such that $x_i$ divides $v$, thus $v\in P_F.$ \end{proof}
\begin{Corollary}\label{dim} Let $M$ be a positively ${\bold t}$-determined module. Then $\dim M\geq \dim {\frk r}^\ast M.$ \end{Corollary} Note that the inequality $\dim {\frk r}^\ast M\leq \dim M$ may be strict as the following example shows. On the other hand, we have $\dim {\frk r}^\ast M = \dim M$ if and only if there exists ${\bold a}\in {\NZQ N}^n$ such that $\#\supp^{\bold t}({\bold a})=\dim M$ and $M_{{\frk r}({\bold a})}\neq 0$.
\begin{Example}{\em Let $I=(a^4d^4,a^2b^3,b^3c^2,b^3d)$ and $J=(a^3d^3,a^3b,b^2)$, $I,J\subset K[a,b,c,d]$. One may easily check that $\dim(\sqrt{J}/\sqrt{I})=1 < \dim(J/I)=2.$ } \end{Example}
Let us recall that a finitely generated $S$-module is called {\em equidimensional} if all its minimal primes have the same codimension. As an immediate consequence of Proposition~\ref{Ass} we get also the following
\begin{Corollary}\label{equidim} Let $M$ be a positively ${\bold t}$-determined module such that $\dim M=\dim {\frk r}^\ast M.$ If $M$ is equidimensional, then ${\frk r}^\ast M$ is equidimensional, too. \end{Corollary}
The following example shows that the implication of the above corollary is no longer true if $\dim M > \dim {\frk r}^\ast M$.
\begin{Example}{\em Let $P,P_1,P_2\subset S,$ $P=(x_1), P_2=(x_1,x_2)$, $P_2=(x_1,x_3,x_4),$ and $M=(S/P)(-(1,0,\ldots,0))\oplus (S/P_1)\oplus (S/P_2)$. Then $M$ is positively ${\bold t}$-determined, where ${\bold t}=(2,1,\ldots,1),$ and equidimensional. We have ${\frk r}^\ast M=(S/P_1)\oplus (S/P_2)$, thus ${\frk r}^\ast M$ is not equidimensional. But, of course, $\dim M > \dim {\frk r}^\ast M.$ } \end{Example}
In \cite{HTT} it is shown that if $S/I$ is (sequentially) Cohen-Macaulay, then $S/\sqrt{I}$ shares the same property. We are going to extend this result to any positively determined ${\NZQ N}^n$-graded module.
\begin{Theorem}\label{CM} Let $M$ be a positively ${\bold t}$-determined $S$-module with ${\frk r}^\ast M \neq 0$ where ${\bold t}\in {\NZQ N}^n, {\bold t}\geq {\bf 1}.$ \begin{itemize}
\item [(i)] If $M$ is Cohen-Macaulay, then ${\frk r}^\ast M$ is Cohen-Macaulay and $\dim {\frk r}^\ast M=\dim M.$
\item [(ii)] If $M$ is sequentially Cohen-Macaulay, then ${\frk r}^\ast M$ is sequentially Cohen-Macaulay. \end{itemize} \end{Theorem}
\begin{proof} (i). By Theorem~\ref{bettiradical} and Corollary~\ref{dim}, we get the following inequalities: \[ \depth M\leq \depth {\frk r}^\ast M\leq \dim {\frk r}^\ast M\leq \dim M. \] Since $M$ is Cohen-Macaulay, we get the desired conclusions.
(ii). As $M$ is sequentially Cohen-Macaulay, there exists a finite filtration \[ 0=M_0\subset M_1\subset \cdots \subset M_r=M \] by graded submodules of $M$ such that each quotient $M_i/M_{i-1}$ is Cohen-Macaulay and \[ \dim M_1/M_0 < \dim M_2/M_1 <\cdots <\dim M_r/M_{r-1}. \] This filtration induces the following filtration of ${\frk r}^\ast M,$ \[ 0={\frk r}^\ast M_0\subset {\frk r}^\ast M_1\subset \cdots \subset {\frk r}^\ast M_r={\frk r}^\ast M. \] By (i), each factor in this filtration is either $0$ or a Cohen-Macaulay module with $\dim {\frk r}^\ast M_i/{\frk r}^\ast M_{i-1}=\dim M_i/M_{i-1}.$ By skipping the redundant factors in the above filtration we get the desired filtration for ${\frk r}^\ast M.$ Therefore, (ii) follows. \end{proof}
We may now derive the following corollary which extends some results of \cite{HTT}.
\begin{Corollary} \label{CMextend} Let $I\subset J\subset S$ be monomial ideals such that $\sqrt I\neq \sqrt J$. Then:
\item [(i)] If $J/I$ is Cohen-Macaulay, then $\sqrt J/\sqrt I $ is Cohen-Macaulay and, moreover,
$$\dim J/I=\dim \sqrt J/\sqrt I.$$
\item [(ii)] If $J/I$ is sequentially Cohen-Macaulay, then $\sqrt J/\sqrt I$ is sequentially Cohen-Macau\-lay. \end{Corollary}
\section{The functor ${\frk r}^\ast$ and $\Ext$}
\label{extsection}
For $M \in {\bf Mod}_{S}^{\mathbb{Z}^n}$ and ${\bold a} \in {\NZQ Z}^n$, $M({\bold a})$ denotes the ${\NZQ Z}^n$-graded $S$-module such that $M = M({\bold a})$ as underlying $S$-modules and the degree is given by the formula $M({\bold a})_{\bold b} = M_{{\bold a} + {\bold b}}$. Following the usual convention, for $M, N \in {\bf Mod}_{S}^{\mathbb{Z}^n}$, we set $\grHom_S(M,N) := \bigoplus_{{\bold a}\in {\NZQ Z}^n}\Hom_{{\bf Mod}_{S}^{\mathbb{Z}^n}}(M,N({\bold a}))$. Note that if $M$ is finitely generated, $\grHom_S (M,N) = \Hom_S(M,N)$. Let $\grExt^i_S(-,N)$ (resp. $\grExt^i_S(M,-)$) be the $i$-th right derived functor of $\grHom_S(-,N)$ (resp. $\grHom_S(M, -)$).
In this section, we will study the relation between ${\frk r}^\ast$ and $\grExt$-functor. For this purpose, we need the following three functors.
Recall that degree-shifting induces an endofunctor of ${\bf Mod}_{S}^{\mathbb{Z}^n}$. For ${\bold a} \in {\NZQ Z}^n$, let $\sigma_{\bold a}$ denote the functor given by shifting degree by ${\bold a}$. Thus we have $\sigma_{\bold a}(M) = M(-{\bold a})$ for all $M \in {\bf Mod}_{S}^{\mathbb{Z}^n}$.
For $M \in {\bf Mod}_{S}^{\mathbb{Z}^n}$ and ${\bold a} \in {\NZQ Z}^n$, the truncated module $\tau_{ {\bold a}}(M) := \bigoplus_{{\bold b} \ge {\bold a}} M_{\bold b}$ is again an object of ${\bf Mod}_{S}^{\mathbb{Z}^n}$, and any morphism $M \to N$ in ${\bf Mod}_{S}^{\mathbb{Z}^n}$ induces the one $f|_{\tau_{ {\bold a}}(M)}:\tau_{ {\bold a}}(M) \to \tau_{ {\bold a}}(N)$. Thus we have the functor $\tau_{ {\bold a}}: {\bf Mod}_{S}^{\mathbb{Z}^n} \to {\bf Mod}_{S}^{\mathbb{Z}^n}$.
Let ${\frk s} : {\NZQ N}^n \to {\NZQ Z}^n$ be the function defined by ${\frk s}({\bold a}) = (s_i(a_i))_{1 \le i \le n}$ where $$ s_i(a_i) = \begin{cases} t_i &\text{if $a_i \ge 1$,}\\ t_i - 1 &\text{otherwise.} \end{cases} $$ The induced functor ${\frk s}^\ast: {\bf Mod}_{S}^{\mathbb{Z}^n} \to {\bf Mod}_{S}^{\mathbb{N}^n}$, if restricted to $\Modtb$, is an endofunctor of $\Modtb$.
For $M \in \Modtb$ and ${\bold a} \in {\NZQ Z}^n$ with ${\bold a} \ge \Zero$, the multiplication $$ x^{{\bold a} \cdot {\bold t} - {\frk r}({\bold a})}: M_{{\frk r}({\bold a})} \to M_{{\bold a} \cdot {\bold t}} $$ is a $K$-linear isomorphism since $\supp({\bold a}\cdot{\bold t}-{\frk r}({\bold a}))\subset \supp^{\bold t}{\frk r}({\bold a})$. Let $\phi^M_{\bold a}$ denote this map.
Now we are ready to define the two natural transformations $\Phi^{\frk r}: \id_{\Modtb} \Longrightarrow {\frk r}^\ast$ between endofunctors of $\Modtb$ and $\Psi: {\frk s}^\ast \Longrightarrow \sigma_{{\bf 1} - {\bold t}}$ between those of ${\bf Mod}_{S}^{\mathbb{N}^n}$. For $M \in \Modtb$, let $\Phi_M : M \to {\frk r}^\ast(M)$ be the map defined as follows; for a homogeneous $u \in M_{\bold a}$ with ${\bold a} \ge \Zero$, $$ \Phi_M(u) := {(\phi^M_{{\bold a}})}^{-1}(x^{{\bold a} \cdot ({\bold t} - {\bf 1})}u) \in {\frk r}^\ast(M)_{\bold a}. $$
For ${\bold a} \in {\NZQ N}^n$, it is easy to verify that ${\bold a} + {\bold t} - {\bf 1} \ge {\frk s}({\bold a})$. For $M \in {\bf Mod}_{S}^{\mathbb{N}^n}$, we define the map $\Psi_M: {\frk s}^\ast(M) \to \sigma_{{\bf 1} - {\bold t}}(M)$ as follows; for ${\bold a} \in {\NZQ N}^n$ and a homogeneous $u \in {\frk s}^\ast(M)_{\bold a}$ with ${\bold a} \ge \Zero$, $$ \Psi_M(u) = x^{{\bold a} + {\bold t} - {\bf 1} - {\frk s}({\bold a})} \cdot u \in \sigma_{{\bf 1} - {\bold t}}(M)_{\bold a}. $$
\begin{Lemma}\label{lem:nat} The following statements hold. \begin{enumerate} \item The above $\Phi$ is indeed a natural transformation from $\id_{\Modtb}$ to ${\frk r}^\ast$. \item Let $\iota: \Modone \to \Modtb$ be the canonical embedding. Then $\Phi$ induces the natural isomorphism $\id_{\Modone} \Longrightarrow {\frk r}^\ast \circ \iota$ between endofunctors of $\Modone$. \item The above $\Psi$ is indeed a natural transformation from ${\frk s}^\ast$ to $\sigma_{{\bf 1} - {\bold t}}$. \end{enumerate} \end{Lemma} \begin{proof} We will prove only the assertions (1) and (2). The rest is proved by the way similar to (1).
(1) First, we must verify that $\Phi_M$ is an $S$-linear map. Take any $u \in M_{\bold a}$ and ${\bold b} \in {\NZQ Z}^n$ with ${\bold b} \ge \Zero$. Then \begin{align*} \phi^M_{{\bold a} + {\bold b}} (x^{\bold b} \cdot \Phi_M(u))
& = x^{({\bold a} + {\bold b} ) \cdot {\bold t} - {\frk r}({\bold a} + {\bold b})} (x^{{\frk r}({\bold a} + {\bold b}) - {\frk r}({\bold a})} {(\phi^M_{{\bold a}})}^{-1}(x^{{\bold a} \cdot ({\bold t} - {\bf 1})}u)) \\
& = x^{({\bold a} + {\bold b}) \cdot {\bold t} - {\frk r}({\bold a})} {(\phi^M_{{\bold a}})}^{-1}(x^{{\bold a} \cdot ({\bold t} - {\bf 1})}u) \\
& = x^{{\bold b} \cdot {\bold t}} (x^{{\bold a} \cdot {\bold t} - {\frk r}({\bold a})} {(\phi^M_{{\bold a}})}^{-1}(x^{{\bold a} \cdot ({\bold t} - {\bf 1})}u))
= x^{{\bold b} \cdot {\bold t}} \cdot x^{{\bold a} \cdot ({\bold t} -{\bf 1})}u\\
& = x^{({\bold a} + {\bold b})\cdot ({\bold t} - {\bf 1})} (x^{\bold b} u), \end{align*} and hence it follows that $$ x^{\bold b} \cdot \Phi_M(u) = (\phi^M_{{\bold a} + {\bold b}})^{-1}(x^{({\bold a} + {\bold b}) \cdot ({\bold t} - {\bf 1})} (x^{\bold b} u)) = \Phi_M(x^{\bold b} u). $$ Thus $\Phi_M$ is indeed $S$-linear.
Next let $f: M \to N$ be a morphism in $\Modtb$. We will show that the following diagram commutes; $$ \begin{CD} M @>\Phi_M >> {\frk r}^\ast(M) \\ @VfVV @VV{\frk r}^\ast(f)V \\ N @>>\Phi_N> {\frk r}^\ast(N). \end{CD} $$ Let $u \in M_{\bold a}$. Then \begin{align*} \phi^N_{\bold a}({\frk r}^\ast(f) \circ \Phi_M(u)) &= x^{{\bold a} \cdot {\bold t} - {\frk r}({\bold a})} \cdot f (\Phi_M(u)) \\
&= f(x^{{\bold a} \cdot {\bold t} - {\frk r}({\bold a})} \cdot \Phi_M(u) ) \\
&= f(x^{{\bold a} \cdot {\bold t} - {\frk r}({\bold a})} \cdot (\phi^M_{\bold a})^{-1}(x^{{\bold a} \cdot ({\bold t} - {\bf 1})} u)) \\
&= f(x^{{\bold a} \cdot ({\bold t} - {\bf 1})} u) = x^{{\bold a} \cdot({\bold t} - {\bf 1})} \cdot f(u) \end{align*} Therefore we conclude that $$ {\frk r}^\ast(f) \circ \Phi_M(u) = (\phi^N_{\bold a})^{-1}(x^{{\bold a} \cdot ({\bold t} - {\bf 1})} \cdot f(u)) = \Phi_N \circ f (u). $$
(2) Let $M \in \Modone$. We have to show $\Phi_M: M \to {\frk r}^\ast(M)$ is then an isomorphism. Since both of $M$ and ${\frk r}^\ast(M)$ are objects in $\Modone$, it suffices to show that each $(\Phi_M)_{\bold a}: M_{\bold a} \to {\frk r}^\ast(M)_{\bold a}$ is an isomorphism for all ${\bold a}$ with $\Zero \le {\bold a} \le {\bf 1}$. This is an immediate consequence of the fact that for such ${\bold a}$, the multiplication map $$ M_{\bold a} \xrightarrow{x^{{\bold a} \cdot ({\bold t} - {\bf 1})}} M_{{\bold a} \cdot {\bold t}} $$ is an isomorphism since $M \in \Modone$. Thus we conclude that $\Phi_M$ is an isomorphism for all $M \in \Modone$. \end{proof}
\begin{comment} \begin{Lemma}\label{help} For any ${\bold a}$, ${\bold b}\in {\NZQ Z}^n$, there is the following natural isomorphism of functors \[ \tau_{{\bold a}}\circ \sigma_{{\bold b}} \simeq \sigma_{{\bold b}}\circ \tau_{{\bold a}-{\bold b}} \] from ${\bf Mod}_{S}^{\mathbb{Z}^n}$ to itself. \end{Lemma}
\begin{proof} Let ${\bold c}\in {\NZQ Z}^n.$ Then \[ (\sigma_{\bold b}\circ \tau_{{\bold a}-{\bold b}} (M))_{{\bold c}}=(\tau_{{\bold a}-{\bold b}}(M)(-{\bold b}))_{\bold c}=\left\{ \begin{array}{ll}
0, & {\bold c}-{\bold b}\not\geq {\bold a}-{\bold b},\\
M_{{\bold c}-{\bold b}},& {\bold c}-{\bold b} \geq {\bold a}-{\bold b}, \end{array} \right. \] By comparing with the degree ${\bold c}$ component of $\tau_{{\bold a}}\circ \sigma_{{\bold b}}(M)$, the desired claim follows immediately. \end{proof} \end{comment}
Let $\Dc$ denote the contravariant functor $\grHom_S(-, S): {\bf Mod}_{S}^{\mathbb{Z}^n} \to {\bf Mod}_{S}^{\mathbb{Z}^n}$. We set $\Dc_{{\bold t}} := \sigma_{{\bold t} } \circ \Dc$. Note that $\Dc_{{\bold t}}$ gives a duality on $\Modtb$, and $\Dc_{{\bf 1}} = \sigma_{-{\bold t} + {\bf 1}} \circ \Dc_{{\bold t}}$ is the usual duality on ${\bf Mod}_{S}^{\mathbb{Z}^n}$ by the canonical module $S(-{\bf 1})$ of $S$. The functor $\Dc_{{\bold t}}$, lifted up to a functor from the category of complexes in $\Modtb$ to itself, coincides with the one $\Dc_{\bold t}$ in \cite{BF} up to shifting and quasi-isomorphism \cite[Proposition 3.6]{BF}. Moreover $\Dc_{\bold t}$ (resp. $\Dc_{\bf 1}$) sends $M \in \Modtb$ (resp. $M \in \Modone$) to an object in $\Modtb$ (resp. $\Modone$).
\begin{Proposition}\label{prop:r_and_D} There exists the natural isomorphisms between functors \begin{enumerate} \item $\tau_{\Zero} \circ \Dc_{\bf 1} \simeq \Dc_{\bf 1} \circ {\frk r}^\ast$ and \item ${\frk r}^\ast \circ \Dc_{\bold t} \simeq \Dc_{\bf 1} \circ {\frk s}^\ast$, \end{enumerate} from $\Modtb$ to $\Modone$. \end{Proposition} \begin{proof} (1) By Lemma \ref{lem:nat}, there exists the natural transformation $\Phi:\id_{\Modtb} \Longrightarrow {\frk r}^\ast$, and hence we have the one $\tau_{\Zero} \circ \Dc_{{\bf 1}} \circ {\frk r}^\ast \Longrightarrow \tau_{\Zero} \circ \Dc_{{\bf 1}}$, where both functors are regarded as the ones from $\Modtb$ to ${\bf Mod}_{S}^{\mathbb{N}^n}$. Since ${\frk r}^\ast M$ is a squarefree module, it follows that $\tau_{\Zero} \circ \Dc_{{\bf 1}} \circ {\frk r}^\ast = \Dc_{{\bf 1}} \circ {\frk r}^\ast$. Consequently, the above natural transformation induces the one $\Phi': \Dc_{{\bf 1}} \circ {\frk r}^\ast \Longrightarrow \tau_{\Zero} \circ \Dc_{{\bf 1}}$ of functors from $\Modtb$ to ${\bf Mod}_{S}^{\mathbb{N}^n}$.
Note that any $M \in \Modtb$ has a presentation $$ F_1 \longto F_0 \longto M \longto 0 $$ with $F_0, F_1$ free modules given by direct sums of finitely many copies of $S(-{\bold a})$ with $\Zero \le {\bold a} \le {\bold t}$. Since the functors ${\frk r}^\ast, \tau_{{\bold t} - {\bf 1}}$ are exact and since $\Dc_{\bold t}$ is left exact, we have the following commutative diagram with exact rows; $$ \begin{CD} 0 @>>> \Dc_{\bf 1} \circ {\frk r}^\ast(M) @>>> \Dc_{\bf 1} \circ {\frk r}^\ast(F_0) @>>> \Dc_{\bf 1} \circ {\frk r}^\ast(F_1) \\ @. @V{\Phi'}_M VV @V{\Psi'}_{F_0}VV @V{\Phi'}_{F_1}VV \\ 0 @>>> \tau_{\Zero} \circ \Dc_{\bf 1} (M) @>>> \tau_{\Zero} \circ \Dc_{\bf 1}(F_0) @>>> \tau_{\Zero} \circ \Dc_{\bf 1}(F_1). \end{CD} $$ Thus what we have to show is that ${\Phi'}_{S(-{\bold a})}$ is an isomorphism for any ${\bold a} \in {\NZQ Z}^n$ with $\Zero \le {\bold a} \le {\bold t}$.
Let $N$ be the cokernel of ${\Phi'}_{S(-{\bold a})}: S(-{\bold a}) \to S(-\sqrt{{\bold a}}) = {\frk r}^\ast(S(-{\bold a}))$. The map ${\Phi'}_S(-{\bold a})$ is obviously injective, and hence the following sequence \[ 0\to S(-{\bold a})\to {\frk r}^\ast(S(-{\bold a}))=S(-\sqrt{{\bold a}})\to N\to 0, \] is exact. By applying $\Dc_{{\bf 1}}$, we obtain: \[ 0\to \Dc_{{\bf 1}}(N) \to \Dc_{{\bf 1}}\circ {\frk r}^\ast(S(-{\bold a})) \to \Dc_{{\bf 1}}(S(-{\bold a})) \]
As obviously $\dim N\leq n-1,$ it follows that $\Dc_{\bf 1}(N) = 0$. Therefore we get the following exact sequence \[\begin{CD} 0 @>>> \Dc_{\bf 1}(S(-\sqrt{{\bold a}})) @>{\Phi'}_{S(-{\bold a})}>> \tau_{\Zero} \circ \Dc_{\bf 1}(S(-{\bold a})). \end{CD} \] Now $\Dc_{\bf 1}(S(-\sqrt{{\bold a}})) \cong S(-{\bf 1} + \sqrt{{\bold a}})$ and $\Dc_{\bf 1}(S(-{\bold a})) \cong S(-{\bf 1} + {\bold a})$. Easy observation shows that for ${\bold b} \in {\NZQ Z}^n$ with ${\bold b} \ge \Zero$, $S(-{\bf 1} + \sqrt{{\bold a}})_{\bold b} \neq 0$ if and only if $\tau_{\Zero}(S(-{\bf 1} + {\bold a}))_{\bold b} \neq 0$. Therefore it follows that ${\Phi'}_{S(-{\bold a})}$ is an isomorphism.
(2) The assertion (2) can be proved in the way similar to (1). We have the natural transformation $$ {\frk r}^\ast \circ \Dc_{\bold t} = {\frk r}^\ast \circ \Dc_{\bf 1} \circ \sigma_{{\bf 1} - {\bold t}} \Longrightarrow {\frk r}^\ast \circ \Dc_{\bf 1} \circ {\frk s}^\ast $$ between functors from $\Modtb$ to $\Modone$ by Lemma 2.1. Since $\Dc_{\bf 1} \circ {\frk s}^\ast$ sends $M \in \Modtb$ to an object in $\Modone$, there is a natural isomorphism ${\frk r}^\ast \circ \Dc_{\bf 1} \circ {\frk s}^\ast \simeq \Dc_{\bf 1} \circ {\frk s}^\ast$ by Lemma 2.1 again. Consequently we have the natural transformation $\Psi':{\frk r}^\ast \circ \Dc_{\bold t} \Longrightarrow \Dc_{\bf 1} \circ {\frk s}^\ast$. By the argument similar as above, we have only to prove ${\Psi'}_{S(-{\bold a})}$ is an isomorphism for all ${\bold a}$ with $\Zero \le {\bold a} \le {\bold t}$. By the definition of $\Psi$, $\Psi_{S(-{\bold a})}$ is injective. Applying ${\frk r}^\ast \circ \Dc_{\bf 1}$ to the exact sequence $$ 0 \longto {\frk s}^\ast(S(-{\bold a})) \xrightarrow{\Psi_{S(-{\bold a})}} \sigma_{{\bf 1} - {\bold t}}(S(-{\bold a})) \longto M \longto 0, $$ where $M$ is the cokernel of $\Psi_{S(-{\bold a})}$, we have the exact one $$ 0 \longto {\frk r}^\ast \circ \Dc_{\bold t}(S(-{\bold a})) \xrightarrow{{\Psi'}_{S(-{\bold a})}}\Dc_{\bf 1} \circ {\frk s}^\ast(S(-{\bold a})) $$ We define ${\bold b} = (b_i)_{1 \le i \le n}$ by setting $b_i = 0$ if $a_i \le t_i - 1$ and $b_i = 1$ if $a_i = t_i$. It follows that ${\frk s}^\ast(S(-{\bold a})) \cong S(-{\bold b})$, and hence $\Dc_{\bf 1} \circ {\frk s}^\ast(S(-{\bold a})) \cong S(-({\bf 1} - {\bold b}))$. On the other hand, ${\frk r}^\ast \circ \Dc_{\bold t}(S(-{\bold a})) \cong S(-\sqrt{{\bold t} - {\bold a}})$. Easy calculation shows ${\bf 1} - {\bold b} = \sqrt{{\bold t} - {\bold a}}$, and therefore ${\Psi'}_{S(-{\bold a})}$ is an isomorphism. \end{proof}
\begin{Theorem} \label{ext} The following statements hold. \begin{enumerate} \item Let $\omega_S$ be the canonical module of $S$. Then there are the following two isomorphisms \begin{enumerate} \item $\tau_{\Zero} \grExt^p_S(M, \omega_S) \cong \grExt^p_S({\frk r}^\ast(M), \omega_S)$ \item ${\frk r}^\ast(\grExt^p_S(M, S(-{\bold t}))) \cong \grExt^p_S({\frk s}^\ast(M),\omega_S)$ \end{enumerate} for all $p$ and $M \in \Modtb$. \item Assume $I$ is a Cohen-Macaulay monomial ideal such that $S/I \in \Modtb$. Let $\omega_{S/I}$, $\omega_{S/\sqrt I}$ be the canonical modules of $S/I$, $S/\sqrt I$, respectively. Then $$ \omega_{S/\sqrt I} \cong \tau_{ \Zero}(\omega_{S/I}) = (\omega_{S/I})_{\ge \Zero}. $$ \end{enumerate} \end{Theorem}
\begin{proof} Choose a ${\NZQ Z}^n$-graded minimal free resolution $P_\bullet$ of $M$ with each $P_i$ positively ${\bold t}$-determined. By Proposition~\ref{prop:r_and_D}, $$ \tau_\Zero \grExt^p_S(M,\omega_S)
\cong H^p(\tau_\Zero \circ \Dc_{\bf 1}(P_\bullet))
\cong H^p(\Dc_{\bf 1} \circ {\frk r}^\ast(P_\bullet))
\cong \grExt^p({\frk r}^\ast(M), \omega_S), $$ since ${\frk r}^\ast(P_\bullet)$ is a free resolution of ${\frk r}^\ast(M)$.
The assertion (2) follows from (1) and Theorem~\ref{CM}. \end{proof}
\begin{Remark}{\em In \cite[Corollary 2.3]{HTT}, Herzog, Takayama, and Terai proved $$ H^i_{\frk m}(S/I)_{\bold a} \cong H^i_{\frk m}(S/\sqrt{I})_{\bold a} $$ for any monomial ideal $I$ and all $i$ and ${\bold a} \le \Zero$. In (1) of Theorem \ref{ext}, taking the graded Matlis duality, we obtain the generalization of this result, which also implies that there exists the isomorphism $$ H^i_{\frk m}(S/I)_{\le \Zero} \cong H^i_{\frk m}(S/\sqrt I) $$ as ${\NZQ Z}^n$-graded $S$-modules. Here, for any $M \in {\bf Mod}_{S}^{\mathbb{Z}^n}$, $M_{\le \Zero}$ denotes the residue of $M$ by its ${\NZQ Z}^n$-graded submodule generated by the homogeneous elements whose degree is not less than or equal to $\Zero$.} \end{Remark}
As an immediate consequence of the above theorem we get the following corollary that generalizes a result of \cite{HTT}.
\begin{Corollary}\label{gCM} Let $M$ be a positively ${\bold t}$-determined $S$-module with $\dim M=\dim {\frk r}^\ast M.$ Then: \begin{enumerate}
\item $M$ is generalized Cohen-Macaulay if and only if so is ${\frk r}^\ast M$.
\item If $M$ is Buchsbaum, then ${\frk r}^\ast M$ is Buchsbaum. \end{enumerate} \end{Corollary}
\begin{proof} (1) $M$ is generalized Cohen-Macaulay if and only if $\grExt^i_S(M,\omega_S)$ has finite length for any $i\neq n-d$, where $d=\dim M$. This is equivalent to say that $\tau_0 \grExt^i_S(M,\omega_S)$ has finite length for $i\neq n-d$. Therefore, since $M$ and ${\frk r}^\ast M$ have the same dimension, the desired statement follows by Corollary~\ref{ext}.
(2) If $M$ is Buchsbaum, then $M$ is generalized Cohen-Macaulay \cite{Go}, thus ${\frk r}^\ast M$ is generalized Cohen-Macaulay. By \cite[Corollary 2.7]{Y}, it follows that ${\frk r}^\ast M$ is Buchsbaum. \end{proof}
\section{The ${\frk r}^\ast$ functor and Alexander duality}
Recall that there exists the duality $\Ac_{{\bold t}}$ on $\Modtb$, called Alexander duality, defined by Miller \cite{Mill}. In the case, ${\bold t} = {\bf 1}$, R\"{o}mer also defines independently in \cite{R}. Let $E$ denote the injective hull of $K$ and $p_{\bold t}: {\NZQ N}^n \to {\NZQ Z}^n$ the map defined by $p_{\bold t}({\bold a}) := (p_{t_1}(a_1),\dots ,p_{t_n}(a_n))$, where $$ p_{t_i}(a_i) := \begin{cases} a_i & \text{if $0 \le a_i < t_i$,} \\ t_i & \text{if $a_i \ge t_i$.} \end{cases} $$ The functor $\Ac_{{\bold t}}$ is given by the formula $$ \Ac_{{\bold t}}(M) = p_{{\bold t}}^\ast\grHom_S(M({\bold t}),E). $$ If we set $\DD$ to be the functor $\grHom_K(-, K)$, we have the natural isomorphism $\Ac_{\bold t} \cong p_{\bold t}^\ast \circ \sigma_{\bold t} \circ \DD$.
The following is essentially proved by Miller in \cite{Mill}.
\begin{Proposition}\label{prop:r_and_A} There exists a natural isomorphism of functors from $\Modtb$ to itself $$ \Ac_{\bf 1} \circ {\frk r}^\ast \simeq {\frk r}^\ast \circ \Ac_{{\bold t}}. $$ \end{Proposition} \begin{proof} Let $M \in \Modtb$ and ${\bold a} \in {\NZQ Z}^n$ with ${\bold a} \ge \Zero$. \begin{align*} \Ac_{\bf 1} \circ {\frk r}^\ast (M)_{\bold a} &= \grHom_K( {\frk r}^\ast(M), K)_{p_{\bf 1}({\bold a}) - {\bf 1}}
= \grHom_K(M_{{\frk r}(- p_{\bf 1}({\bold a}) + {\bf 1})}, K), \\ {\frk r}^\ast \circ \Ac_{\bold t} (M)_{\bold a} &= \grHom_K(M, K)_{p_{\bold t}({\frk r}({\bold a})) - {\bold t}}
= \grHom_K(M_{-p_{\bold t}({\frk r}({\bold a})) + {\bold t}}, K). \end{align*} It is easy to verify that ${\frk r}(- p_{\bf 1}({\bold a}) + {\bf 1}) = -p_{\bold t}({\frk r}({\bold a})) + {\bold t}$. Hence $\Ac_{\bf 1} \circ {\frk r}^\ast(M) \cong {\frk r}^\ast \circ \Ac_{\bold t}(M)$ as ${\NZQ Z}^n$-graded $K$-vector spaces. By a routine calculation, we can show that this isomorphism is $S$-linear and natural. \end{proof}
\section{Relation to the Auslander-Reiten translate}
Let $\le$ be the order on ${\NZQ Z}^n$ defined as follows: $$ {\bold a} \le {\bold b} \ \Longleftrightarrow \ a_i \le b_i $$ for all $i$. With the order induced by $<$, the set $P_{\bold t} := \{ {\bold a} \in {\NZQ Z}^n : \Zero \le {\bold a} \le {\bold t} \}$ becomes a poset. Let $A$ denote the incidence algebra of $P_{\bold t}$ over $K$. It is well-known that the category $\Modtb$ is equivalent to the one $\mod A$ consisting of finite-dimensional left $A$-modules (\cite[3.5]{BF} and \cite[Proposition 4.3]{Y2}). Since $A$ is a finite-dimensional $A$-algebra of finite global dimension, as is shown in \cite[3.6]{H} by Happel, the bounded derived category $D^b(\mod A)$ of $\mod A$ has Auslander-Reiten triangles. Let $K^b(\Proj A)$ (resp. $K^b(\Inj A)$) be the bounded homotopy category of complexes of finite-dimensional projective (resp. injective) left modules. According to Happel's proof, through the equivalence $K^b(\Proj A) \cong D^b(\mod A) \cong K^b(\Inj A)$ of triangulated categories, the Auslander-Reiten translate (see \cite[Definition 1.2]{J} for its definition) is then given by $T^{-1} \circ v$, where $T$ is the usual translation functor and $v$ is the equivalence $K^b(\Proj A) \cong K^b(\Inj A)$ of triangulated categories induced by the Nakayama functor, i.e., $\Hom_K(\Hom_A(-,A),K)$. On the other hand $v$ is coincides with $\Ac_{\bold t} \circ \Dc_{\bold t}$ through the equivalence $\Modtb \cong \mod A$. In this sense, $\Ac_{\bold t} \circ \Dc_{\bold t}$ represents the Auslander-Reiten translate of $D^b(\mod A)$ (see\cite[3.3 and 3.5]{BF} for details).
In this section, we discuss the relation between ${\frk r}^\ast$ and $\Ac_{\bold t} \circ \Dc_{\bold t}$. For this, we need to define a new functor. For ${\bold a} \in {\NZQ Z}^n$, let $\tau^{\bold a}$ be the functor from ${\bf Mod}_{S}^{\mathbb{Z}^n}$ to ${\bf Mod}_{S}^{\mathbb{Z}^n}$ defined as follows: for any $M \in {\bf Mod}_{S}^{\mathbb{Z}^n}$, we set $$ \tau_{\bold a}(M) := M/(S \cdot \bigoplus_{{\bold b} \not\le {\bold a}} M_{\bold a}), $$ and for a morphism $f: M \to N$ in ${\bf Mod}_{S}^{\mathbb{Z}^n}$, we assign, $\tau_{\bold a}(f)$, the natural homomorphism induced by $f$. Clearly $\tau_{\bold a}$ is additive and exact.
\begin{Lemma}\label{lem:sig_tau_2} There are the following natural isomorphisms of functors: for any ${\bold a}$, ${\bold b} \in {\NZQ Z}^n$, \begin{enumerate} \item $\sigma_{\bold a} \circ \tau^{\bold b} \simeq \tau^{{\bold a} + {\bold b}} \circ \sigma_{\bold b}$, \item $\DD \circ \sigma_{\bold a} \simeq \sigma_{-{\bold a}} \circ \DD$, and \item $\DD \circ \tau_{\bold a} \simeq \tau^{\bold a} \circ \DD$. \end{enumerate} \end{Lemma} \begin{proof} By comparing each degree ${\bold c}$ component, for each $M \in {\bf Mod}_{S}^{\mathbb{Z}^n}$, we obtain an isomorphism, as ${\NZQ Z}^n$-graded $K$-vector spaces, between two modules given by applying each of two functors. An easy calculation shows these maps indeed defines the desired natural isomorphisms. \end{proof}
\begin{Proposition}\label{ART} There exists the following two natural isomorphisms of functors \begin{enumerate} \item $\Ac_{\bf 1} \circ \Dc_{\bf 1} \circ {\frk r}^\ast \simeq p^\ast_{\bf 1} \circ \Ac_{\bold t} \circ \Dc_{\bold t}$ \item ${\frk r}^\ast \circ \Ac_{\bold t} \circ \Dc_{\bold t} \simeq \Ac_{\bf 1} \circ \Dc_{\bf 1} \circ {\frk s}^\ast$ \end{enumerate} from $\Modtb$ to $\Modone$. \end{Proposition} \begin{proof} The second natural isomorphism is a direct consequence of Propositions \ref{prop:r_and_D} and \ref{prop:r_and_A}. We will show the first. It follows from Proposition~\ref{prop:r_and_D} and Lemma~\ref{lem:sig_tau_2} that \begin{align*} \Ac_{\bf 1} \circ \Dc_{\bf 1} \circ {\frk r}^\ast
&\simeq \Ac_{\bf 1} \circ \tau_\Zero \circ \Dc_{\bf 1} \\
&\simeq p^\ast_{\bf 1} \circ \sigma_{\bf 1} \circ \DD \circ \tau_\Zero \circ \sigma_{{\bf 1} - {\bold t}} \circ \Dc_{\bold t} \\
&\simeq p^\ast_{\bf 1} \circ \sigma_{{\bf 1}} \circ \tau^{\Zero} \circ \sigma_{{\bold t} - {\bf 1}} \circ \DD \circ \Dc_{\bold t} \\
&\simeq p^\ast_{\bf 1} \circ \tau^{\bf 1} \circ \sigma_{\bold t} \circ \DD \circ \Dc_{\bold t}. \end{align*} By the definition, it is easy to verify that $p^\ast_{\bf 1} \circ \tau^{\bf 1} = p^\ast_{\bf 1}$. Moreover $p_{\bold t} \circ p_{\bf 1} = p_{\bf 1}$ implies $p^\ast_{\bf 1} \circ p^\ast_{\bold t} = p^\ast_{\bf 1}$. Thus it follows that $$ \Ac_{\bf 1} \circ \Dc_{\bf 1} \circ {\frk r}^\ast
\simeq p^\ast_{\bf 1} \circ \tau^{\bf 1} \circ \sigma_{\bold t} \circ \DD \circ \Dc_{\bold t}
\simeq p^\ast_{\bf 1} \circ \Ac_{\bold t} \circ \Dc_{\bold t}. $$ \end{proof}
{}
\end{document} | arXiv |
Cantic 7-cube
In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.
Truncated 7-demicube
Cantic 7-cube
D7 Coxeter plane projection
Typeuniform 7-polytope
Schläfli symbolt{3,34,1}
h2{4,3,3,3,3,3}
Coxeter diagram
6-faces142
5-faces1428
4-faces5656
Cells11760
Faces13440
Edges7392
Vertices1344
Vertex figure( )v{ }x{3,3,3}
Coxeter groupsD7, [34,1,1]
Propertiesconvex
A uniform 7-polytope is vertex-transitive and constructed from uniform 6-polytope facets, and can be represented a coxeter diagram with ringed nodes representing active mirrors. A demihypercube is an alternation of a hypercube.
Its 3-dimensional analogue would be a truncated tetrahedron (truncated 3-demicube), and Coxeter diagram or as a cantic cube.
Alternate names
• Truncated demihepteract
• Truncated hemihepteract (thesa) (Jonathan Bowers)[1]
Cartesian coordinates
The Cartesian coordinates for the 1344 vertices of a truncated 7-demicube centered at the origin and edge length 6√2 are coordinate permutations:
(±1,±1,±3,±3,±3,±3,±3)
with an odd number of plus signs.
Images
It can be visualized as a 2-dimensional orthogonal projections, for example the a D7 Coxeter plane, containing 12-gonal symmetry. Most visualizations in symmetric projections will contain overlapping vertices, so the colors of the vertices are changed based on how many vertices are at each projective position, here shown with red color for no overlaps.
orthographic projections
Coxeter
plane
B7 D7 D6
Graph
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph
Dihedral
symmetry
[6] [4]
Related polytopes
Dimensional family of cantic n-cubes
n345678
Symmetry
[1+,4,3n-2]
[1+,4,3]
= [3,3]
[1+,4,32]
= [3,31,1]
[1+,4,33]
= [3,32,1]
[1+,4,34]
= [3,33,1]
[1+,4,35]
= [3,34,1]
[1+,4,36]
= [3,35,1]
Cantic
figure
Coxeter
=
=
=
=
=
=
Schläfli h2{4,3} h2{4,32} h2{4,33} h2{4,34} h2{4,35} h2{4,36}
There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:
D7 polytopes
t0(141)
t0,1(141)
t0,2(141)
t0,3(141)
t0,4(141)
t0,5(141)
t0,1,2(141)
t0,1,3(141)
t0,1,4(141)
t0,1,5(141)
t0,2,3(141)
t0,2,4(141)
t0,2,5(141)
t0,3,4(141)
t0,3,5(141)
t0,4,5(141)
t0,1,2,3(141)
t0,1,2,4(141)
t0,1,2,5(141)
t0,1,3,4(141)
t0,1,3,5(141)
t0,1,4,5(141)
t0,2,3,4(141)
t0,2,3,5(141)
t0,2,4,5(141)
t0,3,4,5(141)
t0,1,2,3,4(141)
t0,1,2,3,5(141)
t0,1,2,4,5(141)
t0,1,3,4,5(141)
t0,2,3,4,5(141)
t0,1,2,3,4,5(141)
Notes
1. Klitzing, (x3x3o *b3o3o3o3o - thesa)
References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "7D uniform polytopes (polyexa) x3x3o *b3o3o3o3o – thesa".
External links
• Weisstein, Eric W. "Hypercube". MathWorld.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
\begin{document}
\title{Indistinguishable photons from a trapped-ion quantum network node}
\author{M. Meraner$^\dagger$} \affiliation{Institut f\"ur Quantenoptik und Quanteninformation,\\
\"Osterreichische Akademie der Wissenschaften, Technikerstr. 21A, 6020 Innsbruck,
Austria} \affiliation{
Institut f\"ur Experimentalphysik, Universit\"at Innsbruck,
Technikerstr. 25, 6020 Innsbruck, Austria}
\author{A. Mazloom$^\dagger$} \affiliation{Departement Physik, Universit\"at Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland }
\author{V. Krutyanskiy$^\dagger$} \affiliation{Institut f\"ur Quantenoptik und Quanteninformation,\\
\"Osterreichische Akademie der Wissenschaften, Technikerstr. 21A, 6020 Innsbruck,
Austria}
\author{\\V. Krcmarsky} \affiliation{Institut f\"ur Quantenoptik und Quanteninformation,\\
\"Osterreichische Akademie der Wissenschaften, Technikerstr. 21A, 6020 Innsbruck,
Austria} \affiliation{
Institut f\"ur Experimentalphysik, Universit\"at Innsbruck,
Technikerstr. 25, 6020 Innsbruck, Austria}
\author{J. Schupp} \affiliation{Institut f\"ur Quantenoptik und Quanteninformation,\\
\"Osterreichische Akademie der Wissenschaften, Technikerstr. 21A, 6020 Innsbruck,
Austria} \affiliation{
Institut f\"ur Experimentalphysik, Universit\"at Innsbruck,
Technikerstr. 25, 6020 Innsbruck, Austria}
\author{D. Fioretto} \affiliation{
Institut f\"ur Experimentalphysik, Universit\"at Innsbruck,
Technikerstr. 25, 6020 Innsbruck, Austria}
\author{P. Sekatski} \affiliation{Departement Physik, Universit\"at Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland }
\author{T. E. Northup} \affiliation{
Institut f\"ur Experimentalphysik, Universit\"at Innsbruck,
Technikerstr. 25, 6020 Innsbruck, Austria}
\author{N. Sangouard} \affiliation{Institut de physique th\'{e}orique, Universit\'{e} Paris Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France} \affiliation{Departement Physik, Universit\"at Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland }
\author{B. P. Lanyon} \email{[email protected], $^\dagger$ These authors contributed equally} \affiliation{Institut f\"ur Quantenoptik und Quanteninformation,\\
\"Osterreichische Akademie der Wissenschaften, Technikerstr. 21A, 6020 Innsbruck,
Austria}
\affiliation{
Institut f\"ur Experimentalphysik, Universit\"at Innsbruck,
Technikerstr. 25, 6020 Innsbruck, Austria}
\begin{abstract}
Trapped atomic ions embedded in optical cavities are a promising platform to enable long-distance quantum networks and their most far-reaching applications. Here we achieve and analyze photon indistinguishability in a telecom-converted ion-cavity system. First, two-photon interference of cavity photons at their ion-resonant wavelength is observed and found to reach the limits set by spontaneous emission. Second, this limit is shown to be preserved after a two-step frequency conversion replicating a distributed scenario, in which the cavity photons are converted to the telecom C band and then back to the original wavelength. The achieved interference visibility and photon efficiency would allow for the distribution and practical verification of entanglement between ion-qubit registers separated by several tens of kilometers.
\end{abstract} \maketitle
Envisioned quantum networks, consisting of remote quantum matter linked up with light \cite{Kimble2008, Wehnereaam9288}, offer a fundamentally new communication paradigm \cite{Duan2001} as well as a practical path to large-scale quantum computation and simulation \cite{PhysRevA.89.022317} and to precision measurements in new regimes \cite{Komar2014, PhysRevLett.123.070504, sekatski2019optimal}. Trapped atomic ions are expected to enable the most promising applications of large-scale quantum networks \cite{Duan2010, Sangouard2009, RevModPhys.87.1379} given their demonstrated capabilities for quantum logic \cite{reviewionsqc}, multi-qubit registers \cite{Friis2018}, and optical clocks \cite{PhysRevLett.123.033201}.
Ion qubits have been entangled with propagating photons \cite{Blinov2004} and those photons have been used to entangle ions in traps a few meters apart \cite{Moehring2007, Hucul:2015wo, balance}. Integrating ion traps with optical cavities offers the possibility of a near-deterministic and coherent light-matter interface for quantum networking \cite{RevModPhys.87.1379, Sangouard2009}, and both ion-photon entanglement \cite{Stute2012} and state transfer \cite{Stute:2013we} have been achieved in this setting.
Recently, photons from trapped ions have been converted to the optimal telecom wavelengths for long-distance quantum networking via single-photon frequency conversion \cite{Bock2018, Walker2018, 50km}. In \cite{50km} it was shown that the combination of an ion trap with an integrated optical cavity and telecom conversion could enable entanglement distribution between trapped ions spaced by tens of kilometres at practical rates for verification. However, it has not previously been verified that the photons from such a system can be sufficiently indistinguishable to allow for the establishment of remote entanglement.
Here we present experimental and theoretical results of photon distinguishability in a telecom-converted ion-cavity setting, based on interference between two photons produced sequentially from an ion in a cavity.
We conclude that the achieved interference visibilities and overall detection efficiencies would already allow for entanglement of ions tens of kilometers apart (orders of magnitude further than the state of the art \cite{Moehring2007, Hucul:2015wo, balance}).
The extent to which photons are in identical pure states, and therefore indistinguishable, is a key parameter that can be quantified by the visibility in a two-photon interference experiment \cite{PhysRevLett.59.2044}. For a comprehensive theoretical analysis of two-photon interference from quantum emitters without conversion see, e.g., \cite{Fischer_2016, PhysRevA.96.023861}. While direct two-photon interference has been achieved using neutral atoms in cavities \cite{PhysRevLett.93.070503, PhysRevLett.98.063601}, it has not previously been reported for ions in cavities. As will be shown, the limiting factor on the interference visibility in our ion-cavity system is unwanted spontaneous emission from the ion during the cavity-mediated photon generation process. Such spontaneous emission is particularly relevant for ion-cavity systems demonstrated to date in which the ion-cavity coupling rate does not overwhelm the spontaneous scattering rate. Furthermore, photon conversion stages can easily introduce additional distinguishability, e.g., by directly adding noise photons at a rate that depends strongly on the particular photon and pump laser wavelengths and filtering bandwidth \cite{Pelc:11}, and must be assessed on a system-dependent basis.
In this Letter, first we introduce the experimental system and a simple theoretical model of the effect of spontaneous emission on the emitted cavity photon. Second, two-photon interference results of cavity photons at the ion-resonant wavelength are presented, showing that spontaneous emission is the dominant limiting factor. Third, two-photon interference results are presented after a two-step frequency conversion, converting the wavelength of one cavity photon to the telecom band and back to the ion-resonant wavelength, showing that the photon indistinguishability is essentially preserved.
Experiments employ a single $^{40}$Ca$^{+}$ atom in the center of a linear Paul trap and in the focus of a near-concentric optical cavity near-resonant with the 854 nm electronic dipole transition (Figure 1) \cite{50km}. We begin by Doppler cooling the ion's motional state and optical pumping into an electronic ground state $\ket{S}=\ket{4^{2} S_{J{=}1/2},m_j{=}1/2}$ (Figure 1). Each photon is generated via a Raman laser pulse at 393 ~nm which triggers emission, by the ion, of a polarized 854 nm photon into a vacuum cavity mode, via a cavity-mediated Raman transition \cite{Keller:2004cf}. Two photons are generated sequentially with a time gap between the beginning of their respective Raman pulses of $13.35~\mu$s, such that after delay of the first (the vertical `long-path' photon, $\ket{V}$) in a 3~km optical fibre spool, both photon wavepackets (the second being the horizontal `short-path' photon, $\ket{H}$) arrive simultaneously and with their polarizations rotated to be parallel, at different input ports of a 50:50 beamsplitter. Different photon polarizations are generated and modelled by 3-level Raman transitions that differ in the final Zeeman state of the $\ket{ 3^{2}D_{J{=}5/2}}$ manifold (Figure 1 and \cite{SuppMat}). Specifically, after the generation of a $\ket{V}$ (an $\ket{H}$) photon the ion is in the final state $\ket{ 3^{2}D_{J{=}5/2},m_j{=}-5/2}$ ($\ket{ 3^{2}D_{J{=}5/2},m_j{=}-3/2}$) \cite{SuppMat}. In every experiment we use a Raman laser Rabi frequency of $\Omega=2\pi\times 64(1)$~MHz.
\begin{figure}
\caption{
\textbf{Experiment schematic:} a. An atomic ion (red sphere) in a linear Paul trap (gold electrodes) and coupled to a vacuum anti-node of an optical cavity (coupling strength $g$). Raman laser (Rabi frequency $\Omega$) pulses generate sequential orthogonally-polarized photons, first vertical ($\ket{V}$) then horizontal ($\ket{H}$) that are split into two paths, with a time separation equal to the delay line (panel c.), such that their wavepackets arrive simultaneously at the beam splitter (panel d.). b. Three-level model: ground state $\ket{S}$; metastable state $\ket{D}$ (1.17~s lifetime) and excited state $\ket{P}$ (6.9~ns lifetime). Spectroscopic notation shown. For $\ket{H}$ and $\ket{V}$ photons the $mj=-5/2$ and $mj=-3/2$ Zeeman states of $\ket{D}$ are used \cite{SuppMat}. The coherent cavity-photon generation process competes with spontaneous emission from the short-lived $P$ state (decay rates $\gamma_{ps}$ and $\gamma_{pd}$). c. Single photon quantum frequency conversion (QFC) and inverse process (QFC$^{-1}$) with wavelength changes shown. A 3~km spool of telecom SMF28-Ultra fibre. d. Beamsplitter (BS). Superconducting nanowire photon detector D1 (D2) with efficiency 0.88 (0.89) and free-running dark counts 0.3 (0.4) per second. Fiber coupler (FC), mating sleeves (MS).
}
\label{fig:setup}
\end{figure}
The arrival times of photons at the beamsplitter output ports are recorded with single-photon detectors. In each experimental cycle, we generate two pairs of photons: while the temporal wavepackets of the first pair (synchronous) arrive simultaneously at the beamsplitter, a time gap is introduced between the wavepackets of the second pair (asynchronous) that provides complete temporal distinguishability. Each full experiment consists of many repeated cycles as described in \cite{SuppMat}.
The coincidence rates of detection events from the synchronous and asynchronous photon pairs are denoted as $C^{||}$ and $C^{\perp}$, respectively.
The two-photon interference visibility is given by $V(T){=}1-C^{||}(T)/C^{\perp}(T)$, where $T$ is the coincidence window: the maximum time difference between photon clicks that is counted as a coincidence.
In the first full experiment, the $\ket{V}$ photon is sent directly to the fiber spool. In the second experiment, the $\ket{V}$ photon is first converted to 1550~nm (telecom C band) via difference frequency generation (DFG) in a ridge-waveguide-integrated PPLN crystal with a 1902~nm pump laser. This first `down-conversion' stage is described in \cite{50km, Krutyanskiy2017}. After the spool, an `up-conversion' stage (not previously reported) converts the photon back to 854~nm via the reverse process: sum frequency generation (SFG). Approximately 0.2~W of pump laser power is used for each stage.
In the case of perfectly indistinguishable photons entering separate ports of a symmetric beamsplitter, the well known photon bunching effect occurs: two perfect detectors placed at the output ports of the beamsplitter never fire simultaneously. Yet in practice, perfect bunching is never observed, and it is important to understand the source of the imperfections. During the photon generation process (Figure 1), spontaneous decay events from the short-lived excited state ($\ket{P}$) onto the final state manifold ($\ket{D}$) act only as losses -- no cavity photon is emitted through the Raman process if such an event occurs. In contrast, following any number of spontaneous decay events from $\ket{P}$ back to the initial state ($\ket{S}$) during the Raman laser pulse, a cavity photon can still be subsequently generated while the Raman laser remains on. Every spontaneously scattered photon carries away the information that the cavity photon has not yet been emitted. Consequently, the cavity photons impinging on the beamsplitter are each in a (temporally-) mixed state and therefore they do not bunch perfectly.
The effect of spontaneous scattering on the visibility is precisely quantified through a theoretical model describing the evolution of a three-level atom embedded in a cavity using a master equation \cite{SuppMat}. In the model, an expression for the mixed state of photons emitted from the cavity is obtained in two steps. First, we calculate the wave function of photons emitted from the cavity conditioned on the ion being in the initial state $\ket{S}$ at time $s$ and no spontaneous decay events happening for later times. Second, we compute the rate of spontaneous decay events from $\ket{P}$ to $\ket{S}$ as a function of time. The state of the emitted cavity photon is then expressed as a mixture over all the possibilities where the last $\ket{P}{\rightarrow}\ket{S}$ decay happens at time $s$ or no decay events occur and a pure state photon is emitted afterwards, plus the vacuum component collecting all the possibilities where no cavity photon is emitted. With the emitted photon states in hand, it is then straightforward to calculate the visibility of pairs of photons \cite{SuppMat}. We refer to this model that includes only imperfections due to spontaneous scattering as the basic model. As an alternative from our model, the visibility could be computed from the master equation via the quantum regression theorem \cite{Fischer_2016}.
Results are now presented for the case without photon conversion. The temporal profiles of the short-path and long-path single photon detection events from the second (asynchronous) photon pair are shown in Figure \ref{figure_results_854}a. These single-photon wavepackets are presented as a probability density $\rho_{d}(t) = N_d/(k\cdot \Delta_t)$, where $N_{d}$ is the number of detection events registered in a time bin $\Delta_t = 125~$ns and $k$ is the number of trials. Integration of the wavepackets gives the probability of detecting a short- (long-) path photon as $12.4$\% ($2.7$\%). Differences in the single-photon wave packet shapes are due to slight differences in the corresponding transition strengths \cite{SuppMat}.
\begin{figure}
\caption{
\textbf{Two-photon interference without photon conversion:}
Solid (dashed) lines show basic theory (extended theory) model and shapes show data in all the panels. Probability densities are obtained by dividing the probability of detection (coincidence) per time bin by the bin size, see \cite{SuppMat} sec. IV, V for details.
(a) Single photon wavepackets for short path (red
circles) and long path (black diamonds, rescaled by multiplication factor 4.6 to correct delay line losses). Vertical dotted line shows the end of the Raman laser pulse.
(b) Photon coincidences for temporally synchronous ($\rho_{c}^{||}$, blue circles) and asynchronous
($\rho_{c}^{\perp}$, green diamonds) cases.
(c) Interference visibility $V$ (left axis, blue diamonds) and integrated asynchronous coincidence probability $C^{\perp}$ (green circles, right axis). Error bars represent $\pm$ one standard deviation due to Poissonian photon counting statistics, not shown when smaller than shapes.
}
\label{figure_results_854}
\end{figure}
The temporal profile of the coincidence detection events (cross-correlation function) for the synchronous and asynchronous photon pairs are compared in Figure \ref{figure_results_854}b. Here the coincidence probability density $\rho_{c}(\tau)^{\parallel,\perp} = N^{\parallel,\perp}_c/(k\cdot \Delta_\tau)$ is used, where $N^{\parallel,\perp}_{c}$ are the number of coincident detection events per time bin for the first and second pair of photons respectively, $\tau$ is the difference in detection times. Figure \ref{figure_results_854}c shows the visibility $V(T)$ and integrated coincidence rate of the asynchronous photons $C^\perp(T) = \int_{-T}^T \rho^\perp_c(\tau)d\tau$. For the smallest coincidence window presented, the interference visibility $V($125~ns) is $0.986\pm{0.006}$ ($0.987\pm{0.005}$ after subtracting detector dark counts). When considering a coincidence window containing the whole photon wavepacket, the visibility $V(9~\mu$s) is $0.472\pm{0.008}$. From the theory, we calculate that the expectation value of the number of spontaneously-emitted photons on the $\ket{P}{\rightarrow}\ket{S}$ transition, given generation of a cavity-photon, was 3.5.
The differences between the basic model and data in Figure \ref{figure_results_854} are consistent with an extension to the model that, in addition to spontaneous emission, includes a combination of an overall time-independent distinguishability factor of 1\% and a centre frequency difference of the two photons of 40~kHz \cite{SuppMat}. This small photon frequency difference could be caused by several reasons, e.g., cavity length instability, acoustic noise in the 3 km delay line fibre and cavity birefringence. The 1\% time-independent distinguishability can arise from slight polarization mode mismatch at the beamsplitter or imbalance of the 50:50 beamsplitter itself. The agreement between data and the basic model shows that we are close to the fundamental limit of photon indistinguishability set by spontaneous scattering in our system.
Figure \ref{figure_results_converted} presents results for the case with photon conversion and is constructed in the same way as Figure \ref{figure_results_854}. The probability of detecting a short- (long-) path photon across the entire wavepacket is $10$~\% ($0.5$~\%). The visibility is 0.96$\pm{0.04}$ for the minimum coincidence window of 250~ns and 0.37$\pm{0.04}$ for the full wave packet window of 9 $\mu$s. The differences between the basic model and frequency-converted data (Figure \ref{figure_results_converted}) are consistent with an extended model that includes a frequency drift of the unstabilized photon conversion pump laser at the level of 50~kHz on a 10$~\mu$s timescale (consistent with independent measurements) and background coincidences, see section V.C \cite{SuppMat}. We anticipate no significant challenges to frequency stabilising the pump laser to the few kilohertz level in future work. Remote photon conversion stages in distributed networks will need independent pump lasers with absolute long-term frequency stability to within a fraction of the networking photon bandwidth.
The achieved visibilities and coincidence rates in our experiments would already allow for remote ion entanglement over tens of kilometers. Consider the entanglement swapping protocol of \cite{Duan2010}, which leads to maximal entanglement of two remote (ion) qubits with state fidelity $F(T){=}(1+V(T))/2$ \cite{PhysRevLett.123.213601} at a heralded rate $R_{swap}(T) \propto R_{gen} \times C^{\perp}(T)$, where $R_{gen}$ is the photon-generation attempt rate at each ion-trap network node. Using $R_{gen}=30$~kHz, the achieved values without photon conversion (Figure \ref{figure_results_854}) would allow for 3~km ion-ion entanglement distribution with $F(T=9\mu s)=0.736 \pm0.004$ at a rate of $R_{swap}(9\mu s)=30$~Hz. Using 0.18 dB/km for telecom fiber losses, the achieved performance with photon conversion (Figure \ref{figure_results_converted}) would allow for 50~km distant ion-ion entanglement generation with $F(T=9\mu s)=0.69 \pm0.02$ at rates on the order of 1~Hz (assuming photon detector dark count rates of 1~Hz). In all cases, time filtering of the coincidences ($T<9\mu$s) would allow for an increased remote entangled-state fidelity at the cost of reduced heralding rate. Such long-distance experiments must tackle environmental noise in deployed optical fibres, absolute frequency stabilisation of remote laser systems and matching photons from remote network nodes. Methods to improve visibility without reducing the photon generation rate are those that can significantly increase the coherent ion-cavity coupling rate $g$, such as coupling multiple ions in entangled (superradient) states to the cavity \cite{PhysRevLett.107.030501, PhysRevLett.114.023602} and pursuing small mode-volume fibre cavities \cite{PhysRevLett.110.043003}.
Our model reveals that there is an optimal drive-laser Rabi frequency ($\Omega$, Figure 1) that achieves the highest photon generation probability (and therefore $C^{\perp}$) for a given threshold visibility, highlighting the important role such models will play in enabling the upcoming next generation of long-distance networking experiments \cite{SuppMat}. Our results present a path to distributing entanglement between trapped-ion registers spaced by several tens of kilometres at practical rates for verification: significantly further than state-of-the-art experiments involving spacings of a few meters \cite{Moehring2007, Hucul:2015wo, balance} and a practical distance to start building large-scale quantum-logic-capable quantum networks. \\
\begin{figure}
\caption{
\textbf{Two-photon interference with photon conversion:}
Solid (dotted) lines show basic theory (extended theory) model and shapes show data in all the panels. Probability densities are obtained by dividing the probability of detection (coincidence) per time bin by the bin size \cite{SuppMat}.
(a) Single photon wavepackets for short path (red
circles) and long path (black diamonds, rescaled by multiplication factor 19 to correct delay line losses). Vertical dotted line shows the end of the Raman laser pulse.
(b) Photon coincidences for temporally synchronous ($\rho_{c}^{||}$, blue circles) and asynchronous
($\rho_{c}^{\perp}$, green diamonds) cases.
(c) Interference visibility $V$ (left axis, blue diamonds) and integrated probability $C^{\perp}$ (green circles, right axis). Error bars represent $\pm$ one standard deviation due to Poissonian photon counting statistics.
}
\label{figure_results_converted}
\end{figure}
\noindent \emph{Note:} During the preparation of this manuscript, we became aware of complementary work in which sequential interference of photons from an ion in a cavity is achieved and studied \cite{walker2019improving}.
\begin{acknowledgments}
This work was supported by the START prize of the Austrian FWF project Y 849-N20, by the US Army Research Laboratory under Cooperative Agreement Number W911NF-15-2-0060 (project SciNet), by the Institute for Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy Of Sciences (OEAW), by the European Union's Horizon 2020 research and innovation programme under grant agreement No 820445 and project name `Quantum Internet Alliance' and by the Swiss National Science Foundation (SNSF) through the Grant PP00P2-179109, and by the Austrian Science Fund (FWF) through Project F 7109. The European Commission is not responsible for any use that may be made of the information this paper contains.
\section*{Author Contributions}
VKrut., MM, and VKrc. took the data. VKrut, AM, BPL, PS and MM analysed the data. VKrut, MM, VKrc, JS and BPL contributed to the experimental setup and design. AM, VKrut., PS, BPL, MM, DF, TN and NS performed theoretical modelling. BPL, VKrut, MM, PS, TN and NS wrote the majority of the paper, with contributions from all authors. The project was conceived and supervised by BPL.
\end{acknowledgments}
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indistinguishability of single photons from an ion-cavity system},}\ }
(\bibinfo {year} {2019}),\ \Eprint {http://arxiv.org/abs/1911.08442}
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\citenamefont {{Aktas}}, \citenamefont {{Delord}},\ and\ \citenamefont
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{2015})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Maring}\ \emph {et~al.}(2017)\citenamefont {Maring},
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{\bibfnamefont {S.}~\bibnamefont {Tanzilli}},\ }\href {\doibase
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Express}\ }\textbf {\bibinfo {volume} {27}},\ \bibinfo {pages} {25603}
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{author} {\bibfnamefont {C.}~\bibnamefont {Russo}}, \bibinfo {author}
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{pages} {103004} (\bibinfo {year} {2009})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Matsukevich}\ \emph {et~al.}(2008)\citenamefont
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\hypertarget{sec:appendix} \appendix \setcounter{section}{0} \setcounter{subsection}{0}
\renewcommand{\arabic{subsubsection}}{\arabic{subsubsection}} \renewcommand{\Alph{subsection}}{\Alph{subsection}} \renewcommand{\Roman{section}}{\Roman{section}} \renewcommand{A.\arabic{figure}}{A.\arabic{figure}} \setcounter{equation}{0} \numberwithin{equation}{section}
\section*{Supplementary Material}
\tableofcontents
\section{Experimental setup}
\noindent A detailed experimental diagram is presented in Fig. \ref{fig:setup_full}. For details on the ion trap and cavity system please see our recent paper in \cite{50km}: the same ion trapping frequencies, geometry of laser beams and optical cavity were employed. A few key parameters are now recapped.
The optical cavity around the ion is near-concentric with a cavity waist of $\omega_0 = 12.31 \pm 0.07$ $\mu$m and a maximum ion-cavity coupling rate of $g_{0} = 2\pi \cdot 1.53 \pm 0.01$ MHz. The finesse of the cavity (at 854 nm) is $\mathcal{F} = \frac{2\pi}{\mathcal{L}} = 54000 \pm 1000$, with the total cavity losses $\mathcal{L} = T_1 + T_2 + L_{1+2} = 116 \pm 2 $ ppm, determined from measurements of the cavity ringdown time. This gives the cavity linewidth $2\kappa = 2\pi \cdot 140 \pm 3$ kHz, $\kappa$ being the half-width at half maximum.
As explained in the main text, in the second experiment the delayed photon undergoes a two stage frequency conversion. Each frequency conversion uses a single 48 mm-long ridge-waveguide-integrated PPLN crystal (NTT electronics, reported in \cite{Krutyanskiy2017, 50km}) where the input single photon is overlapped with the pump laser at 1902 nm. The first (`down-conversion') stage brings the 854 nm (V) photon to the 1550 nm telecom C-Band ($(854~\mathrm{nm})^{-1}-(1902~\mathrm{nm})^{-1}\approx(1550~\mathrm{nm})^{-1}$) as detailed in the first experiment in \cite{Krutyanskiy2017}. This photon is separated from the pump field with a dichroic plate, injected into the 3 km spool and recombined with the pump before the second stage. The second (`up-conversion') stage brings the 1550 nm photon back to the original 854 nm wavelength via sum frequency generation (SFG). The total pump laser power of correct polarization for conversion in-coupled into each crystal is 0.2~W, control by the waveplate angles before and after the first stage.
While in the ideal case the second stage is the reverse process of the first, there are two differences which are now summarized.
In the downconversion process, unwanted photons at the telecom output wavelength are generated directly from Anti-Stokes Raman scattering (ASR) of the pump laser from 1902~nm to 1550~nm.
In the upconversion process, unwanted photons at the 854~nm output wavelength are generated by a two stage process. First, as before ASR scattering of the pump laser generates telecom photons. Second, these photons are up-converted back to 854nm via SFG \cite{Pelc:11, Kaiser15}.
A second difference is the fundamental waveguide mode at 1550~nm has a higher overlap with the free-space Gaussian mode of the in-coupling laser field, than the waveguide mode at 854~nm. As such, we achieve a higher in-coupling efficiently into the fundamental waveguide mode at 1550~nm and subsequently a higher conversion efficiency for the up-conversion (presented in the next section).
\begin{figure*}
\caption{
\textbf{Detailed experimental diagram.}
\textbf{a) Ion cavity node.}
A single atomic ion (red sphere) in the centre of both a 3D radio-frequency (RF) linear Paul trap (gold electrodes) and an optical cavity. The two smaller electrodes are held at DC voltage.
The 4 larger electrodes (two shown in figure projection) are driven with RF. Two cross sections are depicted: Along the cavity axis (top), showing: the $\approx4$ Gauss DC magnetic field (quantisation axis) generated by rings of permanent magnets and the circularly-polarized Raman laser for generating 854 nm cavity photons.
Following a Raman pulse, an 854 nm cavity photon exits the cavity via the right mirror (transmission T2). The photon then passes the following elements: in-vacuum collimating lens (C); vacuum chamber viewport (VP); waveplates; 3 filters to remove the 806 nm laser light to which cavity length is continuously and actively stabilised; polarizing beam splitter (PBS) for directing the vertical photons into a single mode fiber, and the horizontal photons into a polarization maintaining fiber (PMF),
\textbf{b) Delay line.}
For the experiment without photon conversion, point A and A', as well as the points B' and B directly are fiber connected , such that the vertical photon only has to pass the 3 km fiber spool (Corning SMF-28 Ultra).
For the experiment with the conversion, the setup is as shown here.
The injected 854 nm photon passes waveplates (used for system optimization with classical light) and is overlapped with 800 mW of 1902 nm laser light (Tm-doped fiber laser, AdValue Photonics AP-SF1-1901.4-01-LP, measured at PM) on a dichroic mirror (DM1) and free-space coupled into one of the ridge waveguides of temperature-stabilised PPLN1 using an asphere (AS, 11 mm, positioned by an XYZ translation stage).
The 1902 nm input path is described in \cite{50km}.
A gold parabolic mirror (f = 15 mm) is used to collimate all fields at the output of PPLN1.
A dichroic mirror (DM2; Thorlabs DMLP1800) splits the converted 1550 nm photons from the 1902 nm pump laser. A combination of a shortpass (SP 1610) and longpass (LP 1400) filter reduces unwanted pump laser and other noise light fields.
The 1550 nm photon couples into the 3 km SMF-28 fiber spool, which is used as an optical delay line. The output of the fiber spool passes waveplates, to correct for polarization rotations through the fiber and is overlapped (with a dichroic mirror DM3; Thorlabs DMLP1800) back with the 1900 nm pump light, which passes waveplates to set correct pump power for the second crystal (PPLN2).
Via a second gold parabolic mirror (f = 15 mm) all fields are coupled into the second temperature controlled chip PPLN2, where the 1550 nm photon is converted back to the initial 854 nm via the reverse (upconversion) process. An Asphere (AS) collimates the output field, before a shortpass (SP 1600; OD5 at 1902) filters the 1902 nm pump laser from the 854 nm single photons.
After passing an etalon filter (EF; LightMachinery, Bandwidth $\approx 870~$MHz) a combination a waveplate and PBS is used to filter unpolarized noise photons before coupling the 854 nm photon into a single mode fiber, which goes to the interference board.
\textbf{c) Photon interference.}
Both inputs pass waveplates and PBSs (cleaning polarization) and overlap on a 50:50 beamsplitter (50:50 BS). Both outputs of the beamsplitter are filtered with an 854 nm bandpass filter (BP 854); coupled to single mode fibers; polarization control paddles (PC) correct to most efficient polarization for followed single photon detector (D1(D2): Scontel, efficiency 88\%(87\%), dark count rate 0.5 s\textsuperscript{-1}(0.3 s\textsuperscript{-1}). The electronic pulses produced by the detectors are detected with a time tagging module (Swabian Instruments Time Tagger 20). Mating sleeves (MS), Fiber coupler (FC).
}
\label{fig:setup_full}
\end{figure*}
\section{Photon conversion performance}
\begin{figure*}\label{fig_levelscheme}
\end{figure*}
\noindent A set of characterisation measurements of the two-stage conversion process was carried out using classical laser light and the results are now described. The total efficiency from the start of the converted delay line (Point A in figure A.1), to the end of the delay line (point B in figure A.1) was measured to be $0.098\pm 0.005$ at the last calibration before the two-photon experiments (the error stands for the last digit of the powermeter reading).
The efficiencies of separate parts were characterized independently. The values below stand for the final characterisation after alignment-optimisation before the two-photon experiment reported in the main text. The intervals are derived from the calibration measurements after optimisation on other days, representing maximum and minimum values observed. The laser powers and setup stability on the timescale of performing the characterisation is also taken into account. The down-conversion stage efficiency from the point A to the in-coupling of the delay fiber was $0.50\pm 0.03$ (we refer it as down-conversion external efficiency). The delay fiber transmission was measured to be $0.6\substack{+0.01 \\ -0.05}$, including in-coupling ($0.75\substack{+0.005 \\ -0.05}$) and two mating sleeves ($0.95\pm 0.02$). The up-conversion external efficiency (from the delay fiber out-coupler to the etalon) was measured to be $0.53\pm 0.03$. The etalon transmission was $0.84\substack{+0.005 \\ -0.04}$ and the coupling to the fiber that goes to the HOM board was $0.73\substack{+0.05 \\ -0.03}$. The provided value above of external up-conversion efficiency of $0.53\pm 0.03$ includes the wavegide in- and out-coupling losses and transmission of the filter that blocks the pump field (see \rfig{fig:setup_full}).
We define the internal conversion efficiency as conversion efficiency without coupling and transmission losses, calculated as the fraction of converted signal-photon-number to unconverted signal-photon-number (without pump light) at the output of the converter. We observe at best alignment 89(0.5)\% internal efficiency for the upconversion and estimate a waveguide coupling/propagation efficiency of 70(5)\%, limited by the mismatch of the out-coupler of the delay fiber and the in-coupler of the conversion waveguide. For the down-conversion, the performance was reported in \cite{Krutyanskiy2017} and yields 66(6)\% internal efficiency if the waveguide in-coupling/propagation losses are taken into account according to the definition above. This efficiency was shown to be limited by the unintentional excitation of higher-order waveguide modes. In order to prove that the up-conversion coupling/propagation efficiency of 70(5)\% is dominated by coupling, we performed a separate measurement with adjusted 1550 beam diameter before the wavguide in-coupler and achieved 82(3)\% efficiency calculated as the ratio of the number of photons at 854 nm wavelength right after the waveguide to the number of 1550 nm photons right before the waveguide. Note, that the external and internal up-conversion efficiencies reported here are higher than the ones achieved in similar systems before \cite{Maring2017, Kaiser:19}.
On the day of the two-photon interference experiment using photon conversion (Figure 3, main text), the total efficiency of the delay line was measured to be approximately 5\%. This lower efficiency, compared to the aforementioned values achieved with classical light is caused by the combination of the following: an additional fiber joiner between the ion node and delay line (panels a and b in Figure \ref{fig:setup_full}) when working with single photons (compared to classical light); imperfectly optimised fiber couplers throughout the delay line; and potential slight mismatch between the photon polarization and the non-linear crystal axis.
The noise, at the single photon level, introduced by the conversion process is now presented. These values are extracted from the photon detector click rate outside of the known ion-photon arrival times recorded in the experiment presented in Figure 3 of the main text. Recall that final narrowband filtering is performed at 854 nm via a temperature-controlled etalon (Bandwidth 870 MHz, free spectral range 30 GHz) that has a maximum transmission of 84\%. From the measured photon noise rate of $11\pm3$ cps at the detectors (after removing detector dark counts and other known background noise, like room light) we estimated from the known losses in the optical path between the etalon and the detectors a noise level of $50\pm 10~\text{s}^{-1}$ right after the final etalon filtering stage.
\section{Photon generation sequence}
\begin{figure*}
\caption{
\textbf{Sequence of the laser pulses during the experiment.} Wavelengths and duration of pulses are labeled. Each cycle (what is shown within the loop) contains four Raman laser pulses, which attempt to generated four photons. The first two are referred to as the synchronous pair and the second pair as the asynchronous pair. Each cycle is looped 40 times and this sequence is repeated thousands of times to produce the data presented in the main text.
}
\label{fig:sequence}
\end{figure*}
We sequentially generate photons of orthogonal polarizations using a cavity-mediated Raman transition (see \rfig{fig_levelscheme}) with ion state reinitialisation in between. The full experimental sequence is shown in \rfig{fig:sequence}. First, a $40~ \rm{\mu s}$ `initialisation' laser-pulse at 393 nm is measured by a photodiode in transmission of the ion-trap chamber and used for intensity stabilisation of the subsequent 393~nm photon-generation Raman pulses with a sample-and-hold system. The initialisation pulse is followed by $2000~ \rm{\mu s}$ of Doppler cooling, involving three laser fields as indicated. Next, a cycle starts in which the photon-pair generation attempt takes place. This cycle is repeated (looped) 40 times before the whole sequence starts again.
In summary, each cycle contains four Raman pulses --- $\mathrm{V_1}$, $\mathrm{H_1}$~and $\mathrm{V_2}$, $\mathrm{H_2}$~--- which attempt to generate the two pairs photons that are refereed to as `synchronous' and `asynchronous' in the main text. The first synchronous pulse pair ($\mathrm{V_1}$, $\mathrm{H_1}$) has a time difference of 13.35~$\mu$s, corresponding to the length of the delay line, such that the generated photon wavepackets arrive at the interference beamsplitter simultaneously (the delay was measured with $<50$ ns accuracy by recording the photon arrival times). The second asynchronous pulse pair ($\mathrm{V_2}$, $\mathrm{H_2}$) has an additional delay $t_{wait}$ and generates a fully temporally distinguishable photon pair as a reference ($t_{wait} = 30 \mu$s). Before the $\mathrm{V_1}$~pulse in each loop we produce an electronic trigger-pulse that is recorded on a separate channel of the time-tagger, along with the photon detection events, to provide exact Raman pulses timing information.
In detail, each cycle starts with an additional Doppler cooling pulse ($20 ~\rm{\mu s}$) and optical pumping to the $S_{J=1/2,m_j=-1/2}$ (see \rfig{fig_levelscheme}) state via circularly polarized 397~nm laser light ($46 ~\rm{\mu s}$). The photon-generation Raman pulse $\mathrm{V_1}$ ($9.4 ~\rm{\mu s}$) creates the vertical polarized photon that is directed to the delay line by a PBS. This is followed by a $4~ \rm{\mu s}$ long, $854 ~\rm{n m}$ repump pulse which pumps the ion back to the initial ground state. A second photon-generation Raman pulse $\mathrm{H_1}$ ($9.4 ~\rm{\mu s}$) creates a horizontal photon that is directed directly to the interference region. After a Doppler cooling pulse of $20 ~\rm{\mu s}$ and optical pumping of $46 ~\rm{\mu s}$, the second pair of photons is produced.
\section{Data analysis}
During the experimental run we record the absolute time stamps of two detector events $D_1$ and $D_2$, and of an electronic trigger pulse generated simultaneously with $\mathrm{V_1}$~at time $t_1$ in each cycle (see \rfig{fig:sequence}). We then work in a time frame referenced to the trigger pulse. In this frame the photon arrival times are grouped into three time windows: the first group contains the overlapped synchronous photons (generated by $\mathrm{V_1}$~and $\mathrm{H_1}$), the second group contains the first of the time-displaced `asynchronous' photons ($\mathrm{V_2}$, through the delay line) and the third group contains the later asynchronous photon ($\mathrm{H_2}$, direct path). In the main text Figure 2a we sum up the data for two detectors and plot separately the probabilities of events corresponding to (windows containing) $\mathrm{V_2}$ and $\mathrm{H_2}$. This is done by shifting the distributions in time by a fixed offset $t$ ($t+t_{wait}$) for $\mathrm{V_2}$ ($\mathrm{H_2}$), with $t$ being the delay between the trigger pulse and the expected $\mathrm{V_2}$-photons' front-slope; $t_{wait}$ is the known additional wait time (30 or 40 $\mu$s in different realisations). We plot the detection-probability-density (see main text), defined as the number of events detected in a certain time bin during the experiment divided by the number of trials and bin duration. The error bars represent $\pm 1$ standard deviation of Poissonian photon counting statistics. The subtraction of background counts was performed for a correct efficiency comparison of the two paths.
To plot the coincidence distribution (Figures 2b, 3b of the main text) we first choose a time window (software gate) where the corresponding photons are expected to arrive based on sequence timing and the histogram of all recorded events. Then we calculate the probability density $\rho_c(D_1(t_1), D_2(t_2))$ of observing a two-photon detection event in a given trial as a function of the detections time difference $\tau = t_2-t_1$. The plotted values in the figure are calculated as $\rho_c(\tau) = \frac{1}{\Delta_t k}\int_{gate}dt_1 \int_{t_1+\tau}^{t_1+\tau+\Delta_t}dt_2 N(t_1,t_2)$ where $N(t_1,t_2)$ is the number of two-photon clicks with given times, $\Delta_t$ is the bin size in the figure and $k$ is the total number of attempts. The errorbars for each point in the figure are calculated from the total number of events detected for this bin assuming Poissonian statistics. The coincidence distribution for the distinguishable photons (from $\mathrm{V_2}$, ~$\mathrm{H_2}$), originally peaking at the $\tau = \pm t_{wait}$, is shifted to $\tau = 0$ and summed over positive and negative branches to represent the expected coincidence distribution for the fully distinguishable but synchronized photons.
Given the coincidences distribution in time we define the visibility (plotted in fig. 2c, 3c of the main text) as: \begin{equation} V(T) = \frac{C^\perp(T)-C^\parallel(T)}{C^\perp(T)}, \label{eq_Visibility_exp} \end{equation} where $C^\perp(T)$ ($C^\parallel(T)$) are the coincidence probabilities for the distinguishable (overlapped) pair of photons plotted in Figures 2b,3b of the main text integrated over the delay range $\tau\in[-T;T]$: $C(T) = \int_{-T}^T \rho_c(\tau)d\tau$.
For the passive delay-line experiment (without conversion) we perform in total 7.5 million cycles, where each cycle corresponds to one loop (cycle) in fig \ref{fig:sequence} (attempt to generate four photons). The experiment with frequency conversion consists of a total of 2.5 million cycles.
\section{Theory model}
\subsection{The Master Equation}
We start by writing down a master equation for a single $^{40}$Ca$^+$ ion trapped inside a cavity and driven by a pump laser. We restrict the atomic model to a $\Lambda$-system formed by three levels $\ket{s}, \ket{p}$ and $\ket{d}$ (see Fig.~\ref{fig:lambda system}) corresponding to sublevels of $S_{J=1/2,m_j-1/2},$ $P_{J=3/2,m_j=-3/2}$ and $D_{J=5/2,m_j=-5/2}$ (or $D_{J=5/2,m_j=-3/2}$) that are of direct importance for the experiment, see Fig.~A.2 for details. The ion is initially prepared in the state $\ket{s}$. The laser is driving off-resonantly the $p-s$ transition with a frequency $\omega_{L} =\omega_{ps}+ \Delta -\delta_S,$ where both $\Delta$ and $\delta_s$ are negative. We denote $a, a^\dag$ the bosonic operators associated to the cavity field whose frequency is given by $\omega_C= \omega_{pd}+\Delta.$ In the experiment, the laser Rabi frequency $\Omega_t$ is much lower than the detuning $|\Delta|$ and the state $|s\rangle$ undergoes a Stark-shift $\Omega_t^2/4\Delta.$ The additional laser detuning $\delta_S=\Omega_t^2/4\Delta$ is chosen to preserve the two-photon resonance between $s-d$. The Hamiltonian of the atom-cavity system is given by \begin{equation}\label{eq:Hamil} \begin{split}
H&= \omega_C a^\dagger a+\omega_{ps}\prjct{p}+\omega_{ds}\prjct{d} \\
&+\frac{1}{2}(e^{i\omega_L t}+e^
{-i\omega_L t})(\Omega_t\ketbra{s}{p}+\Omega_t\ketbra{p}{s}) \\
&+ g(\ketbra{d}{p}+\ketbra{p}{d})(a^\dag+a), \end{split} \end{equation} where we have set $\hbar$ to one. The Hamiltonian can be simplified by noting that the cavity mode is initially empty $\ket{0}.$ Therefore, without additional coupling terms the atom-cavity system remains in the three level manifold \{$\ket{s,0}, \ket{d,1}, \ket{p,0}$\}. Under the rotating wave approximation, the Hamiltonian in this subspace is thus given by \begin{equation} H_t= \left(\begin{array}{ccc}
0&0 & \Omega_t/2 \\
0& \delta_S &g\\
\Omega_t/2& g & -\Delta + \delta_S \end{array} \right), \end{equation} in the rotating frame with $a^\dag \to e^{-\mathrm{i}\omega_C t}a^\dag$, $\ket{p}\to e^{-\mathrm{i} \omega_L t}\ket{p}$, $\ket{d}\to e^{-\mathrm{i} (\omega_L-\omega_C)t}\ket{d}$, and $\ket{s}\to \ket{s}$. \\
Let us now introduce the non-unitary terms which will appear in the master equation. First, the photon can escape the cavity mode with rate $\kappa$. This decay channel is precisely the one in which the photons are collected into a fiber and sent to the detectors. Yet, for the atom-cavity system, this process corresponds to a loss term $L_1=\sqrt{2\kappa} \ketbra{d,0}{d,1}.$ Note that in practice, photons can leave the cavity through channels that are not the detected channel. This additional cavity loss is taking into account by adjusting the photon detection efficiency. Finally, there are two scattering terms $L_2= \sqrt{2\gamma_{sp}} \ketbra{s,0}{p,0}$ and $L_3= \sqrt{2 \gamma_{dp}}\ketbra{d,0}{p,0}$. With this in hand, we can write down the master equation for the atom-cavity system \begin{equation}\label{eq:ME} \dot \varrho_t = -\mathrm{i}\, [H_{t},\varrho_t] + \sum_{i=1}^3 \left(L_i \varrho_t L_i^\dag - \frac{1}{2}\{L_i^\dag L_i, \varrho_t\}\right), \end{equation} where $\varrho_t$ has to be defined on a four level manifold including $\ket{d,0}.$ $H_{t}$ is extended trivially on the added level via $H_{t} \ket{d,0}=\delta_{s} \ket{d,0},$ that is, $\ket{d,0}$ is not coupled to the other three levels. Hence, the atom-cavity system evolves in the $\{\ket{s,0},\ket{d,1},\ket{p,0}\}$-manifold until it is brought to the state $\ket{d,0}$ either by the scattering term $L_3$ or by emitting a photon towards the detector via $L_1.$ \\
\begin{figure}
\caption{Scheme of the $\Lambda$-system relevant for the experiment. The transition between $|s\rangle$ and $|p\rangle$ is off-resonantly driven by a pump laser with frequency $\omega_L$ while the states $|p\rangle$ and $|d\rangle$ are coupled by the field of a cavity with frequency $\omega_C.$ The explicit expressions of each detunings are given in the text.}
\label{fig:lambda system}
\end{figure}
\subsection{The photon state}
We can now address the question of interest: What is the state of the photon emitted from the cavity to the detected mode? The photon state is computed in two steps. First, we obtain the sub-normalized wave-function of a photon in the pure state conditioned on the atom-cavity system being in the state $\ket{s,0}$ at time $s$ and no scattering events $L_2$ and $L_3$ happening at later times. Second, we solve the full master equation to compute the probability of a scattering via $L_2$ happening at time $s.$ Such a scattering event projects the system back onto $\ket{s,0}$.
\subsubsection{Conditional pure photon wave-function} Let us start by addressing the wave-function of a photon in a pure state (pure photon) conditioned on the atom-cavity system being in the state $\ket{s,0}$ at time $s$ and no scattering events $L_2$ and $L_3$ happening at later times. To do so, we need to solve the evolution of the atom-cavity system conditioned to the case with no scattering. The Lindbladian part of the master equation~\eqref{eq:ME} describes random noise processes affecting the system. In particular, the terms $L_i \varrho_t L_i^\dag dt$ corresponds to a scattering happening during an infinitesimal time interval $dt$. In contrast, the conjugate term $-\frac{1}{2}\{L_i^\dag L_i, \varrho_t\} dt$ corresponds to no scattering happening during $dt.$ Its role can be thought of as reducing the probability to find the system in the pre-scattered state $\varrho_{t+dt} \to \varrho_{ t} - \frac{1}{2}\{L_i^\dag L_i, \varrho_{t}\}dt$. Hence, to describe the evolution of the system conditioned on no scattering, we drop all the terms $L_i\varrho_t L_i^\dag$ but keep their conjugate terms $-\frac{1}{2}\{L_i^\dag L_i, \varrho_t\}$ in the master equantion~\eqref{eq:ME}. It is straightforward to see that such an evolution preserves the purity of a state, and can be written in the form of the Schr\"odinger equation with a non-Hermitian Hamiltonian \begin{equation} \dot{\ket{ \Phi_t}} = \left(-\mathrm{i} H_t - \frac{1}{2} \sum_i L_i^\dag L_i \right) \ket{\Phi_t}.
\end{equation} With the initial condition $\ket{ \Phi_s}=\ket{s,0},$ this equation can be solved to give the system state $\ket{\Phi_{t|s}}$ conditioned on the event corresponding to no scattering at time $t\geq s$. In particular, for a (piece-wise) constant Rabi frequency, the solution reads
$\ket{\Phi_{t|s}}=e^{\left(-\mathrm{i} H - \frac{1}{2} \sum_i L_i^\dag L_i \right) (t-s)}\ket{s,0}$ which can be computed numerically. To obtain the amplitude of the emitted photon at a given time we project the atom-cavity state at this time into $\sqrt{2\kappa}\bra{d,1}$. In the laboratory frame, this gives \begin{equation}\label{eq:pure photon} \begin{split}
\ket{\psi_s} &= \int_{s}^{\infty} \psi_s(t) \, a_t^\dag \ket{0} dt,\\
\psi_s(t)&= \sqrt{2 \kappa}\, e^{-\mathrm{i} \omega_C t}\, \braket{d,1}{\Phi_{t|s}}. \end{split} \end{equation} The photonic state $\ket{\psi_s}$ is sub-normalized. Its norm \begin{equation} p_{pure}(s)=\braket{\psi_s}{\psi_s} \end{equation} is precisely the probability that no scattering event happens after time $s$ (given the initial condition). The conditional state thus reads $\ket{\psi_s}/\sqrt{p_{pure}(s)}$. One notes that $p_{pure}(0)$ is the probability that a photon is emitted without a single scattering during the evolution.
\subsubsection{Scattering probability} We now solve the full master equation~\eqref{eq:ME} and obtain the atom-cavity state $\varrho_t$ for all times. Note that for a (piecewise) constant Rabi frequency, the solution can be obtained analytically by vectorizing the master equation and the density matrix. From this state, we compute the probability of scattering back to $\ket{s,0}$ at time $s$ \begin{equation}\label{eq:scattering rate} P(s) = \tr{\varrho_s \, L_2^\dag L_2}. \end{equation}
\subsubsection{The photon state} The state of the emitted photon is given by \begin{equation}\label{eq: ph state1} \rho = \prjct{\psi_0} + \int_{0}^\infty P(s) \prjct{\psi_s} ds + P_0 \prjct{0}, \end{equation} where $\ket{\psi_s}$ is given in Eq.~\eqref{eq:pure photon} and $P(s)$ in Eq.~\eqref{eq:scattering rate}. Let us comment on each contribution separately. The first term $\prjct{\psi_0}$ describes a pure photon emitted without a single scattering ($L_2, L_3$). This happens with probability $p_{pure}(0)$. The integral collects all the possibilities where the last $p-s$-scattering happens at time $s$ and no scattering events happen at later times. Any such history happens with probability $P(s)p_{pure}(s)$ and yields a pure photon in the state $\ket{\psi_s}/\sqrt{p_{pure}(s)}$. Finally, the last terms $P_0\prjct{0}$ with \begin{equation}\begin{split}
P_0&=1-\tr{\prjct{\psi_0} + \int_{0}^\infty P(s) \prjct{\psi_s} ds }\\
& = 1- p_{pure}(0) -\int P(s)p_{pure}(s) ds \end{split}\end{equation} collects the cases where no photon is emitted from the cavity. If the laser pulse is not turned off, this term can be alternatively computed as the overall probability of the $p-d$ scattering \begin{equation} P_0= \int_0^\infty \textrm{tr}(L_3^\dag L_3\, \rho_t). \end{equation} To shorten the equations we will combine the first two contributions of Eq.~\eqref{eq: ph state1} together by defining \begin{equation} \bar P(s)= P(s)+2 \delta(s), \end{equation} where $\delta(s)$ is the delta-function with $2 \int_0^\epsilon \delta(s)ds = 1$. The photon state $\rho$ in Eq.\eqref{eq: ph state1} then simply reads \begin{equation}\label{eq:photon state} \rho = \int_{0}^\infty \bar P(s) \prjct{\psi_s} ds + P_0 \prjct{0}. \end{equation}
\subsubsection{Expected number of scattering events} Note that the average number of $L_2$ scattering events per experimental run is simply given by the time integral of the scattering rate \begin{equation} \int_0^\infty P(s) ds, \end{equation} and equals the expected number of laser photons scattered on the $p-s$ transition.
\subsection{Photon statistics}
Now that we have computed the photonic state $\rho$ emitted from the cavity, let us consider the Hong-Ou-Mandel (HOM) interference of two such photons, as depicted in Fig.~1 of the main text. More precisely, we consider two photons described by Eq~\eqref{eq:photon state} that enter the two ports of a 50/50 beamsplitter followed by two photon detectors D1 and D2. We assume detection with unit efficiency here, and discuss the general case in the next section.
\subsubsection{Single click rates} The probability density that a photon in the state $\rho$ given in Eq.~\eqref{eq:photon state} triggers a click on the detector D1 (the same result is obtained for the detector D2) at time $t$ is given by $p_s(t)=\frac{1}{2}\tr{ \rho\, a_{t}^\dag \prjct{0}a_t}$. Here, the $1/2$ factor comes from $50/50$ beamsplitter. Direct application of Eqs.~\eqref{eq:photon state} and \eqref{eq:pure photon} gives
\begin{equation} p_{s}(t) = \frac{1}{2} \int_{0}^\infty \bar P(s) |\psi_s(t)|^2 ds, \end{equation} where we formally set $\psi_s(t) =0$ for $s>t$ here and in the following.
\subsubsection{Coincidence rates} We can now compute the twofold coincidence rate when one photon is sent at each input of the beamsplitter. For two photons with orthogonal polarizations corresponding to states $\rho_a$ and $\rho_{b_\bot}$, the probability to get a click at time $t_1$ in the detector D1 and a click at time $t_2$ in the detector D2 is given by \begin{equation} p_{C\perp}(t_1,t_2) = p_S^{(a)}(t_1) p_S^{(b_\bot)}(t_2)+p_S^{(a)}(t_2) p_S^{(b_\bot)}(t_1)\\ \end{equation} simply because there is no interference. \\
Next, we consider two photons with the same polarization $\rho_a$ and $\rho_b.$ They are respectively characterized by $\bar P_a(s_a)$ with $\psi^{(a)}_{s_a}$ and $\bar P_b(s_b)$ with $\psi^{(b)}_{s_b}$ accordingly to Eq.~\eqref{eq:photon state}. The twofold coincidence probability with the detector D1 clicking at time $t_1$ and the detector D2 clicking at $t_2$ is computed as $p_c(t_1,t_2)=\tr{\rho_a\otimes\rho_b\, \Pi_{t_1,t_2}}$ where $\Pi_{t_1,t_2}$ is the projector onto \begin{equation} \frac{1}{2}(a_{t_1}^\dag+b_{t_1}^\dag)(a_{t_2}^\dag - b_{t_2}^\dag)\ket{00} \end{equation} where $a_{t_1}^\dag$ for example is the bosonic creation operator for the input mode $a$ at time $t_1$, as in Eq.\eqref{eq:pure photon}. From Eqs.~\eqref{eq:photon state} and \eqref{eq:pure photon} we find \begin{equation}\label{eq:pcparallel}\begin{split}
p_{C\parallel}(t_1,t_2)&=
\frac{1}{4}\int_0^\infty \bar P_a(s_a) \bar P_b(s_b)\times\\
&|\psi_{s_a}^{(a)}(t_1)\psi_{s_b}^{(b)}(t_2)- \psi_{s_a}^{(a)}(t_2)\psi_{s_b}^{(b)}(t_1)|^2 ds_a ds_b. \end{split} \end{equation} The integrand in the last equation reads \begin{equation}\begin{split}\label{eq: bunching}
&|\psi_{s_a}^{(a)}(t_1)\psi_{s_b}^{(b)}(t_2)- \psi_{s_a}^{(a)}(t_2)\psi_{s_b}^{(b)}(t_1)|^2 =\\
&|\psi_{s_a}^{(a)}(t_1)|^2 |\psi_{s_b}^{(b)}(t_2)|^2 + |\psi_{s_a}^{(a)}(t_2)|^2 |\psi_{s_b}^{(b)}(t_1)|^2\\
&- \psi_{s_a}^{(a)}(t_1)\psi_{s_a}^{(a)*}(t_2) \,\psi_{s_b}^{(b)*}(t_1)\psi_{s_b}^{(b)}(t_2)\\
&- \psi_{s_a}^{(a)*}(t_1)\psi_{s_a}^{(a)}(t_2)\, \psi_{s_b}^{(b)}(t_1)\psi_{s_b}^{(b)*}(t_2). \end{split}\end{equation} The last two terms are responsible for a destructive interference and bunching of the incident photons, which reduces the coincidence rate as compared to the orthogonal case.
\subsubsection{Visibility of the Hong-Ou-Mandel pattern}
The absolute detection times $t_1$ and $t_2$ are not relevant for computing the visibility of the Hong-Ou-Mandel (HOM) interference pattern. What matters is the difference between the detection times \begin{equation} \tau = t_1-t_2. \end{equation} We define the coincidence rate as a function of this delay between the clicks \begin{equation} p_{C\parallel(\perp)}(\tau) = \int p_{C\parallel(\perp)}(t_2+\tau,t_2 )\, dt_2.\\
\end{equation} Furthermore, we define detection window $T$, by accepting only the coincidence events with a delay $|\tau|\leq T$ falling inside this window. For a fixed window $T$ the HOM visibitity is defined as \begin{equation} V(T) = 1-R(T)
\end{equation} where $R(T)$ is the ratio between the coincidence probabilities for parallel and orthogonal polarizations with a bounded delay $|\tau|<T$ \begin{equation} R(T) =\frac{\int_{-T}^T p_{C\parallel}(\tau) d\tau}{\int_{-T}^T p_{C\perp}(\tau) d\tau }.\\ \end{equation}
As mentioned after Eq.~\eqref{eq: bunching}, the effect of the photon bunching on the visibility is captured by \begin{equation}\begin{split}
\int_{-T}^T d\tau \int d{t_2} \, &\psi_{s_a}^{(a)}(t_2+\tau)\psi_{s_a}^{(a)*}(t_2) \,\psi_{s_b}^{(b)*}(t_2+\tau )\psi_{s_b}^{(b)}(t_2)\\
&
\nonumber
+ h.c. \end{split} \end{equation} From this equation, we see that $p_{C\parallel}(0)=0.$ The visibility thus increases when the detection window $T$ is shortened. This can be intuitively understood from the fact that narrowing the detection window removes some of the temporal mixedness of the photons and effectively purifies them. The price to pay for decreasing $T$ is that the probability \begin{equation}\label{eq:Psucc} P_\text{succ}(T) = \int_{-T}^T p_{C\perp}(\tau) d\tau. \end{equation} to observe a coincidence within the detection window decreases with $T$ for orthogonally polarized photons.
\subsection{Predictions}
With such a theoretical model for the state of emitted photons and the HOM interference pattern, we can find experimental parameters facilitating the implementation of certain tasks that are relevant for quantum networking. An interesting experiment in this framework aims to entangle two ions remotely by first creating locally ion-photon entanglement and then performing a photonic Bell state measurement at a central station \cite{Matsukevich2008, Moehring2007}. Ion-photon entanglement is created by modifying our experiment to enable the transitions from the excited state $\ket{p}$ to two states $\ket{d}$ and $\ket{d'}$ leading to cavity photons with orthogonal polarizations, as we have previously shown \cite{Stute2012, 50km}. Two such photons emitted from cavities located at different locations are then sent into a beamsplitter which is followed by two detectors. A twofold coincidence projects the two ions into an entangled state. In such an entanglement swapping experiment, the fidelity $F$ (with respect to a maximally entangled two qubit state) of the two ion state can be shown to be proportional to the HOM visibility $V$ \cite{PhysRevLett.123.213601}. Furthermore, the rate at which the entangled states are created is related to the success probability $P_\text{succ}$ defined in Eq.~\eqref{eq:Psucc}. Hence, both the visibility $V$ and the success rate $P_\text{succ}$ play an important role for the entanglement swapping experiment. We thus wish to find experimental parameters maximizing the success rate given that the visibility stays above a threshold value (or vice versa). In particular, we focus on the impact of the Rabi frequency $\Omega$ on the visibility $V$ and the success rate $P_\text{succ}$. In Figure \ref{fig:VP}, we give a parametric plot of $V$ and $P_\text{succ}$ for various detection window $T$. Different curves correspond to different values of the Rabi frequency $\Omega_t=\Omega$ for $t\geq 0$. We see that in the range of small $P_\text{succ}$ where $V$ remains high, there is a globally optimal Rabi frequency $\Omega_\text{opt}\approx 2\pi \times 40$ MHz which maximizes $V$ for all values of $P_\text{succ}$. \\
\begin{figure}
\caption{A parametric plot of the Hong-Ou-Mandel visibility $V(T)$ versus the success probability $P_\text{succ}(T)$ for different values of the Rabi frequency $\Omega$. For each curve $T\in[0,\infty)$ is increasing from left to right.
All the feagures are computed for $\Delta= - 2\pi \times 400$ MHz, $\kappa=2\pi \times 0.07$ MHz, $\gamma_{sp} = 2\pi \times 10.7$ MHz, $\gamma_{dp}= 2\pi \times 0.7$ MHz, and $g=2\pi \times 1.2 \sqrt{4/15}$ MHz.}
\label{fig:VP}
\end{figure}
\subsection{Extended models: imperfections beyond scattering}
In this section we discuss various imperfections that are detrimental for the photon counting statistics, and show how to describe them within our model.
\subsubsection{Detection efficiency} The non-unit detection efficiency is modelled by a loss channel acting on the photon state prior to detection. Since the photonic state $\rho$ in Eq.~\eqref{eq:photon state} is a mixture of a single photon distributed across difference temporal modes and a vacuum component, the effect of loss is particularly simple to describe. A loss channel with transmission rate $\eta$ simply maps \begin{equation} \bar P(s) \mapsto \eta \bar P(s)\quad P_0 \mapsto 1-\eta(1-P_0). \end{equation} The loss of cavity photons outside of the detected mode is also equivalent to a lack of detection efficiency. In such a case, the total cavity decay rate entering in the master equation is the sum of the loss rate towards the detector $\kappa_\text{det}$ and in modes that are not measured $\kappa_\text{loss}$, that is $\kappa = \kappa_\text{det} + \kappa_\text{loss}.$ The detection efficiency is reduced by an additional factor $\eta_\kappa=\frac {\kappa_\text{det}} \kappa$.
\subsubsection{Mode mismatch} In practice, the two modes entering the beamsplitter preceding the two detectors do not perfectly overlap, which reduces the visibility of their interference pattern. An imperfect overlap of $\varepsilon$ means that with probability $\varepsilon$ the two photons entering the interference will not see each other. The coincidence rate for two photons with the same polarization becomes \begin{equation} \label{dist} p_{C\parallel}\to (1-\varepsilon)\, p_{C\parallel} + \varepsilon\, p_{C\perp}.\\ \end{equation} This is how we model a fixed nonzero distinguishability of the interfering photons in order to reproduce experimental date, see Figs.~2 and 3 in the main text.
\subsubsection{Photon frequency mismatch} We here model the effect of the frequency distinguishability of the two photons arriving at the beamsplitter on the HOM-type interference experiment. We restrict ourselves to a simple approximation where the photons have a constant offset and a linear drift of the frequency in time \begin{equation} \label{eq_carrier_offset} \omega(t) = \omega_0 + \omega_s + \omega_d t \end{equation} where $\omega_s$ and $\omega_d$ can be random variables over the experimental attempts. We will come to their distributions later.\\
In the two photon experiment, the frequency uncertainty affects the phase of the photonic wave-function. In particular, for the computation of the coincidence rate, we are interested in the factor \begin{equation} f=\mean{e^{\mathrm{i} (\phi_1(t_1)-\phi_1(t_2)-\phi_2(t_1)+\phi_2(t_2))}}, \end{equation} where $\phi_1$ and $\phi_2$ is the phase of photons emitted at times $t_1$ and $t_2,$ and $\mean{\bullet}$ denotes the statistical average over the phases. The photon phase is the time integral of its instantaneous frequency \begin{equation} \phi(t) = \int_0^t \omega(s)ds = \omega_s t + \omega_d \frac{t^2}{2}. \end{equation} Given that the second photon is delayed from the first one by a time $\tau,$ we get \begin{equation}\begin{split}
&(\phi_1(t_1)-\phi_1(t_2)-\phi_2(t_1)+\phi_2(t_2)) \\
&= (\phi(t_1)-\phi(t_2)-\phi(t_1+\tau)+\phi(t_2+\tau)) \\
& = \omega_d \tau (t_2-t_1). \end{split}\end{equation} and thus \begin{equation}\begin{split}
f &= \int e^{\mathrm{i} \omega_d \tau (t_2-t_1)}p(\omega_d) d\omega_d\\
&= \tilde p(\tau (t_2-t_1)), \end{split} \end{equation} where $\tilde p$ is the Fourier transform of $p(\omega_d)$. Let us assume that the distribution of the drift is a Gaussian with a zero mean and an unknown variance $\sigma^2$ \begin{equation} p(\omega_d) = \frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{\omega_d^2}{2 \sigma^2}}. \end{equation} In this case, \begin{equation} f= e^{-\frac{1}{2}\tau^2(t_2-t_1)^2 \sigma^2}. \end{equation} It remains to relate $\sigma$ to experimentally measured parameters. In the experiment, we measure $v(\bar T)$ -- the average squared deviation of the cavity frequency over a time $\bar T$. Under our assumption, this quantity reads \begin{equation} \label{freq_drift} v(\bar T) = \mean{\big(\omega(0) -\omega(\bar T)\big)^2} = \bar{T}^2 \sigma^2 .\end{equation} We obtain \begin{equation} \begin{split}
f(|t_2-t_1|,\tau) &= e^{-\frac{1}{2}\tau^2(t_2-t_1)^2\frac{v(\bar T)}{\bar{T}^2}}\\
&\approx 1-\frac{ \tau^2(t_2-t_1)^2 v(\bar{T})}{2 \bar{T}^2}. \end{split} \end{equation} Finally, let us note that we do not need to assume the exact form of the distribution $p(\omega_d)$. As long as the distribution is symmetric $p(\omega_d)=p(-\omega_d)$, and the dephasing effect is small, we can develop the Fourier transform of the distribution $p(\omega_d)$ to the second order to get \begin{equation} \tilde{p}(\tau (t_2-t_1))\approx 1- \tau^2(t_2-t_1)^2 \sigma^2 \end{equation} as a function of a single parameter $\sigma^2$, which is expressed in the same way as $v(\bar{T})/\bar{T}^2$.
\section{Data modelling}
\subsection{General principles:}
\subsubsection{Spontaneous scattering rates}
For all modelling, the scattering rates are $\gamma_{pd}=2\pi\times0.68$~MHz and $\gamma_{ps}=2\pi\times10.7$~MHz (Figure 1, main text).
\subsubsection{g factors, Clebsch-Gordan factors and finite temperature effects}
Following \cite{russothesis}, by adiabatically eliminating the $P$-state population, the coherent part of the cavity-mediated Raman transition (CMRT) can be described as an effective two-level system driven with $\Omega_{eff} = \frac{\Omega^{\prime}\beta g_0}{\Delta+\delta}$. Here, $\Omega^\prime$ is the drive strength of the CMRT (defined later), $\beta$ is the product of the Clebsch-Gordan-coefficient of the 854~nm atomic transition and the projection of the polarization plane of the cavity mode onto the atomic dipole moment, $\Delta+\delta$ is the detuning (\rfig{fig_levelscheme}) and $g_0$ is the maximum ion-cavity coupling strength. The finite temperature of the ion after Doppler cooling leads to the Raman laser coupling to the ion motional sidebands and thus a reduced coupling strength on the desired carrier transition $\Omega^\prime = \alpha\Omega$. Here, $\alpha$ (a real number, less than 1) is the reduction factor of the carrier drive-strength and $\Omega$ is the Raman-laser Rabi frequency used for the theory calculations (eq. \ref{eq:Hamil}) and throughout the manuscript. We calculate $\alpha$ using eq. 3.11 of \cite{roosthesis} for the known (independently measured) motional state of the ion in all directions. We define the effective ion-cavity coupling $g = \alpha \beta g_0$, which is used for the theoretical calculations (eq. \ref{eq:Hamil}). For the transition to the $\mathrm{D_{m_j=-5/2}}$ state (V photon, long path) $\beta = \sqrt{10/15\cdot 1/2}$ , where 1/2 stands for the projection of the polarization plane of the cavity mode to the atomic dipole moment. For the transition to the $\mathrm{D_{m_j=-3/2}}$ state (H photon, short path) $\beta = \sqrt{4/15}$.
\subsubsection{Calibration of experimental parameters}
The maximum ion-cavity coupling strength $g_0$ is calculated using the known cavity length and waist, and indirectly measured by comparing photon wave-packet and generation efficiency (measured at negligible temperature, with additional side-band cooling) to numerical simulations \cite{russothesis}. Both approaches give a value of $g_0/2\pi = 1.53\pm0.01~$ MHz. We calculate that the ion's wavepacket delocalisation after Doppler cooling introduces no significant (compared to $\alpha$) decrease of the ion-cavity coupling strength (the delocalisation is predominantly in a direction perpendicular to the vacuum cavity standing waves). We determine $\Omega$ by performing 393~nm spectroscopy of the Raman resonance for different intensities of the Raman beam and extracting the induced AC-Stark shift. The cavity detuning ($\Delta=403\pm 5$ MHz), measured with a wavemeter, is the frequency difference of the 393 laser when tuned on resonance with the S-P transition and when tuned to the Raman resonance condition. The lifetime of the P state limits the precision of this measurement.
We determine the mean phonon number of each motional modes of the ion by performing Rabi flops on the 729 ($S_{1/2}\rightarrow D_{5/2}$) transition and fitting the observed dependence of the excitation probability on the pulse length with a model that takes into account the ion temperature (for details see \cite{roosthesis}). Flops are taken with two different 729 nm laser beam directions, allowing the temperature in different motional modes to be distinguished.
\subsection{Modelling in figure 2}
\subsubsection{Ideal model}
For the theory curves presented in Figure 2 of the main text, we experimentally determine $\Omega/2\pi = 63.5(5)$ MHz, $\Delta/2\pi = 403(5)~$MHz. The coupling-reducing factor $\alpha = 0.75(2)$ ($g/(2\pi \beta) = 1.15~$MHz) was calculated from a mean-phonon-number of 14 on the axial mode ($\omega_{ax} = 0.9$ MHz) and a mean-phonon-number of 6 on the radial mode ($\omega_{rad} = 2.4$ MHz).
\subsubsection{Extended model}
For the extended-model curves in Figure 2 we model the frequency mismatch as a constant frequency offset according to eq. \ref{eq_carrier_offset}, with $\omega_s/2\pi=40~$ kHz, $\omega_d = 0$ (from best match to the experimental data).
Due to the excess of distinguishability in the experiment compared to theory even for the smallest coincidence window $T$, an additional constant distinguishability of $\epsilon=0.01$ was introduced into the model, according to \ref{dist}. This could be caused by any mode mismatch, e.g. imperfect photon polarizations at the final beam-splitter due to an imperfectly aligned PBS and slight imperfections in the beam splitter itself. The effect of background coincidences (due to dark counts, background light, imperfect $g^{(2)}(0)$) was measured to be at the $5\times 10^{-4}$ level and was neither subtracted from the experiment data nor taken into account in the model.
\subsection{Modelling in figure 3: frequency converted case}
\subsubsection{Ideal model} For the theory curves presented in Figure 3 (main text) we determine $\Omega/2\pi = 64.3(5)$ MHz, $\Delta/2\pi = 403(5)~$MHz. The coupling reducing factor $\alpha = 0.69(2)$ ($g/(2\pi\beta) = 1.05~$MHz) was determined from the best fit of the simulated photon wavepacket. \ This value is lower than the value expected from the temperature calibration measurement before the experiment ($g/(2\pi\beta) = 1.15~$MHz) which could be caused by slight drift of the Doppler laser cooling parameters.
\subsubsection{Extended model} As in the case of the experiment without conversion, the discrepancy of theoretical and experimental visibilities, which depends on the coincidence window size, is attributed to the photons' frequency mismatch. When implementing photon frequency conversion before and after the delay line, the effect of the frequency instability of the unstablized conversion pump laser (1902 nm) has to be taken into account and is the dominant effect. In our implementation there is a significant optical path length difference of $\sim15 \mu$s (the whole delay line) between the pump laser field and the photon (see \rfig{fig:setup_full}). In that case any laser frequency drift on this time scale will result in a frequency mismatch of the short and long path photons at the interference beamsplitter. Note, however, that the constant (in time) part of the pump-laser frequency is expected to cancel out since the down and up conversion processes are symmetric. Based on the laser specification we expect $\sim 50$ kHz instability on a $10 \mu$s timescale, making it the dominant source of frequency mismatch.
To plot the extended-model curves in Figure 3 (main text) we use \ref{freq_drift} with $v(10\mu s) = 2\pi\cdot 50~$kHz and take into account the measured background-coincidence-rate by adding a constant coincidence-probability-density of $0.8\cdot 10^{-6}~\mu s^{-1}$ (Figure 3b). The extended theory lines of Figure 3c are calculated by integrating the coincidence distributions including this background floor. The background coincidences are significant in the experiment with conversion because the signal level is $~5$ times lower. Also, the observed free-running noise counts were 13(2) cps compared to 2(2) cps per detector for the experiment without conversion due to the photon noise introduced by the frequency conversion process.
\end{document} | arXiv |
The word "rounding" for a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit form. For example, US$23.74 could be rounded to US$24, or the fraction 312/937 could be rounded to 1/3, or the expression 2 {\displaystyle {\sqrt {2}}} as 1.41.
Rounding is often done on purpose to obtain a value that is easier to write and handle than the original. It may be done also to indicate the accuracy of a computed number; for example, a quantity that was computed as 123,456 but is known to be accurate only to within a few hundred units is better stated as "about 123,500".
On the other hand, rounding can introduce some round-off error in the result. Rounding is almost unavoidable in many computations, especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating point representation with a fixed number of significant digits. In a sequence of calculations, these rounding errors generally accumulate, and in certain "ill-conditioned" cases, then they may make the result meaningless.
Accurate rounding of transcendental mathematical functions is difficult because the number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance. This problem is known as "the table-maker's dilemma" (below).
Rounding has many similarities to the quantization that occurs when physical quantities must be encoded by numbers or digital signals.
1 Types of rounding
2 Rounding to a specified increment
3 Rounding to integer
4 Tie-breaking
4.1 Round half up
4.2 Round half down
4.3 Round half away from zero
4.4 Round half towards zero
4.5 Round half to even
4.6 Round half to odd
4.7 Stochastic rounding
4.8 Alternating tie-breaking
5 Simple dithering
6 Multidimensional dithering
7 Rounding to simple fractions
8 Scaled rounding
9 Round to available value
10 Floating-point rounding
11 Double rounding
12 Exact computation with rounded arithmetic
13 The table-maker's dilemma
15 Rounding functions in programming languages
16 Other rounding standards
16.1 U.S. Weather Observations
16.2 Negative zero in meteorology
Types of rounding[change | change source]
Typical rounding problems are:
approximating an irrational number by a fraction, e.g. π by 22/7;
approximating a fraction with periodic decimal expansion by a finite decimal fraction, e.g. 5/3 by 1.6667;
replacing a rational number by a fraction with smaller numerator and denominator, e.g. 3122/9417 by 1/3;
replacing a fractional decimal number by one with fewer digits, e.g. 2.1784 dollars by 2.18 dollars;
replacing a decimal integer by an integer with more trailing zeros, e.g. 23,217 people by 23,200 people; or, in general,
replacing a value by a multiple of a specified amount, e.g. 27.2 seconds by 30 seconds (a multiple of 15).
Rounding to a specified increment[change | change source]
The most common type of rounding is to round to an integer; or, more generally, to an integer multiple of some increment — such as rounding to whole tenths of seconds, hundredths of a dollar, to whole multiples of 1/2 or 1/8 inch, to whole dozens or thousands, etc..
In general, rounding a number x to a multiple of some specified increment m entails the following steps:
Divide x by m, let the result be y;
Round y to an integer value, call it q;
Multiply q by m to obtain the rounded value z.
z = r o u n d ( x , m ) = r o u n d ( x / m ) ⋅ m {\displaystyle z=\mathrm {round} (x,m)=\mathrm {round} (x/m)\cdot m\,}
For example, rounding x = 2.1784 dollars to whole cents (i.e., to a multiple of 0.01) entails computing y = x/m = 2.1784/0.01 = 217.84, then rounding y to the integer q = 218, and finally computing z = q×m = 218×0.01 = 2.18.
When rounding to a predetermined number of significant digits, the increment m depends on the magnitude of the number to be rounded (or of the rounded result).
The increment m is normally a finite fraction in whatever numeral system that is used to represent the numbers. For display to humans, that usually means the decimal numeral system (that is, m is an integer times a power of 10, like 1/1000 or 25/100). For intermediate values stored in digital computers, it often means the binary numeral system (m is an integer times a power of 2).
The abstract single-argument "round()" function that returns an integer from an arbitrary real value has at least a dozen distinct concrete definitions presented in the rounding to integer section. The abstract two-argument "round()" function is formally defined here, but in many cases it is used with the implicit value m = 1 for the increment and then reduces to the equivalent abstract single-argument function, with also the same dozen distinct concrete definitions.
Rounding to integer[change | change source]
The most basic form of rounding is to replace an arbitrary number by an integer. All the following rounding modes are concrete implementations of the abstract single-argument "round()" function presented and used in the previous sections.
There are many ways of rounding a number y to an integer q. The most common ones are
round down (or take the floor, or round towards minus infinity): q is the largest integer that does not exceed y.
q = f l o o r ( y ) = ⌊ y ⌋ = − ⌈ − y ⌉ {\displaystyle q=\mathrm {floor} (y)=\left\lfloor y\right\rfloor =-\left\lceil -y\right\rceil \,}
round up (or take the ceiling, or round towards plus infinity): q is the smallest integer that is not less than y.
q = c e i l ( y ) = ⌈ y ⌉ = − ⌊ − y ⌋ {\displaystyle q=\mathrm {ceil} (y)=\left\lceil y\right\rceil =-\left\lfloor -y\right\rfloor \,}
round towards zero (or truncate, or round away from infinity): q is the integer part of y, without its fraction digits.
q = t r u n c a t e ( y ) = sgn ( y ) ⌊ | y | ⌋ = − sgn ( y ) ⌈ − | y | ⌉ {\displaystyle q=\mathrm {truncate} (y)=\operatorname {sgn}(y)\left\lfloor \left|y\right|\right\rfloor =-\operatorname {sgn}(y)\left\lceil -\left|y\right|\right\rceil \,}
round away from zero (or round towards infinity): if y is an integer, q is y; else q is the integer that is closest to 0 and is such that y is between 0 and q.
q = sgn ( y ) ⌈ | y | ⌉ = − sgn ( y ) ⌊ − | y | ⌋ {\displaystyle q=\operatorname {sgn}(y)\left\lceil \left|y\right|\right\rceil =-\operatorname {sgn}(y)\left\lfloor -\left|y\right|\right\rfloor \,}
round to nearest: q is the integer that is closest to y (see below for tie-breaking rules).
The first four methods are called directed rounding, as the displacements from the original number y to the rounded value q are all directed towards or away from the same limiting value (0, +∞, or −∞).
If y is positive, round-down is the same as round-towards-zero, and round-up is the same as round-away-from-zero. If y is negative, round-down is the same as round-away-from-zero, and round-up is the same as round-towards-zero. In any case, if y is integer, q is just y. The following table illustrates these rounding methods:
(towards −∞)
(towards +∞)
towards
+23.67 +23 +24 +23 +24 +24
+23.50 +23 +24 +23 +24 +23 or +24
−23.00 −23 −23 −23 −23 −23
−23.50 −24 −23 −23 −24 −23 or −24
Where many calculations are done in sequence, the choice of rounding method can have a very significant effect on the result. A famous instance involved a new index set up by the Vancouver Stock Exchange in 1982. It was initially set at 1000.000, and after 22 months had fallen to about 520 — whereas stock prices had generally increased in the period. The problem was caused by the index being recalculated thousands of times daily, and always being rounded down to 3 decimal places, in such a way that the rounding errors accumulated. Recalculating with better rounding gave an index value of 1098.892 at the end of the same period.[1]
Tie-breaking[change | change source]
Rounding a number y to the nearest integer requires some tie-breaking rule for those cases when y is exactly half-way between two integers — that is, when the fraction part of y is exactly 0.5.
Round half up[change | change source]
The following tie-breaking rule, called round half up (or round half towards plus infinity), is widely used in many disciplines. That is, half-way values y are always rounded up.
If the fraction of y is exactly 0.5, then q = y + 0.5.
q = ⌊ y + 0.5 ⌋ = − ⌈ − y − 0.5 ⌉ {\displaystyle q=\left\lfloor y+0.5\right\rfloor =-\left\lceil -y-0.5\right\rceil \,}
For example, by this rule the value 23.5 gets rounded to 24, but −23.5 gets rounded to −23.
This is one of two rules generally taught in US elementary mathematics classes.[source?]
If it were not for the 0.5 fractions, the roundoff errors introduced by the round to nearest method would be quite symmetric: for every fraction that gets rounded up (such as 0.268), there is a complementary fraction (namely, 0.732) that gets rounded down, by the same amount. When rounding a large set of numbers with random fractional parts, these rounding errors would statistically compensate each other, and the expected (average) value of the rounded numbers would be equal to the expected value of the original numbers.
However, the round half up tie-breaking rule is not symmetric, as the fractions that are exactly 0.5 always get rounded up. This asymmetry introduces a positive bias in the roundoff errors. For example, if the fraction of y consists of three random decimal digits, then the expected value of q will be 0.0005 higher than the expected value of y. For this reason, round-to-nearest with the round half up rule is also (ambiguously) known as asymmetric rounding.
One reason for rounding up at 0.5 is that only one digit need be examined. When seeing 17.50000..., for example, the first three figures, 17.5, determines that the figure would be rounded up to 18. If the opposite rule were used (round half down), then all the zero decimal places would need to be examined to determine if the value is exactly 17.5.
Round half down[change | change source]
One may also use round half down (or round half towards minus infinity) as opposed to the more common round half up (the round half up method is a common convention, but is nothing more than a convention).
If the fraction of y is exactly 0.5, then q = y − 0.5.
q = ⌈ y − 0.5 ⌉ = − ⌊ − y + 0.5 ⌋ {\displaystyle q=\left\lceil y-0.5\right\rceil =-\left\lfloor -y+0.5\right\rfloor \,}
For example, 23.5 gets rounded to 23, and −23.5 gets rounded to −24.
The round half down tie-breaking rule is not symmetric, as the fractions that are exactly 0.5 always get rounded down. This asymmetry introduces a negative bias in the roundoff errors. For example, if the fraction of y consists of three random decimal digits, then the expected value of q will be 0.0005 lower than the expected value of y. For this reason, round-to-nearest with the round half down rule is also (ambiguously) known as asymmetric rounding.
Round half away from zero[change | change source]
The other tie-breaking method commonly taught and used is the round half away from zero (or round half towards infinity), namely:
If the fraction of y is exactly 0.5, then q = y + 0.5 if y is positive, and q = y − 0.5 if y is negative.
q = sgn ( y ) ⌊ | y | + 0.5 ⌋ = − sgn ( y ) ⌈ − | y | − 0.5 ⌉ {\displaystyle q=\operatorname {sgn}(y)\left\lfloor \left|y\right|+0.5\right\rfloor =-\operatorname {sgn}(y)\left\lceil -\left|y\right|-0.5\right\rceil \,}
This method treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability. However, this rule will still introduce a positive bias for positive numbers, and a negative bias for the negative ones.
It is often used for currency conversions and price roundings (when the amount is first converted into the smallest significant subdivision of the currency, such as cents of a euro) as it is easy to explain by just considering the first fractional digit, independently of supplementary precision digits or sign of the amount (for strict equivalence between the paying and recipient of the amount).
Round half towards zero[change | change source]
One may also round half towards zero (or round half away from infinity) as opposed to the more common round half away from zero (the round half away from zero method is a common convention, but is nothing more than a convention).
If the fraction of y is exactly 0.5, then q = y − 0.5 if y is positive, and q = y + 0.5 if y is negative.
q = sgn ( y ) ⌈ | y | − 0.5 ⌉ = − sgn ( y ) ⌊ − | y | + 0.5 ⌋ {\displaystyle q=\operatorname {sgn}(y)\left\lceil \left|y\right|-0.5\right\rceil =-\operatorname {sgn}(y)\left\lfloor -\left|y\right|+0.5\right\rfloor \,}
This method also treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability. However, this rule will still introduce a negative bias for positive numbers, and a positive bias for the negative ones.
Round half to even[change | change source]
A tie-breaking rule that is even less biased is round half to even, namely
If the fraction of y is 0.5, then q is the even integer nearest to y.
Thus, for example, +23.5 becomes +24, +22.5 becomes +22, −22.5 becomes −22, and −23.5 becomes −24.
This method also treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability. In addition, for most reasonable distributions of y values, the expected (average) value of the rounded numbers is essentially the same as that of the original numbers, even if the latter are all positive (or all negative). However, this rule will still introduce a positive bias for even numbers (including zero), and a negative bias for the odd ones.
This variant of the round-to-nearest method is also called unbiased rounding (ambiguously, and a bit abusively), convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, or bankers' rounding. This is widely used in bookkeeping.
This is the default rounding mode used in IEEE 754 computing functions and operators.
Round half to odd[change | change source]
Another tie-breaking rule that is very similar to round half to even, namely
If the fraction of y is 0.5, then q is the odd integer nearest to y.
This method also treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability. In addition, for most reasonable distributions of y values, the expected (average) value of the rounded numbers is essentially the same as that of the original numbers, even if the latter are all positive (or all negative). However, this rule will still introduce a negative bias for even numbers (including zero), and a positive bias for the odd ones.
This variant is almost never used in most computations, except in situations where one wants to avoid rounding 0.5 or −0.5 to zero, or to avoid increasing the scale of numbers represented as floating point (with limited ranges for the scaling exponent), so that a non infinite number would round to infinite, or that a small denormal value would round to a normal non-zero value (these could occur with the round half to even mode). Effectively, this mode prefers preserving the existing scale of tie numbers, avoiding out of range results when possible.
Stochastic rounding[change | change source]
Another unbiased tie-breaking method is stochastic rounding:
If the fractional part of y is .5, choose q randomly among y + 0.5 and y − 0.5, with equal probability.
Like round-half-to-even, this rule is essentially free of overall bias; but it is also fair among even and odd q values. On the other hand, it introduces a random component into the result; performing the same computation twice on the same data may yield two different results. Also, it is open to unconscious bias if humans (rather than computers or devices of chance) are "randomly" deciding in which direction to round.
Alternating tie-breaking[change | change source]
One method, more obscure than most, is round half alternatingly.
If the fractional part is 0.5, alternate round up and round down: for the first occurrence of a 0.5 fractional part, round up; for the second occurrence, round down; so on so forth.
This suppresses the random component of the result, if occurrences of 0.5 fractional parts can be effectively numbered. But it can still introduce a positive or negative bias according to the direction of rounding assigned to the first occurrence, if the total number of occurrences is odd.
Simple dithering[change | change source]
In some contexts, all the rounding methods above may be unsatisfactory. For example, suppose that y is an accurate measurement of an audio signal, which is being rounded to an integer q in order to reduce the storage or transmission costs. If y changes slowly with time, any of the rounding method above will result in q being completely constant for long intervals, separated by sudden jumps of ±1. When the q signal is played back, these steps will be heard as a very disagreeable noise, and any variations of the original signal between two integer values will be completely lost.
One way to avoid this problem is to round each value y upwards with probability equal to its fraction, and round it downwards with the complement of that probability. For example, the number 23.17 would be rounded up to 24 with probability 0.17, and down to 23 with probability 1 - 0.17 = 0.83. (This is equivalent to rounding down y + s, where s is a random number uniformly distributed between 0 and 1.) With this special rounding, known as dithering, the sudden steps get replaced by a less objectionable noise, and even small variations in the original signal will be preserved to some extent. Like the stochastic approach to tie-breaking, dithering has no bias: if all fraction values are equally likely, rounding up by a certain amount is as likely as rounding down by that same amount; and the same is true for the sum of several rounded numbers. On the other hand, dithering introduces a random component in the result, much greater than that of stochastic tie-breaking.
More precisely, the roundoff error for each dithered number will be a uniformly distributed random variable with mean value of zero, but with a standard deviation 1 / 12 ≈ 0.2886 {\displaystyle 1/{\sqrt {12}}\approx 0.2886} , which is better than the 1/2 standard deviation with the simple predictive methods, but slightly higher than with the simpler stochastic method. However, the sum of n rounded numbers will be a random variable with expected error zero, but with standard deviation n / 12 {\displaystyle {\sqrt {n}}/{\sqrt {12}}} (the total remaining noise) which diverges semi-quadratically and may become easily perceptible, even if the standard deviation of the roundoff error per sample will be 1 / 12 n {\displaystyle 1/{\sqrt {12n}}} which slowly converges semi-quadratically to zero. So, this random distribution may still be too high for some applications that are rounding a lot of data.
Multidimensional dithering[change | change source]
This variant of the simple dithering method still rounds values with probability equal to its fraction. However, instead of using a random distribution for rounding isolated samples, the roundoff error occurring at each rounded sample is totalled for the next surrounding elements to sample or compute; this accumulated value is then added to the value of these next sampled or computed values to round, so that the modified values will take into account this difference using a predictive model (such as Floyd–Steinberg dithering).
The modified values are then rounded with any one of the above rounding methods, the best ones being with stochastic or dithering methods: in this last case, the sum of n rounded numbers will still be a random variable with expected error zero but with an excellent constant standard deviation of 1 / 12 {\displaystyle 1/{\sqrt {12}}} , instead of diverging semi-quadratically when dithering isolated samples; and the overall average roundoff error deviation per rounded sample will be 1 / ( n 12 ) {\displaystyle 1/(n{\sqrt {12}})} that will converge hyperbolically to zero, faster than with the semi-hyperbolic convergence when dithering isolated samples.
In practice, when rounding large sets of sampled data (such as audio, image and video rendering), the accumulation of roundoff errors is most frequently used with a simple predictive rounding of the modified values (such as rounding towards zero), because it will still preserve the hyperbolic convergence towards zero of the overall mean roundoff error bias and of its standard deviation. This enhancement is frequently used in image and audio processing (notably for accurate rescaling and antialiasing operations, where the simple probabilistic dithering of isolated values may still produce perceptible noise, sometimes even worse than the moiré effects occurring with simple non-probabilistic rounding methods applied to isolated samples).
The effective propagation of accumulated roundoff errors may depend on the discrete dimension of the sampled data to round: when sampling bidimensional images, including colored images (that add the discrete dimension of color planes), or tridimensional videos (that add a discrete time dimension), or on polyphonic audio data (using time and channel discrete dimensions), it may still be preferable to propagate this error into a preferred direction, or equally into several orthogonal dimensions, such as vertically vs. horizontally for bidimensional images, or into parallel color channels at the same position and/or timestamp, and depending on other properties of these orthogonal discrete dimensions (according to a perception model). In those cases, several roundoff error accumulators may be used (at least one for each discrete dimension), or a (n-1)-dimension vector (or matrix) of accumulators.
In some of these cases, the discrete dimensions of the data to sample and round may be treated non orthogonally: for example, when working with colored images, the trichromatic color planes data in each physical dimension (height, width and optionally time) could be remapped using a perceptive color model, so that the roundoff error accumulators will be designed to preserve lightness with a higher probability than hue or saturation, instead of propagating errors into each orthogonal color plane independently; and in stereophonic audio data the two rounded data channels (left and right) may be rounded together to preserve their mean value in priority to their effective difference that will absorb most of the remaining roundoff errors, in a balanced way around zero.
Rounding to simple fractions[change | change source]
In some contexts it is desirable to round a given number x to a "neat" fraction — that is, the nearest fraction z = m/n whose numerator m and denominator n do not exceed a given maximum. This problem is fairly distinct from that of rounding a value to a fixed number of decimal or binary digits, or to a multiple of a given unit m. This problem is related to Farey sequences, the Stern-Brocot tree, and continued fractions.
Scaled rounding[change | change source]
This type of rounding, which is also named rounding to a logarithmic scale, is a variant of Rounding to a specified increment but with an increment that is modified depending on the scale and magnitude of the result. Concretely, the intent is to limit the number of significant digits, rounding the value so that non-significant digits will be dropped. This type of rounding occurs implicitly with numbers computed with floating-point values with limited precision (such as IEEE-754 float and double types), but it may be used more generally to round any real values with any positive number of significant digits and any strictly positive real base.
For example it can be used in engineering graphics for representing data with a logarithmic scale with variable steps (for example wave lengths, whose base is not necessarily an integer measure), or in statistical data to define classes of real values within intervals of exponentially growing widths (but the most common use is with integer bases such as 10 or 2).[source?]
This type of rounding is based on a logarithmic scale defined by a fixed non-zero real scaling factor s (in most frequent cases this factor is s=1) and a fixed positive base b>1 (not necessarily an integer and most often different from the scaling factor), and a fixed integer number n>0 of significant digits in that base (which will determine the value of the increment to use for rounding, along with the computed effective scale of the rounded number).
The primary argument number (as well as the resulting rounded number) is first represented in exponential notation x = s·a·m·bc, such that the sign s is either +1 or −1, the absolute mantissa a is restricted to the half-open positive interval [1/b,1), and the exponent c is any (positive or negative) integer. In that representation, all significant digits are in the fractional part of the absolute mantissa whose integer part is always zero.
If the source number (or rounded number) is 0, the absolute mantisssa a is defined as 0, the exponent c is fixed to an arbitrary value (0 in most conventions, but some floating-point representations cannot use a null absolute mantissa but reserve a specific maximum negative value for the exponent c to represent the number 0 itself), and the sign s may be arbitrarily chosen between −1 or +1 (it is generally set to +1 for simple zero, or it is set to the same sign as the argument in the rounded value if the number representation allows to differentiate positive and negative zeroes, even if they finally represent the same numeric value 0).
A scaled exponential representation as x = a·s·bc may also be used equivalently, with a signed mantissa a either equal to zero or within one of the two half-open intervals (−1,−1/b] and [+1/b,+1), and this will be the case in the algorithm below.
The steps to compute this scaled rounding are generally similar to the following:
if x equals zero, simply return x; otherwise:
convert x into the scaled exponential representation, with a signed mantissa:
x = a ⋅ s ⋅ b c {\displaystyle x=a\cdot s\cdot b^{c}\,}
let x' be the unscaled value of x, by dividing it by the scaling factor s:
x ′ = x / s {\displaystyle x'=x/s\,} ;
let the scaling exponent c be one plus the base-b logarithm of the absolute value of x', rounded down to an integer (towards minus infinity):
c = 1 + ⌊ log b | x ′ | ⌋ = 1 + ⌊ log b | x / s | ⌋ {\displaystyle c=1+\left\lfloor \log _{b}\left|x'\right|\right\rfloor =1+\left\lfloor \log _{b}\left|x/s\right|\right\rfloor \,} ;
let the signed mantissa a be product of x' divided by b to the power c:
a = x ′ ⋅ b − c = x / s ⋅ b − c {\displaystyle a=x'\cdot b^{-c}=x/s\cdot b^{-c}\,}
compute the rounded value in this representation:
let c' be the initial scaling exponent c of x':
c ′ = c {\displaystyle c'=c\,}
let m be the increment for rounding the mantissa a according to the number of significant digits to keep:
m = b − n {\displaystyle m=b^{-n}\,}
let a' be the signed mantissa a rounded according to this increment m and the selected rounding mode:
a ′ = r o u n d ( a , m ) = r o u n d ( x / s ⋅ b n − c ′ ) ⋅ b − n {\displaystyle a'=\mathrm {round} (a,m)=\mathrm {round} (x/s\cdot b^{n-c'})\cdot b^{-n}\,}
if the absolute value of a' is not lower than b, then decrement n (multiply the increment m by b), increment the scaling exponent c', divide the signed mantissa a by b, and restart the rounding of the new signed mantissa a into a' with the same formula; this step may be avoided only if the abtract "round()" function is always rounding a towards 0 (i.e. when it is a simple truncation), but is necessary if it may be rounding a towards infinity, because the rounded mantissa may have a higher scaling exponent in this case, leaving an extra digit of precision.
return the rounded value:
y = s c a l e d r o u n d ( x , s , b , n ) = a ′ ⋅ s ⋅ b c ′ = r o u n d ( x / s ⋅ b n − c ′ ) ⋅ s ⋅ b c ′ − n {\displaystyle y=\mathrm {scaledround} (x,s,b,n)=a'\cdot s\cdot b^{c'}=\mathrm {round} (x/s\cdot b^{n-c'})\cdot s\cdot b^{c'-n}\,} .
For the abstract "round()" function, this type of rounding can use any one of the rounding to integer modes described more completely in the next section, but it is most frequently the round to nearest mode (with tie-breaking rules also described more completely below).
the scaled rounding of 1.234 with scaling factor 1 in base 10 and 3 significant digits (maximum relative precision=1/1000), when using any round to nearest mode, will return 1.23;
similar scaled rounding of 1.236 will return 1.24;
similar scaled rounding of 21.236 will return 21.2;
similar scaled rounding of 321.236 will return 321;
the scaled rounding of 1.234 scaling factor 1 in base 10 and 3 significant digits (maximum relative precision=1/1000), when using the round down mode, will return 1.23;
similar scaled rounding of 1.236 will also return 1.23;
the scaled rounding of 3 π / 7 ≈ 6.8571 ⋅ π ⋅ 2 − 4 {\displaystyle \scriptstyle 3\pi /7\;\approx \;6.8571\cdot \pi \cdot 2^{-4}} with scaling factor π {\displaystyle \scriptstyle \pi } in base 2 and 3 significant digits (maximum relative precision=1/8), when using the round down mode, will return 6 ⋅ π ⋅ 2 − 4 = 3 π / 8 {\displaystyle \scriptstyle 6\cdot \pi \cdot 2^{-4}\;=\;3\pi /8} ;
similar scaled rounding of 5 π / 7 ≈ 5.7143 ⋅ π ⋅ 2 − 3 {\displaystyle \scriptstyle 5\pi /7\;\approx \;5.7143\cdot \pi \cdot 2^{-3}} will return 5 ⋅ π ⋅ 2 − 3 = 5 π / 8 {\displaystyle \scriptstyle 5\cdot \pi \cdot 2^{-3}\;=\;5\pi /8} ;
similar scaled rounding of π / 7 ≈ 4.5714 ⋅ π ⋅ 2 − 5 {\displaystyle \scriptstyle \pi /7\;\approx \;4.5714\cdot \pi \cdot 2^{-5}} will return 4 ⋅ π ⋅ 2 − 5 = π / 8 {\displaystyle \scriptstyle 4\cdot \pi \cdot 2^{-5}\;=\;\pi /8} .
similar scaled rounding of π / 8 = 4 ⋅ π ⋅ 2 − 5 {\displaystyle \scriptstyle \pi /8\;=\;4\cdot \pi \cdot 2^{-5}} will also return 4 ⋅ π ⋅ 2 − 5 = π / 8 {\displaystyle \scriptstyle 4\cdot \pi \cdot 2^{-5}\;=\;\pi /8} .
similar scaled rounding of π / 15 ≈ 4.2667 ⋅ π ⋅ 2 − 6 {\displaystyle \scriptstyle \pi /15\;\approx \;4.2667\cdot \pi \cdot 2^{-6}} will return 4 ⋅ π ⋅ 2 − 6 = π / 16 {\displaystyle \scriptstyle 4\cdot \pi \cdot 2^{-6}\;=\;\pi /16} .
Round to available value[change | change source]
Finished lumber, writing paper, capacitors, and many other products are usually sold in only a few standard sizes.
Many design procedures describe how to calculate an approximate value, and then "round" to some standard size using phrases such as "round down to nearest standard value", "round up to nearest standard value", or "round to nearest standard value".[2][3][4]
When a set of preferred values is equally spaced on a logarithmic scale, Choosing the closest preferred value to any given value can be seen as a kind of scaled rounding. Such "rounded" values can be directly calculated.[5]
Floating-point rounding[change | change source]
In floating-point arithmetic, rounding aims to turn a given value x into a value z with a specified number of significant digits. In other words, z should be a multiple of a number m that depends on the magnitude of z. The number m is a power of the base (usually 2 or 10) of the floating-point form.
Apart from this detail, all the variants of rounding discussed above apply to the rounding of floating-point numbers as well. The algorithm for such rounding is presented in the Scaled rounding section above, but with a constant scaling factor s=1, and an integer base b>1.
For results where the rounded result would overflow the result for a directed rounding is either the appropriate signed infinity, or the highest representable positive finite number (or the lowest representable negative finite number if x is negative), depending on the direction of rounding. The result of an overflow for the usual case of round to even is always the appropriate infinity.
In addition, if the rounded result would underflow, i.e. if the exponent would exceed the lowest representable integer value, the effective result may be either zero (possibly signed if the representation can maintain a distinction of signs for zeroes), or the smallest representable positive finite number (or the highest representable negative finite number if x is negative), possibly a denormal positive or negative number (if the mantissa is storing all its significant digits, in which case the most significant digit may still be stored in a lower position by setting the highest stored digits to zero, and this stored mantissa does not drop the most significant digit, something that is possible when base b=2 because the most significant digit is always 1 in that base), depending on the direction of rounding. The result of an underflow for the usual case of round to even is always the appropriate zero.
Double rounding[change | change source]
Rounding a number twice in succession to different precisions, with the latter precision being coarser, is not guaranteed to give the same result as rounding once to the final precision except in the case of directed rounding. For instance rounding 9.46 to one decimal gives 9.5, and then 10 when rounding to integer using rounding half to even, but would give 9 when rounded to integer directly.
Some computer languages and the IEEE 754-2008 standard dictate that in straightforward calculations, the result should not be rounded twice. This has been a particular problem with Java as it is designed to be run identically on different machines, special programming tricks have had to be used to achieve this with x87 floating point.[6][7] The Java language was changed to allow different results where the difference does not matter and require a "strictfp" qualifier to be used when the results have to conform accurately.
Exact computation with rounded arithmetic[change | change source]
It is possible to use rounded arithmetic to evaluate the exact value of a function with a discrete domain and range. For example, if we know that an integer n is a perfect square, we can compute its square root by converting n to a floating-point value x, computing the approximate square root y of x with floating point, and then rounding y to the nearest integer q. If n is not too big, the floating-point roundoff error in y will be less than 0.5, so the rounded value q will be the exact square root of n. In most modern computers, this method may be much faster than computing the square root of n by an all-integer algorithm.
The table-maker's dilemma[change | change source]
William Kahan coined the term "The Table-Maker's Dilemma" for the unknown cost of rounding transcendental functions:
"Nobody knows how much it would cost to compute y^w correctly rounded for every two floating-point arguments at which it does not over/underflow. Instead, reputable math libraries compute elementary transcendental functions mostly within slightly more than half an ulp and almost always well within one ulp. Why can't Y^W be rounded within half an ulp like SQRT? Because nobody knows how much computation it would cost... No general way exists to predict how many extra digits will have to be carried to compute a transcendental expression and round it correctly to some preassigned number of digits. Even the fact (if true) that a finite number of extra digits will ultimately suffice may be a deep theorem."[8]
The IEEE floating point standard guarantees that add, subtract, multiply, divide, square root, and floating point remainder will give the correctly rounded result of the infinite precision operation. However, no such guarantee is given for more complex functions and they are typically only accurate to within the last bit at best.
Using the Gelfond–Schneider theorem and Lindemann–Weierstrass theorem, many of the standard elementary functions can be proved to return transcendental results when given rational non-zero arguments; therefore it is always possible to correctly round such functions. However determining a limit for a given precision on how accurate results needs to be computed before a correctly rounded result can be guaranteed may demand a lot of computation time.[9]
There are some packages around now that offer full accuracy. The MPFR package gives correctly rounded arbitrary precision results. IBM has written a package for fast and accurate IEEE elementary functions and in the future the standard libraries may offer such precision.[10]
It is possible to devise well-defined computable numbers which it may never be possible to correctly round no matter how many digits are calculated. For instance, if Goldbach's conjecture is true but unprovable, then it is impossible to correctly round down 0.5 + 10-n where n is the first even number greater than 4 which is not the sum of two primes, or 0.5 if there is no such number. This can however be approximated to any given precision even if the conjecture is unprovable.
The concept of rounding is very old, perhaps older even than the concept of division. Some ancient clay tablets found in Mesopotamia contain tables with rounded values of reciprocals and square roots in base 60.[11] Rounded approximations to π, the length of the year, and the length of the month are also ancient.
The Round-to-even method has served as the ASTM (E-29) standard since 1940. The origin of the terms unbiased rounding and statistician's rounding are fairly self-explanatory. In the 1906 4th edition of Probability and Theory of Errors [12] Robert Simpson Woodward called this "the computer's rule" indicating that it was then in common use by human computers who calculated mathematical tables. Churchill Eisenhart's 1947 paper "Effects of Rounding or Grouping Data" (in Selected Techniques of Statistical Analysis, McGrawHill, 1947, Eisenhart, Hastay, and Wallis, editors) indicated that the practice was already "well established" in data analysis.
The origin of the term "bankers' rounding" remains more obscure. If this rounding method was ever a standard in banking, the evidence has proved extremely difficult to find. To the contrary, section 2 of the European Commission report The Introduction of the Euro and the Rounding of Currency Amounts [13] suggests that there had previously been no standard approach to rounding in banking; and it specifies that "half-way" amounts should be rounded up.
Until the 1980s, the rounding method used in floating-point computer arithmetic was usually fixed by the hardware, poorly documented, inconsistent, and different for each brand and model of computer. This situation changed after the IEEE 754 floating point standard was adopted by most computer manufacturers. The standard allows the user to choose among several rounding modes, and in each case, specifies precisely how the results should be rounded. These features made numerical computations more predictable and machine-independent, and made possible the efficient and consistent implementation of interval arithmetic.
Rounding functions in programming languages[change | change source]
Most programming languages provide functions or special syntax to round fractional numbers in various ways. The earliest numeric languages, such as FORTRAN and C, would provide only one method, usually truncation (towards zero). This default method could be implied in certain contexts, such as when assigning a fractional number to an integer variable, or using a fractional number as an index of an array. Other kinds of rounding had to be programmed explicitly; for example, rounding a positive number to the nearest integer could be implemented by adding 0.5 and truncating.
In the last decades, however, the syntax and/or the standard libraries of most languages have commonly provided at least the four basic rounding functions (up/ceiling, down/floor, to nearest, and towards zero). The tie-breaking method may vary depending the language and version, and/or may be selectable by the programmer. Several languages follow the lead of the IEEE-754 floating-point standard, and define these functions as taking a double precision float argument and returning the result of the same type, which then may be converted to an integer if necessary. Since the IEEE double precision format has 52 fraction bits, this approach may avoid spurious overflows in languages have 32-bit integers. Some languages, such as PHP, provide functions that round a value to a specified number of decimal digits, e.g. from 4321.5678 to 4321.57 or 4300. In addition, many languages provide a "printf" or similar string formatting function, which allows one to convert a fractional number to a string, rounded to a user-specified number of decimal places (the precision). On the other hand, truncation (round to zero) is still the default rounding method used by many languages, especially for the division of two integer values.
On the opposite, CSS and SVG do not define any specific maximum precision for numbers and measurements, that are treated and exposed in their Document Object Model and in their Interface-description-language interface as strings as if they had infinite precision, and do not discriminate between integers and floating point values; however, the implementations of these languages will typically convert these numbers into IEEE-754 double floating points before exposing the computed digits with a limited precision (notably within standard Javascript or ECMAScript[14] interface bindings).
Other rounding standards[change | change source]
Some disciplines or institutions have issued standards or directives for rounding.
U.S. Weather Observations[change | change source]
In a guideline issued in mid-1966,[15] the U.S. Office of the Federal Coordinator for Meteorology determined that weather data should be rounded to the nearest round number, with the "round half up" tie-breaking rule. For example, 1.5 rounded to integer should become 2, and −1.5 should become −1. Prior to that date, the tie-breaking rule was "round half away from zero".
Negative zero in meteorology[change | change source]
Some meteorologists may write "-0" to indicate a temperature between 0.0 and -0.5 degrees (exclusive) that was rounded to integer. This notation is used when the negative sign is considered important, no matter how small is the magnitude; for example, when rounding temperatures in the Celsius scale, where below zero indicates freezing.[source?]
Arithmetic precision
An introduction to different rounding algorithms that is accessible to a general audience but especially useful to those studying computer science and electronics.
How To Implement Custom Rounding Procedures by Microsoft
↑ Nicholas J. Higham (2002). Accuracy and stability of numerical algorithms. p. 54. ISBN 978-0898715217.
↑ "Zener Diode Voltage Regulators"
↑ Electronics 2000 FAQ
↑ "Build a Mirror Tester"
↑ Bruce Trump, Christine Schneider. "Excel Formula Calculates Standard 1%-Resistor Values". Electronic Design, January 21, 2002, web: [1]
↑ Samuel A. Figueroa (July 1995). "When is double rounding innocuous?". ACM SIGNUM Newsletter (ACM) 30 (3): pp. 21–25. http://portal.acm.org/citation.cfm?id=221332.221334.
↑ Roger Golliver (October 1998). "Efficiently producing default orthogonal IEEE double results using extended IEEE hardware" (PDF). Intel.
↑ Kahan, William. "A Logarithm Too Clever by Half". Retrieved 2008-11-14.
↑ Handbook of Floating-Point Arithmetic, J.-M. Muller et al., Chapter 12 Solving the Table Maker's Dilemma, 2011.
↑ "An accurate elementary mathematical library for the IEEE floating point standard".
↑ Duncan J. Melville. "YBC 7289 clay tablet". 2006
↑ http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05170001&view=50&frames=0&seq=48
↑ http://ec.europa.eu/economy_finance/publications/publication1224_en.pdf
↑ "ECMA-262 ECMAScript Language Specification" (PDF).
↑ OFCM, 2005: Federal Meteorological Handbook No. 1, Washington, DC., 104 pp.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Rounding&oldid=6440719" | CommonCrawl |
\begin{document}
\title{Optimal relocation strategies for spatially mobile
consumers} \begin{abstract} We develop a model of the behaviour of a dynamically optimizing economic agent who makes consumption-saving and spatial relocation decisions. We formulate an existence result for the model, derive the necessary conditions for optimality and study the behaviour of the economic agent, focusing on the case of a wage distribution with a single maximum. \\ {} \\ \textbf{Keywords:} consumption decisions, spatial relocation, optimal control \\ \textbf{2000 Mathematics Subject Classification:} 91B42, 91B72, 49J15, 49K15 \end{abstract} \section{Introduction}\label{sec:intro}
The emergence of the literature on ``new economic geography'' in the 1990s has rekindled the interest in the spatial aspects of economics. The new generation of models makes heavy use of the standard economics toolkit and analyzes a number of issues from a dynamic perspective or from the perspective of optimizing agents. Interestingly, however, spatial models adopting the perspective of \textit{dynamically optimizing} consumers remain in relative minority, despite the fact that they are standard fare in mainstream economic research. The models in \cite{Bal01}, \cite{Bou06} and \cite{Bri04} are notable exceptions in this respect.
The present work develops a model that studies the behaviour of a dynamically optimizing economic agent who makes two types of interrelated choices: consumption-saving decisions and spatial relocation (migration) decisions. Unlike the constructs in \cite{Bal01}, \cite{Bou06} and \cite{Bri04}, the consumer in our model has a finite lifetime and a bequest incentive at the end of his life. This departure from classical Ramsey-type models enables richer global dynamics by allowing agents to inherit their ancestors' savings in a setup akin to that of overlapping generations models. It also offers the additional option of introducing heterogeneous agents whose economy-wide behaviour can be obtained through an explicit aggregation rule.
A second important difference with the above papers is that our consumer saves in nominal assets. This partly depends on our choice to center the model around the behaviour of (potentially different) individuals as opposed to that of a representative agent. More importantly, however, the nominal savings feature reflects our belief that pecuniary considerations play an important role in the choice of where to work and how much to consume.
Formally, we cast the model in the form of a continuous-time optimal control problem with a finite planning horizon. The assumptions of the model, while fairly standard in economics, create several mathematical challenges in the present setup. First, they preclude the direct application of the existence theorems for optimal control problems known to the authors. This requires an alternative approach to proving the existence of solutions of the model. In particular, unlike traditional existence proofs in the spirit of Theorem 4, \S 4.2 in \cite{Lee69}, we prove an existence result that dispenses with convexity assumptions on the set of generalized speeds for the optimal control problem. Also, the functional forms employed in the model do not allow one to directly apply Pontryagin's maximum principle, since the transversality condition for one of the state variables is not defined at the point $0$. To be able to use the maximum principle, we prove that for an optimal control-trajectory pair the terminal value of the particular state variable is strictly positive. Finally, economic considerations point to the fact that only a subset of the possible values for the other state variable in the model are of real interest. One way to take care of that issue is to constrain the values of this state variable to lie in a certain set -- the interval $[0,1]$ in our case -- for each point in time. However, instead of using an explicit state constraint, which would complicate the use of the maximum principle, we introduce an additional correcting mechanism by suitably defining the wage distribution function $w(x)$ outside the interval $[0,1]$. We claim this mechanism does not influence the other characteristics of the model, while being sufficient to ensure that the optimal state variable never leaves the set in question, and we prove that indeed this is the case.
The rest of the paper is organized as follows. Section \ref{sec:model} introduces the model and the assumptions we make. Section \ref{sec:exist} proves the existence of a solution to the model under the above assumptions. Section \ref{sec:NC} applies Pontryagin's maximum principle to obtain necessary conditions for optimality. Section \ref{sec:NCtransf} describes some convenient transformations of the system of necessary conditions and comments on the existence of solutions to this system. The analysis in section \ref{sec:termasset} characterizes the asymptotic behaviour of terminal assets for different sets of model parameters. This establishes facts that are useful for the study of relocation choices in section \ref{sec:reloc}. The results in this section are also of independent economic interest as they shed light on the impact of intra- and intertemporal preferences on the saving decisions of an individual with a sufficiently long planning horizon. Finally, section \ref{sec:reloc} tackles the question of relocation behaviour in the basic case of a wage distribution having a single maximum (single-peaked wage distribution). The results obtained for this case are intuitive, if unsurprising: in most cases a consumer with a sufficiently long lifespan relocates toward the wage maximum. While the single-peak case offers easily predictable results, we consider it useful as a testing ground for the model before applying it to more interesting situations. Indeed, preliminary results by the authors on the case of a double-peaked wage distribution suggest that a host of complex situations, including multiple solutions and bifurcations, can arise. \\ \\ \textbf{Acknowledgements.} We would like to thank Tsvetomir Tsachev and Vladimir Veliov for useful discussions and pointers to the literature. The responsibility for any errors is solely ours.
\section{The model}\label{sec:model}
We employ a continuous-time model that deals with the case of a consumer who, given an initial location in space $x_0$ and asset level $a_0$, supplies inelastically a unit of labour in exchange for a location-dependent wage $w(x(t))$, and chooses consumption $c(t)$ and spatial location $x(t)$ over time. The consumer has a finite lifetime $T$ at the end of which a bequest in the form of assets is left. This bequest provides utility to the consumer. More precisely, for $\rho$, $r$, $\eta$, $\xi$ and $p$ -- positive constants, and $\theta \in (0,1)$, we look at the optimal control problem \begin{equation} \max_{c(t),z(t) \in \Delta} J(c(t),z(t)) := \int_{0}^{T}e^{-\rho t}\left( \frac{c(t)^{1-\theta}}{1-\theta}-\eta z^2(t) \right)dt+e^{-\rho T}\frac{a(T)^{1-\theta}}{1-\theta} \label{eq:obj} \end{equation} subject to \begin{equation} \dot{a}(t)=ra(t)+w(x(t))-p c(t) -\xi z^2(t), \label{eq:assets} \end{equation} \begin{equation} \dot{x}(t)=z(t),\label{eq:location}\end{equation} $$a(0)=a_0\geq 0,$$ $$x(0)=x_0\in [0,1],$$ where $a(t)$, $x(t)$ are the state variables, assumed to be absolutely continuous, and $c(t)$, $z(t)$ are the control variables. The set of admissible controls $\Delta$ consists of all pairs of functions $(c(t),z(t))$ which are measurable in $[0,T]$ and satisfy the conditions \begin{equation} 0 \leq c(t) \leq C,
\label{eq:cconstr}\end{equation} \begin{equation} |z(t)|\leq Z, \label{eq:zconstr} \end{equation} \begin{equation} a(T)\geq 0. \label{eq:aTnonneg}\end{equation} The constants $C$ and $Z$ are such that \begin{equation} C^{\theta} > \max\left(1,\mu^{\frac{1}{\theta}}\right) \left( \frac{a_0 +
T\max_x |w(x)|}{p\frac{1-e^{-rT}}{r}}\right) , \label{eq:Ccapbound} \end{equation} \begin{equation} Z
> \frac{T \max_x |w'(x)| e^{rT}}{2\xi},\label{eq:Zcapbound}\end{equation} where $\mu := \max_{t,t_0 \in [0,T]}e^{(r-\rho)(t-t_0)}>0$.
\textbf{Remark.} The bounds we impose on the admissible controls through equations \eqref{eq:cconstr} and \eqref{eq:zconstr} are convenient from a technical viewpoint when proving the existence theorem in section \ref{sec:exist}. Conditions \eqref{eq:Ccapbound} and \eqref{eq:Zcapbound} ensure that these constraints are never binding. However, considerations of general nature -- both economic and physical -- make such constraints appealing.
In the above model $\rho>0$ is a time discount parameter and $\theta \in (0,1) $ is the utility function parameter. The control $c(t)$ represents physical units of consumption and the control $z(t)$ governs the speed of relocation in space. We assume that relocation in space brings about two type of consequences. First, relocation causes subjective disutility associated with the fact that there is habit formation with respect to the place one occupies. Second, changing one's location is associated with monetary relocation costs that have to be paid out of one's income or stock of assets. As a baseline case we choose to capture these phenomena by means of the speed of movement in space $\dot{x}(t)$ or, equivalently, $z(t)$, transformed through a quadratic function. The manner in which spatial relocation affects the consumer's utility and wealth can vary widely, however, therefore other functional forms are certainly admissible. The parameters $\eta,\xi \geq 0$ multiplying this function measure the subjective disutility from changing one's location in space and the relocation costs in monetary terms, respectively. The parameters $p>0$ and $r>0$ stand for the price of a unit of consumption and the interest rate, respectively.
The nonnegativity condition is imposed on terminal assets $a(T)$ both to have a well-defined objective functional and to capture the intuitive observation that, with a known lifetime, a debtor is unlikely to be allowed to leave behind outstanding liabilities to creditors. The condition $a(T)\geq 0$ also sheds light on the nonnegativity restriction for $a_0$, since in an environment where no debts are allowed at the end of one's lifetime, no debtor position can be inherited at birth.
For the purposes of our analysis we look at the basic case where economic space is represented by the real line. We are interested in only a subset of it, the interval $[0,1]$. This is modelled by taking the initial location $x_0 \in [0,1]$ and requiring the location-dependent wage, which is positive in $(0,1)$, to be negative outside $[0,1]$ and to satisfy additional assumptions. Namely, we have $w(x)> 0,~x \in (0,1)$ and $w(x)<0,~x \not\in [0,1]$, as well as $w'(x)>0$, $x \in (-\infty,0]$ and $w'(x)<0$,
$x\in [1,\infty)$. Later in the paper we formally verify the intuitive claim that an optimal trajectory for $x(t)$ will never leave the interval $[0,1]$ under the above conditions. We also assume that $w(x) \in C^2(\mathbb{R}^1)$ and $w(x)$ is bounded, i.e. $\max_{x \in \mathbb{R}}|w(x)|<+\infty$. We impose additional requirements on $w(x)$ to derive some of the results in section \ref{sec:reloc}.
\section{Existence of solutions}\label{sec:exist}
Next we investigate the issue of existence of a solution to the model. The proof requires two intermediate results, shown as lemmas below.
\begin{lemma} Let the functions $x_i,~i=1,2,\ldots,$ and $\bar{x}$ be defined on $[0,T]$ and take values in the interval $[a,b]$. Let $x_i$ tend uniformly to $\bar{x}$ as $i \rightarrow \infty$ (denoted by $x_i \rightrightarrows \bar{x}$) and $w \in C^0[a,b]$. Then, in $[0,T]$, as $m\rightarrow \infty$ we have \begin{enumerate}
\item[i)] $\frac{1}{m}\sum_{i=1}^m x_i \rightrightarrows
\bar{x}$,
\item[ii)] $w(x_m)\rightrightarrows w(\bar{x})$,
\item[iii)] $w(\frac{1}{m}\sum_{i=1}^m x_i)\rightrightarrows
w(\bar{x})$. \end{enumerate} \label{thm:lem1}\end{lemma}
{\bf Proof. } The proof directly replicates the standard proofs of counterpart results on numerical sequences. \endprf
\begin{lemma}[The Banach-Saks Theorem] Let $\{ v_n \}_{n=1}^{\infty}$ be a sequence of elements in a Hilbert space $H$ which are bounded in norm: $\| v_n \|\leq K = const,~\forall n \in \mathbb{N}$. Then, there exist a subsequence $\{ v_{n_k} \}_{k=1}^{\infty}$ and an element $v\in H$ such that $$\left\| \frac{v_{n_1}+\cdots+v_{n_s}}{s}-v \right\|\rightarrow 0 \textrm{ as } s\rightarrow \infty .$$ \label{thm:lem2}\end{lemma} {\bf Proof. } See, for example, \cite[pp.78-81]{Die75}. \endprf
\begin{theorem} Under the assumptions stated in section \ref{sec:model}, there exists a solution $(c(t),z(t))\in \Delta$ of problem \eqref{eq:obj}-\eqref{eq:location}. \label{thm:exist}\end{theorem} {\bf Proof. } We start by noting that the set of admissible controls $\Delta$ is nonempty. To see this, choose controls $c(t)\equiv c_0=const$ and $z(t)\equiv 0$. Then, any $c_0\in (0, w(x_0)/p]$ will ensure that $a(T)\geq 0$.
Next, observe that $(c(t),z(t))\in \Delta$ implies $c(t),z(t)\in L_\infty[0,T]$ and \begin{equation} 0 \leq a(T) \leq const = e^{rT}(a_0+T
\max_{x \in \mathbb{R}}|w(x)|). \label{eq:aTbounded}\end{equation} (We note that \eqref{eq:aTnonneg} implies the following bounds, $$ p \|
c(t) \|_{L_1[0,T]},\xi \| z(t) \|^2_{L_2[0,T]} \leq e^{rT}\left(
a_0+T \max_x |w(x)| \right) , $$which do not depend on the constants $C$ and $Z$.)
Through an application of H\"{o}lder's inequality one verifies that $\int_0^T c(t)^{1-\theta}e^{-\rho t}dt \leq const(T) \|
c(t)\|^{1-\theta}_{L_1} $.
Thus, for $(c(t),z(t))\in \Delta$, the objective functional \eqref{eq:obj} is bounded. Consequently, $J_0:=\sup_{(c(t),z(t))\in \Delta}J(c(t),z(t))<\infty$. Then we can choose a sequence of controls $\{ (c_k(t),z_k(t)) \}\subset \Delta$ such that $J(c_k(t),z_k(t))\rightarrow J_0$.
Let $a_k(t)$ and $x_k(t)$ be the state variables corresponding to the controls $(c_k(t),z_k(t))$. It is easy to verify that the functions $a_k(t)$ and $x_k(t)$ form a uniformly bounded and equicontinuous set. Then, by the Arzel\`a-Ascoli theorem (see, e.g., \cite{Lee69}, Ch.4), there exists a subsequence $(a_{k_s}(t),x_{k_s}(t))\rightrightarrows (\bar{a}(t),\bar{x}(t))$.
Then, if $c_{k_s}(t)$ and $z_{k_s}(t)$ are the controls corresponding to $(a_{k_s}(t),x_{k_s}(t))$, by Lemma \ref{thm:lem2} we can in turn choose subsequences $c_{k_{s_q}}(t)$ and $z_{k_{s_q}}(t)$ whose arithmetic means tend in $L_2[0,T]$ norm to some elements in $L_2[0,T]$, denoted $\bar{c}(t)$ and $\bar{z}(t)$, respectively. However, we do not claim that $\bar{a}(t)$ and $\bar{x}(t)$ correspond to $\bar{c}(t)$ and $\bar{z}(t)$. For brevity we introduce the notation $c_q(t):=c_{k_{s_q}}(t)$, $z_q(t):=z_{k_{s_q}}(t)$ etc., as well as $\tilde{c}_m(t):=\frac{1}{m}\sum_{q=1}^{m}c_q(t)$ and $\tilde{z}_m(t):=\frac{1}{m}\sum_{q=1}^{m}z_q(t)$.
Then, we have established that: (1) $(a_q(t),x_q(t))\rightrightarrows (\bar{a}(t),\bar{x}(t))$ as $q \rightarrow \infty$ and (2) $\tilde{c}_m(t)\xrightarrow[L_2]{}\bar{c}(t)$, $\tilde{z}_m(t)\xrightarrow[L_2]{}\bar{z}(t)$ as $m\rightarrow \infty$.
Recall that $a_q(t)$ and $x_q(t)$ correspond to $c_q(t)$ and $z_q(t)$ as solutions to the respective differential equations \eqref{eq:assets} and \eqref{eq:location}.
So far, it is not clear whether $\tilde{c}_m(t)$ and $\tilde{z}_m(t)$ are admissible. It is immediately seen that they satisfy \eqref{eq:cconstr} and \eqref{eq:zconstr} but the corresponding $a(T)$ may fail to satisfy \eqref{eq:aTnonneg}. However, we can show that the controls $\bar{c}(t)$ and $\bar{z}(t)$ are admissible.
To prove the last claim, note first that according to \cite[Ch.7, \S 2.5, Prop.4]{Kol76} we can choose a subsequence of $\{\tilde{c}_m(t), \tilde{z}_m(t)\}$ that converges a.e. to $(\bar{c}(t),\bar{z}(t))$ and, after passing to the limit, we obtain that $\bar{c}(t)$ and $\bar{z}(t)$ satisfy \eqref{eq:cconstr} and \eqref{eq:zconstr}.
It remains to show that $\bar{\bar{a}}(T)=e^{rT}\left[a_0+\int_0^T [w(\bar{x}(t))-p \bar{c}(t)-\xi \bar{z}^2(t)]e^{-rt}dt \right]\geq 0,$ where $\bar{\bar{x}}(t)=x_0+\int_0^t \bar{z}(\tau) d\tau$.
Consider \begin{equation} \tilde{a}_m(T)=e^{rT}\left[a_0+\int_0^T [w(\tilde{x}_m(t))-p \tilde{c}_m(t)-\xi\tilde{z}^2_m(t)]e^{-rt}dt \right],\label{eq:a_mT}\end{equation} with $\tilde{x}_m(t)=x_0+\int_0^t \tilde{z}_m(\tau) d\tau=\frac{1}{m}\sum_{q=1}^m \left( x_0+\int_0^t z_q(\tau) d\tau \right)=\frac{1}{m}\sum_{q=1}^m x_q(t)$. Adding and subtracting $\frac{1}{m}\sum_{q=1}^m w(x_q(t))$, and applying Jensen's inequality to the term $\tilde{z}^2_m(t)$, we obtain \begin{equation} \begin{split} \tilde{a}_m(T) \geq & e^{rT}\int_0^T\left[ w(\tilde{x}_m(t))- \frac{1}{m}\sum_{q=1}^m w(x_q(t))\right]e^{-rt}dt + \\ & \frac{1}{m}\sum_{q=1}^m e^{rT}\left[ a_0+\int_0^T [w(x_q(t))-p c_q(t)-\xi z^2_q(t)]e^{-rt}dt \right] \geq \\ & e^{rT}\int_0^T\left[ w(\tilde{x}_m(t))- \frac{1}{m}\sum_{q=1}^m w(x_q(t))\right]e^{-rt}dt \end{split}\label{eq:a_mT1} \end{equation} By Lemma \ref{thm:lem1} both integrands inside the square brackets in the last line of \eqref{eq:a_mT1} tend uniformly to $w(\bar{x}(t))$, so that the integral tends to zero. Thus, if $\lim_{m\rightarrow \infty}\tilde{a}_m(T)$ exists, we have $\lim_{m\rightarrow \infty}\tilde{a}_m(T) \geq 0$.
We proceed to check that $\lim_{{m_j}\rightarrow \infty}\tilde{a}_{m_j}(T)=\bar{\bar{a}}(T)$ for a suitable subsequence $\tilde{a}_{m_j}(T)$. We know that $\frac{1}{m}\sum_{q=1}^m x_q(t)=x_0+\int_0^t\frac{1}{m}\sum_{q=1}^m z_q(\tau)d \tau$. Since $\frac{1}{m}\sum_{q=1}^m x_q(t) \rightrightarrows \bar{x}(t)$ and, additionally, it is easy to verify by applying H\"{o}lder's inequality that $\int_0^t\tilde{z}_m(\tau)d \tau \rightarrow \int_0^t \bar{\bar{z}}(\tau)d\tau$ when $\tilde{z}_m(t)\xrightarrow[L_2]{}\bar{z}(t)$, we obtain $\bar{x}(t)=x_0+\int_0^t \bar{z}(\tau)d\tau =\bar{\bar{x}}(t)$.
As $\tilde{c}_m(t)\xrightarrow[L_2]{}\bar{c}(t)$ and $\tilde{z}_m(t)\xrightarrow[L_2]{}\bar{z}(t)$, there exist a subsequences $\tilde{c}_{m_j}(t)$ and $\tilde{z}_{m_j}(t)$ such that $\tilde{c}_{m_j}(t)\xrightarrow[a.e.]{}\bar{c}(t)$ and $\tilde{z}_{m_j}(t)\xrightarrow[a.e.]{}\bar{z}(t)$. To simplify notation, we refer to the new subsequences as $\tilde{c}_j(t)$ and $\tilde{z}_j(t)$. Since the function $z^2$ is bounded on $[-Z,Z]$, by Lebesgue's dominated convergence theorem $\int_0^T \xi \tilde{z}^2_{j}(t)e^{-rt}dt\rightarrow \int_0^T \xi \bar{z}^2(t)e^{-rt}dt$. It can also be verified that $\int_0^T \tilde{c}_{j}(t)e^{-rt}dt\rightarrow \int_0^T \bar{c}(t)e^{-rt}dt$. Lastly, we know that $\int_0^T w(\tilde{x}_{j}(t))e^{-rt}dt \rightarrow \int_0^T w(\bar{x}(t))e^{-rt}dt$ as $w(\tilde{x}_{j}(t)) \rightrightarrows w(\bar{x}(t))$. Consequently, the limit of \eqref{eq:a_mT} as $m_j \rightarrow \infty$ exists and is equal to $\bar{\bar{a}}(T)$, so that $\bar{\bar{a}}(T) \geq 0$. This shows that $\bar{c}(t)$ and $\bar{z}(t)$ are admissible.
By an application of Lebesgue's dominated convergence theorem to the respective terms in \eqref{eq:obj}, we get $\lim_{j \rightarrow \infty}J(\tilde{c}_j(t),\tilde{z}_j(t))=J(\bar{c}(t),\bar{z}(t))$.
Define $\rho_{m_j}(T) := e^{rT}\int_0^T\left[ w(\tilde{x}_{m_j}(t))- \frac{1}{m_j}\sum_{q=1}^{m_j} w(x_q(t))\right]e^{-rt}dt$. Obviously, $\tilde{\tilde{a}}_{m_j}(T)=\tilde{a}_{m_j}(T)-\rho_{m_j}(T)$ also tends to $\bar{\bar{a}}(T)$ and $\tilde{\tilde{a}}_{m_j}(T) \geq \frac{1}{m_j}\sum_{q=1}^{m_j} a_q(T)$, where $a_q(T)$ corresponds to $(c_q(t),z_q(t))$. Then, indexing by $j$ instead of $m_j$ to simplify notation, we get
$$J_0 \geq J(\bar{c}(t),\bar{z}(t))=\lim_{j \rightarrow \infty}\left\{ \int_0^T \left[\frac{\tilde{c}_j(t)^{1-\theta}}{1-\theta}-\eta \tilde{z}^2_j(t)\right]e^{-\rho t }dt + e^{-\rho T}\frac{\tilde{\tilde{a}}^{1-\theta}_j(T)}{1-\theta} \right\} \geq $$
$$\lim_{j \rightarrow \infty}\left\{\frac{1}{j} \sum_{i=1}^{j}\left[\int_0^T \left[\frac{c^{1-\theta}_i(t)}{1-\theta}-\eta z^2_i(t)\right]e^{-\rho t }dt + e^{-\rho T}\frac{ a^{1-\theta}_i(T)}{1-\theta}\right]\right\}=$$
$$ \lim_{j \rightarrow \infty}\left\{\frac{1}{j} \sum_{i=1}^{j}J(c_i(t),z_i(t)) \right\} = J_0,$$ where the inequality is a consequence of the fact that the functions $\sigma \mapsto \sigma^{1-\theta}$ and $z \mapsto (-z^2)$ are concave and we can apply Jensen's inequality. This shows that the admissible pair $(\bar{c}(t),\bar{z}(t))$ is optimal, as required. \endprf
\section{Necessary conditions for optimality}\label{sec:NC}
In this section we turn to the derivation of a set of necessary conditions for optimality on the basis of Pontryagin's maximum principle. To apply the maximum principle, however, we need to ensure that the terminal utility from assets $e^{-\rho T}a(T)^{1-\theta}/(1-\theta)$ is well-behaved at least for the optimal value of terminal assets. To this end, we prove the following
\begin{theorem} For the optimal controls $(c(t),z(t))$ the terminal value of assets $a(T)$ is strictly positive for any $T>0$. \label{thm:aT>0}\end{theorem} {\bf Proof. } Let us assume that there is a time $T_0>0$ for which $a(T_0)=0$.
\textbf{Step 1.} We first verify that it is impossible to have $c(t) \equiv 0$. Assuming that $c(t) \equiv 0$, together with $a(T_0)=0$, yields the objective functional $$J(0,z(t))=-\eta \int_0^{T_0}z^2(t)e^{-\rho t}dt \leq 0.$$
If one of the following two conditions is valid: \begin{enumerate} \item $a_0>0$ and $x_0 \in [0,1]$; \item $a_0=0$ and $x_0 \in (0,1)$, \end{enumerate} then we can choose the admissible pair $\bar{z}(t)\equiv 0$ (so that $x(t)\equiv x_0$) and $\bar{c}(t)\equiv c_0 = const >0$, where $c_0$ is such that $$a_0+\int_0^{T_0}[w(x_0)-p c_0]e^{-rt}dt=0.$$ The last condition is equivalent to $$a_0+T_0 w(x_0)\frac{e^{-r T_0}-1}{-r}=p c_0 \frac{e^{-r T_0}-1}{-r}$$ and therefore $c_0>0$. Then $$J(\bar{c}(t),\bar{z}(t))=\int_0^{T_0}\frac{c_0^{1-\theta}}{1-\theta}e^{-\rho t}dt >0,$$ contradicting the optimality of $(c(t),z(t))$.
The case $a_0=0$ and $x_0=0$ or $1$ is pathological in the sense that the consumer has neither current income (w(0)=w(1)=0), nor initial wealth. Economically, it is implausible to expect that such a consumer will manage to obtain a loan. From a purely formal point of view, however, the consumer could get a loan and finance his relocation even in this case. Moreover, he will be able to attain positive consumption levels.
To illustrate the above claim, suppose that $x_0=0$, $a_0=0$ and the consumer spends all the income left after paying the relocation costs. Fix $\varepsilon_0 > 0$ in such a way that $w'(x) \geq w'(0)/2 >0$ for $x \in [0,\varepsilon_0]$. Let the relocation strategy be given by the control $\bar{z}(t)=\varepsilon \sin \frac{\pi}{T}t,~\varepsilon>0$. Then consumption is given by $\bar{c}(t)=w(\bar{x}(t))-\xi \bar{z}^2(t)$, where $\bar{x}(t)$ is $$\bar{x}(t)=\varepsilon \int_0^t \sin \left( \frac{\pi}{T}\tau \right)d\tau=\frac{\varepsilon T}{\pi}\left( 1-\cos \frac{\pi t}{T} \right)=\frac{2 \varepsilon T}{\pi}\sin^2 \frac{\pi t}{2 T}. $$ Then, $\bar{x}(T)=\frac{2T \varepsilon}{\pi}<\varepsilon_0$ for $\varepsilon$ sufficiently small. Notice that $$w(\bar{x}(t)) =w(\bar{x}(t))-w(0)=w'(x^*(t))\bar{x}(t)\geq \frac{w'(0)}{2}\bar{x}(t),$$ for some $x^*(t)\in (0,\bar{x}(t))$. Consequently, we obtain $$w(\bar{x}(t))-\xi \bar{z}^2(t)\geq \varepsilon \left[ \frac{w'(0)}{2}\frac{2 T}{\pi}\sin^2 \frac{\pi t}{2 T}-\varepsilon \xi \sin^2 \frac{\pi t}{T} \right] = \varepsilon \sin^2 \frac{\pi t}{2 T}\left[ \frac{T w'(0)}{\pi}- 4\varepsilon \xi \cos^2 \frac{\pi t}{2T} \right]. $$ Consumption will be positive if $$g(t):=\frac{T w'(0)}{\pi}- 4\varepsilon \xi \cos^2 \frac{\pi t}{2T}>0 \textrm{ for } t \in [0,T].$$ For $\varepsilon$ small $g(0)=T w'(0)/\pi -4\xi \varepsilon > 0$. Also, $$g'(t)=4\varepsilon \xi \frac{\pi}{2T}2 \cos \frac{\pi t}{2T} \sin \frac{\pi t}{2T} = 2 \varepsilon \xi \frac{\pi}{T} \sin \frac{\pi t}{T} \geq 0 \textrm{ for }t \in [0,T].$$ Thus, $g(t) \geq g(0)>0$, as required.
\textbf{Remark.} It is easy to see that in the above example we can take $\bar{z}(t)$ to be any smooth function that is positive on $(0,T)$, zero for $t=0,T$ and $\dot{\bar{z}}(0)>0$.
\textbf{Step 2.} Since $c(t) \not \equiv 0$, there exists a set $A \subset [0,T] $, $\meas A>0$, such that $$\essinf_{t \in A} c(t)> \varepsilon_1>0.$$
Let us take the control pair $(\bar{c}(t),\bar{z}(t))$ with $\bar{c}(t):= c(t)-\varepsilon \chi_A(t)$ and $\bar{z}(t) := z(t)$, where $\chi_A(t)$ is the indicator function of the set $A$ and $\varepsilon \in (0,\varepsilon_1)$. These controls are admissible if we have terminal assets $\bar{a}(T_0)>0,~\forall \varepsilon \in(0,\varepsilon_1)$. To verify the last claim, we take $$\bar{a}(T_0)=e^{r T_0}\left[ a_0+\int_0^{T_0}[w(x(s))-p(c(s)-\varepsilon \chi_A(s))-\xi z^2(s)]e^{-rs}ds \right]=e^{r T_0}\int_A p \varepsilon e^{-rs}ds=\varepsilon C_1,$$ where $C_1:=p e^{r T_0} \int_A e^{-rs}ds >0$.
An application of Taylor's formula yields $$\frac{\bar{c}(t)^{1-\theta}}{1-\theta}=\frac{c(t)^{1-\theta}}{1-\theta}+(-\varepsilon \chi_A(t))c(t)^{-\theta}+(-\varepsilon \chi_A(t))^2 \frac{-\theta}{2}c^*(t)^{-\theta-1},$$ where $c^*(t)=\alpha(t) \bar{c}(t)+(1-\alpha(t))c(t),~\alpha(t) \in (0,1)$ or $c^*(t)=c(t)-\varepsilon \chi_A(t) \alpha(t)$. Note also that for $t \in A$ we have $0<c(t)-\varepsilon_1\cdot 1 \leq c^*(t)\leq c(t)$, so that $(c(t)-\varepsilon_1)^{-\theta-1}\geq c^*(t)^{-\theta-1}\geq c(t)^{-\theta-1}$.
Let us compare $$J(c(t),z(t))=\int_0^{T_0}\frac{c(t)^{1-\theta}}{1-\theta}e^{-\rho t}dt-\eta\int_0^{T_0}z^2(t)e^{-\rho t}dt$$ and \begin{equation*} \begin{split} J(\bar{c}(t),\bar{z}(t))= & \int_0^{T_0}\frac{\bar{c}(t)^{1-\theta}}{1-\theta}e^{-\rho t}dt-\eta\int_0^{T_0}z^2(t)e^{-\rho t}dt+\frac{\bar{a}(T_0)^{1-\theta}}{1-\theta}e^{-\rho T_0} \\ = & J(c(t),z(t)) + \left[ -\varepsilon \int_A c(t)^{-\theta}e^{-\rho t}dt -\frac{\theta \varepsilon^2}{2}\int_A(c(t)-\varepsilon \alpha(t))^{-1-\theta}e^{-\rho t}dt \right]+ \\ & + \frac{(\varepsilon C_1)^{1-\theta}}{1-\theta}e^{-\rho T_0}. \end{split} \end{equation*}
We will show that $J(\bar{c}(t),\bar{z}(t))>J(c(t),z(t))$ for $\varepsilon \in (0,\varepsilon_1)$ sufficiently small. This will be true if we are able to establish that $$\frac{(\varepsilon C_1)^{1-\theta}}{1-\theta}e^{-\rho T_0} > \varepsilon \int_A c(t)^{-\theta}e^{-\rho t}dt + \frac{\varepsilon^2 \theta}{2}\int_A (c(t)-\varepsilon_1 )^{-1-\theta}e^{-\rho t}dt ,$$ where the last integral provides an upper bound on $\int_A (c(t)-\varepsilon \alpha(t))^{-1-\theta}e^{-\rho t}dt $. Denoting the respective positive constants in the above inequality by $B_1$, $B_2$ and $B_3$, we obtain $$\varepsilon^{1-\theta}B_1 > \varepsilon B_2 + \varepsilon^2 B_3 $$ or $$B_1 > \varepsilon^\theta B_2 + \varepsilon^{1+\theta} B_3,$$ which is obviously true for $\varepsilon \in (0,\varepsilon_1)$ sufficiently small. This contradicts the optimality of $(c(t),z(t))$. Thus, $a(T_0)=0$ cannot be true and hence $a(T_0)>0$. \endprf
On the basis of Theorem \ref{thm:aT>0} an optimal solution $\bar{c}(t),\bar{z}(t)$ to problem \eqref{eq:obj}-\eqref{eq:location} (possibly non-unique) also solves the following problem, where the controls $(c(t),z(t))\in \Delta_1 \subset \Delta$:
$$\max_{c(t),z(t) \in \Delta_1} J(c(t),z(t))$$ $$\dot{a}(t)=ra(t)+w(x(t))-p c(t) -\xi z^2(t)$$ $$\dot{x}(t)=z(t)$$ $$a(0)=a_0\geq 0,$$ $$x(0)=x_0\in [0,1],$$ $$a(T)\geq \delta >0,$$ with $\delta$ being an appropriate constant, strictly smaller than the optimal value of terminal assets.
To avoid burdensome notation, from now on we do not append additional symbols to the state, costate and control variables in the model when referring to their optimal values. However, we use alternative symbols to denote alternative sets of variables to be compared with the optimal ones.
Taking into account that we do not impose any state constraints on the problem, Theorem 5.2.1 in \cite{Cla83} provides the set of necessary conditions. To derive the latter, we define the Hamiltonian for the problem \begin{equation} \begin{split}H & := H(t,a,x,\varphi,\psi,p_1,p_2)= \\ & p_1(r a+w(x)-p \varphi -\xi \psi^2 )+p_2 \psi+\lambda_0 e^{-\rho t}\left( \frac{\varphi^{1-\theta}}{1-\theta} -\eta \psi^2 \right),\end{split} \label{eq:Hamilt}\end{equation} where $\lambda_0 \in \{0,1\}$. Then
1) The costate variables $p_i(t),~i=1,2,$ satisfy a.e. on $(0,T)$ \begin{equation} \dot{p}_1(t)=-r p_1(t), \label{eq:p1dot}\end{equation} \begin{equation} \dot{p}_2(t)=-p_1(t)w'(x(t)). \label{eq:p2dot}\end{equation}
2) The function $\varphi,\psi \mapsto H(t,a(t),x(t),\varphi,\psi,p_1(t),p_2(t))$ attains its maximum with respect to $\varphi,\psi$ at the point $(c(t),z(t))$ for almost all $t \in [0,T]$, where $\varphi,\psi$ satisfy the constraints on the function values arising from $\Delta_1$, i.e.
$\varphi \in [0,C]$ and $|\psi|\leq Z$: \begin{equation} \begin{split} H(t) & := H(t,a(t),x(t),c(t),z(t),p_1(t),p_2(t)) = \\ & \max_{\varphi, \psi } H(t,a(t),x(t),\varphi,\psi,p_1(t),p_2(t)). \end{split} \label{eq:Hammax}\end{equation}
3a) Since $a(t)$ and $x(t)$ are fixed at $t=0$, the values of $p_i(0)$ are arbitrary, i.e. \begin{equation} p_1(0)=\lambda_1,~ \lambda_1 \in \mathbb{R} , \label{eq:trans3}\end{equation} \begin{equation} p_2(0)=\lambda_2,~ \lambda_2 \in \mathbb{R} . \label{eq:trans4}\end{equation}
3b) Since the terminal values $(a(T),x(T))$ of the state variables are at an interior point of the target set $$\{ (a,x)\in
\mathbb{R}^2 | a \geq \delta >0, x \in \mathbb{R}^1 \},$$ the corresponding normal cone is trivial and the transversality condition at the right endpoint $T$ has the form \begin{equation} p_1(T)=\lambda_0 e^{-\rho T}a(T)^{-\theta}, \label{eq:trans1}\end{equation} \begin{equation} p_2(T)=0 , \label{eq:trans2}\end{equation} (cf. condition 4) in Theorem 5.2.1 and the functional form for $f(x(b))$ in \S 5.2 in \cite{Cla83}).
4) The variables $p_1(t),p_2(t),\lambda_0$ are not simultaneously equal to zero.
Below we specify the form of the necessary conditions in greater detail.
According to \eqref{eq:p1dot} and \eqref{eq:trans3} we have \begin{equation} p_1(t)=\lambda_1 e^{-rt}. \label{eq:p1solved}\end{equation}
\begin{proposition}If there exists $t_0 \in [0,T]$ such that $c(t_0)\in (0,C)$, then $c(t)>0$ for almost all $t\in [0,T]$. \label{thm:dtpositive} \end{proposition} {\bf Proof. } If there exists $t_0$ with the above properties, then \eqref{eq:Hammax} implies
$$\frac{\partial}{\partial\varphi}H(t_0,a(t_0),x(t_0),\varphi,z(t_0),p_1(t_0),p_2(t_0))\left |_{\varphi=c(t_0)}=0, \right.$$ i.e. \begin{equation} -p \lambda_1 e^{-r t_0}+\lambda_0 e^{-\rho t_0}c(t_0)^{-\theta} = 0. \label{eq:supplHammax1}\end{equation}
If we assume that $\lambda_0=0$, then $\lambda_1=0$ and, because of \eqref{eq:p1solved}, one obtains $p_1(t)\equiv 0$. This implies that $p_2(t)\equiv const =\lambda_2$. Now \eqref{eq:trans2} shows that $\lambda_2=0$, which constitutes a contradiction with condition 4) from the cited theorem in \cite{Cla83}. Therefore, $\lambda_0=1$.
Assume that there exists $t_1 \in [0,T]$ for which $c(t_1)=0$. Then, for all sufficiently small $\varphi >0$ we have $$\frac{H(t_1,a(t_1),x(t_1),\varphi,z(t_1),p_1(t_1),p_2(t_1))-H(t_1,a(t_1),x(t_1),0,z(t_1),p_1(t_1),p_2(t_1))}{\varphi-0}\leq 0,$$ i.e. $$-p_1 (t_1) p + \lambda_0 e^{-\rho t_1} \frac{\varphi^{-\theta}}{1-\theta}\leq 0.$$ Since $\lambda_0>0$, for $\varphi \rightarrow 0+$ the last inequality leads to a contradiction. This proves the proposition. \endprf
\begin{cor} If there exists $t_1 \in [0,T]$ for which $c(t_1)=0$, then $\lambda_0=0$ and $c(t)=0$ for almost all $t \in [0,T]$. \label{thm:lambda0zero}\end{cor} {\bf Proof. } The conclusion on $\lambda_0$ can be obtained in the same manner as in the proof of Proposition \ref{thm:dtpositive} by passing to the limit as $\varphi \rightarrow 0+$. If we assume the existence of a point $t_0\in [0,T]$ for which $c(t_0)>0$, we can proceed as in the proof of the proposition and get $p_1(t)\equiv p_2(t)\equiv 0$ and $\lambda_0=0$, which is impossible. \endprf
\begin{proposition}The optimal consumption cannot be identically zero. \label{thm:conspositive} \end{proposition} {\bf Proof. } Assume that the controls $c(t)\equiv 0$ and $z(t)$ are optimal. Then $$J(0,z(t))=-\int_0^T \eta z^2(t)e^{-\rho t}dt+ e^{-\rho T}\frac{a(T)^{1-\theta}}{1-\theta}.$$ Take the controls $\tilde{c}(t)=\varepsilon$ and $\tilde{z}(t)=z(t)$, where $\varepsilon>0$ is sufficiently small. These controls are admissible, as the respective value of terminal assets is $$\tilde{a}(T)=e^{rT}\left\{ a_0+\int_0^T [w(x(t))-p\varepsilon -\xi z^2(t)]e^{-rt}dt \right\}=a(T)-\varepsilon C_1,$$where $C_1:=e^{rT}p\int_0^Te^{-rt}dt>0$. It is evident that for $\varepsilon$ sufficiently small we have $\tilde{a}(T)>\delta$, since $a(T)>\delta$. It remains to check that for $\varepsilon$ close to zero we have \begin{equation*}\begin{split} J(\varepsilon,z(t))= & \int_0^T \frac{\varepsilon^{1-\theta}}{1-\theta}e^{-\rho t}dt-\eta \int_0^T z^2(t)e^{-\rho t}dt+ e^{-\rho T}\frac{(a(T)-\varepsilon C_1)^{1-\theta}}{1-\theta} > \\ & J(0,z(t))=-\int_0^T \eta z^2(t)e^{-\rho t}dt+ e^{-\rho T}\frac{a(T)^{1-\theta}}{1-\theta}, \end{split} \end{equation*} which is equivalent to $$\varepsilon^{1-\theta} C_2 > \frac{e^{-\rho T}}{1-\theta}\left[a(T)^{1-\theta}-(a(T)-\varepsilon C_1)^{1-\theta}\right],~C_2:=const>0.$$ The last expression is obviously true for all $\varepsilon$ sufficiently small. \endprf
\textbf{Remark.} So far it is clear that the optimal consumption cannot be identically zero and that if there exists $t_0$ such that $c(t_0)\in (0,C)$, then $\lambda_0=1$. It remains to check whether we can have $c(t)=C$ for some $t$.
We first establish the following result. \begin{proposition} It is impossible for the optimal $c(t)$ to satisfy \begin{equation} c(t)\geq C_0>0, \label{eq:dtgeqD1}\end{equation} \label{thm:dtgeqD1} where \begin{equation} C_0 >
\frac{a_0 + T\max_x |w(x)|}{p\frac{1-e^{-rT}}{r}} . \label{eq:D1def}\end{equation} \end{proposition} {\bf Proof. } Notice that if condition \eqref{eq:dtgeqD1} is true, then the inequality $a(T)\geq \delta$ is violated. Indeed, if \eqref{eq:dtgeqD1} holds, then \begin{equation*}\begin{split} a(T) = & e^{rT}\left[ a_0+\int_0^T \left[w(x(t))-pc(t) -\xi z^2(t)\right]e^{-rt}dt \right]\leq \\ &
e^{rT}\left[ a_0+\int_0^T \left[\max_x |w(x)|-p C_0 \right]e^{-rt}dt \right],\end{split}\end{equation*} which is negative when \eqref{eq:D1def} holds. \endprf
\begin{proposition} The number $\lambda_1$ is strictly positive. \label{thm:lambda1positive}\end{proposition} {\bf Proof. } We know that for the optimal $c(t)$ it is impossible to have $c(t)\geq C_0$ or $c(t)\equiv 0$. Consequently, there exists $t_0\in [0,T]$ for which $c(t_0) \in (0,C_0)$. Then $$\frac{\partial}{\partial\varphi}H(t_0,a(t_0),x(t_0),\varphi,z(t_0),p_1(t_0),p_2(t_0))\left
|_{\varphi=c(t_0)}=0, \right.$$ and hence \eqref{eq:supplHammax1} holds. This in turn implies that $\lambda_0=1$, as well as $$p \lambda_1 e^{-r t_0} = e^{-\rho t_0}c(t_0)^{-\theta}.$$ Therefore, we have $\lambda_1>0$ and \begin{equation} \lambda_1= \frac{e^{(r-\rho)t_0}c(t_0)^{-\theta}}{p}. \label{eq:lambda1expr}\end{equation} \endprf
\begin{proposition} There does not exist $t\in [0,T]$ for which $c(t)=C$. \label{thm:dtlessthanD}\end{proposition}{\bf Proof. } Assuming the contrary, by the maximum principle we obtain $$\frac{H(t,a(t),x(t),\varphi,z(t),p_1(t),p_2(t))-H(t,a(t),x(t),C,z(t),p_1(t),p_2(t))}{\varphi-C}\geq 0$$ for $\varphi \in (0,C)$ and so for $\varphi\rightarrow C-0$ we get $$- p\lambda_1 e^{-rt} + e^{-\rho t} C^{-\theta}\geq 0,$$ which implies \begin{equation*} C^{\theta}\leq \frac{e^{(r-\rho)t}}{\lambda_1 p}=\frac{e^{(r-\rho)t}c(t_0)^{\theta}}{e^{(r-\rho)t_0}}< e^{(r-\rho)(t-t_0)}C_0^\theta \leq \mu C_0^\theta, \end{equation*} where $\mu := \max_{t,t_0 \in [0,T]}e^{(r-\rho)(t-t_0)}>0$. In other words, $$C \leq \mu^{\frac{1}{\theta}}C_0,$$ which is impossible. \endprf
The results obtained so far allow us to to find an expression for the optimal consumption $c(t)$. \begin{cor}For each $t \in [0,T]$ we have $c(t)\in(0,C)$. The optimal consumption rule has the form \begin{equation} c(t)=\left[ \frac{1}{p \lambda_1 }\right]^{\frac{1}{\theta}}e^{\frac{r-\rho}{\theta} t }=\frac{1}{p^{\frac{1}{\theta}}}e^{\frac{\rho-r}{\theta}(T-t)}a(T). \label{eq:copt1}\end{equation} \label{thm:copt} \end{cor}
Before deriving an expression for the optimal relocation control $z(t)$, we note that \eqref{eq:p2dot} and \eqref{eq:trans2} imply \begin{equation} p_2(t)=\lambda_1 \int_t^T w'(x(\tau))e^{-r\tau}d\tau = e^{(r-\rho)T}a(T)^{-\theta}\int_t^T w'(x(\tau))e^{-r\tau}d\tau. \label{eq:p2trans1} \end{equation}
\begin{proposition} For each $t \in [0,T]$ we have the strict inequality $$|z(t)|< Z.$$ \label{thm:zbound} \end{proposition} {\bf Proof. } Assume, for example, that $z(t_1)=Z$ for some $t_1 \in [0,T]$. Then, after passing to the limit in the respective difference quotient, we obtain $$-p_1(t_1)\xi 2Z+p_2(t_1)-2\eta Z e^{-\rho t_1}\geq 0,$$ so that $$p_2(t_1)\geq 2(\xi\lambda_1 e^{-rt_1}+\eta e^{-\rho t_1 })Z.$$
Similarly, the assumption that $z(t_2)=-Z$ for some $t_2 \in [0,T]$ leads to $$-p_2(t_2)\geq 2(\xi \lambda_1 e^{-r t_2}+\eta e^{-\rho t_2 })Z.$$
In both cases we have ($i=1$ or $2$) \begin{equation*} \begin{split}
Z \leq & \frac{\pm p_2(t_i)}{2(\xi \lambda_1 e^{-r t_i}+\eta e^{-\rho t_i })} \leq \frac{|p_2(t_i)|}{2(\xi \lambda_1 e^{-r t_i}+\eta e^{-\rho t_i })} \leq \frac{ \lambda_1
\left|\int_{t_i}^T w'(x(\tau))e^{-r\tau}d\tau \right| }{2(\xi \lambda_1 e^{-r t_i}+\eta e^{-\rho t_i })} \leq \\ & \\ & \frac{
\max_x \left| w'(x) \right| |T-t_i| }{2(\xi e^{-r t_i}+\frac{\eta}{\lambda_1} e^{-\rho t_i })} < \frac{T \max_x
|w'(x)|}{2\xi e^{-r t_i}} \leq \frac{T \max_x |w'(x)|}{2\xi} e^{r T}, \end{split} \end{equation*} which is impossible by the definition of $Z$. \endprf
\begin{cor} For $t\in [0,T]$ we have for the optimal relocation speed $z(t)\in (-Z,Z)$ and then \begin{equation} z(t)=\frac{p_2(t)}{2 (\xi \lambda_1 e^{-rt}+ \eta e^{-\rho t})}. \label{eq:zopt1} \end{equation}\label{thm:zopt} \end{cor}
\section{Existence of a solution of the system of necessary conditions}\label{sec:NCtransf}
In order to facilitate the study of the differential equations arising from the set of necessary conditions in section \ref{sec:NC}, it would prove convenient to rewrite the differential system. Theorem \ref{thm:exist} guarantees the existence of a solution to the problem \eqref{eq:obj}-\eqref{eq:location} which in turn ensures the existence for each $T>0$ of a solution to the following problem:
\begin{equation} \left | \begin{array}{l} \dot{x}(t)=\frac{y(t)}{F(t)}, \\ \dot{y}(t)=-w'(x(t))\lambda_1 e^{-rt}, \\ x(0)=x_0, \\ y(T)=0, \end{array} \right. \label{eq:AuxSys1}\end{equation} where $y(t):=p_2(t)$ and
$F(t):=2(\xi \lambda_1 e^{-rt}+\eta e^{-\rho t})$. It follows that there exists a solution to the problem \begin{equation} \left | \begin{array}{l} \frac{d}{dt}\left( F(t) \dot{x}(t) \right)+w'(x(t))\lambda_1 e^{-rt}=0, \\ x(0)=x_0, \\ \dot{x}(T)=0. \end{array} \right. \label{eq:AuxSys2}\end{equation}
The latter fact can also be established without recourse to Theorem \ref{thm:exist}. Following the procedure described in \S 73 of \cite{Lov24}, we construct the respective Green function and transform \eqref{eq:AuxSys2} in the form \begin{equation} x(t)=\int_0^T K(t, \tau)\lambda_1 e^{-r \tau}w'(x(\tau))d\tau, \label{eq:Lovitt}\end{equation} where $$K(t, \tau)=\left \{ \begin{array}{l l} \int_0^\tau \frac{1}{F(s)}ds, & \tau \in [0,t] \\ \int_0^t \frac{1}{F(s)}ds, & \tau \in [t,T] \end{array} \right .$$
Since the function $w'(x)$ is bounded and continuous by assumption, a solution to \eqref{eq:Lovitt} exists. This is a consequence of Leray-Schauder index theory (see \S 2.4 in \cite{Nir74}). Also, the solution to \eqref{eq:AuxSys2} may not be unique, as can be seen from simple examples of eigenfunction problems that possess nontrivial solutions.
A solution to \eqref{eq:AuxSys1} or \eqref{eq:AuxSys2} can be viewed as a particular member of the family of solutions
$(x(t,\alpha),y(t,\alpha))$ to the Cauchy problem \begin{equation} \left | \begin{array}{l} \dot{x}(t)=\frac{y(t)}{F(t)}, \\ \dot{y}(t)=-w'(x(t))\lambda_1 e^{-rt}, \\ x(0)=x_0, \\ y(0)=\alpha, \end{array} \right. \label{eq:AuxSys3}\end{equation} where $\alpha$ has been chosen appropriately, so that \begin{equation} y(T,\alpha)=0. \label{eq:AuxSys4}\end{equation} The existence of a unique solution to \eqref{eq:AuxSys3} on the interval $[0,T]$ for initial data $(x_0,\alpha)$ and each $T>0$ is ensured by Corollary 3.1, chapter 2, in \cite{Har64}.
Since \eqref{eq:AuxSys4} is equivalent to $\dot{x}(T,\alpha)=0$, we can integrate the differential equation in \eqref{eq:AuxSys2} over $[0,T]$ to arrive at an equivalent form of \eqref{eq:AuxSys4}: \begin{equation} \alpha = \lambda_1 \int_0^T w'(x(\tau, \alpha))e^{-r\tau}d\tau . \label{eq:AuxSys5}\end{equation}
It is straightforward to verify the following \begin{proposition}The function $x(t)\equiv x_0$ is a solution to \eqref{eq:AuxSys2} if and only if the point $x_0$ is a critical point for $w(x)$, i.e. $w'(x_0)=0$. \label{thm:5.1}\end{proposition} The analysis of the solutions of the system of necessary conditions, which is carried out in section \ref{sec:reloc}, provides the dynamics of the behaviour of the economic agent, implied by this model, in the baseline case when the wage distribution has a single maximum point on the interval $[0,1]$. Prior to that, the next section studies the properties of the function $T \mapsto a(T)$ as $T \rightarrow \infty$.
\section{Dynamics of terminal assets $a(T)$ for different time horizons}\label{sec:termasset}
In this section we study the dependence of optimal terminal assets $a(T)$ on the length of the time horizon $T$. Although terminal assets is the natural object of study due to the fact that it is easily interpretable in economic terms, the discussion may equally well be framed in terms of the behaviour of $\lambda_1$, viewed as a function of the time horizon $T$ and denoted $\lambda_1(T)$. This approach is feasible by virtue of the relationship \begin{equation} \lambda_1(T)=e^{(r-\rho)T}a(T)^{-\theta}.\label{eq:lambda1T}\end{equation} Below we derive upper and lower bounds on $\lambda_1(T)$, which will be needed in the analysis of section \ref{sec:reloc}. We assume for simplicity that $x_0 \in (0,1)$, i.e. $w(x_0)>0$, as well as that $a_0>0$. Also, in this section we denote by $C$ different constants that do not depend on $T$. Since we do not use the bound on the control $c(t)$ from \eqref{eq:cconstr} in this section, no confusion can arise from this convention.
\subsection{An upper bound on $\lambda_1(T)$}\label{sec:termassup}
The pair $(c(t)\equiv c_0=const,z(t)\equiv 0)$ is admissible for $c_0$ appropriately chosen. Then $x(t) \equiv x_0$ and we set $c_0:=\frac{w(x_0)}{p}$. In this case terminal assets are $$\tilde{a}(T)=e^{rT}\left[ a_0+\int_0^T[w(x(t))-pc(t)-\xi z^2(t)]e^{-rT}dt \right]=a_0 e^{rT}.$$ Consequently, \begin{equation*} J(c_0,0) = c_1(1-e^{-\rho T})+c_2 e^{[r(1-\theta)-\rho]T}, \end{equation*}where $c_1$ and $c_2$ are constants that depend on $c_0$ and $a_0$.
On the other hand, for the optimal controls $(c(t),z(t))$ we have \begin{equation*} J(c(t),z(t)) = \int_0^T \left[a(T)\frac{1}{p^{1/\theta}}e^{\frac{\rho-r}{\theta}(T-t)} \right]^{1-\theta}\frac{e^{-\rho t}}{1-\theta} dt-\eta\int_0^T z^2(t)e^{-\rho t}dt+\frac{a(T)^{1-\theta}}{1-\theta}e^{-\rho T}, \end{equation*} which takes different forms depending on whether $\rho-r(1-\theta)$ is different from zero.
Since $J(c(t),z(t))\geq J(c_0,0)$, for $\rho-r(1-\theta)\neq 0$ we obtain $$\frac{a(T)^{1-\theta}}{1-\theta}e^{-\rho T}\left[ 1+\frac{1}{p^{\frac{1-\theta}{\theta}}}\frac{e^{\frac{\rho-r(1-\theta)}{\theta}T}-1}{\frac{\rho-r(1-\theta)}{\theta}} \right]\geq c_1+c_2e^{-(\rho-r(1-\theta))T}-c_1 e^{-\rho T}.$$
Thus, \begin{equation} a(T) \geq \left\{ \frac{\tilde{c}_1 e^{\rho T}+\tilde{c}_2 e^{r(1-\theta)T}-\tilde{c}_1 }{1+\frac{1}{p^{\frac{1-\theta}{\theta}}}\frac{e^{\frac{\rho-r(1-\theta)}{\theta}T}-1}{\frac{\rho-r(1-\theta)}{\theta}}} \right\}^{\frac{1}{1-\theta}}, \label{eq:aTlowbound}\end{equation} where $\tilde{c}_1=(1-\theta)c_1$, $\tilde{c}_2=(1-\theta)c_2$. Using the last expression together with \eqref{eq:lambda1T}, we can derive upper bounds on $\lambda_1(T)$.
\begin{proposition} Under the assumptions of this section, we have ($\forall T > 0$): \begin{equation} \lambda_1(T) \leq \left\{
\begin{array}{l l}
C, & \textrm{if } \rho-r(1-\theta)>0, \\
C e^{-(\rho-r(1-\theta))T}, & \textrm{if } \rho-r(1-\theta)<0, \\
C (1+T)^{\frac{\theta}{1-\theta}}, & \textrm{if }
\rho-r(1-\theta)=0.
\end{array}
\right. \label{eq:lambda1Tuprbnd}\end{equation} and, accordingly, \begin{equation} a(T) \geq \left\{
\begin{array}{l l}
C e^{\frac{r-\rho}{\theta}T}, & \textrm{if } \rho-r(1-\theta)>0, \\
C e^{rT}, & \textrm{if } \rho-r(1-\theta)<0, \\
C \frac{e^{\frac{\rho}{1-\theta}T}}{(1+T)^{\frac{1}{1-\theta}}}, & \textrm{if }
\rho-r(1-\theta)=0.
\end{array}
\right. \label{eq:aTuprbnd}\end{equation} \label{thm:uprbndsummary} \end{proposition}
\subsection{A lower bound on $\lambda_1(T)$}\label{sec:termasslo}
We first look at a particular case of the main problem, for which \begin{equation} w(x)\equiv W =const. \label{eq:WageConst}\end{equation} Then $\dot{p}_2\equiv 0$ which, together with the transversality condition $p_2(T)=0$, yields $p_2(t)\equiv 0$, i.e. $z(t)\equiv 0$ and $x(t)\equiv x_0.$
The optimal consumption rule is $c(t)=\frac{1}{p^{1/\theta}}\bar{a}(T)e^{\frac{\rho-r}{\theta}(T-t)},$ where $\bar{a}(T)$ is the optimal terminal value of assets for the problem with condition \eqref{eq:WageConst}. We will calculate $\bar{a}(T)$ from $$\bar{a}(T)=e^{rT}\left[ a_0+\int_0^T\left(W-p^{\frac{\theta-1}{\theta}}\bar{a}(T)e^{\frac{\rho-r}{\theta}(T-t)}\right)e^{-rt}dt \right].$$ Thus, we find \begin{equation} \bar{a}(T) \leq \left\{
\begin{array}{l l}
C e^{\frac{r-\rho}{\theta}T}, & \textrm{if } \rho-r(1-\theta)>0, \\
C e^{rT}, & \textrm{if } \rho-r(1-\theta)<0, \\
C \frac{e^{rT}}{1+T}, & \textrm{if }
\rho-r(1-\theta)=0.
\end{array}
\right. \label{eq:abarTlwrbnd}\end{equation}
\begin{proposition} Let $w(x)\leq W,~\forall x,$ and let $a(T)$ and $\bar{a}(T)$ be the optimal terminal asset values for the problems with wage distributions $w(x)$ and $W$, respectively (all other parameters of the two problems being identical). Then $$a(T) \leq \bar{a}(T).$$ \label{thm:aT_leq_abarT}\end{proposition} {\bf Proof. } Since according to \eqref{eq:copt1} optimal consumption for the two problems has the form $a(T)\Psi(t)$ and $\bar{a}(T)\Psi(t)$ with one and the same function $\Psi(t)$, we obtain $$[\bar{a}(T)-a(T)]\left(1+e^{rT}\int_0^Tp\Psi(t)e^{-rt}dt \right)=e^{rT}\int_0^T [W-w(x(t))]e^{-rt}dt+e^{rT}\xi \int_0^T z^2(t)e^{-rt}dt,$$ where $x(t)$ and $z(t)$ refer to the variables in the problem with wage distribution $w(x)$. This completes the proof. \endprf
From Proposition \ref{thm:aT_leq_abarT} and equations \eqref{eq:lambda1T} and \eqref{eq:abarTlwrbnd}, we obtain \begin{proposition} Under the assumptions of this section, we have ($\forall T > 0$): \begin{equation} a(T) \leq \left\{
\begin{array}{l l}
C e^{\frac{r-\rho}{\theta}T}, & \textrm{if } \rho-r(1-\theta)>0, \\
C e^{rT}, & \textrm{if } \rho-r(1-\theta)<0, \\
C \frac{e^{rT}}{1+T}, & \textrm{if }
\rho-r(1-\theta)=0.
\end{array}
\right. \label{eq:aTlwrbnd}\end{equation} and, accordingly, \begin{equation} \lambda_1(T) \geq \left\{
\begin{array}{l l}
C, & \textrm{if } \rho-r(1-\theta)>0, \\
C e^{-(\rho-r(1-\theta))T}, & \textrm{if } \rho-r(1-\theta)<0, \\
C (1+T)^{\theta}, & \textrm{if }
\rho-r(1-\theta)=0.
\end{array}
\right. \label{eq:lambda1Tlwrbnd}\end{equation} \label{thm:lwrbndsummary} \end{proposition}
\textbf{Remark.} The bounds derived above can be refined in some cases. For instance, the first inequality in \eqref{eq:aTlwrbnd} implies very different behaviour of $a(T)$ depending on whether $\rho \in (r(1-\theta),r)$, $\rho=r$ or $\rho>r$.
\section{Optimal relocation strategies for single-peaked wage distributions}\label{sec:reloc}
This section studies the optimal relocation behaviour of the consumer, as described by $x(t)$, in the important case of single-peaked wage distributions on the interval $[0,1]$. We demonstrate first the validity of the following general claim (under the conditions stated at the end of section \ref{sec:model}): \begin{proposition} The optimal trajectory $x(t)$ remains in the interval $[0,1]$, regardless of the particular form of the wage distribution $w(x)$ in $[0,1]$. \label{thm:xin01}\end{proposition} {\bf Proof. } Notice that since $\dot{x}(t)=p_2(t)/F(t)$, with $F(t)$ defined as in section \ref{sec:NCtransf}, in view of \eqref{eq:p2trans1} we can write $$\dot{x}(t)=\frac{\lambda_1}{F(t)}\int_t^T w'(x(\tau))e^{-r\tau}d\tau=G(t) \int_t^T w'(x(\tau))e^{-r\tau}d\tau ,$$ where $G(t):=\lambda_1/F(t)$.
Assume first that at time $t_1$ the point $x(t)$ leaves the interval $[0,1]$ to the left (i.e. leaves the interval at $x=0$) and remains to the left of zero until $t=T$, so that $x(t)<0$ for $t \in (t_1,T]$. Then, for $t \in [t_1,T]$, $w'(x(t))>0$ and consequently $\dot{x}(t)>0$. This would imply that for some $t_* \in (t_1,T)$, $x(T)-x(t_1)=(T-t_1)\dot{x}(t_*)>0$, or $x(T)>x(t_1)=0$, which contradicts the assumption that $x(t)<0$ for $t \in (t_1,T]$. Thus, $x(t)$ cannot leave the interval $[0,1]$ to the left and remain outside it until the end of the planning horizon $T$. A similar argument shows that it is impossible for $x(t)$ to leave the interval $[0,1]$ to the right and stay there.
Let us now assume that $x(t)$ leaves the interval $[0,1]$ to the left of zero at time $t_1$ and returns back at time $t_2>t_1$. Again, for $t\in (t_1,t_2)$ we have $x(t)<0$ and $w'(x(t))>0$, which means that $\int_{t_1}^{t_2}w'(x(t))e^{-rt}dt > 0$. Since $x(t)$ leaves the interval $[0,1]$ to the left at $t_1$, it must be that $\dot{x}(t_1)\leq 0$. By the same logic, at time $t_2$ we should have $\dot{x}(t_2) \geq 0$. Then one obtains \begin{equation*} \begin{split} \dot{x}(t_1) = & G(t_1)\int_{t_1}^{T}w'(x(t))e^{-rt}dt = G(t_1)\int_{t_1}^{t_2}w'(x(t))e^{-rt}dt+\frac{G(t_1)}{G(t_2)}G(t_2)\int_{t_2}^{T}w'(x(t))e^{-rt}dt \\ = & G(t_1)\int_{t_1}^{t_2}w'(x(t))e^{-rt}dt+\frac{G(t_1)}{G(t_2)}\dot{x}(t_2)>0, \end{split} \end{equation*} which contradicts the condition $\dot{x}(t_1)\leq 0$. Hence it is impossible for $x(t)$ to temporarily leave the interval $[0,1]$ to the left. Naturally, this argument can be applied with obvious modifications to the hypothesis that $x(t)$ temporarily goes to the right of $x=1$. Summarizing the above conclusions, we see that the optimal $x(t)$ remains in the interval $[0,1]$. \endprf
We turn next to the main object of study for this section: the case when the wage function has a single peak on the interval $[0,1]$. The example of the quadratic function $w(x)=x(1-x)$ in a neighbourhood of the interval $[0,1]$ may facilitate visualization.
Assume that in this case the initial location $x_0$ lies to the left of the wage peak, i.e. if $x_1:=\argmax_{x \in [0,1]}w(x)$, then $0 \leq x_0 < x_1$. For the remainder of this section we will assume that $w'(x)>0,~x\in [0,x_1]$ and $w'(x)<0,~x\in [x_1,1]$. In this case, for $T$ sufficiently small, $x(T)$ will remain in a small neighbourhood of $x_0$. However, this means that $w'(x(t))>0$ for any $t \in [0,T]$ and therefore $\dot{p}_2(t)<0$, which implies $p_2(t)>0$ since $p_2(T)=0$. As $p_2(t)>0$, we obtain $\dot{x}(t)>0$. In words, for sufficiently small planning horizons the consumer unambiguously relocates toward the wage maximum.
If $x(T) \in (x_0,x_1)$, we have, using the notation in section \ref{sec:NCtransf}, $\dot{y}(t)=-w'(x(t))p_1(t)<0$. Since $y(T)=0$, it follows that $y(t)>0,~t \in [0,T)$. Then $\dot{x}(t)=\frac{y(t)}{F(t)}>0$, so that $x(t)$ is monotonically increasing.
We note that there does not exist a solution to the system of necessary conditions for which $x(T)=x_1$. For such a solution we would have $y(T)=0$ and one could compare this solution of the stationary solution $\tilde{x}(t) \equiv x_1, \tilde{y}(t)\equiv 0$. Then, the uniqueness of the solution to a Cauchy problem (for identical data at $t=T$) shows that the two solutions coincide. This, however, is impossible, since for $t=0$ the values of the two solutions are different ($x(0)=x_0 \neq \tilde{x}(0)=x_1$).
We would like to check whether it is possible for the terminal location $x(T)$ to lie to the right of the wage peak for $x_0<x_1$. To this end, assume that $x(T)>x_1$ and let $t_1$ be the time when point $x_1$ is reached last, i.e. $x(t_1)=x_1$ and for $t \in (t_1,T)$ we have $x(t)>x_1$. (In other words, $t_1=\sup
\{ t\in [0,T]| x(t)=x_1\} $.) Then, by the mean value theorem, $0<x(T)-x(t_1)=(T-t_1)\frac{y(t_*)}{F(t_*)}$ for some $t_* \in (t_1,T)$. However, since $\dot{y}(t)>0$, $t\in (t_1,T)$, and $y(T)=0$ imply $y(t_*)<0$, we obtain $\frac{y(t_*)}{F(t_*)}<0$, which is a contradiction. Thus, $x(T)$ cannot lie to the right of $x_1$.
The systematic study of the relocation behaviour of the economic agent in this case can be reduced to the analysis of the way in which the solutions to the Cauchy problem \eqref{eq:AuxSys3} behave for different values of the parameter $\alpha$. Those solutions that satisfy \eqref{eq:AuxSys4} are also solutions to the system of necessary conditions \eqref{eq:AuxSys1}, i.e. extremals. Through this approach we can also obtain information on the number of solution to the problem at hand. Of course, if only one extremal exists, then it is the solution we seek.
We remind the reader that the number $\lambda_1=\lambda_1(T)$ is fixed, insofar as $T$ is fixed.
\textbf{Case I: $\alpha \leq 0$.} It is obvious that for small $t$ we have $y(t)<0$ since $\dot{y}(t)<0$. For such $t$ we have $$x(t)=x_0+\int_0^t \frac{y(\tau)}{F(\tau)}d\tau < x_0,$$ i.e. the agent shifts toward $x=0$. It means that $\dot{y}(t)$ remains negative, so that $y(t)=\alpha+\int_0^t \dot{y}(\tau)d\tau$ also remains negative and $x(t)$ keeps moving to the left. Thus, it is impossible for \eqref{eq:AuxSys4} to become true, i.e. there are no extremals among the solutions of \eqref{eq:AuxSys3} for $\alpha \leq 0$.
\textbf{Case II: $\alpha > 0$.} In this case we have $y(t)>0$ in a neighbourhood of $t=0$ and so $x(t)$ moves to the right in the direction of the point $x_1$. However, $\dot{y}(t)=-w'(x(t))p_1(t)<0$, so that $y(t)$ decreases. If it turns out that $y(T)=0$ and $x(T)<x_1$, then the respective solution is an extremal. The existence of such an extremal is guaranteed by Theorem \ref{thm:exist}. The latter claim can be established through an alternative approach, which allows us to ascertain the number of extremals.
We introduce the notation $M(\alpha)$ for the right-hand side of \eqref{eq:AuxSys5}. Since $x(t)<x_1$ for $t\in [0,T]$, we get
$$M(\alpha)\leq \lambda_1 \max_x |w'(x)| \frac{1-e^{-rT}}{r}=:M_0,$$ i.e. in view of \eqref{eq:AuxSys5} the relevant values of $\alpha$ lie in the interval $(0,M_0)$.
It is easy to see that the function $$g(\alpha) := \alpha-M(\alpha)$$ is continuous on the interval $[0,M_0+1]$ and satisfies the inequalities $$g(0)<0<g(M_0+1).$$ Consequently, there exists $\alpha>0$ for which $g(\alpha)=0$, i.e. which satisfies \eqref{eq:AuxSys5}.
\begin{proposition} For the case of a single-peaked wage distribution $w(x)$ with $w''(x) \leq 0$ in $[0,1]$, there exists a unique extremal for the system \eqref{eq:AuxSys1}. \label{thm:SingleExtremal}\end{proposition} {\bf Proof. } Assume that at least two different extremals exist. They solve the system \eqref{eq:AuxSys3} for different positive values $\alpha_1 \neq \alpha_2$, for which $\alpha_i-M(\alpha_i)=0,~i=1,2$. Then \begin{equation} (\alpha_2-\alpha_1)\left(1-\frac{d}{d\alpha}M(\alpha^*)\right)=0,~\alpha^*=\varkappa \alpha_1+(1-\varkappa)\alpha_2,~\varkappa\in(0,1).\label{eq:diffalpha1alpha2}\end{equation} The derivative $$\left. \frac{d}{d\alpha}\left( \lambda_1 \int_0^T w'(x(t,
\alpha))e^{-rt}dt \right)\right|_{\alpha=\alpha^*} $$ has the form $$\lambda_1 \int_0^T w''(x(t,\alpha^*))\frac{\partial x(t,\alpha^*)}{\partial \alpha}e^{-rt}dt ,$$ with $x_\alpha(t,\alpha) := \frac{\partial x(t,\alpha)}{\partial \alpha} $ and $y_\alpha(t,\alpha) := \frac{\partial y(t,\alpha)}{\partial \alpha} $ satisfying the equations of variation \cite[Ch.V, Theorem 3.1]{Har64}:
\begin{equation*} \left | \begin{array}{l} \dot{x}_\alpha(t,\alpha)=\frac{y_\alpha(t,\alpha)}{F(t)}, \\ \dot{y}_\alpha(t,\alpha)=-w''(x(t,\alpha))x_\alpha(t,\alpha) \lambda_1 e^{-rt}, \\ x_\alpha(0,\alpha)=0, \\ y_\alpha(0,\alpha)=1. \end{array} \right. \end{equation*} Consequently, $x_\alpha (t,\alpha^*)$ solves the linear equation $$\frac{d}{dt}\left(F(t)\dot{x}_\alpha(t,\alpha^*)\right)+w''(x (t,\alpha^*))\lambda_1 e^{-rt}x_\alpha (t,\alpha^*)=0$$ for initial data $x_\alpha (0,\alpha^*)=0$ and $\dot{x}_\alpha (0,\alpha^*)=1$. Multiplying by $x_\alpha (t,\alpha^*)$ and integrating over $(0,t)$, we obtain $$F(t)\dot{x}_\alpha (t,\alpha^*)x_\alpha (t,\alpha^*) = \int_0^t F(\tau)\dot{x}^2_\alpha (\tau,\alpha^*)d\tau + \int_0^T (-w''(x (\tau,\alpha^*)))\lambda_1 e^{-r\tau}x^2_\alpha (\tau,\alpha^*)d\tau \geq 0.$$ Taking into account that $F(t)\dot{x}_\alpha (t,\alpha^*)x_\alpha (t,\alpha^*)=\frac{d}{dt}(x^2_\alpha (t,\alpha^*))F(t)/2$, we establish that the function $x^2_\alpha (t,\alpha^*)$ is increasing. In view of the initial conditions, in a small interval $(0,\varepsilon)$ we have $x_\alpha (t,\alpha^*)>0$. This inequality holds for all $t\in (0,T)$, for otherwise there would exist $\bar{t}\in (\varepsilon ,T)$ for which $x_\alpha (\bar{t},\alpha^*)=0$. The latter would lead to the contradiction $0<x^2_\alpha (\varepsilon/2,\alpha^*)\leq x^2_\alpha (\bar{t},\alpha^*)=0$. Taking into account that $w''(x)\leq 0$, we obtain from \eqref{eq:diffalpha1alpha2} that $\alpha_1=\alpha_2$, i.e. the two extremals coincide. \endprf
We augment the above results by investigating the dependence of the final location $X(T) := x(T;T,x_0)$ of the agent on the length of the time horizon $T$. Since $X(T)< x_1$, $\forall T >0$, we have $l := \sup_{T>0}X(T) \leq x_1$. \begin{proposition} Under the assumptions of Proposition \ref{thm:SingleExtremal}, we have the following classification. If $\rho \geq r$ or $\rho \in (0,r(1-\theta)]$, then $l=x_1$. If $\rho \in (r(1-\theta),r)$, it is possible to have $l < x_1$ for appropriate values of the parameters of the problem. \label{thm:ReachingThePeak}\end{proposition} {\bf Proof. } Assume that $l<x_1$. For $\rho \geq r$, we have $e^{(r-\rho)t}\leq 1$ and so $$X(T)=x_0+ \int_0^T \frac{ \int_\tau^T \lambda_1(T) w'(x(s))e^{-rs}ds}{2 (\xi \lambda_1(T) e^{-r\tau}+ \eta e^{-\rho \tau})} d\tau \geq x_0+ \frac{ \lambda_1(T) w'(l)}{2 (\xi \lambda_1(T)+ \eta )} \int_0^T e^{r \tau} \left(\int_\tau^Te^{-rs}ds \right)d\tau,$$ where the integral evaluates to $\frac{1}{r}\left[ T-\frac{1}{r}+\frac{e^{-rT}}{r} \right]$.
According to the results from section \ref{sec:termasset}, the expression $\frac{ \lambda_1(T)}{2 (\xi \lambda_1(T)+ \eta )}$ does not tend to zero as $T \rightarrow \infty$, so $\lim_{T \rightarrow \infty}X(T)=\infty$, which contradicts the fact that $X(T)$ is bounded.
For $\rho < r$, we study three cases according to the behaviour of $\lambda_1(T)$: \begin{enumerate}
\item[i)] $\rho \in (r(1-\theta),r)$. In this case $\lim_{T \rightarrow \infty}\lambda_1(T) = const$,
\item[ii)] $\rho \in (0,r(1-\theta))$. In this case $\lim_{T \rightarrow \infty}\lambda_1(T) = \infty$.
\item[iii)] $\rho = r(1-\theta)$. In this case $C_1 (1+T)^\theta \leq
\lambda_1(T)$. \end{enumerate} For case i) we have $$X(T) \leq x_0+ \frac{ \lambda_1(T) w'(x_0)}{2 \eta} \int_0^T e^{\rho \tau} \left(\int_\tau^Te^{-rs}ds \right)d\tau \leq x_0 + \frac{ \lambda_1(T) w'(x_0)}{2 \rho(r-\rho)\eta},$$ after taking into account that the integral evaluates to $\frac{1}{r}\left[\frac{1}{r-\rho}-\frac{r}{(r-\rho)\rho}e^{-(r-\rho)T}+\frac{e^{-rT}}{\rho}\right]$. It is clear that, for instance, for large values of $\eta$ this upper bound on $X(T)$ can be strictly smaller than $x_1$.
In case ii), defining $A := r(1-\theta)-\rho > 0$, we have from section \ref{sec:termasset} $$C_1 e^{AT}\leq \lambda_1(T) \leq C_2 e^{AT} \textrm{ with } C_i>0,~i=1,2.$$ Consequently, assuming that $l<x_1$, we have \begin{equation} X(T) \geq x_0+\frac{C_1 w'(l)}{2r}\int_0^T e^{AT} \frac{e^{-r\tau}-e^{-rT}}{\zeta e^{AT}e^{-r\tau}+\eta e^{-\rho \tau}}d\tau ,\label{eq:XTlowbound}\end{equation} where $\zeta := \xi C_2 >0 $. Denote the integral in \eqref{eq:XTlowbound} by $I$. We have $$I = \int_0^T \frac{1-e^{r\tau}e^{-rT}}{\zeta +\eta \frac{e^{(A+r\theta)\tau}}{e^{AT}}}d\tau \geq \int_0^{\frac{A}{A+r\theta}T} \frac{1-e^{r\tau}e^{-rT}}{\zeta +\eta \frac{e^{(A+r\theta)\tau}}{e^{AT}}}d\tau .$$ Since $e^{-AT}e^{(A+r\theta)\tau}\leq 1$ for $\tau \in [0, AT/(A+r\theta)]$, we have from the last expression $$I \geq \int_0^{\frac{A}{A+r\theta}T} \frac{1-e^{r\tau}e^{-rT}}{\zeta +\eta }d\tau = \frac{A}{(A+r\theta)(\zeta+\eta)}T -\frac{e^{-rT}}{\zeta + \eta}\cdot \frac{e^{\frac{A}{A+r\theta}rT}-1}{r}.$$ The last expression tends to infinity as $T \rightarrow \infty$, implying that $X(T)$ is unbounded. This contradiction shows that $l = x_1$.
In case iii) the condition from section \ref{sec:termasset} is equivalent to $$\frac{1}{\lambda_1(T)}\leq \frac{1}{C_1 (1+T)^\theta}.$$ Assume that $l < x_1$. Then \begin{equation*} \begin{split} X(T) \geq & x_0+\frac{w'(l)\lambda_1(T)}{2r}\int_0^T \frac{1-e^{-rT}e^{r\tau}}{\xi \lambda_1(T)+\eta e^{r\theta\tau}}d\tau \geq x_0 + \frac{w'(l)}{2r}\int_0^T \frac{1-e^{-rT}e^{r\tau}}{\xi + \frac{\eta e^{r\theta\tau}}{C_1 (1+T)^\theta}}d\tau = \\ & x_0 + \frac{w'(l)}{2r\eta}C_1 (1+T)^\theta \int_0^T \frac{1-e^{-rT}e^{r\tau}}{\xi \frac{C_1 (1+T)^\theta}{\eta} + e^{r\theta\tau}}d\tau . \end{split} \end{equation*} Set $B=B(T):= \xi \frac{C_1 (1+T)^\theta}{\eta}$ and introduce the change of variables $\mu=e^{r\theta\tau}$ in the last expression to obtain $$X(T) \geq x_0+const (1+T)^\theta \int_1^{e^{r\theta T}} \frac{1-e^{-rT}\mu^{\frac{1}{\theta}}}{r\theta \mu (\mu+B)}d\tau.$$ This requires us to study the behaviour of two expressions.
First, we have $$ (1+T)^\theta \int_1^{e^{r\theta T}} \frac{d \mu}{\mu(\mu + B)} = (1+T)^\theta \frac{1}{B}\left[ \ln\left( \frac{1}{1+\frac{const (1+T)^\theta}{e^{r\theta T}}} \right) + \ln \left( 1+const (1+T)^\theta \right) \right].$$ When $T \rightarrow \infty$, the first logarithm tends to zero and the second one tends to infinity, i.e. the whole expression tends to infinity.
Second, note that $$0 \leq (1+T)^\theta e^{-rT} \int_1^{e^{r\theta T}} \frac{\mu^{\frac{1}{\theta}}}{\mu(\mu + B)}d \mu \leq \frac{(1+T)^\theta}{ e^{rT}} \int_1^{e^{r\theta T}} \frac{\mu^{\frac{1}{\theta}}}{\mu^2}d \mu = \frac{\theta}{1-\theta}\left[ \frac{(1+T)^\theta}{e^{r\theta T}}-\frac{(1+T)^\theta}{e^{r T}} \right].$$ The last expression tends to zero as $T \rightarrow \infty$.
Combining the above results, we obtain $X(T) \rightarrow \infty$, which contradicts the fact that $X(T)$ is bounded. Thus, in this case $l = x_1$. \endprf
\end{document} | arXiv |
3rd Grade /
Unit 2: Multiplication and Division, Part 1 /
Multiplication and Division, Part 1
Unit 2: Multiplication and Division, Part 1
Topic A: The Meaning of Multiplication and Division
Identify and create situations involving equal groups and describe these situations using the language and notation of multiplication.
Identify and create situations involving arrays and describe these situations using the language and notation of multiplication.
Identify and create situations involving unknown group size and find group size in situations.
Identify and create situations involving an unknown number of groups and find the number of groups in situations.
3.OA.B.6
Relate multiplication and division and understand that division can represent situations of unknown group size or an unknown number of groups.
Topic B: Multiplication and Division by 2, 5, and 10
Build fluency with multiplication facts using units of 2, 5, and 10.
Demonstrate the commutativity of multiplication.
Build fluency with division facts using units of 2, 5, and 10.
Solve one-step word problems involving multiplication and division using units of 2, 5, and 10.
Topic C: Multiplication and Division by 3 and 4
Build fluency with multiplication and division facts using units of 3.
Solve one-step word problems involving multiplication and division using units of 3 and 4.
Topic D: More Complex Multiplication and Division Problems
3.OA.D.8
Determine the unknown whole number in a multiplication or division equation relating three whole numbers, including equations with a letter standing for the unknown quantity.
Solve one-step word problems involving multiplication and division and write problem contexts to match expressions and equations.
Solve two-step word problems involving multiplication and division and assess the reasonableness of answers.
Solve two-step word problems involving all four operations and assess the reasonableness of answers.
Anchor Tasks
Problem Set & Homework
Additional Practice
3.OA.A.1 — Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
3.OA.C.7 — Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
2.NBT.A.2 — Count within 1000; skip-count by 5s, 10s, and 100s.
2.OA.C.3 — Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
2.OA.C.4 — Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
Skip-count by twos, fives, and tens.
Solve multiplication problems involving twos, fives, and tens by skip-counting, keeping track on their papers or on their fingers of how many twos, fives, or tens have been counted, stopping the skip-counting sequence when they reach the number of twos, fives, or tens they intend to count, knowing that the last number said in the skip-counting sequence is the solution.
Students should have mastered counting by 2s, 5s, and 10s in Grade 2 (2.OA.3). If they haven't, however, you can help students toward mastery of counting by twos with the following process. Instead of having students say all the numbers in the count sequence, have them hum, clap, or make some sort of noise in place of the number names that aren't part of the count sequence, just thinking about what those numbers are instead. For example, for the skip-count sequence for twos, students would do: <clap> "two!" <clap> "four!" <clap> "six!" <clap> "eight!" etc. Then, they can remove the interspersed sounds and just say the skip-counting sequence. This provides a nice scaffold for students to use a placeholder for those numbers not in the count sequence before removing them entirely.
Students will only see multiplication problems where 2, 5, and/or 10 is the second factor, i.e., the size of the group. Students will explore the commutative property in Lesson 7, allowing them to solve multiplication problems where 2, 5, and 10 are the first factor, i.e., the number of groups.
Skip-counting to solve a multiplication problem can be more challenging than skip-counting to solve a division problem. As the OA Progression notes, "for $$8\times 3$$, you know the number of 3s and count by 3 until you reach 8 of them. For $$24\div 3$$, you count by 3 until you hear 24, then look at your tracking method to see how many 3s you have. Because listening for 24 is easier than monitoring the tracking method for 8 3s to stop at 8, dividing can be easier than multiplying" (OA Progression, p. 25). Thus, giving students ample time to practice will be greatly beneficial, and practice can be even more targeted in Lesson 7 when students apply the commutative property to solve even more multiplication problems.
As a supplement to the Problem Set, we recommend 2 additional games you can play with students.
"Double Up" from Building Conceptual Understanding and Fluency Through Games by the North Carolina Department of Public Instruction to review twos facts. (You'll need to modify it so that students are focusing on their twos facts within $$10\times 2$$, and you could modify the game so that it instead focuses on fives facts or tens facts.)
"Charlotte Speedway Race" from Building Conceptual Understanding and Fluency Through Games by the North Carolina Department of Public Instruction to review twos and fives. (The directions can be changed to have students give a multiplication fact using 2, 5, or 10 (instead of just 2 or 5).
If you need to adapt or shorten this lesson for remote learning, we suggest prioritizing Anchor Task 2 (benefits from worked example). Find more guidance on adapting our math curriculum for remote learning here.
Solve.
a. $$9\times 2 =$$ _____
b. _____ $$= 5 \times 2$$
c. $$2\times 5 = $$ _____
d. _____ $$= 9\times 5$$
e. $$4\times 10 =$$ _____
f. _____ $$= 10\times 10$$
Problem Set Answer Key
Answer keys for Problem Sets and Homework are available with a Fishtank Plus subscription.
Homework Answer Key
Discussion of Problem Set
Look at #3d and #6d. What do you notice? What do you wonder?
Why do you think it is that when you skip-count by twos you say all the even numbers?
Is Rob's reasoning correct in #11? Why is place value understanding helpful when multiplying by ten?
What do you notice about #12a and #12b? What does it make you wonder?
Look at #12c. Why do you think 4 x 5 is equivalent to 2 x 10?
1. $$8\times10=$$ _____ 2. _____ $$= 9\times 5$$ 3. $$7\times 2 =$$ _____
Karen counts, "5, 10, 15, 20, 25, 30." Write a multiplication sentence that corresponds with this count-by.
Unit Practice
With Fishtank Plus you can access our Daily Word Problem Practice and our content-aligned Fluency Activities created to help students strengthen their application and fluency skills. | CommonCrawl |
Kimbune's theorem (on The Tyrant Baru Cormorant , part 6)
home criticism Baru Cormorant
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This is the sixth part of a series of articles on The Tyrant Baru Cormorant —part review, part meta, part commentary. For intro and links to the others, go here!
Depending on how much you'd like to read about Euler's formula, this is either a short article or a long one.
The number of interest
Kimbune's Theorem
The first ingredient: exponential functions and calculus
The second ingredient: Taylor series
The third ingredient: trigonometry
The fourth ingredient: complex numbers and the path of inventing new numbers
Cooking Kimbune's theorem
The consequence of Kimbune's thoerem
From Kimbune's theorem to ecology
The danger of over-abstraction
In taking this ecological reading of culture, I am reminded of something Seth once wrote on their Twitter—unfortunately I can't get the direct quote—about the parallels between cancer and colonialism: a part of an organism or community ceasing to participate in symbiosis, but attempting to gather all resources to itself. (Seth said it better.)
So let's look at the thing that unites all the powers, and evils, explored by Baru Cormorant books. Powers such as…
disease epidemics
radioactive materials
What do these have in common? In mathematical terms, there's one striking point of unity: exponential growth and decay.
What's that, then? Exponential growth is a pattern which manifests itself whenever the rate of growth of something is proportionate to how much is already there.
That may sound a bit confusing, so let's consider an example. Imagine a population of bacteria (with plenty of tasty sugars to eat). Every hour, each bacterium splits into two bacteria. The more bacteria there are, the more will be born each generation. When you only have two bacteria, the population grows very slowly… but as the colony gets bigger, the bacteria appear faster and faster. If we count the bacteria every hour, over a number of hours we'll see a curve like this:
This is the the exponential function, \(\exp(x)\), or \(e^x\). Though there are many exponential functions, like \(2^x\) or \(10^x\), they can all be related back to this one.
Exponential functions are of deep importance to capitalist economics, because the terrifyingly powerful engine at the heart of capitalism—at least, once the brutal primitive accumulation phase has seized control of land and materials through direct violence—is the cycling of profits back into making more stuff than the last time around ('expanded reproduction' in the language of Marx).
All societies must reproduce themselves, and all growing societies produce more than they need to just continue to exist—but capitalism made this growth the core of what a society is. Everything in capitalism is keyed to an assumption of exponential economic growth, from the interest on a loan to the expectation of annual profits from a company. So if any part of the system—a company, for example—can't keep up with the pace of growth, it will be crowded out by its competitors, starved of funds or bought up and remade.
But it's not just capitalism. Diseases, too, spread exponentially at first through a large population—something we're all too aware of in the age of COVID-19. Cancer cells, not subject to programmed cell death, reproduce themselves on a similar trajectory (although, pedantically) the specifics vary for different cancers, and some only grow at the surface of a tumour).
And radiation? Radioactive decay is something of the opposite: every atom has a random chance to decay in every instant, so the more atoms there are, the more quickly they disappear. In the specific case of uranium, so beloved of the Cancrioth, the most common isotope (by far) has a half life in the billions of years—slow enough that it hasn't all decayed already, fast enough that, in abundance, it creates some serious activity.
(While we're at it… in a nuclear weapon, which even the Cancrioth have yet to imagine, the nuclear chain reaction exploits exponential growth in the other direction: one neutron becomes two, becomes four, each time releasing more energy until, in an instant, most of the fissile material has transformed and all that energy is ready to incinerate a city.)
Perhaps alone among the cast, the Brain is aware of the terror of exponential growth on the scale of societies:
"They understand the secret of power, Baru."
"The ability to improve one's own power, no matter how slowly, triumphs in the long run over any other power. Time magnifies small gains into great advantages. If you are hungry, then it is better, in the long run, to plant one seed than to steal a pound of fruit. Falcrest applies this logic in all their work. They do not conquer. They make themselves irresistible as trading partners. They do not keep their wealth in a royal hoard. They send it out among their people, stored in banks and concerns, where it helps the whole empire grow. They do not wait to treat the sick. They inoculate against the disease before it spreads. All their power sacrifices brute strength in the present for the ability to capture a piece of the future."
The thing about exponential growth is that, though it starts out apparently slow, once it gets into motion it is the fastest-growing mathematical function we routinely encounter. This is what makes disease outbreaks so scary—and it's what makes capitalism have such force.
Exponential economic growth is Falcrest's meta-weapon, but—Barhu eventually comes to believe—it is the weapon that can be turned against them too.
But the interest in exponentials comes at a different angle, too. While visiting the Cancrioth, Baru runs into a mathematician who is determined to track down Abdumasi Abd for a different reason than most: in Abdumasi's tumour is, she believes, the soul of her husband, who died before she could win an argument. And what's this argument about?
It's about Euler's formula. You know:
\[e^{i\pi} + 1 = 0\]
When I saw that formula on the page, I was like… Seth you absolute dork. Oh, sure, she's invented the "most beautiful theorem in mathematics" (as decided by vote)…
Then, I ended up spending a very fun afternoon introducing some friends to the significance of this formula, and started thinking about why it would be here.
In the book, Kimbune's formula comes across to Baru (who can't understand the proof) as a bizarre connection between unrelated, but important numbers: an indication of the numerical structure of the universe, that Falcrest can't perceive. But Baru, not a pure mathematician, does not grasp the proof, nor the heart of the formula, which is better rendered in the more general form:
\[e^{i\theta} = \cos\theta + i \sin \theta\]
Naturally I came up with a reading of the book's broader themes in relation to Euler's formula. But first, to get everyone on the same page, I need to explain the recipe. Since this is a long aside not exactly about Baru Cormorant , it gets the box.
A recipe for Kimbune's Theorem
Euler's/Kimbune's formula requires a lot of conceptual leaps to understand. We've talked about exponential growth and decay: but what this formula creates is a connection between that growth and circular, or more generally repeating, motion. And in so doing, it creates the fundamental tool that we use to calculate with complex numbers, which ended up becoming vital to just about every branch of physics and maths.
You can read my attempt to introduce this subject below. Or, if you prefer, you can watch one of the excellent videos by Grant of 3blue1brown, such as this short one or this longer one, which present it perhaps more visually and intuitively than I can do here.
We have a few topics to (re)introduce: complex numbers, exponential functions, trigonometry and Taylor series. Or, in Barhu's world, the Impossible Number, the Number of Interest, and the Round Number. (She doesn't mention calculus—it's not clear if the Masquerade has it!—but the usual proof of Euler's/Kimbune's theorem is through a tool of calculus called a Taylor series.)
To begin: a little more on the 'number of interest'. We spoke of the breeding elephants: but to understand what makes \(e^x\) special compared to, say, \(2^x\), we need to deal with not discrete, but continuous functions.
So let's follow the path of Jacob Bernoulli, studying compound interest in 1683. We imagine a bank account, accruing interest over time. Let's imagine an extremely generous bank awards interest at 100%. If you have £100 in your bank account, at the end of the year, they give you 100% more money, and you have £200.
Another, even more generous bank might offer 50% after 6 months, and another 50% (of whatever's in your account after the first payment) after the full year. Despite seeming to add up to the same 100%, this is a better deal. After the first payment, you have £150; after the second payment, you get an extra 50% of the first payment, so the total you have is:
\[£150\times1.5=£225\]
A pretty nice increase! And over time, this increase will turn into a bigger increase. The same goes for 'turnover' of a capitalist's goods as they're sold, and the proceeds reinvested into production.
Now, you can imagine slicing up the year into smaller and smaller slices, until the amount of money is continually changing by infinitesimal amounts. So if you get \(N\) interest payments each year, the amount you'll have after \(i\) payments will be
\[£100 \times \left(1+\frac{1}{N}\right)^i\]
Interest payments
And the limit of getting "infinitely small payments all the time" gives you a special, smooth curve. In this limit, by the end of the first year, the money has grown by a factor of
\[e=2.71281828459\dots\]
This number is special: its decimal expansion goes on forever, without repeating itself. It can't be written as a fraction: the best we can do is describe a process, like the limiting process above, which slowly gets closer to the value of \(e\).
Why is this number important? Instead of a complicated limit, Bernoulli's fancy "infinitely frequent compound interest" formula can be calculated by this number, raised to the power of the number of years that pass.
\[\text{money}(t)=£100 \times e^\frac{t}{1\text{year}}\]
We've just done some calculus, by taking a 'limit', and ended up with an exponential function: a number raised to the power of the input.
But there's another, much nicer way to look at this particular exponential function than Bernouilli's method. Calculus tells us a way to measure how quickly something is changing. If we have something called \(f\) which depends on time, then we can do a similar process of slicing time into smaller and smaller parts, and looking at how \(f\) changes. This leads us to a number called the rate of change or first differential of \(f\), which we might write
\[f'(t)\]
if we like Newton, or
\[\df{f}{t}\]
if we're more into Leibniz.
This particular exponential function has one very special unique property. Its rate of change is always equal to the function itself:
\[\df{}{t} e^t = e^t\]
This makes it incredibly convenient, expecially when it comes to 'differential equations'. A differential equation comes up in a situation where we know the rules that describe how something changes: for example, the number of new elephants born in a year will always be a certain fraction of the number of living elephants. Or, the average number of particles that radioactively decay per unit time will be a certain proportion of the total. The hotter something is, the faster its heat spreads to the surroundings.
All those situations I just mentioned are variants on this equation:
\[x'(t)=kx(t)\]
This says that the rate that something, measured as \(x\), is changing—growing or shrinking—is proportional to the size of the thing itself (and \(k\) is just a number to say how strong the connection is). A bigger thing grows or shrinks faster than a smaller thing. We can always solve this kind of equation with the exponential function, scaled up or down by some factor.
Calculus gives us a special trick: we can approximate smoothly varying functions, with something that is (often) easier to calculate. We take a starting point—the time \(t=0\), say—and build up a 'power series', so that the rate of change, the rate of change of the rate of change, the rate of change of the rate of change of the rate of change, etc., all matches our function.
For example, a power series might look like:
\[1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}\dots\]
At \(x=0\), the value of this function is 1, the rate of change is 1, the rate of change of the rate of change is 1, and so on all the way down. Which is exactly the same as the exponential function… and that's not a coincidence, this power series is just another way of writing the exponential function!
Most useful functions can be approximated by a power series as long as we're near enough to the point where we make the approximation. But \(e^x\) is very special, because the approximation is good for the entire number line! The further away from \(x=0\), the more terms in the power series you need to add up, but add up enough terms and you can get as close as you like to \(e^x\).
Which tells us why exponentials grow so quickly, faster than every 'polynomial' (function of powers, such as \(x^5 + x^4+3x^2\)). No matter how high the powers in a polymonial, \(e^x\) contains a higher one.
How did we find this Taylor series, which I seemingly pulled out of a hat? We use the rate of change. If you have a power of \(x\), such as \(x^5\), you can easily find its rate of change: multiply by the exponent, then drop the exponent by one. For example, the rate of change of \(x^5\):
\[\df{}{x} x^5 = 5 x^4\]
and in general…
\[\df{}{x} x^k = k x^{k-1}\]
This is all we need. Suppose we have some function, and we know the value of the function, its rate of change, the rate of change of the rate of change, etc. In less cumbersome language, its first, second, third, fourth… differentials. And suppose we suspect it's equal to a power series:
\[f(x) = k_0 + k_1 x + k_2 x^2 + k_3 x^3 + \dots\]
where the \(k_0\), \(k_1\), etc. are numbers whose values we do not know. If we calculate the first, second, etc. differential of this power series, we get another power series.
\[\begin{align*} f'(x) &= k_1 + 2 k_2 x + 3 k_3 x^2 + 4 k_4 x^3 + \dots\\ f''(x) &= 2 k_2 + (3 \times 2) k_3 x + (4 \times 3) k_4 x^2 + (5 \times 4) k_5 x^5 \dots \\ \end{align*}\]
Now, when we set \(x\) to zero, all the terms of every power series disappear… except the one at the beginning! This gives us an expression with just one \(k\) constant in it:
\[\begin{align*} f(0) &= k_0\\ f'(0) &= k_1\\ f''(0) &= 2 k_2\\ f^{(3)}(0) &= 6 k_2\\ f^{(n)}(0) &= n! k_n \end{align*}\]
So if we know all the differentials at zero, we can find a general expression for the power series equivalent to any function (assuming certain technicalities hold up…)
\[f(x) = f(0) + f'(0) x + f''(0) \frac{x^2}{2} + \dots + f^{(n)}(0) \frac{x^n}{n!} + \dots\]
This is known as the Taylor series around zero. For \(e^x\), we've seen that all the differentials are equal to 1:
\[f^{(n)}(0)=1\]
So our power series is nice and simple. This form of \(e^x\) is the easiest way to discover Kimbune's theorem.
Now, let's look at another branch of maths which Baru would surely have encountered: the geometry of triangles, circles, and trigonometric functions.
The trigonometric functions are, essentially, ways to describe points on a circle. Let's say you start walking along a circle of radius 1—we call this the 'unit circle'. To walk half the way round the circle is a special distance, which we give the name \(\pi\) (the Greek letter pi). Like \(e\), \(\pi\) goes on forever… it is, somehow, deeply baked into the geometry of flat space. Its value is approximately…
\[\pi=3.141592653589793238...\]
And if we keep going around the circle, we'll have eventually walked a distance \(2\pi\). In the meantime, we will move through a series of \((x,y)\) positions. Assuming we start at \((x,y)=(1,0)\), and walk anticlockwise, then after we've walked a distance \(\theta\), we will be at some specific position \((x(\theta),y(\theta))\) on the circle. For example, if we walk a distance \(\frac{\pi}{6}\), we'll be at position
\[\left(x\left(\frac{\pi}{6}\right),y\left(\frac{\pi}{6}\right)\right)=\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)\]
The positions we go through around the circle are have been given names, defining two functions, called the 'cosine' and the 'sine' function for historical reasons. So after we've walked a distance \(\theta\), we're at position
\[(x,y)=(\cos\theta,\sin\theta)\]
What's so important about this? These functions, \(\cos \theta\) and \(\sin \theta\), turn out to have far more uses than just examining circles. (Kimbune's theorem helps tell us why). We can understand this by looking at the rate of change. While the rate of change of \(e^x\) was just \(e^x\), calculating the rate of change of \(\cos\) and \(\sin\) turns one into the other:
\[\begin{align*} \df{}{t}\cos t &= - \sin t\\ \df{}{t}\sin t &= \cos t \end{align*}\]
So if we find the rate of change of the rate of change—acceleration, as opposed to velocity—we get back to where we started, with a minus sign:
\[\begin{align*} \df{^2}{t^2}\cos t &= - \cos t\\ \df{^2}{t^2}\sin t &= - \sin t \end{align*}\]
And this fact has a useful consequence. Let's think about a spring. The further we stretch the spring, the stronger the force pulling it back to its original length. The acceleration of the end of the spring depends on the force. So we get another kind of differential equation:
\[\df{^2x}{t^2}=-k x\]
This kind of differential equation is called the harmonic oscillator, and it's tremendously important to physics—to the point that it's been joked that physics is just finding new ways to solve the harmonic oscillator. This is because all sorts of problems can be approximated by harmonic oscillators.
Because their differentials are so nice, and because \(\cos 0=1\) and \(\sin 0=0\), the trigonometric functions also have very helpful Taylor series. They essentially divide up the terms of the exponential function's Taylor series between each other. Sine takes all the odd powers, and cos, the even. Only, the signs (plus or minus) alternate.
\[\begin{align*} \cos x &= 1 - \frac{x^2}{2} + \frac{x^4}{24} \dots\\ \sin x &= x - \frac{x^3}{6} + \frac{x^5}{120}\dots \end{align*}\]
You might be wondering… if their power series are so similar, is there some deeper relationship between the trigonometric functions and the exponential function? There absolutely is, but to find it, we need to make one of the most profound leaps in mathematical history.
The thing we're about to describe is known as the 'imaginary number'. The name betrays the deep discomfort of the mathematicians who called it into being, but in fact the imaginary number is no more imaginary than any other kind of number.
To discover the imaginary number, let's pretend we know nothing but counting, and adding things up. All the numbers we know about are the natural numbers:
\[\mathbb{N}=\{0,1,2,\dots\}\]
Given two natural numbers, we can 'add' them together and get a third natural number: for example,
\[2+3=5\]
Now, let's suppose we want to ask a question like: what number, added to 3, will give 5? We can write this as:
\[3+x=5\]
and we want to discover the value of \(x\). Sometimes, there is an answer: in this case it's \(x=2\). But we want a general method that will always get us the answer; and moreover, we want to be able to handle cases like
which have no valid answer in the numbers we know.
To solve this problem, we create new numbers. For every natural number \(n\), we create a matching number written \(-n\), with the property that:
\[n+(-n)=0\]
We call these numbers the 'additive inverses' of the natural numbers, because by adding them to the natural numbers, they cancel out to the 'additive identity', zero.
These new numbers mean that we're no longer just working with the set of natural numbers. Now we have the integers:
\[\mathbb{Z}=\{\dots,-3,-2,-1,0,1,2,3,\dots\}\]
Suitably armed with these new numbers, we can solve all equations along the line of "if \(a+x=b\), what is \(x\)?"—at least, assuming \(a\) and b$$ are all integers. Good trick. But you know that's not the only kind of numbers.
Turns out you can build the entire number system wtih variants on this one weird trick!
For example, let's invent multiplication. The most basic kind of multiplication is just an instruction to repeatedly add up natural natural numbers:
\[x \times 5 = x + x + x + x + x\]
And we can work out how to multiply negative numbers, in order to make multiplication work in a way that is consistent. We find interesting rules like \((-x)\times(-y)=+(x\times y)\), which is pretty sweet.
But we soon hit a roadbump. Suppose we want to solve questions like, is there a number \(x\) which will make an equation like
\[x \times 3 = 6\]
turn out to be true? (Yes, in this case! If \(x=2\), the equation is true.)
Soon, we'll hit on the problem that, with the numbers we currently have, we can't solve certain equations. For example, try
We don't have any integer to solve this one!
So let's pull out the same trick, and create new numbers. Now, for every integer \(x\), we create a new number called its multiplicative inverse, written \(\frac{1}{x}\) with the property :
\[x \times \frac{1}{x}=1\]
Then, we multiply all these inverses with the existing integers to get even more new numbers, such as \(\frac{3}{2}=3\times\frac{1}{2}\). In this way, we build up the whole system of 'fractions' (which are just all the possible products of an integer with the inverse of an integer)—and now we're in yet another new set of numbers, called the rational numbers.
While we're building out the rational numbers, we might notice something interesting. If we work out the rules, we find that \(\frac{3}{2}\) is the exact same number as \(\frac{6}{4}\). Suddenly, we have infinitely many different ways of writing a particular number. \(2\) could also be written \(\frac{2}{1}\) or \(\frac{14}{7}\) etc. etc. But that's not all that much of a problem.
We've come quite a long way from our original natural numbers!
Why this long detour? Well, by looking back like this at the most 'basic' kinds of number, we can see that inventing new numbers to solve a problem is really nothing new. However, because all of these ideas were invented pretty early in human history, nobody doubts that fractions or negative numbers are 'real' kinds of number.
Now, let's consider one more operation: squaring, i.e. multiplying a number by itself. As before, we might want to solve certain kinds of equation, such as:
\[x^2=4\]
This one can be solved in two different ways: \(x=2\) and \(x=-2\). And other cases, such as
\[x^2=\frac{9}{4}\]
can be solved with fractions, in this case \(\frac{3}{2}\) and \(\frac{-3}{2}\).
But surprisingly enough, we often find equations which can't be answered using a fraction. For example:
can't be solved by any rational number! There's no fraction which we can multiply by itself, and end up with 2.
We have to invent yet more new numbers to solve this problem. These numbers, called 'radicals', lie 'in between' the rational numbers. Although they aren't rational numbers, we can still approximate them by adding together a series of rational numbers. For example, the positive square root of 2—the number which, when multiplied by itself, gives 2—has the 'decimal expansion' starting:
\[\sqrt{2}=1.41421356\dots\]
A decimal expansion is just a list of rational numbers to add up: \(1\), \(\frac{4}{10}\), \(\frac{1}{100}\), etc. But when you have an infinitely long list, you can find your way to numbers that aren't rational.
By adding the radicals to our number system, we've found our way to the 'algebraic numbers'. Although the algebraic numbers aren't fractions, which caused consternation to ancient Greeks trying to build them with rulers and compasses, they at least have a clear place in between the fractions. Given any fraction \(f\), we definitely can say whether \(\sqrt{2}\) is larger or smaller than \(f\).
So have we now discovered all the numbers? Not quite! When we added the radicals, we opened the door to numbers which aren't fractions, but which we can get to by adding infinitely long lists of fractions together. Turns out, there are a whole lot of decimal expansions which aren't given by square roots (or cube roots, etc.),
Well, we can go ahead and patch this hole. Let's declare that every decimal expansion now leads us to a number. This includes the numbers like \(\pi\) and \(e\) which we've met already. This does mean there are, once again, multiple ways to write down the same number: the number \(1\) can now also be written \(0.99999\dots\), because
\[\frac{9}{10}+\frac{9}{100}+\frac{9}{1000}+\dots=1\]
Like a wizard, we've called a whole lot of numbers into being. We can draw strict conclusions about what these numbers do, and how they relate to other numbers. (Incidentally, doing this is actually a pretty profound step in itself. If we ask 'how many real numbers are there', the answer is obviously 'infinitely many'. But in a very bizarre twist, it turns out there's different kinds of infinity, and the number of real numbers is a bigger kind of infinity than all the other sets of numbers we've seen so far.)
Have we, at last, found all the numbers we're likely to need? Unfortunately, not quite! And this is where we get towards the amazing insight that Baru sees in Kimbune's theorem.
Sometimes, the solution to an equation like
\[x^2=-9\]
…is not somewhere in the 'gaps' in the number line. It's not 3, since \(3^2=9\). But it also can't be \(-3\), since \((-3)^2=9\) as well.
Hopefully we're now comfortable with what we have to do: we invent! more! numbers!
In fact, the first step is to just invent one new number, the square root of \(-1\). We call this number \(i\), or the imaginary unit. All the other square roots of negative numbers are just multiples of \(i\). The square roots of \(-4\), for example, are \(2i\) and \(-2i\).
That name, 'imaginary number' suggests there's something distinct about \(i\) that makes it different from other kinds of number. But as we've seen, we made up just about every other kind of number to close a gap in the number system.
With \(i\), we can build all sorts of new numbers, such as \(5+3i\). These numbers don't live anywhere on the number line, but we can imagine they live in a 2D plane, known as the complex plane. We call them 'complex numbers', and the nice thing about them is that we can solve any polynomial with a complex number, even ones without real roots. We can still invent new systems of number if we like, such as quaternions, but we've finally plugged all the gaps.
And now, for our next trick, let's put all our ingredients together…
So we have our ingredients:
the exponential function
From here, it's easy. Plug an imaginary number—let's say \(i\theta\) where \(theta\) is some real number—into the power series for the exponential function, and apply the rule that \(i^2=-1\):
\[e^{i\theta}=1+i\theta - \frac{\theta^2}{2} - i \frac{\theta^3}{6} + \frac{\theta^4}{24} + i \frac{\theta^5}{120} \dots\]
Can you see the power series for the trigonometric functions hiding in there? Let's rearrange the power series a little…
\[e^{i\theta}=\left(1-\frac{\theta^2}{2} + \frac{\theta^4}{24}+\dots\right)+i\left(\theta - \frac{\theta^3}{6} + \frac{\theta^5}{120} + \dots\right)\]
The power series for this exponential function is the same as the power series for \(\cos \theta\), plus the imaginary unit times the power series for \(\sin \theta\). Brushing over some technical details regarding convergence, we've proven the following amazing thing…
\[e^{i\theta}=\cos \theta + i \sin \theta\]
This is Kimbune's formula (or Euler's, in the boring world). It does a few things. It tells us how to handle complex numbers in exponents, which was not at all obvious. It gives a fascinating geometric interpretation of complex numbers, as a kind of polar coordinates. It leads into the whole terrifying world of complex functions, which add some fascinating headaches. But we're getting away from the main point…
That relentless exponential growth and decay? Kimbune's theorem turns it 'sideways', transforming real numbers into imaginary numbers and back, in an endless circle.
This short video does a good job of illustrating this concept:
Now, let's imagine a point roaming the complex plane. At each instant, we can draw a little arrow to see which way it's going (its rate of change). The bigger the arrow, the faster the change.
Exponential growth means that arrow points away from the origin. Exponential decay, back towards the origin.
And now we have a way to circle the origin, neither growing nor decaying, but always changing, ever faster the larger it is.
In Barhu's world, the number \(e\) is called the 'number of interest'. Interest is the result of a loan: it is money turned to make more of itself. But through Kimbune's theorem, the process of interest is turned aside: not to expand but to transform…
Perhaps this metaphor is kind of a stretch? I think it's a fun reading though, given what Barhu plans to do.
Although we've talked about how bacterial growth gives rise to economic growth, in the real world, very few lifeforms get to happily grow their population without limit… as often as not, there's a shortage of food… or something out there to eat them.
One of the simplest models involves a population of predators, and a population of prey animals. We can model this as a coupled system of differential equations: the population of predators rises as their food source increases, but predation wipes out the prey, and the starving predators die out. This video discusses a stripped down version of the problem:
Mathematicians have a tool for analysing this kind of problem: they move to a 'phase space', not so different from the complex numbers we've explored so far. One axis represents the number of prey, the other the number of predators, and the system 'moves' through phase space as the two populations vary.
Many choices of parameter result in orbits around a 'fixed point'. At the 'fixed point', the predators devour the prey as quickly as they are born, and themselves die off as quickly as they breed, and the two populations remain stable. Everywhere else, waves of prey breed without fear of predators, only to precipitate an explosion in the predator population which slaughters most of the prey… and then the predators starve, and the cycle begins again.
When we humans want to control a population of animals—for example, of deer—one of the most effective ways we've found is to introduce a population of predators to the region. And when we humans disrupt these cycles by, for example, killing off wolves for the sake of agriculture, the cycles of the ecosystem are disrupted.
Unfortunately, it's rarely as simple as this simple, two-component system. The explosion of prey populations will affect all the things the prey eat. One fascinating example is the reintroduction of wolves into Yellowstone. Prior to the reintroduction of wolves, elk populations had grown rapidly, threatening the reproduction of the things the elks ate, such as aspen trees. The failure of the aspen crashed the beaver population, and the lack of beaver dams caused further effects, increasing the variability of water runoff and removing places for fish to breed.
Reintroducing the wolves, on the other hand, made the elk start moving around a lot more—to avoid getting eaten! The aspen could bounce back, the beavers could bounce back, and so on; the effect was termed a 'trophic cascade'. The ecosystem started to edge back towards its rhythm of self-reproduction.
So, then, in this rather strained metaphor, the Masquerade's economic engine is like the population of elk. But there is no predator to keep it in check… not yet, anyway.
Yet at the same time, we don't want to disappear up our own asshole with this. The masquerade's problems aren't merely that its number (money) is growing faster than, say, the Mbo's number: it is that, on the strength of its ever-growing production, it can impose a powerfully self-replicating, horrifying eugenic terror regime on the people living in its shadow. Treating everything on the level of an abstract phase space is to ignore what is actually happening to the living beings inside that system.
Consider: a wolf chases a terrified elk, tears it limb from limb, and drags its corpse back to its cubs. The death of a deer, as Disney demonstrated with Bambi , can inspire a lot of empathy even in us humans. We've just explained how that is a really good thing for the perspective of a stable ecosystem, but it's hardly a good thing for the elk! The humans ruling over Yellowstone made the decision that they valued 'restoring' the ecosystem to a particular function enough that the painful death of a certain number of elk was an acceptable price to pay.
The same decision is made by those who set hunt quotas: the course of evolution has produced a world whose stability deeply relies on most lives teetering on the edge of sudden annihilation. Thinking too hard about this is what leads people to making grand declarations that the long term mission of humanity is to ascend to some kind of transhumanist omnipotence, and then reengineer death out of nature… or else to a radical break from the values of capitalist 'civilisation'. (Tain Shir says hi.)
What of Baru? Her plan is a little less abstract, but as we'll shortly see, the end she's pursuing is, for now, merely the economic destruction of Falcrest.
Will it be enough? Perhaps history can help tell us… | CommonCrawl |
If $\times$ means -, $\div$ means + , - means $\div$, + means $\times$, then 64 $\div$ 32 - 8 $\times$ 4 + 6 = ?
In the following list of English alphabets, one alphabet has not been used. Identify the same.
If south-east becomes north, then what will north-east become ?
Poining towards a girl Sameer said, She is the daughter of only son of my paternal grandfather. How is the girl related to Sameer ?
5 . In questions, a series is given, with one term missing. Choose the correct alternative from the given ones that will complete the series.
6 . In questions, a series is given, with one term missing. Choose the correct alternative from the given ones that will complete the series.
7 . In questions, a series is given, with one term missing. Choose the correct alternative from the given ones that will complete the series.
BCB, DED, FGF, HIH, ?
8 . In questions, a series is given, with one term missing. Choose the correct alternative from the given ones that will complete the series.
PBQ, QCR, RDS, SET, ?
9 . In questions, a series is given, with one term missing. Choose the correct alternative from the given ones that will complete the series.
O, 6, M, 13, K, 19, I , 26, G, ?, ?
If REASON is coded as 5 and BELIEVED as 7, what is the code for GOVERNMENT ? | CommonCrawl |
# Supervised Learning: Overview and Goals
Supervised learning is a type of machine learning where the algorithm learns from labeled data. The goal of supervised learning is to build a model that can predict the output based on the input data. This is useful in various applications, such as predicting stock prices, spam detection, and image recognition.
In supervised learning, the data is divided into two sets: a training set and a test set. The training set is used to train the model, while the test set is used to evaluate its performance.
Consider a dataset of housing prices with features such as the size of the house, the number of rooms, and the location. The goal of the supervised learning algorithm is to learn the relationship between these features and the price of the house.
## Exercise
Instructions:
- Use Python and the scikit-learn library to train a linear regression model on a dataset of housing prices.
- Evaluate the model's performance on a test set.
### Solution
```python
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
# Load the dataset
data = pd.read_csv('housing_prices.csv')
# Split the dataset into training and test sets
X = data.drop('price', axis=1)
y = data['price']
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Train the linear regression model
model = LinearRegression()
model.fit(X_train, y_train)
# Evaluate the model's performance on the test set
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
print(f'Mean Squared Error: {mse}')
```
# Linear Regression: Model, Evaluation, and Optimization
Linear regression is a supervised learning algorithm that models the relationship between a dependent variable and one or more independent variables. It is widely used for predicting continuous values.
The linear regression model can be represented by the following equation:
$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_n x_n$$
Where $y$ is the dependent variable, $x_1, x_2, ..., x_n$ are the independent variables, and $\beta_0, \beta_1, ..., \beta_n$ are the coefficients.
To evaluate the performance of a linear regression model, we can use metrics such as mean squared error (MSE) and R-squared.
To optimize the linear regression model, we can use techniques such as gradient descent and stochastic gradient descent. These techniques are used to find the best values for the coefficients that minimize the error between the predicted values and the actual values.
## Exercise
Instructions:
- Use Python and the scikit-learn library to train a linear regression model on a dataset of housing prices.
- Evaluate the model's performance on a test set using the mean squared error (MSE) and R-squared metrics.
### Solution
```python
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score
# Load the dataset
data = pd.read_csv('housing_prices.csv')
# Split the dataset into training and test sets
X = data.drop('price', axis=1)
y = data['price']
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Train the linear regression model
model = LinearRegression()
model.fit(X_train, y_train)
# Evaluate the model's performance on the test set
y_pred = model.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)
print(f'Mean Squared Error: {mse}')
print(f'R-squared: {r2}')
```
# Classification: Logistic Regression, Decision Trees, and Random Forest
Classification is a type of supervised learning where the goal is to predict a categorical output. There are several classification algorithms, including logistic regression, decision trees, and random forest.
Logistic regression is a classification algorithm that models the relationship between a dependent variable and one or more independent variables. It is used to predict binary outcomes.
Decision trees are a type of supervised learning algorithm that can be used for both classification and regression tasks. They work by recursively splitting the data into subsets based on the values of the input features.
Random forest is an ensemble learning method that combines multiple decision trees to make predictions. It is a popular algorithm for classification tasks.
## Exercise
Instructions:
- Use Python and the scikit-learn library to train a logistic regression model on a dataset of customer churn.
- Evaluate the model's performance on a test set using the accuracy metric.
### Solution
```python
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LogisticRegression
from sklearn.metrics import accuracy_score
# Load the dataset
data = pd.read_csv('customer_churn.csv')
# Split the dataset into training and test sets
X = data.drop('churn', axis=1)
y = data['churn']
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Train the logistic regression model
model = LogisticRegression()
model.fit(X_train, y_train)
# Evaluate the model's performance on the test set
y_pred = model.predict(X_test)
accuracy = accuracy_score(y_test, y_pred)
print(f'Accuracy: {accuracy}')
```
# Unsupervised Learning: Clustering Algorithms
Unsupervised learning is a type of machine learning where the algorithm learns from unlabeled data. The goal of unsupervised learning is to discover patterns or structures in the data.
Clustering is a type of unsupervised learning algorithm that is used to group similar data points together. There are several clustering algorithms, such as k-means and hierarchical clustering.
K-means is a popular clustering algorithm that works by iteratively assigning data points to clusters based on their similarity. The algorithm aims to minimize the within-cluster sum of squares.
## Exercise
Instructions:
- Use Python and the scikit-learn library to perform k-means clustering on a dataset of customer segmentation.
- Visualize the clusters using a scatter plot.
### Solution
```python
import pandas as pd
from sklearn.cluster import KMeans
import matplotlib.pyplot as plt
# Load the dataset
data = pd.read_csv('customer_segmentation.csv')
# Perform k-means clustering
k = 3
model = KMeans(n_clusters=k)
model.fit(data)
# Visualize the clusters
plt.scatter(data['feature1'], data['feature2'], c=model.labels_, cmap='viridis')
plt.xlabel('Feature 1')
plt.ylabel('Feature 2')
plt.title('K-means Clustering')
plt.show()
```
# Evaluation and Validation of Machine Learning Models
Evaluating and validating machine learning models is crucial to ensure their reliability and effectiveness. There are several evaluation metrics and validation techniques that can be used to assess the performance of a model.
Cross-validation is a validation technique that involves splitting the data into multiple subsets and using each subset as a test set. The model's performance is then averaged over all the subsets.
Confusion matrix is a table that is used to evaluate the performance of a classification model. It shows the true positive, true negative, false positive, and false negative counts.
## Exercise
Instructions:
- Use Python and the scikit-learn library to evaluate a logistic regression model on a dataset of customer churn using cross-validation and a confusion matrix.
### Solution
```python
import pandas as pd
from sklearn.model_selection import cross_val_score
from sklearn.metrics import confusion_matrix
from sklearn.linear_model import LogisticRegression
# Load the dataset
data = pd.read_csv('customer_churn.csv')
# Split the dataset into training and test sets
X = data.drop('churn', axis=1)
y = data['churn']
# Train the logistic regression model
model = LogisticRegression()
# Evaluate the model using cross-validation
scores = cross_val_score(model, X, y, cv=5)
print(f'Cross-validation scores: {scores}')
# Evaluate the model using a confusion matrix
y_pred = model.fit(X, y).predict(X)
cm = confusion_matrix(y, y_pred)
print(f'Confusion matrix: \n{cm}')
```
# Hands-on Examples and Applications
We will start by building a linear regression model to predict housing prices based on features such as the size of the house, the number of rooms, and the location.
Next, we will build a logistic regression model to predict customer churn based on features such as customer demographics and purchase history.
Finally, we will perform k-means clustering on a dataset of customer segmentation to group similar customers together.
## Exercise
Instructions:
- Follow the hands-on examples and applications in this section to build a linear regression model, a logistic regression model, and a k-means clustering model.
### Solution
```python
# Linear regression example
# ...
# Logistic regression example
# ...
# K-means clustering example
# ...
```
# Advanced Topics: Neural Networks and Reinforcement Learning
Neural networks are a type of machine learning algorithm that is inspired by the structure and function of the human brain. They are widely used for tasks such as image recognition, natural language processing, and game playing.
Reinforcement learning is a type of machine learning algorithm where an agent learns to make decisions by interacting with an environment. It is used in various applications, such as robotics, game playing, and self-driving cars.
## Exercise
Instructions:
- Use Python and a deep learning library such as TensorFlow or PyTorch to build a neural network model for a specific task, such as image classification or natural language processing.
- Implement a reinforcement learning algorithm, such as Q-learning or Deep Q-Networks (DQN), to train an agent to solve a specific problem, such as playing a game or navigating an environment.
### Solution
```python
# Neural network example
# ...
# Reinforcement learning example
# ...
```
# Table Of Contents
1. Supervised Learning: Overview and Goals
2. Linear Regression: Model, Evaluation, and Optimization
3. Classification: Logistic Regression, Decision Trees, and Random Forest
4. Unsupervised Learning: Clustering Algorithms
5. Evaluation and Validation of Machine Learning Models
6. Hands-on Examples and Applications
7. Advanced Topics: Neural Networks and Reinforcement Learning | Textbooks |
Regulation of Fat and Fatty Acid Composition in Beef Cattle
Smith, Stephen B.;Gill, Clare A.;Lunt, David K.;Brooks, Matthew A. 1225
https://doi.org/10.5713/ajas.2009.r.10 PDF KSCI
Fat composition of beef, taken here to mean marbling, can be manipulated by time on feed, finishing diet, and breed type. These three factors also strongly influence the fatty acid composition of beef. Both the amount of marbling and the concentration of monounsaturated fatty acids (MUFA) increase with time on feed in grain-fed and pasture-fed cattle, but much more dramatically in grain-fed cattle. High-concentrate diets stimulate the activity of adipose tissue stearoyl-CoA desaturase (SCD), which is responsible for the conversion of saturated fatty acids (SFA) to their $\Delta{9}$ desaturated counterparts. Also, grain feeding causes a depression in ruminal pH, which decreases those populations of ruminal microorganisms responsible for the isomerization and hydrogenation of polyunsaturated fatty acids (PUFA). The net result of elevated SCD activity in marbling adipose tissue and depressed ruminal isomerization/hydrogenation of dietary PUFA is a large increase in MUFA in beef over time. Conversely, pasture depresses both the accumulation of marbling and SCD activity, so that even though pasture feeding increases the relative concentration of PUFA in beef, it also increases SFA at the expense of MUFA. Wagyu and Hanwoo cattle accumulate large amounts of marbling and MUFA, and Wagyu cattle appear to be less sensitive to the effects of pastures in depressing overall rates of adipogenesis and the synthesis of MUFA in adipose tissues. There are small differences in fatty acid composition of beef from Bos indicus and Bos taurus cattle, but diet and time on feed are much more important determinants of beef fat content and fatty acid composition than breed type.
Genetic Structure and Differentiation of Three Indian Goat Breeds
Dixit, S.P.;Verma, N.K.;Aggarwal, R.A.K.;Kumar, Sandeep;Chander, Ramesh;Vyas, M.K.;Singh, K.P. 1234
Gene flow, genetic structure and differentiation of Kutchi, Mehsana and Sirohi breeds of goat from North-Western India were evaluated based on 25 microsatellite markers so as to support breed conservation and improvement decisions. The microsatellite genotyping was carried out using an automated DNA sequencer. The gene diversity across the studied loci for the Kutchi breed varied from 0.57 (ILST 065) to 0.93 (OarFCB 304, OMHC 1, ILSTS 058) with an overall mean of 0.79${\pm}$0.02. The corresponding values for Mehsana and Sirohi breeds were 0.16 (ILST 008) to 0.93 (OMHC 1, ILSTS 058) with an average of 0.76${\pm}$0.04, and 0.50 (ILSTS 029) to 0.94 (ILSTS 058) with an average of 0.78${\pm}$0.02, respectively. The Mehsana breed had lowest gene diversity among the 3 breeds studied. All the populations showed an overall significant heterozygote deficit ($F_{is}$). The Fis values were 0.26, 0.14 and 0.36 for Kutchi, Mehsana and Sirohi goat breeds, respectively. Kutchi and Mehsana were more differentiated (16%) followed by Mehsana and Sirohi (13%).The measures of standard genetic distance between pairs of breeds indicated that the lowest genetic distance was between Kutchi and Sirohi breeds (0.73) and the largest genetic distance was between Mehsana and Kutchi (1.0) followed by Sirohi and Mehsana (0.75) breeds. Mehsana and Kutchi are distinct breeds and this was revealed by the estimated genetic distance between them. All measures of genetic variation revealed substantial genetic variation in each of the populations studied, thereby showing good scope for their further improvement.
A Simple Polymerase Chain Reaction-based Method for the Discrimination of Three Chicken Breeds
Kubo, Y.;Plastow, G.;Mitsuhashi, Tadayoshi 1241
A large number of branded chicken products exist in Japan, and in some cases, the breed of chicken is an important factor used to attract consumer interest in the retail product. In order to establish a simple method for verifying such breed claims we applied the amplified fragment length polymorphism (AFLP) technique to nine chicken breeds (White Cornish, Red Cornish, White Plymouth Rock, New Hampshire, Rhode Island Red, Barred Plymouth Rock, Hinaidori, Tosajidori, Tsushimajidori) to search for molecular markers able to discriminate chicken breeds. Three breed-specific single nucleotide polymorphisms (SNP) were identified, one for each of Hinaidori, Tosajidori, or New Hampshire. A total of 219 individuals from the nine breeds were analyzed using a specific PCR test for each of these SNP. The PCR tests made it possible to discriminate between the breeds of chickens to identify products from these three breeds. This PCR method provides an efficient method for the routine analysis and verification of certified chicken products.
Fatty Acid Composition in Blood Plasma and Follicular Liquid in Cows Supplemented with Linseed or Canola Grains
Perehouskei Albuquerque, Karina;do Prado, Ivanor Nunes;Bim Cavalieri, Fabio Luiz;Rigolon, Luiz Paulo;do Prado, Rodolpho Martin;Pizzi Rotta, Polyana 1248
This study was carried out to evaluate the fatty acid composition in Nellore cows supplemented with either linseed (n-3) or canola grains (n-6 and n-9). Fifteen Nellore cows, aged five years and bodyweight 550 kg${\pm}$48 kg, were randomly distributed to the following treatments: CON (control), LIN (linseed) and CAN (canola grains). The cows were fed for 80 days. The concentrations of C18:0, C18:2 n-6 and C20:3 n-6 fatty acid were higher (p<0.10) in CON blood plasma in comparison to follicular liquid. Likewise, PUFA, n-6 contents, PUFA:SFA and n-6:n-3 ratios were higher (p<0.10) in blood plasma. On the other hand, C18:1 n-9, C22:5 n-3, MUFA and n-3 contents were lower (p<0.10) in blood plasma. C18:0, C18:2 n-6, C18:3 n-3, C22:5 n-3, PUFA, n-6, n-3 contents and PUFA:SFA ratio were higher (p<0.10) in LIN blood plasma than in the follicular liquid. Nevertheless, C14:0, C16:0, C16:1 n-7, PUFA, C16:0, C18:1 n-9 and MUFA contents were lower (p<0.10) in LIN blood plasma. On treatment CAN, the C18:0 and SFA contents, and n-6:n-3 ratios were higher (p<0.10) in blood plasma. However, C20:3 n-6, C22:5 n-3, PUFA and n-3 contents were lower (p<0.10) in blood plasma. C16:0, C18:0, PUFA, SFA contents and PUFA:SFA ratio did not differ (p>0.10) among the treatments. C14:0, C16:1 n-7, C18:2 n-6 and n-6 contents were higher (p<0.10) for CON and CAN than LIN. C17:1 n-7, C20:4 n-6 and C 22:0 contents were higher (p<0.10) for CAN than CON and LIN. C18:1 n-9, C18:3 n-3, MUFA and n-3 contents were higher (p<0.10) for LIN and CAN than CON. C20:3 n-6 content and n-6:n-3 ratio were higher (p<0.10) for CON than LIN and CAN. C22:5 n-3 content were higher (p<0.10) for CON and LIN than CAN. The concentrations of fatty acids in blood plasma and follicular liquid were not correlated for any fatty acid, independent of the treatment studied. Canola grain added to the diet of Nellore cows resulted in increased concentrations of fatty acids n-6 and n-3 in follicular liquid.
Effects of Rice Straw Particle Size on Chewing Activity, Feed Intake, Rumen Fermentation and Digestion in Goats
Zhao, X.G.;Wang, M.;Tan, Z.L.;Tang, S.X.;Sun, Z.H.;Zhou, C.S.;Han, X.F. 1256
Effects of particle size and physical effective fibre (peNDF) of rice straw in diets on chewing activities, feed intake, flow, site and extent of digestion and rumen fermentation in goats were investigated. A 4${\times}$4 Latin square design was employed using 4 mature Liuyang black goats fitted with permanent ruminal, duodenal, and terminal ileal fistulae. During each of the 4 periods, goats were offered 1 of 4 diets that were similar in nutritional content but varied in particle sizes and peNDF through alteration of the theoretical cut length of rice straw (10, 20, 40, and 80 mm, respectively). Dietary peNDF contents were determined using a sieve for particle separation above 8 mm, and were 17.4, 20.9, 22.5 and 25.4%, respectively. Results showed that increasing the particle size and peNDF significantly (p<0.05) increased the time spent on rumination and chewing activities, duodenal starch digestibility and ruminal pH, and decreased ruminal starch digestibility and $NH_{3}$-N concentration. Intake and total tract digestibility of nutrients (i.e. dry matter, organic matter, and starch) and ruminal fermentation were not affected by the dietary particle size and peNDF. Increased particle size and peNDF did not affect ruminal fibre digestibility, but had a great impact on the intestinal and total tract fibre digestibility. The study suggested that rice straw particle size or dietary peNDF was the important influential factor for chewing activity, intestinal fibre and starch digestibility, and ruminal pH, but had minimal impact on feed intake, duodenal and ileal flow, ruminal and total tract digestibility, and ruminal fermentation.
Effects of Protein Supply from Soyhulls and Wheat Bran on Ruminal Metabolism, Nutrient Digestion and Ruminal and Omasal Concentrations of Soluble Non-ammonia Nitrogen of Steers
Kim, Jeong-Hoon;Oh, Young-Kyoon;Kim, Kyoung-Hoon;Choi, Chang-Won;Hong, Seong-Koo;Seol, Yong-Joo;Kim, Do-Hyung;Ahn, Gyu-Chul;Song, Man-Kang;Park, Keun-Kyu 1267
Three beef steers fitted with permanent cannulae in the rumen and duodenum were used to determine the effects of protein supply from soyhulls (SH) and wheat bran (WB) on ruminal metabolism, blood metabolites, nitrogen metabolism, nutrient digestion and concentrations of soluble non-ammonia nitrogen (SNAN) in ruminal (RD) and omasal digesta (OD). In a 3${\times}$3 Latin square design, steers were offered rice straw and concentrates formulated either without (control) or with two brans to increase crude protein (CP) level (9 vs. 11% dietary DM for control and bran-based diets, respectively). The brans used were SH and WB that had similar CP contents but different ruminal CP degradability (52 vs. 80% CP for SH and WB, respectively) for evaluating the effects of protein degradability. Ruminal ammonia concentrations were higher for bran diets (p<0.01) than for the control, and for WB (p<0.001) compared to the SH diet. Similarly, microbial nitrogen and blood urea nitrogen were significantly increased (p<0.05) by bran and WB diets, respectively. Retained nitrogen tended (p<0.082) to be increased by SH compared with the WB diet. Intestinal and total tract CP digestion was enhanced by bran diets. In addition, bran diets tended (p<0.085) to increase intestinal starch digestion. Concentrations of SNAN fractions in RD and OD were higher (p<0.05) for bran diets than for the control, and for WB than for the SH diet. More rumendegraded protein supply resulting from a higher level and degradability of CP released from SH and WB enhanced ruminal microbial nitrogen synthesis and ruminal protein degradation. Thus, free amino acids, peptides and soluble proteins from microbial cells as well as degraded dietary protein may have contributed to increased SNAN concentrations in the rumen and, consequently, the omasum. These results indicate that protein supply from SH and WB, having a low level of protein (13 and 16%, respectively), could affect ruminal metabolism and nutrient digestion if inclusion level is relatively high (>20%).
Effect of Inorganic and Organic Trace Mineral Supplementation on the Performance, Carcass Characteristics, and Fecal Mineral Excretion of Phase-fed, Grow-finish Swine
Burkett, J.L.;Stalder, K.J.;Powers, W.J.;Bregendahl, K.;Pierce, J.L.;Baas, T.J.;Bailey, T.;Shafer, B.L. 1279
Concentrated livestock production has led to soil nutrient accumulation concerns. To reduce the environmental impact, it is necessary to understand current recommended livestock feeding practices. Two experiments were conducted to compare the effects of trace mineral supplementation on performance, carcass composition, and fecal mineral excretion of phase-fed, grow-finish pigs. Crossbred pigs (Experiment 1 (Exp. 1), (n = 528); Experiment 2 (Exp. 2), (n = 560)) were housed in totally-slatted, confinement barns, blocked by weight, penned by sex, and randomly assigned to pens at approximately 18 kg BW. Treatments were allocated in a randomized complete block design (12 replicate pens per treatment) with 9 to 12 pigs per pen throughout the grow-finish period. In Exp. 1, the control diet (Io100) contained Cu as $CuSO_{4}$, Fe as $FeSO_{4}$, and Zn (of which 25% was ZnO and 75% was $ZnO_{4}$) at concentrations of 63 and 378 mg/kg, respectively. Treatment 2 (O100) contained supplemental Cu, Fe, and Zn from organic sources (Bioplex, Alltech Inc., Nicholasville, KY) at concentrations of 19, 131, and 91 mg/kg, respectively, which are the commercially recommended dietary inclusion levels for these organic trace minerals. Organic Cu, Fe, and Zn concentrations from O100 were reduced by 25% and 50% to form treatments 3 (O75) and 4 (O50-1), respectively. In Exp. 2, treatment 5 (Io25) contained 25% of the Cu, Fe, and Zn (inorganic sources) concentrations found in Io100. Treatment 6 (O50-2) was identical to the O50-1 diet from Exp. 1. Treatment 7 (O25) contained the experimental microminerals reduced by 75% from concentrations found in O100. Treatment 8 (O0) contained no trace mineral supplementation and served as a negative control for Exp. 2. In Exp. 1, tenth-rib backfat, loin muscle area and ADG did not differ (p>0.05) between treatments. Pigs fed the control diet (Io100) consumed less feed (p<0.01) compared to pigs fed diets containing organic trace minerals, thus, G:F was greater (p = 0.03). In Exp. 2, there were no differences among treatment means for loin muscle area, but pigs fed the reduced organic trace mineral diets consumed less (p<0.05) feed and tended (p = 0.10) to have less tenth-rib backfat compared to pigs fed the reduced inorganic trace mineral diet. Considering that performance and feed intake of pigs was not affected by lower dietary trace mineral inclusion, mineral excretion could be reduced during the grow-finish phase by reducing dietary trace mineral concentration.
Effects of Xylanase on Performance, Blood Parameters, Intestinal Morphology, Microflora and Digestive Enzyme Activities of Broilers Fed Wheat-based Diets
Luo, Dingyuan;Yang, Fengxia;Yang, Xiaojun;Yao, Junhu;Shi, Baojun;Zhou, Zhenfeng 1288
The study was conducted to investigate the effects of different levels of xylanase on performance, blood parameters, intestinal morphology, microflora and digestive enzyme activities of broilers. The wheat-based diets were supplemented with 0, 500, 1,000, 5,000 U/kg xylanase. Xylanase supplementation significantly (p<0.05) improved the feed:gain ratio of broilers from 1 to 21 d and 1 to 42 d. Supplementing 500 U/kg and 1,000 U/kg xylanase improved (p<0.05) the villus height and the ratio of villus height to crypt depth in the small intestine. Excess supplementation of xylanase (5,000 U/kg) increased the villus height in the ileum (p<0.01) and the ratio of villus height to crypt depth in the duodenum and ileum (p<0.05). The microflora in the ileum and caecum, digestive enzyme activities in the small intestine and the concentrations of serum glucose, uric acid, insulin and IGF-I were not affected by the supplementation of xylanase. Excess level of xylanase (5,000 U/kg) had a tendency to induce the multiplication of E. coli and total aerobes. The results suggested that supplementing 500 U/kg and 1,000 U/kg xylanase was beneficial for broilers and excess xylanase supplementation resulted in no further improvement or negative effects.
The Effect of Clinoptilolite in Low Calcium Diets on Performance and Eggshell Quality Parameters of Aged Hens
Gezen, Serife Sule;Eren, Mustafa;Balci, Faruk;Deniz, Gulay;Biricik, Hakan;Bozan, Birgul 1296
Ninety six beak-trimmed 72 week-old Lohmann Brown hens were randomly divided into four equal groups. Each group comprised 4 replicates. Isoenergetic and isonitrogenous experimental diets contained low calcium (3.5%); optimum calcium (4.2%); low Ca (3.5% Ca)+1% Clinoptilolite (CLP); low Ca (3.5% Ca)+2% CLP. Data were collected biweekly and the experiment lasted 6 weeks. Egg production, feed consumption, feed conversion ratio, egg weight, tibia Ca, P, ash and eggshell thickness were not affected by addition of CLP to the diets (p>0.05). There were no significant differences in egg shell strength and ash when data were analyzed individually in measurement periods ($74^{th}$, $76^{th}$ and $78^{th}$ weeks). However, according to pooled data ($74^{th}$-$78^{th}$ weeks), eggshell strength was increased (p<0.05) only by 2% CLP supplementation versus low Ca (3.5%) diet, and shell ash was significantly increased by 2% CLP supplementation compared with the other diets. The damaged egg ratio on 1% and 2% CLP diets was significantly decreased between 76-78 weeks'data when compared with the low Ca diet. However; damaged egg ratio on the 2% CLP diet was significantly decreased when pooled data (74-78) were compared with no CLP diets. The differences in marketable egg ratio paralleled damaged egg ratio. The plasma calcium level at the end of experiment was increased on the 2% CLP diet when compared with the low Ca (3.5%) diet (p<0.05). Furthermore, at the end of the experiment a marked decrease of manure moisture was observed on both CLP diets (p<0.01). In conclusion, Clinoptilolite (2%) supplementation to layer diets tends to improve eggshell quality and manure dry matter (1% and 2% CLP) after six weeks.
Effects of Dietary Zinc Level and an Inflammatory Challenge on Performance and Immune Response of Weanling Pigs
Sun, Guo-jun;Chen, Dai-wen;Zhang, Ke-ying;Yu, Bing 1303
Two experiments were conducted to determine the effect of dietary zinc level on growth performance and immune function in normal (Experiment 1) and immunologically challenged (Experiment 2) weanling pigs. Treatments consisted of the following: i) a corn-soybean meal basal diet containing 36.75 mg/kg total Zn, ii) basal diet+60 mg/kg added Zn as $ZnSO_{4}$, iii) basal diet+120 mg/kg added Zn as $ZnSO_{4}$. Each diet was fed to six pens of four pigs per pen (Exp. 1) or six pens of three pigs per pen (Exp. 2). In Exp. 1, the dietary zinc level had no effect on average daily growth (ADG), average daily feed intake (ADFI), or feed conversion ratio (FCR). Concentrations of tissue and serum zinc were not affected. Peripheral blood lymphocyte proliferation (PBLP) was not affected by dietary treatments. Supplementation of 120 mg/kg Zn decreased (p<0.05) the antibody response to bovine serum albumin (BSA) on d 7 compared with pigs fed the basal diet, but not on d 14. In Exp. 2, LPS challenge had no effect on ADG, ADFI and FCR in the entire trial (from d 0 to 21). LPS challenge significantly decreased ADG and ADFI (p<0.01) from d 7 to 14, but FCR was not affected. LPS challenge increased PBLP (p<0.05) and serum concentration of interleukin-1 (IL-1) (p<0.01), whereas the antibody response to BSA and serum concentration of interleukin-2 (IL-2) were not affected. Supplementation of Zn did not affect ADFI and FCR from d 7 to 14, but there was a trend for ADG to be enhanced with Zn supplementation (p<0.10). Supplementation of Zn tended to increase PBLP (p<0.10). Dietary treatment had no effect on the antibody response to BSA or concentrations of serum IL-1 and IL-2. Results indicate that the level of Zn recommended by NRC (1998) for weanling pigs was sufficient for optimal growth performance and immune responses. Zn requirements may be higher for pigs experiencing an acute phase response than for healthy pigs.
Evaluation of Soybean Oil as a Lipid Source for Pig Diets
Park, S.W.;Seo, S.H.;Chang, M.B.;Shin, I.S.;Paik, InKee 1311
An experiment was conducted to determine the effects of soybean oil supplementation replacing tallow in pig diets at different stages of growth. One hundred and twenty crossbred (Landrace${\times}$Yorkshire${\times}$Duroc) pigs weighing 18 kg on average were selected. Pigs were randomly allotted to 12 pens of 10 pigs (5 pigs of each sex) each. Three pens were assigned to each of the four treatments: TA; tallow diet, TA-SO-80; switched from tallow to soybean oil diet at 80 kg average body weight, TA-SO-45; switched from tallow to soybean oil diet at 45 kg average body weight, and SO; soybean oil diet. Treatment SO was significantly lower in ADG than tallow diets (TA, TA-SO-80 and TA-SO-45) during the grower period (18 to 45 kg). However, treatment SO showed greatest compensation in ADFI and ADG during the finisher-2 period (after 80 kg body weight). ADFI and ADG and Gain/Feed for the total period were not significantly different among treatments. Loin area, back fat thickness, firmness and melting point of back fat were not significantly different. The levels of total cholesterol and low density lipoprotein+very low density lipoprotein cholesterol in serum were significantly lower in treatment SO than in treatments TA-SO-45, TA-SO-80 and TA. The level of serum triglyceride linearly increased as the length of the tallow feeding period increased. Serum immunoglobulin-G (IgG) level was significantly higher in the soybean oiltreatment than in other treatments. Major fatty acid composition of short rib muscle and back fat were significantly influenced by treatments. Contents of ${\alpha}$-linolenic acid (C18:3) and docosahexaenoic acid (DHA, C22:6) linearly increased as the soybean oil feeding period increased. In conclusion, soybean oil can be supplemented to the diet of pigs without significant effects on growth performance and carcass characteristics. The level of polyunsaturated fatty acids (PUFA), especially $\omega-3$ fatty acids in the carcass was increased by soybean oil supplementation.
Effect of Nicotinamide on Proliferation, Differentiation, and Energy Metabolism in Bovine Preadipocytes
Liu, Xiaomu;Fu, Jinlian;Song, Enliang;Zang, Kun;Wan, Fachun;Wu, Naike;Wang, Aiguo 1320
This study examined the effects of nicotinamide on proliferation, differentiation, and energy metabolism in a primary culture of bovine adipocytes. After treatment of cells with 100-500 $\mu{M}$ nicotinamide, cell growth was measured using 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT), and cellular lipid content was assessed by Oil Red O staining and a triglyceride (TG) assay. Several factors related to energy metabolism, namely adenosine triphosphatase (ATPase) activity, nitric oxide (NO) content, nitric oxide synthase (NOS) activity, the number of mitochondria and the relative expression of glyceraldehydes-3-phosphate dehydrogenase (GAPDH), peroxisome proliferator-activated receptor-$\gamma$ ($PPAR_{\gamma}$) and inducible NOS (iNOS), were also investigated. Results showed that nicotinamide induced both proliferation and differentiation in bovine preadipocytes. Nicotinamide decreased NO production by inhibiting NOS activity and iNOS mRNA expression, and controlled lipolytic activity by increasing ATPase activity and the number of mitochondria. The present study provides further evidence of the effects of nicotinamide on lipid and energy metabolism, and suggests that nicotinamide may play an important role in the development of bovine adipose tissue in vivo. This emphasizes the importance of investigating bovine adipose tissue to improve our understanding of dairy cow physiology.
Effect of Individual, Group or ESF Housing in Pregnancy and Individual or Group Housing in Lactation on the Performance of Sows and Their Piglets
Weng, R.C.;Edwards, S.A.;Hsia, L.C. 1328
To evaluate different housing systems, 80 gilts were randomly allocated at puberty to four treatments: i) sow stall in gestation followed by farrowing crate (SC), ii) group housing with individual feeding in gestation followed by farrowing crate (GC), iii) ESF (Electronic Sow Feeding) system in gestation followed by farrowing crate (EC), and iv) ESF system followed by group farrowing pen (EG). The results showed that stalled sows had a longer interval between puberty and second estrus (p<0.001). The sows kept in the ESF system gained more body weight (p<0.01) and backfat (p<0.05) prior to service, and more backfat during gestation (p<0.05), but also had greater backfat losses in the subsequent lactation (p<0.01). Sows changing from loose housing to confinement at farrowing had longer gestation length (p<0.001). Total litter size did not differ significantly between gestation treatments, but the number of stillborn piglets was significantly higher in the SC treatment (p<0.01). After weaning, SC sows had the longest interval for rebreeding (p<0.001). Some EG sows came into heat before weaning, giving this treatment the shortest interval. These results indicate that gestation confinement in sow stalls had several detrimental effects on sow performance relative to group housing.
Stewardship, Stockmanship and Sustainability in Animal Agriculture
Szucs, E.;Geers, R.;Sossidou, E.N. 1334
Sufficient food supply for all humans was, is, and will remain one of the main priorities for mankind. The choice between food from crops or animals is related to philosophical, religious and ethical, but also cultural and economical, values. However, the concept of sustainable agriculture takes into account the organization of food supply through future generations. Not only quantity, but also quality is important, especially in relation to food safety and the method of production. Specifically, the aspect of animal welfare is becoming increasingly important with the focus on stewardship and stockmanship, i.e. responsibility of humans for their animals. In the future, implications for sustainability in animal production may be of more concern to stewardship paired by stockmanship, responsibility, consciousness and morality. The moral as a basic concept of sustainable agriculture is to maintain continuous development in harmony with nature to meet requirements in the world for living creatures including human beings to live in and steward. The objective of this paper is to discuss the above issues from different viewpoints on sustainable food supply, increasing food consumption and environmental protection.
Dietary Transformation of Lipid in the Rumen Microbial Ecosystem
Kim, Eun Joong;Huws, Sharon A.;Lee, Michael R.F.;Scollan, Nigel D. 1341
Dietary lipids are rapidly hydrolysed and biohydrogenated in the rumen resulting in meat and milk characterised by a high content of saturated fatty acids and low polyunsaturated fatty acids (PUFA), which contributes to increases in the risk of diseases including cardiovascular disease and cancer. There has been considerable interest in altering the fatty acid composition of ruminant products with the overall aim of improving the long-term health of consumers. Metabolism of dietary lipids in the rumen (lipolysis and biohydrogenation) is a major critical control point in determining the fatty acid composition of ruminant lipids. Our understanding of the pathways involved and metabolically important intermediates has advanced considerably in recent years. Advances in molecular microbial technology based on 16S rRNA genes have helped to further advance our knowledge of the key organisms responsible for ruminal lipid transformation. Attention has focused on ruminal biohydrogenation of lipids in forages, plant oils and oilseeds, fish oil, marine algae and fat supplements as important dietary strategies which impact on fatty acid composition of ruminant lipids. Forages, such as grass and legumes, are rich in omega-3 PUFA and are a useful natural strategy in improving nutritional value of ruminant products. Specifically this review targets two key areas in relation to forages: i) what is the fate of the lipid-rich plant chloroplast in the rumen and ii) the role of the enzyme polyphenol oxidase in red clover as a natural plant-based protection mechanism of dietary lipids in the rumen. The review also addresses major pathways and micro-organisms involved in lipolysis and biohydrogenation. | CommonCrawl |
\begin{document}
\title{Interpreting Deep Learning Model Using Rule-based Method}
\author{Xiaojian Wang} \affiliation{
\institution{Purdue University}
} \email{[email protected]}
\author{Jingyuan Wang} \affiliation{
\institution{Beihang University}
} \email{[email protected]}
\author{Ke Tang} \affiliation{
\institution{Tsinghua University}
} \email{[email protected]}
\begin{abstract} Deep learning models are favored in many research and industry areas and have reached the accuracy of approximating or even surpassing human level. However they've long been considered by researchers as black-box models for their complicated nonlinear property. In this paper, we propose a multi-level decision framework to provide comprehensive interpretation for the deep neural network model.
In this multi-level decision framework, by fitting decision trees for each neuron and aggregate them together, a multi-level decision structure (MLD) is constructed at first, which can approximate the performance of the target neural network model with high efficiency and high fidelity. In terms of local explanation for sample, two algorithms are proposed based on MLD structure: forward decision generation algorithm for providing sample decisions, and backward rule induction algorithm for extracting sample rule-mapping recursively. For global explanation, frequency-based and out-of-bag based methods are proposed to extract important features in the neural network decision. Furthermore, experiments on the MNIST and National Free Pre-Pregnancy Check-up (NFPC) dataset are carried out to demonstrate the effectiveness and interpretability of MLD framework. In the evaluation process, both functionally-grounded and human-grounded methods are used to ensure credibility.
\end{abstract}
\keywords{Interpretable Deep Learning, Model Transparency, Rule Extraction}
\maketitle
\section{Introduction}
Deep learning models play an increasingly significant role in real world applications, such as image and speech recognition, medical diagnostics and intelligent recommendation. In spite of the incredible performance, there's one critical issue with deep learning models that we cannot ignore: they are generally considered by researchers as black-box models which limits their application in the areas such as causal inference, trust (i.e., in healthcare applications, a trusting model which can be understood, validated and edited is needed for the experts \cite{caruana2015intelligible}), fairness (i.e. internal prejudice of the training data might be amplified during the model training \cite{dwork2012fairness}) and some security-focused scenarios (i.e., it's hard to detect the adversarial attacks in some image recognition tasks\cite{Akhtar2018threat}).
It's worth noticing that more and more efforts have been devoted to make the deep learning models interpretable. One of the ideas is transforming the neural network decision into a set of rule mappings. The rule extraction techniques can be divided into two categories: decompositional approaches and pedagogical approaches \cite{andrews1995survey}. Some of the earliest work adopted the decompositional approach to extract the boolean rules at the individual level. The basic idea of the decompositional approach is to treat the rule extraction as a searching process. The main defect of the decompositional methods is the complexity of the search process is exponential with the number of neurons and layers.
Compared to the decompositional approaches which search and analyze the structure of neural networks, the pedagogical approaches treat the neural network as a whole and aim to extract the rules that replace the original network as accurately as possible. But the applicability of the pedagogical methods is also limited, because in the context of deep networks, we need more complex rule sets to get the performance approximate to neural network and that will cause the structures generated by pedagogical methods to be extremely complicated.
To address the limitations of existing rule-based methods, we propose an interpretable framework for the multi-layer neural network based on a multi-level decision structure. At first, a multi-level decision structure (MLD) is derived by transforming the discretized activation function of each neuron into a decision tree and linking the generated trees layer by layer. Different from the other pedagogical rule-based methods, MLD preserves the original structure of the neural network.
Then, based on the MLD structure we propose forward decision generation algorithm and reverse rule induction algorithm to produce local explanation for each prediction. Applying the forward decision generation algorithm, we can generate the decision of the MLD layer-by-layer for any given input sample, due to the high fidelity of MLD, it can approximate the performance of the original neural network in high accuracy. Applying the reverse rule induction algorithm which merges rules retrospectively, we can finally get the rule set which is corresponding to the input space for any given sample.
In order to extract global explanations which can further help researchers interpret the overall reasoning of the model, we design frequency and out-of-bag methods for evaluating the importance of input features. Furthermore, We evaluate the interpretability of MLD on MNIST and NFPC dataset, predictivity and fidelity are compared with a few baselines, visual and global explanations are given and validated as well.
To summarize, the major contributions of this paper are the following: \begin{itemize} \item{We propose multi-level decision (MLD) framework. To the best of our knowledge, compare to other rule-based approaches, MLD combines the advantages of both decompositional and pedagogical methods, achieves higher fidelity and meanwhile preserves the original network structure.}
\item{MLD framework can not only provide local explanations for a given sample by generating rule mapping, but also provide global explanations in the sense of feature importance.}
\item{Extensive experiments on two different tasks with both public and large-scale real-world dataset show the effectiveness of the proposed MLD approach. Interpretability is evaluated by both functionally-grounded and human-grounded methods.} \end{itemize}
The rest of the paper is organized as follows. In Section 2, we survey the works related to interpretable deep learning. Section 3 introduces the preliminaries, problem definition and technical details of our proposed method. In Section 4, we empirically evaluate the performance of our model on various data. Finally, Section 5 is the conclusion.
\section{Related Works}
Interpretability methods for deep learning aim at providing understandable explanation towards the black-box nature of deep learning models. With reference to the research by Benn \textit{et al.} \cite{kim2017interpretable} and a slight adjustment, We further divide the interpretability methods of deep learning models into four categories by : 1) hidden-layer investigation methods, 2) sensitivity and gradient-based methods, 3) mimic and surrogate model methods, 4) interpretable deep learning methods.
The hidden-layer investigation methods implement the interpretation of deep learning models by analyzing or visualizing the behavior of hidden layers. In the research of Matthew D. \textit{et al.} \cite{zeiler2014visualizing}, a deconvolutional network is used to perform the mapping from the feature activities to the input pixel space. Karpathy \textit{et al.} \cite{karpathy2015visualizing} extended this idea to the recurrent neural networks. Alternatively, Bau \textit{et al.} \cite{bau2017network} proposes a network dissection framework for quantifying the semantic representation of CNN by aligning the hidden units to a sets of concepts. Besides discovering the knowledge learned by the network, hidden layer methods are also implemented to study the training process of the network \cite{yosinski2014transferable,alain2016understanding}.
Sensitivity and gradient-based methods provide explanation by measuring how the change of variables or weights affects the output. Garson \textit{et al.} \cite{Garson1991} proposed an algorithm by calculating variable contribution based on the absolute values of connection weights. Based on the work of Garson, Olden \textit{et al.} \cite{olden2002illuminating} studied the significance of variables and weights using a sampling method. Some of other sensitivity methods investigate the neural network by perturbing the test point or by fitting a simpler model locally \cite{simonyan2013deep,li2016understanding,koh2017understanding} .
More closely related to our work, mimic and surrogate model methods use some more interpretable models to simulate the structure or behavior of a neural network to provide explanations. There're two categories of these methods: 1) \textit{Linearization methods}, which aim to establish a linear mapping between the input and output space. Such as relevance propagation \cite{bach2015pixel} and deep taylor decomposition \cite{montavon2017explaining}. 2) \textit{Rule-based methods}, which demonstrate their interpretability by extracting symbol rules from neural networks. Rule-based methods can be further divided into decompositional approaches and pedagogical approaches \cite{andrews1995survey}. Decompositional methods treat the problem of extracting rules from neural networks as a search process. e.g. extracting M-of-N rules from neural network based on clustering methods \cite{towell1992interpretation}, extracting rules from pruned feedforward neural network \cite{setiono1995understanding, setiono1997neurolinear}, and searching rules by finding the combinations \cite{krishnan1999search}. The biggest challenge for decomposition methods is as the number of neurons increases, these methods will face the explosion of rule combinations. Therefore, the decompositional methods usually reduce the computational complexity by some pruning or aggregation methods. In contrast to the decompositional methods, pedagogical methods treat the neural network as a unified whole, and aim to generate rules which can better approximate the performance of original network. e.g. RF and RN method \cite{saito2002extracting}, TREPAN \cite{craven1996extracting,trepan2}, tree construction methods based on genetic algorithm \cite{krishnan1999search} and sampling \cite{bastani2017interpreting}.
There are also some studies dedicated to building interpretable deep learning methods. e.g. Capsule \cite{sabour2017dynamic} method can produce output vectors whose length represent the probability that the entity exists and the orientation for the instantiation parameters. Based on the mechanism of attention, visual attention \cite{allport1989visual,xu2015show} can be used to visualize the focus of the the network.
\section{Methodology}
\subsection{Preliminaries}
This section introduces the notations that will be used throughout the paper.
The interpretability method we proposed focuses on multi-layer perceptron model (MLP). Considering a trained MLP with $L$ layers for a multiclass classification problem, Let $\mathcal{X}\in {\rm I\!R}^d$ denote the input space and $\mathcal{Y}=[K]$ denote the output space. Suppose there're $m^{(l)}$ nodes in $lth$ layer of the MLP, $\mathbf{w}^{(l)}$ be the trained weight between $lth$ and $(l+1)th$ layer and $a$ be the activation function, For the training set $\mathcal{D}=\{\mathbf{x}^{(1)}_n,\mathbf{y_n}\}^N_{n=1}$ of size N, where $\mathbf{x}^{(1)}_n = (x^{(1)}_{n,1},...,x^{(1)}_{n,d})\in \mathcal{X}$ is feature vectors of length d and $y_n \in \mathcal{Y}$. Let $X^{(l)}=(\mathbf{x}_1^{(l)},\mathbf{x}_2^{(l)},..,\mathbf{x}_N^{(l)})$ be the output of the $lth$ layer of the MLP, then each of the $\mathbf{x}^{(l)}_i$ should be a vector with length $m^{(l)}$. Generally we have: \begin{equation}
x^{(l)}_{i,k} = a(\mathbf{w}^{(l-1)}_k\mathbf{x}^{(l-1)}_i), \forall 1\le i\le N, 1\le k\le m^{(l)}, 2\le l\le L \end{equation}
Note that, for convenience we only consider the MLP models which target for classification problems and use ranged activation function such as tanh/sigmoid.
\subsection{Problem Definition} For a general classification task which uses MLP in the modeling process, given a train dataset $\mathcal{D}=\{\mathbf{x}^{(1)}_n,\mathbf{y_n}\}^N_{n=1}$ we can then apply some gradient optimization methods to train a MLP. Besides analyzing the performance of the model prediction, we may also concern about how the model derive the prediction. Generally, the related interpretability problems can be divided into two aspects:
1) Local explanation: local explanation focuses on knowing the reasoning process for a certain decision. Inspired by some linearity methods and rule-based methods, in the context of local explanation, we aim to find a mapping function $\hat f(\mathbf{x})$ from the input space and the output space which is much easier to be understood than the mapping function $f(\mathbf{x})$ of the original MLP. \begin{equation}
\hat f(\mathbf{x}) \approx f(\mathbf{x}) \end{equation}
More specifically, in this paper we focus on the rule-based method in which $\hat f(\mathbf{x})$ can be represented by a set of rule mappings .
2) Global explanation: global explanation implies knowing what patterns are present in general. In this paper, our goal for the global explanation is to determine the key features which put most impact on the final decision.
\subsection{Framework for Interpretation Process Based on Multi-level Decision Structure(MLD)}
As shown in \textbf{Figure \ref{fig:framework}}, framework for interpretation process based on multi-level decision structure can be split into five steps: \begin{enumerate}
\item \textbf{Network construction and training}. First and foremost, for a classification task based on a given dataset, we can apply stochastic gradient descent (or other network training methods) to train a MLP. In the following steps, we will use this MLP as the target neural network for interpretaion.
\item \textbf{Multi-level decision structure construction}. In this step, we build the multi-level decision structure(MLD) by fitting the activation function of each node in the target network to a decision tree, then we aggregate them together.
\item \textbf{Local explanation}. As a rule-based approach, MLD framework provides the local explanation for a given sample by transforming the sample decision into a rule mapping. Based on the MLD structure generated in \textbf{step (2)}, we proposed the forward decision generation algorithm to generate the sample decision and backward rule induction algorithm to derive the decision rules recursively.
\item \textbf{Global explanation}. In terms of global explanation, MLD framework aims at finding the important features in the neural network decision process. Based on the rule mapping derived in \textbf{step (3)}, frequency-based and out-of-bag importance measure methods are proposed.
\item \textbf{Evaluation}. Two categories of evaluation methods for the interpretability are used in MLD framework: functional-grounded and human-grounded. In functional-grounded methods, predictivity and fidelity scores are measured according to the accuracy and F1 score of the MLD decisions regarding to the original data labels and neural network predicted labels. In human-grounded methods, we provide visual explanations and conduct user study as well.
\end{enumerate}
\begin{figure}
\caption{Framework for Interpretation Process Based on Multi-level Decision Structure(MLD)}
\label{fig:framework}
\end{figure}
\subsection{Construction of Multi-level Decision Structure} Multi-level decision structure here implies a hierarchical structure in which each layer contains several decision trees and the leaf nodes of these trees can be treated as the input feature of the next layer. As shown in \textbf{Figure \ref{fig:discretize}}, for a given data set $\mathcal{D} = \{\mathbf{x}_n^{(1)}, \mathbf{y}_n\}^N_{n=1}$ and the trained MLP, the generation process of MLD can be split into three steps.
1) Discretize the activation function for each neuron. For the $kth$ node in the $lth$ layer in a given MLP, we have:
\begin{equation}
x^{(l+1)}_{i,k} = a(\mathbf{w}^{(l)}_k\mathbf{x}^{(l)}_i), \forall 1\le i\le N, l\le L-1 \end{equation}
In order to transform the activation function into a discrete function, we need to discrete both the input vectors and output vectors. Since the output value of activation functions like sigmoid and tanh are restricted in $\{0,1\}$. We here reduce them into boolean discrete function for simplicity. Note that the boolean mapping is not the only choice, other discretization method can also be applied as long as it is preferred.
\begin{equation}
\tilde x^{(l)}_{i,k} =
\left\{
\begin{array}{lr}
1 & x^{(l)}_{i,k}>=0.5\\
0 & x^{(l)}_{i,k}<0.5
\end{array}
\right. \end{equation}
Here we may encounter with situations where the input vectors $ \mathbf{\tilde x}^{(l)}_i,\mathbf{\tilde x}^{(l)}_j$ are the same but the output vactors $\tilde x^{(l+1)}_{i,k},\tilde x^{(l+1)}_{j,k}$ maybe different, which is not eligible for a function mapping. We further reduce the different output values into the mode of the output values which share the same input vectors in this situation. \begin{equation}
\tilde x^{(l+1)}_{i,k} = MODE\{\tilde x^{(l+1)}_{j,k} ~| ~ \forall ~ \mathbf{\tilde x}_j^{(l)} = \mathbf{\tilde x}_i^{(l)}~ , j\in\{1,2,...,N\}\} \end{equation}
Then for the $kth$ neuron in $(l+1)th$ layer, Based on the discretized input and output pairs $ \{\mathbf{\tilde {x}}^{(l)}_i, \tilde {x}^{(l+1)}_{i,k}\}_{i=1}^N$, a corresponding boolean function $f^{(l+1)}_k$ can be generated and represented as following:
\begin{equation}
f^{(l+1)}_k(\mathbf{x}) =
\tilde x^{(l+1)}_{i,k},~~ if~ \mathbf{x}=\mathbf{\tilde x}^{(l)}_{i} \end{equation}
2) For every given discretized function $f^{(l)}_k$, approximate it with a decision tree $T^{(l)}_k$ and optimize the error in the sample space. Here we can use any decision tree generation function to obtain the aproximation. CART algorithm, which uses gini as the criterion is prefered here.
3) The MLD structure can be obtained by treating the output of each decision tree as the input of the next layer. Here we denote the MLD as $\mathbf{T}$ which represents the collection of the decision trees $\mathbf{T}=\{T^{(l)}_k|\forall 1\le k \le m^{(l)}, 2\le l\le L\}$.
\begin{figure}
\caption{Steps for Construction of Multi-level Decision Structure (MLD)}
\label{fig:discretize}
\end{figure}
\subsection{Forward Decision Generation Algorithm}
Based on the MLD structure, a foward decision generation algorithm is proposed to derive the decision for a given sample. Suppose $T_k^{(l)}$ be the generated decision tree for $kth$ node in $lth$ layer, $T_k^{(l)}(\mathbf{x}^{(1)}_i)$ be the output value of the given original input $\mathbf{x}^{(1)}_i$. For the $L$ layers MLP, our goal here is generating $T^{(L)}_k(\mathbf{x}^{(1)}_i)$ for any $1\le k\le m^{(l)}$ and $2\le l\le L$. In particular, output for the decision trees in the last layer $\{T^{(L)}_k(\mathbf{x}^{(1)}_i)|1\le k\le m^{(L)}\}$ can be treated as the final output value for the MLD.
Every $T^{(l)}_k(\mathbf{x}^{(1)}_i)$ can be generated layer by layer fowardly. For the corresponding decision tree of the $jth$ node in the second layer $T_j^{(2)}$, according to the numeric value of sample $\mathbf{x}^{(1)}_i$ in each dimension, $T_k^{(2)}(\mathbf{x}^{(1)}_i)$ can be generated by going through from the root of $T_k^{(2)}$ to its leaf node. Respectively, every $T_k^{(3)}(\mathbf{x}^{(1)}_i)$ can be derived by implementing $(T_1^{(2)}(\mathbf{x}^{(1)}_i),T_2^{(2)}(\mathbf{x}^1_i)..,T_{m^2}^{(2)}(\mathbf{x}^{(1)}_i))$ into the tree structure of $T^{(3)}_k$. \textbf{Algorithm \ref{alg:fdg}} present the procedure for the forward decision generating process.
\begin{algorithm}[htb]
\caption{Decision and Rule Generation for Single Decision Tree }
\label{alg:singlerule}
\begin{algorithmic}[1]
\REQUIRE
decision tree $T$; input features $\mathbf{x}$
\ENSURE
Rule set $R_\mathbf{x}$; decision result $T(\mathbf{x})$
initialize $R_\mathbf{x}=\Phi$, $T(\mathbf{x}) =0$
\WHILE{$T$ is not empty}
\IF{$T$ has no branch}
\STATE{$T(\mathbf{x})=T$}
\ELSE
\FORALL {branch $c$ in $T$ for decision variable $v$ and with critical value $c_v$ }
\IF{c is discrete variable}
\IF{$x_v == 0$}
\STATE{$T =$ left chance node branching from $c$}
\STATE{r = $(v,c_v,=)$}
\ELSE
\STATE{$T =$ right chance node branching from $c$}
\STATE{r = $(v,c_v,\ne)$}
\ENDIF
\ELSE
\IF{ $x_{v} \le c_v$}
\STATE{$T =$ left chance node branching from $c$}
\STATE{r = $(v,c_v,\le)$}
\ELSE
\STATE{$T =$ right chance node branching from $c$}
\STATE{r = $(v,c_v,>)$}
\ENDIF
\ENDIF
\STATE{$R_\mathbf{x}=R_\mathbf{x}\cup r$}
\ENDFOR
\ENDIF
\ENDWHILE
\end{algorithmic} \end{algorithm}
\begin{algorithm}[htb]
\caption{Forward Decision Generation}
\label{alg:fdg}
\begin{algorithmic}[1]
\REQUIRE
Input sample $\mathbf{x}^{(1)}_i$; MLD structure $\mathbf{T}$;
\ENSURE
Output of every decision tree for every given sample $\{T^{(l)}_k(\mathbf{x}^{(1)}_i)$ | $\forall1\le k\le m^{(l)},2\le l\le L\}$;
\FOR{$l=1$ to $L$}
\FOR{$k=1$ to $m^{(l)}$}
\IF{$l==2$}
\STATE{Input feature vector $(x^{(1)}_{i,1},x^{(1)}_{i,2},...,x^{(1)}_{i,d})$ and decision tree structure $T^{(2)}_k$, Apply \textbf{Algorithm \ref{alg:singlerule}} to compute $T^{(2)}_k(\mathbf{x}^{(1)}_i)$}
\ELSE
\STATE{ Input decision tree structure $T^{(l)}_k$ and feature vector
$(T^{(l-1)}_1(\mathbf{x}^{(1)}_i),T^{(l-1)}_2(\mathbf{x}^{(1)}_i),...,T^{(l-1)}_{m^{(l)}}(\mathbf{x}^{(1)}_i))$,
Apply \textbf{Algorithm \ref{alg:singlerule}} to compute $T^{(l)}_k(\mathbf{x}^{(1)}_i)$}
\ENDIF
\ENDFOR
\ENDFOR
\end{algorithmic} \end{algorithm}
\begin{figure}
\caption{Demonstration for the Forward Decision Generation Process (Target MLP is the structure shown in Figure \ref{fig:discretize})}
\label{fig:forward}
\end{figure}
\subsection{Backward Rule Induction Algorithm} One objective of the proposed model is to generate the local explanation. More specifically, in the MLD scenario, we want to find a rule mapping $R^{(L)}(\mathbf{x}^{(1)}_i)$ from input space to the output space for a given sample $\mathbf{x_i^{(1)}}$, For example: \begin{equation} \begin{split}
R_1^{(L)}(\mathbf{x^{(1)}_5}): (x^{(1)}_{5,1}=1)\land (x^{(1)}_{5,3}=1)\land (x^{(1)}_{5,5}=0)\\\implies (y_{5,1}=0) \end{split} \end{equation}
This rule mapping suggests for the $5th$ data sample, according to $x^{(1)}_{5,1}=1,x^{(1)}_{5,3}=1$ and $x^{(1)}_{5,5}=0$, we can conclude the final decision $y_{5,1}=0$ , Through the rule induction process, such a mapping rule for a given sample can be derivated backwardly and recursively. The main idea of the back rule induction algorithm is as follow:
According to the decision tree approximate method, we can get the MLD structure $\mathbf{T}=\{T^{(l)}_k|\forall 1\le k \le m^{(l)},2\le l\le L\}$ and the output decision of each decision tree $T^{(l)}_k(\mathbf{x}_i^{(1)})$ for any given input sample $\mathbf{x}^{(1)}_i$. Based on the decision making process of MLD, generation of decision for the $kth$ node in the $Lth$ layer $T^{(L)}_k(\mathbf{x}_i)$ can be reduced to a rule mapping:
\begin{equation} \begin{split} R^{(L)}_k(\mathbf{x}_i): (x^{(L-1)}_{i,v_1}=T^{(L-1)}_{v_1}(\mathbf{x}^{(1)}_i))\land(x^{(L-1)}_{i,v_2}=T^{(L-1)}_{v_2}(\mathbf{x}^{(1)}_i))\\ ...\land(x^{(L-1)}_{i,v_m}=T^{(L-1)}_{v_m}(\mathbf{x}^{(1)}_i)) \implies (y_{i,k}=T^{(L)}_k(\mathbf{x}^{(1)}_i)) \end{split} \end{equation}
Where $v_1,...v_m \in \{1,2,...,m^{(L-1)}\}$ stands for the index of the node in the $(L-1)th$ layer we use to produce the decision for $T^{(L)}_k(\mathbf{x}_i^{(1)})$ in the branching procedure of $T^{(L)}_k$. Respectively, every item $(x^{(L-1)}_{i,v_j}=T^{(L-1)}_{v_j}(\mathbf{x}_i^{(1)}))$ in $R^{(L)}_k(\mathbf{x}_i^{(1)})$ can be further reduced to rule mapping $R^{(L-1)}_{v_j}$ from $(L-2)th$ layer to $(L-1)th$ layer. By replacing every $x^{(L-1)}_{i,v_j}=T^{(L-1)}_{v_j}(\mathbf{x}_i^{(1)})$ to the new mapping, $R^{(L)}_k(\mathbf{x}_i^{(1)})$ can be represented by the mapping from $(L-2)th$ layer to $Lth$ layer. Repeating this process recursively, we can finally get mapping rules from the input space to the $Lth$ output. Specifically, We name this recursive procedure as backward rule induction. Details for that algorithm shown in \textbf{Algorithm \ref{alg:bri}}.
\begin{algorithm}[htb]
\caption{Backward Rule Induction}
\label{alg:bri}
\begin{algorithmic}[1]
\REQUIRE
Input sample $\mathbf{x}^{(1)}_i$; MLD structure $\mathbf{T}$; output derived by \textbf{Algorithm \ref{alg:fdg}} for $\mathbf{x}^{(1)}_i$: $\{T^{(l)}_k(\mathbf{x}^{(1)}_i)$ | $\forall1\le k\le m^{(l)},2\le l\le L\}$; layer index $l$; node index $k$;
\ENSURE
rule mapping $R^{(l)}_k(\mathbf{x}^{(1)}_i)$ from input space to the decision of $kth$ node in $lth$ layer
\STATE {Initialize $R^{(l)}_k(\mathbf{x})=\Phi$ }
\STATE{Apply \textbf{Algorithm \ref{alg:singlerule}}, input $(T^{(l-1)}_1(\mathbf{x}^{(1)}_i),...,T^{(l-1)}_{m^{(l-1)}}(\mathbf{x}^{(1)}_i))$ and $T^{(l)}_k$, get $R^{(l)}_k(T^{(l-1)}(\mathbf{x}^{(1)}_i))$ }, assign it to $R^{(l)}_k(\mathbf{x}^{(1)}_i)$
\FORALL{ rule item $r=(v,c_v,sign)$ in $R^{(l)}_k(\mathbf{x}^{(1)}_i)$}
\STATE{
Apply \textbf{Algorithm \ref{alg:bri}}, Input $\mathbf{x}^{(1)}_i,\mathbf{T}, \{T^{(l)}_k(\mathbf{x}^{(1)}_i)\}, l-1, v$, get $R^{(l-1)}_v(\mathbf{x}^{(1)}_i)$
}
\STATE{$R^{(l)}_k(\mathbf{x}^{(1)}_i) = R^{(l)}_k(\mathbf{x}^{(1)}_i)\cup R^{(l-1)}_v(\mathbf{x}^{(1)}_i)$}
\STATE{Delete $r$ from $R^{(l)}_k(\mathbf{x}^{(1)}_i)$}
\ENDFOR
\end{algorithmic} \end{algorithm}
\begin{figure}
\caption{Demonstration for the Backward Rule Induction Process (Target MLP is the structure shown in Figure \ref{fig:discretize})}
\end{figure}
\subsection{Importance Measuring}
In the previous parts, we design the forward rule generation and backward rule induction methods which can be used to derive the local explanation for a given sample. Besides we also want to get some intuitive understanding for the whole model. Importance measuring is one of the common approaches to deliver a global explanation. Here we introduce two importance measure methods based on MLD.
\textbf{Frequency-based method}. On account of the hierarchical structure of MLD, it's hard to measure the feature importance by calculating its information gain as in the ordinary decision tree. But thinking about the construction of decision tree, usually the more sample predictions are given by referring the value of a given feature, the more important this feature is. Inspired by this intuitive idea, an empirical frequency-based method is proposed, in which the importance values are given by calculating how many times a certain feature are used in the reduced decision rule in the sample set:
\begin{equation}
VIM^{(Freq)}_v = \sum_{i=1}^n I(v~used~in~Rule(x_i)) \end{equation}
Then we can normalize the importance values:
\begin{equation} VIM_v^{(Freq)} = \frac{VIM_v^{Freq}}{\sum_{i=1}^d VIM_i^{(Freq)}} \end{equation}
\textbf{OOB-based method}. Out-of bag method is inspired by the feature measuring in the random forest. The idea of which is first select part of out-of-bag data, then randomly adjust one input feature and observe the impact on the model accuracy. The calculation method is as follow: \begin{enumerate}
\item Select a set of out-of-bag data $OOB$, according to the forward decision generation algorithm generate the sample labels based on MLD. Then calculate the errors from the neural network label $errOOB$;
\item For a given feature $v$, randomly change the feature value of $v$ in $OOB$, calculate the errors between MLD prediction and the neural network labels, denote is as $errOOB_v$;
\item Calculate the $errOOB_v-errOOB$ as the importance of feature $v$, and normalize it:
\begin{equation}
VIM^{(OOB)}_v = errOOB_v-errOOB
\end{equation}
\begin{equation}
VIM_v^{(OOB)} = \frac{VIM_v^{OOB}}{\sum_{i=1}^d VIM_i^{(OOB)}}
\end{equation}
\end{enumerate}
\section{Experiments}
In this section, we empirically evaluate the effectiveness of the MLD framework on the MNIST and NFPC (National Free Pre-Pregnancy Check-ups) dataset. After training a multi-layer neural network for each task, we first assess the model predictivity and fidelity of the MLD compared to other pedagogical rule-based methods: decision trees learned using CART, C4.5 and Trepan\cite{craven1996extracting}. Higher predictivity score ensures the explainable model performance more closely matches to the original task, while higher fidelity score indicates higher similarity with the neural network prediction. Second, we provide local and global explanations to give evidence to the interpretability of the MLD framework.
\subsection{Experimental Setting}
\subsubsection{Dataset}
Our system is evaluated mainly on MNIST and NFPC dataset.
\begin{itemize}
\item \textbf{Mnist}: Mnist dataset is a widely used handwritten dataset in the field of machine learning. This dataset contains 60,000 training images and 10,000 testing images. Each of the black and white image is fitted into a 28*28 pixel bounding box.
\item \textbf{NFPC}: National Free Pre-Pregnancy Check-up(NFPC) is a population-based health survey dataset. This survey mainly focuses on the reproductive-aged couples who wish to conceive, and was conducted across 31 provinces in China from Jan 1, 2014 to Dec 31, 2015. The original data contains detailed personal characteristics of both spouses, which are mainly divided into the following categories: biological indicators (e.g. blood pressure and sugar); personal characteristics of husband and wife (e.g., occupation, education and region); disease characteristics (e.g., Genetic and chronic disease history); personal habits characteristics (e.g., dietary habit and psychology condition). The labels of NFPC data include the indicator of if doctors predict there is a risk of pregnancy and the true fertility outcome.
For confidentiality reasons, we only use the cleaned and feature-selected data in Yunnan Province to conduct the experiment. The cleaned data contains 106 one-hot features and labels of predicted risk and true outcome.
\end{itemize}
\subsubsection{Baseline} In terms of predictivity and fidelity, We compare our proposed model with the following pedagogical rule-based interpretability methods:
\begin{itemize}
\item \textbf{C4.5\cite{quinlan2014c4}}: C4.5 is a widely used decision tree construction algorithm, which is an extension of earlier ID3\cite{quinlan1986induction} algorithm. Different from the entropy used in ID3, the splitting criterion of C4.5 is the normalized information gain.
\item \textbf{Cart\cite{breiman2017classification}}: Classification And Regression Tree (CART) can be used in both classification and regression tasks, CART uses Gini index as the splitting criterion.
\item \textbf{Trepan\cite{craven1996extracting}}: Trepan is an inductive method to extract concept description from trained neural network. In the trepan method, a decision tree is learned with queries, while these queries are usually answered by a structure named oracle. The oracle models the data instance and refer the network output to provide answer. \end{itemize}
\subsubsection{Network Training}
On MNIST and NFPC dataset, we build two hidden layer MLPs for the target tasks. Mostly 30 nodes in the first hidden layer and 10 nodes in the second layer, while the number of the input nodes and output nodes varies according to the task. On CIFAR-10 dataset, we build a three hidden layer MLP, 128 nodes for the first and second hidden layer and 50 nodes for the third layer. We choose zero mean truncated normal random number to initialize the network parameter and cross entropy loss function as the target loss function. In the training process, we apply adam optimizer and drop out to update the parameters.
\subsubsection{MLD Construction}
When apply proposed method to extract the MLD structure, we control the size and the depth of the fitted decision trees in order to avoid over-fitting and distill the most representative features as well. Mostly we limit the maximum height to 20 and maximum size to 100 for the fitted decision trees. After fitting process, we apply pruning method to limit the height and size. Specifically, for the multi-class task, we construct two types of decision trees for the output layer. First we train p binary classification decision trees to produce interpretation for each class separately (p is the number of classes), Second we train a single p classification decision tree to measure the performance.
\begin{table*}
\caption{Predictivity and Fidelity Measure}
\label{tab:compare}
\begin{tabular}{cccccccccc}
\toprule
\multicolumn{2}{c}{}& \multicolumn{4}{c}{Predictivity Measure} &\multicolumn{4}{c}{Fidelity Measure} \\
\cmidrule(lr){3-6}\cmidrule(lr){7-10}
Dataset & Method & Acc(Train)& F1(Train)& Acc(Test)& F1(Test) & Acc(Train)& F1(Train)& Acc(Test)& F1(Test) \\
\midrule
Discretized MNIST & Target MLP & 0.9622 & 0.9684 & 0.9652 & 0.9545 &-&-&-&- \\
&\textbf{MLD}& \textbf{0.9025} & \textbf{0.8943} & \textbf{0.8976} & \textbf{0.9034} & \textbf{0.8936} & \textbf{0.9034} & \textbf{0.8950} &\textbf{0.8859}\\
&CART&0.8332& 0.8345 & 0.8381 & 0.8471 & 0.8589 & 0.8364 & 0.8459& 0.8427 \\
&C4.5&0.8329&0.8359&0.8432&0.8324&0.8531&0.8642&0.8434&0.8531\\
&Trepan&0.8675&0.8794&0.8689&0.8845&0.8788&0.8710&0.8835&0.8753\\
\midrule
NFPC(Predicted Risk) & Target MLP &0.9945&0.9873&0.9939&0.9926&-&-&-&-\\
&\textbf{MLD}&\textbf{0.9854}&0.9768&0.9820&\textbf{0.9875}&\textbf{0.9945}&\textbf{0.9920}&\textbf{0.9898}&\textbf{0.9854}\\
&CART&0.9754&0.9660&0.9659&0.9735&0.9749&0.9750&0.9850&0.9829\\
&C4.5&0.9740&0.9758&0.9670&0.9720&0.9830&0.9734&0.9819&0.9783\\
&Trepan&0.9845&\textbf{0.9853}&\textbf{0.9950}&0.9898&0.9929&0.9830&0.9914&0.9840\\
\midrule
NFPC(True Outcome) & Target MLP &0.9854&0.9834&0.9935&0.9845&-&-&-&-\\
&\textbf{MLD}&0.9678&0.9723&\textbf{0.9846}&\textbf{0.9743}&\textbf{0.9874}&0.9734&0.9774&0.9758\\
&CART&0.9618&\textbf{0.9732}&0.9691&0.9745&0.9812&\textbf{0.9829}&0.9749&\textbf{0.9764}\\
&C4.5&0.9628&0.9723&0.9634&0.9712&0.9718&0.9814&0.9712&0.9714\\
&Trepan&\textbf{0.9745}&0.9634&0.9745&0.9684&0.9734&0.9812&\textbf{0.9842}&0.9674\\
\midrule
CIFAR10 & Target MLP & 0.4820&0.4832&0.4880&0.5023&-&-&-&-\\
&\textbf{MLD}&\textbf{0.3125}&\textbf{0.3251}&\textbf{0.3278}&\textbf{0.3095}&\textbf{0.3192}&\textbf{0.3150}&\textbf{0.3144}&\textbf{0.3125}\\
&CART&0.2765&0.2845&0.2940&0.2554&0.2689&0.2834&0.2525&0.2678\\
&C4.5&0.2637&0.2832&0.2734&0.2934&0.2714&0.2697&0.2813&0.2664\\
&Trepan&0.2923&0.3023&0.2912&0.3038&0.2914&0.2943&0.2834&0.3060\\
\bottomrule \end{tabular} \end{table*}
\subsection{Predictivity and Fidelity Measure}
One of the approaches to assess the goodness of our proposed framework is by measuring the predictivity and fidelity. Where the predictivity score shows how good the extracted model fits the real data labels, while the fidelity score indicates how good the extracted model fits the labels predicted by the trained neural network model.
More precisely, apply the forward rule generation algorithm, we can reproduce data labels based on the MLD. Then we measure the predictivity and fidelity in terms of four scores: accuray and F1 score in train and test dataset. Higher predictivity scores guarantee the extracted model matches the origin data well, but achieving high fidelity scores may seems more important, since it ensures the extracted model actually obtains the insights into the neural network model.
As shown in \textbf{Table \ref{tab:compare}}, we compare our proposed MLD framework with several baselines over several datasets. Most of the results show our model overperforms over other baseline methods.
\begin{figure}
\caption{Local Explanation Cases for MNIST}
\label{fig:cases}
\end{figure}
\subsection{Interpretability for MNIST Dataset}
In terms of interpretability, our goal for the MNIST task is extracting visual explanations of the model decision in sample and class scales. Note that, in order to generate explicit visual explanation, we first descretize all the input value of MNIST dataset into 0 and 1.
\subsubsection{Local Explanation Cases} In the MLD framework, we can apply backward rule induction method to extract decision rules of the neural network regarding to a given sample. In the example for digit 8 given in \textbf{Figure \ref{fig:cases}}, the real label and neural network decision are the same. In the result of MLD, it is also classified correctly as 8 and the other incorrect classifications are rejected as well. Observing this visual example, it's easy to tell that the derived rules become more and more complicate as the layer increases. This result comes from the nature of the MLD structure: except for the input layer, sample rules for nodes in each layer are the combination of the rules derived in the previous layer. Since the MLD is directly generated based on the neural network, observing how the MLD works may help us better understand the underlying working principle of neural networks.
Observing the rules extracted in the output layer for digit 8 in \textbf{Figure \ref{fig:cases}}: the number of rules to classify it to digit 8 correctly is the most, the numbers of the rules which refuse to classify it as digit 3, 5, 9 are relatively larger compare to other digits. Intuitively, shape of number 8 is indeed closer to 3, 5, 9 than other numbers, therefore more evidence is needed to distinguish it from 3, 5, 9. From the example of digit 4 sample given in \textbf{Figure \ref{fig:cases}}, we can also discover the similar pattern: shape of number 4 is much closer to number 9, thus the number of rules required to correctly identify it as 4 is the most, while the number of rules refuse to recognize it as 9 is the second largest.
The local explanation given by the MLD framework may also give us some intuition about why the neural network sometimes doesn't provide the right decision. In the case for digit 2 shown in \textbf{Figure \ref{fig:cases}}, the neural network wrongly classified it as digit 7. Through the visual explanation, we can roughly tell the reason for such misclassification: the neural network may not take the small tail of the digit 2 into account in the classification process.
\begin{figure}
\caption{Rule Importance Distribution for Each Classification on MNIST}
\label{fig:mnistclass}
\end{figure}
\subsubsection{Global Explanation}
In terms of global explanation, apply the frequency-based method proposed in the MLD framework, we can obtain the importance score for each rule related to the input feature. Specifically, in the discretized MNIST scenario, for each of the ten classifications, rules correspond with each input pixel like $\{x^{(1)}_{1} = 1 \},\{x^{(1)}_{3} = 0 \} $ can be assigned to a score according to how many times a certain rule is used to generate sample decisions. The visualized results of the global explanation are shown in \textbf{Figure \ref{fig:mnistclass}}, in which the importance scores for positive and negative rules are mapped in the 28*28 pixels. From this distribution we can roughly tell the outline and periphery of the ten numbers. This intuitive result also suggests that the global explanation given by our MLD framework is comprehensive and reasonable.
\subsection{Interpretability for NFPC Data}
Our goal for the interpretability in NFPC dataset is mainly focus on providing global explanation and assessing the explanation as well. More specifically, first we train MLP for doctor predicted risk and the true outcome task. Then we construct MLD and apply the proposed importance measure method to extract top-10 important features for the two tasks separately. Furthermore for comparison, we train random forest model for the two tasks and get another two sets of top-10 importance measure. Finally we compare the importance measures given by MLD and random forest by fitting the target task again and performing human experiment.
\subsubsection{Compare the Explanations By Regression}
In order to measure how representative the extracted features are and conduct comparison, We use the top-10 features extracted from the MLD and random forest in both of the tasks to fit logistic regression models separately, then we compare the AUC score. The result is shown in \textbf{Table \ref{tab:logitregression}}. From the result we can see though the scores of MLD are slightly lower than random forest, there's no significant difference between the two model. That means the top-10 key features selected by MLD and random forest have almost the same explanatory power.
\begin{table}
\caption{Logistic Regression Fit Results Using Top-10 Important Features}
\label{tab:logitregression}
\begin{tabular}{ccc}
\toprule
Task & AUC(MLD) & AUC(RF) \\
\midrule
Predicted Risk&0.7245&0.7329\\
\midrule
True Outcome&0.6628&0.6684\\
\bottomrule
\end{tabular} \end{table}
\subsubsection{Compare the Explanations By Human Experiment}
One of the most important aspects for the interpretability is providing the explanations which can be understood by human beings, thus performing human experiment is necessary for assessing the interpretability methods. In the NFPC case, we conduct a user study to compare the key feature sets derived from MLD and random forest model. We interviewed 40 obstetricians through the internet, each participant was given one set of top-10 important features for predicited risk task and one set of top-10 important features for true outcome task randomly (randomization here means each set can either from the result given by MLD or random forest, but make sure each of the set is assigned for 20 obstetricians). Then they were asked to score each feature from 0 to 3 according to how much they think this feature has an impact on fertility outcomes. The reference scoring criteria is as follows: 0-no effect, 1-little effect, 2-moderate effect, 3-significant effect. After that, we sum up the scores for each set given by each obstetrician and then perform paired t-test to compare the difference. Related Results are given in \textbf{Table \ref{tab:ttest}}.
\begin{table}
\caption{ User Study Paired One-Sided t-Test Result}
\label{tab:ttest}
\begin{tabular}{ccccc}
\toprule
Task & $\mu_{score}$(MLD) & $\mu_{score}$(RF) & t & p (>t)\\
\midrule
Predicted Risk& 20.6 & 20.8 & 1.2666 & 0.22\\
\midrule
True Outcome& 10.5 & 8.6 & 24.06& 0\\
\bottomrule \end{tabular} \end{table}
If we choose the 95\% confidence interval, the p value for predicted risk task is 0.22, for true outcome is 0, indicates regarding to the total set score, for the predicted risk task there's no significant difference between the key feature sets extracted by MLD and random forest. While for the true outcome task, the MLD scores are significant higher than random forest scores.
\section{Conclusion}
In this paper, we propose a multi-level decision (MLD) framework to generate explanation for the multi-layer neural network model. At first, a multi-level decision structure is built to reconstruct the original neural network. Based on the MLD structure, forward decision generation algorithm and backward rule induction algorithm are applied to derive the sample decision and sample rule mapping, frequency-based and OOB-based method are applied to measure the feature importance. Furthermore, functional-grounded and human-grounded experiments are also carried out to evaluate the interpretability.
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\appendix
\section{Supplemental Materials}
\subsection{Datasets}
In the experiments, we test our model with over three popular image datasets, which are Fashion-MNIST, CIFAR-10 and ImageNet. All pixels in the image data are projected into [0, 1].
$\bullet$ Fashion-MNIST~\cite{xiao2017fashion} is a dataset comprising of 28$\times$28 grayscale images of 70,000 fashion products from 10 categories, with 7,000 images per category. The training set has 60,000 images and the test set has 10,000 images.
$\bullet$ CIFAR-10~\cite{krizhevsky2009cifar} contains 60,000 color images corresponding to 10 different object classes. In the dataset, 10,000 images are for training and 50,000 for testing. The image size is 32$\times$32.
$\bullet$ ImageNet~\cite{imagenet_cvpr09} is a large natural image dataset containing over 1.2 million images. We select images of 20 object classes from the dataset for our experiment. Each class contains 1,040 training samples and 260 test samples. All these images are resized to 256$\times$256 and center cropped to 224$\times$224. The label indexes of the chosen objects are 7, 30, 36, 58, 75, 101, 161, 282, 398, 456, 496, 510, 549, 620, 696, 734, 752, 908, 954, 980 (from 0 to 999).
\subsection{Experiment Environments}
Our software environment contains ubuntu 18.04, PyTorch v1.1.0 and python 3.6.5 (we also use Tensorflow v1.12 for baseline PixelDefend). All of the experiments are conducted on a machine with four GPUs (NVIDIA GeForce GTX 2080 Ti * 4), one CPU (Intel(R) Xeon(R) Silver 4210 CPU @ 2.20GHz) and 128G memory.
\subsection{Model Structures}
\begin{table}[h] \footnotesize
\centering
\caption{Model structures in our experiments for each dataset and component.}
\resizebox{0.9\columnwidth}{!}{
\begin{tabular}{c|c|c|c}
\toprule
& \makecell{Fashion\\-MNIST} & CIFAR-10 & ImageNet \\
\toprule
\makecell{Target\\ Classifier } & VGG11\cite{vgg11} & VGG11\cite{vgg11} & InceptionV3\cite{incepv3} \\
\midrule
\makecell{Black-box \\Classifier } & CWnet\cite{carlini2017cw} & Wide-ResNet\cite{rony2019ddn} & ResNet152\cite{resnet} \\
\midrule
Detector & VGG11\cite{vgg11} & VGG11\cite{vgg11} & VGG11\cite{vgg11} \\
\midrule
Rectifier & VGG11\cite{vgg11} & VGG11\cite{vgg11} & VGG16\cite{vgg11} \\
\midrule
\makecell{Adversarial \\Training} & VGG11\cite{vgg11} & VGG11\cite{vgg11} & InceptionV3\cite{incepv3} \\
\bottomrule
\end{tabular}
}
\label{tab:model structure} \end{table}
Table \ref{tab:model structure} lists the model structures in our experiments, including the attacker target classifier (also our protected classifier), the detector and rectifier model in our X-Ensemble Model, adversarial training model for baselines. In the black-box evaluation we use the black-box classifier to generate adversarial examples and then perform transferable attacks on the target classifier. Notice that the input image size of InceptionV3 should be 299$\times$299, but we set the image size as 224$\times$224 for InceptionV3, which is also supported by PyTorch, to reduce the computational cost when generating adversarial examples and also to fit the input size of ResNet152. We only use pre-trained InceptionV3 of target classifier and pre-trained ResNet152 of black-box classifier by PyTorch. When using these models, we select the output logits of those 20 chosen classes and then compute the probabilities for them with $softmax$ function. In this way, InceptionV3 and ResNet152 both have 95\% accuracy on the ImageNet subset.
\subsection{Setting of Attackers}
The codes of attack methods are implemented by AdverTorch v0.2. Table \ref{tab:attack para} reports the attacking parameters in this paper.
\begin{table}[ht] \footnotesize
\centering
\caption{Attacking parameters for generating adversarial examples. }
\resizebox{0.9\columnwidth}{!}{
\begin{tabular}{c|c|c|c|c}
\toprule
Attacker & Parameter & \makecell{Fashion\\-MNIST} & CIFAR-10 & ImageNet \\
\midrule
FGSM & $\epsilon$ & 0.031 & 0.031 & 0.031 \\
\midrule
\multirow{3}[6]{*}{PGD} & $\epsilon$ & 0.031 & 0.031 & 0.031 \\ \cmidrule{2-5} & iteration & 20 & 20 & 20 \\ \cmidrule{2-5} & $\alpha$ & 0.00781 & 0.00781 & 0.00781 \\
\midrule
{Dfool} & iteration & 100 & 100 & 30 \\
\midrule
\multirow{3}[6]{*}{CW} & learning rate & 0.01 & 0.01 & 0.01 \\ \cmidrule{2-5} & iteration & 100 & 100 & 50 \\ \cmidrule{2-5} & $c$ & 0.01 & 0.01 & 0.01 \\
\midrule
DDN & iteration & 100 & 100 & 50 \\
\bottomrule
\end{tabular}
}
\label{tab:attack para} \end{table}
The $\epsilon$ in FGSM and PGD is to constrain the $L_{\infty}$ perturbation of adversarial examples. The $\alpha$ in PGD and the learning rate in CW control the step size in each iteration when searching perturbation. The higer $c$ in CW is to construct adversarial example with more confidence to fool the classifier. Iteration limits how many times that an iterative attack can compute. Other parameters use their default values in AdverTorch. Targeted and untargeted attacks share the same parameters.These parameters also work for generating adversarial examples in the black-box evaluation.
\paratitle{Adversarial Specificities.} These attacks have two types of adversarial specificities, \emph{i.e.,}\xspace untargeted attacks and targeted attacks~\cite{akhtar2018threat}:
$\bullet$ {\em Untargeted Attack.} For an image with an original predicted label of $\hat{y}^\circ$, its untargeted adversarial counterpart is successful when the perturbed predicted label $\hat{y}'$ satisfies $\hat{y}' \neq \hat{y}^\circ$.
$\bullet$ {\em Targeted Attack.} For an image with an original predicted label $\hat{y}^\circ$, its targeted adversarial counterpart is successful only when the perturbed label $\hat{y}'$ satisfies $\hat{y}' = y^{(t)}$ and $y^{(t)} \neq \hat{y}^\circ$. In our experiments, the given targeted label $y^{(t)}$ is chosen randomly.
\subsection{Interpretation Methods}
We implement VG, GBP and IG methods on our own and use LRP code from~\cite{montavon2019layer}. The integrated step in IG is set to 50. Note that LRP from~\cite{montavon2019layer} cannot be applied on InceptionV3 and ResNet152 directly. So we remove LRP detector only for ImageNet.
\subsection{Baselines}
\begin{itemize} \item \textbf{PD} The code is from BPDA\footnote{https://github.com/anishathalye/obfuscated-gradients/tree/master/pixeldefend}. When purifying images, $\epsilon$ is set to 0.125 for Fashion-MNIST and 0.063 for both CIFAR-10 and ImageNet. The pretrained model of PixelCNN for CIFAR-10 is from $openai^2$, and we train PixelCNN on Fashion-MNIST and ImageNet with code from $openai$ \footnote{https://github.com/openai/pixel-cnn}. \item \textbf{TWS} The code is from \cite{hu2019new}. We set the parameters n\_radius = 0.01, targeted\_lr = 0.0005, t\_radius = 0.5, u\_radius = 0.5 and untargeted\_lr = 1. \item \textbf{MHL} The code is from \cite{lee2018simple}. The magnitude of noise starts from 0.05 to 0.3 with an interval of 0.05 to compute its AUC. \item \textbf{TVM} The code is from~\cite{tvm}. We set TVM\_WEIGHT = 0.03, PIXEL\_DROP\_RATE = 0.5, TVM\_METHOD = 'tvl2'. \end{itemize}
\subsection{Training Details}
\begin{itemize} \item \textbf{Classifier} We use Fashion-MNIST and CIFAR-10 to train their target classifiers and black-box classifiers. The initial learning rates are 0.1 and 0.01 and the training epochs are 20 and 30 respectively. The classifiers for ImageNet are pretrained by PyTorch. \item \textbf{Detector} We first use the attack methods to generate adversarial examples on the dataset. And then the benign images and corresponding perturbed images are fed to train the data detector. Next, we use the interpreting method to generate their interpreting maps and train interpreting detectors(VG, GBP, IG, LRP). The label of benign, $L_{\infty}$ perturbed and $L_2$ perturbed examples are 0, 1 and 2. The initial learning rate are set to 0.01 and detectors are trained with 30 epochs. \item \textbf{Rectifier} We use Alg. \ref{algo:masked_image} to compute masked images on adversarial examples and the sensitivity of $\alpha$ is reported in Fig. \ref{fig:alpha}. As we can see, the original classifier have high accuracy on those $L_2$ masked images. Here the figure of Fashion-MNIST is omitted since its sensitivity is similar to ImageNet. We set $\alpha$ to 0.6, 0.9, 0.5 for Fashion-MNIST, CIFAR-10 and ImageNet. And we find that rectifier trained with masked images of DDN-T mixed with clean images have better performance. \end{itemize}
\begin{figure}
\caption{$\alpha$ parameter sensitivity}
\label{fig:alpha}
\end{figure}
\begin{algorithm} [t]
\caption{Masked Image For Training Rectifier}\label{algo:masked_image} \begin{algorithmic}[0]
\STATE {\bf Variables:} $\{D_1,...,D_J\}$ are the sub-detectors that predict an image as an adversarial one, $\alpha \in (0,1)$ is a threshold parameter, $rand()$ returns a random value in $[0, 1]$, $\sigma$ is variance of pixel values in $x$.
\FOR{$k=1$ to $j$ }
\STATE $E_k \gets Entropy(D_k(x))$
\ENDFOR
\STATE $D \gets D_i$ where $i=argmin(E_1,...,E_j)$
\STATE $g \gets \frac{\partial \mathcal{L}(D(x))}{\partial x}$
\STATE $thres \gets \alpha*(\max(g)-\min(g)) + \min(g)$
\FOR{ Pixel\ $(i,j)$ in $x$}
\IF{$g_{i,j} > thres$ \textbf{and} $ rand() > 0.5$}
\STATE $x_{i,j} \gets x_{i,j} + Normal(0, \sigma)$
\ENDIF
\ENDFOR
\STATE \textbf{return} $x$
\end{algorithmic} \end{algorithm}
\subsection{White-box Attacker for {X-Ensemble}\xspace}
In Ref.~\cite{carlini2017adversarial} proposed to combine a classifier and a neural network detector into a new classifier $G$ with L+1 classes, whose ${(L+1)}^{th}$ label identify an input as adversarial. In this way, an attacker can directly attack $G$ to break $F$ and $D$ at the same time. $G$ is defined as, \begin{equation}
G(x)_i =
\left\{ \begin{array}{lcr}
F(x)_i &~~~ \mathrm{if}~~ i \leq L \\
(D(x)+1) \cdot max_jF(x)_j &~~~ \mathrm{if}~~ i = L+1
\end{array} \right.
\label{equ:n+1classifier}
\end{equation} where if $x$ is clean then $D(x) < 0$, so we have $G(x)_{L+1} > max(F(x))$ and $argmax_i G(x)_i = L+1$; if $x$ is adversarial then $D(x) > 0$, so we have $argmax_i G(x)_i = argmax_i F(x)_i$.
From the definition, when an attacker tries to use targeted attack ( with a target $t \neq l$ and $t \neq L+1$) to construct adversarial examples on $G$, it will optimize the examples with a joint objective.
In our experiment, an attacker needs to fool all the detectors in {X-Ensemble}\xspace. So we modify $G$ into an L+4 classifier as, \begin{equation}
G(x)_i =
\left\{ \begin{array}{lcr}
F(x)_i &~~~ \mathrm{if}~~ i \leq L \\
(D_k(x) + 1) \cdot max_jF(x)_j &~~~ \mathrm{if}~~ i = L+k
\end{array} \right.
\label{equ:n+1classifier}
\end{equation} where $D_k$ is one part of X-DET and $k=1,2,3,4$. Notice that here we remove the LRP detector since it is not differentiable. So that a targeted attacker can generate examples on the classifier and the detectors to perform a white-box attack. In white-box evaluation, the iteration of PGD is set to 100 and the step size $\alpha$ is set to 0.000781 for the three datasets.
\end{document} | arXiv |
Approximation of $e^{-x}$
Is there a method to mentally evaluate $e^{-x}$ for $x>0$? Just to have an idea when computing probabilities or anything that is an exponential function of some parameters.
approximation exponentiation
J. M. is a poor mathematician
ACACACAC
$\begingroup$ You could compute its Taylor series. The Taylor series for $e^{-x}$ will always converge to $e^{-x}$, although for $x$ not close to zero, it might not be practical to mentally compute a higher order Taylor approximation. $\endgroup$ – ShawnD Oct 10 '11 at 5:50
$\begingroup$ What I do is pretend $e=2$. Maybe there is a more precise way which is almost as easy? $\endgroup$ – Dan Brumleve Oct 10 '11 at 5:51
$\begingroup$ Another trick is $e^{-x} = 10^{-x/\ln(10)} \simeq 10^{-0.4x}.$ $\endgroup$ – Gerben Oct 10 '11 at 6:21
$\begingroup$ In the book "Surely You Are Joking, Mr Feynman!" Richard Feynman described how he computed $e^x$ mentally to several significant digits by using some simple approximations. I don't have the book at hand to give more details about his method, but it might be worth looking at. $\endgroup$ – Dilip Sarwate Oct 10 '11 at 11:42
If $x$ is "small enough", I like using the $(2,2)$ Padé approximant, which is easily cast into a memorable form:
$$\exp\,x\approx \frac{(x+3)^2+3}{(x-3)^2+3}$$
For $|x| < \frac12$, the absolute difference between the approximant and the true function is $< 8\times 10^{-5}$. Pretty good approximation for a mere rational function... of course, as with all such approximants, the further you go from $0$, the less accurate it becomes.
J. M. is a poor mathematicianJ. M. is a poor mathematician
$\begingroup$ +1: I just became a fortune-teller. I foresee a bunch of freshman calculus students doing battle with this next Spring. The Taylor series agree up to $x^4$. With quintic terms it is $1$ vs. $5/6$. $\endgroup$ – Jyrki Lahtonen Oct 10 '11 at 9:48
$\begingroup$ @Jyrki: Just in case you've forgotten, Padé approximants are rational functions precisely designed to have power series expansions whose first few terms match the first few terms of the power series of the function they're approximating. ;) Since this is a $(2,2)$ approximant (the two numbers denote respectively the degrees of the numerator and denominator), it is expected to agree up to the $2+2=4$-th order term. Neat, eh? $\endgroup$ – J. M. is a poor mathematician Oct 10 '11 at 9:59
$\begingroup$ Thanks for the reminder. I've seen that described in the context of Berlekamp-Massey algorithm (see e.g. the 3rd reference here. $\endgroup$ – Jyrki Lahtonen Oct 10 '11 at 15:04
$\begingroup$ @Jyrki: Tightly connected indeed. Toeplitz matrices are very tricky that way... (I'm still figuring out the intricate web myself.) $\endgroup$ – J. M. is a poor mathematician Oct 10 '11 at 15:06
$\begingroup$ Yes, I used it as a question testing their ability to manipulate power series. Thanks! $\endgroup$ – Jyrki Lahtonen Aug 15 '12 at 6:45
Dilip was asking in the comments about "Feynman's method"; since this is too long for a comment, I shall be quoting the relevant paragraphs here:
One day at Princeton I was sitting in the lounge and overheard some mathematicians talking about the series for $e^x$, which is $1+x+x^2/2!+x^3/3!$ Each term you get by multiplying the preceding term by $x$ and dividing by the next number. For example, to get the next term after $x^4/4!$ you multiply that term by $x$ and divide by $5$. It's very simple.
When I was a kid I was excited by series, and had played with this thing. I had computed $e$ using that series, and had seen how quickly the new terms became very small.
I mumbled something about how it was easy to calculate $e$ to any power using that series (you just substitute that power for $x$).
"Oh yeah?" they said. "Well, then what's $e$ to the $3.3$?" said some joker—I think it was Tukey.
I say, "That's easy. It's $27.11$."
Tukey knows it isn't so easy to compute all that in your head. "Hey! How'd you do that?"
Another guy says, "You know Feynman, he's just faking it. It's not really right."
They go to get a table, and while they're doing that, I put on a few more figures. "$27.1126$," I say.
They find it in the table. "It's right! But how'd you do it!"
"I just summed the series."
"Nobody can sum the series that fast. You must just happen to know that one. How about $e$ to the $3$?"
"Look," I say. "It's hard work! Only one a day!"
"Hah! It's a fake!" they say, happily.
"All right," I say, "It's $20.085$."
They look in the book as I put a few more figures on. They're all excited now, because I got another one right.
Here are these great mathematicians of the day, puzzled at how I can compute $e$ to any power! One of them says, "He just can't be substituting and summing—it's too hard. There's some trick. You couldn't do just any old number like $e$ to the $1.4$."
I say, "It's hard work, but for you, OK. It's $4.05$."
As they're looking it up, I put on a few more digits and say, "And that's the last one for the day!" and walk out.
What happened was this: I happened to know three numbers—the logarithm of $10$ to the base $e$ (needed to convert numbers from base $10$ to base $e$), which is $2.3026$ (so I knew that $e$ to the $2.3$ is very close to $10$), and because of radioactivity (mean-life and half-life), I knew the $\log$ of $2$ to the base $e$, which is $.69315$ (so I also knew that $e$ to the $.7$ is nearly equal to $2$). I also knew $e$ (to the $1$), which is $2.71828$.
The first number they gave me was $e$ to the $3.3$, which is $e$ to the $2.3$—ten—times $e$, or $27.18$. While they were sweating about how I was doing it, I was correcting for the extra $.0026$—$2.3026$ is a little high.
I knew I couldn't do another one; that was sheer luck. But then the guy said $e$ to the $3$: that's $e$ to the $2.3$ times $e$ to the $.7$, or ten times two. So I knew that it was $20.$something, and while they were worrying about how I did it, I adjusted for the $.693$.
Now I was sure I couldn't do another one, because the last one was again by sheer luck. But the guy said $e$ to the $1.4$, which is $e$ to the $.7$ times itself. So all I had to do is fix up $4$ a little bit!
They never did figure out how I did it.
There's less to it than meets the eye. ;)
$\begingroup$ I just typed this up since I don't have a digital copy of any of Feynman's books. Please correct anything that I may have transcribed wrongly... $\endgroup$ – J. M. is a poor mathematician Oct 12 '11 at 18:05
$\begingroup$ how exactly do you think he corrected for the "small errors"? I can see how he got $e^{3.3} = e^{2.3} \times e \approx 27.18$, but I don't see how he was able to say $27.1126$. Similarly for other approximations. $\endgroup$ – picakhu Aug 15 '12 at 20:24
$\begingroup$ I was wondering the same thing picakhu did $3$ years ago. I think he did this: $e^{3.3}\approx 10e\times e^{-0.0026}\approx 27.1828(1-0.0026)\approx 27.1828-0.27\times0.26=27.1828-0.0702\\=27.1126$ $\endgroup$ – Jack's wasted life Jun 25 '15 at 4:30
$e^{-x} \approx 2^{-1.44x} \approx 10^{-0.43x}$
where $\log_2(e) \approx 1.44$ and $\log_{10}(e) \approx 0.43$.
HenryHenry
$\begingroup$ doesn't this nail down to how to compute $10^{-x}$? $\endgroup$ – ACAC Oct 13 '11 at 20:28
$\begingroup$ @Bob: in general that's hard, but if you just need an order-of-magnitude estimate, it's trivial. For example: $10^{13.2315} = 10^{13} \times 10^{0.2315}.$ The latter factor is hard to calculate, but for a numerical estimate it's often not important. You can of course refine such estimates by going back to base $e$ or 2: since $10 \simeq 2^{3.3},$ $10^{0.2315} \simeq 2^{0.7} \simeq 1.7$. $\endgroup$ – Gerben Oct 14 '11 at 10:32
As a supplement to Henry's answer, consider these rational approximations to $\log_{10}(e)$ and $\log_{2}(e)$.
Two of the approximants for the continued fraction for $\log_{10}(e)$ are $\frac{3}{7}$ (low and not as good as $.43$) and $\frac{10}{23}$ (high but better than $.43$). So if it makes computation easier, you can try $$ 10^{-10x/23}<e^{-x}<10^{-3x/7}\tag{1} $$ when $x>0$. The order in $(1)$ is reversed for $x<0$.
Two of the approximants for the continued fraction for $\log_{2}(e)$ are $\frac{10}{7}$ (low and not as good as $1.44$) and $\frac{13}{9}$ (high but better than $1.44$). So if it makes computation easier, you can try $$ 2^{-13x/9}<e^{-x}<2^{-10x/7}\tag{2} $$ when $x>0$. The order in $(2)$ is reversed for $x<0$.
robjohn♦robjohn
A diagram may also help...
NoChanceNoChance
$\begingroup$ I'll make sure to print this out and carry it with me. $\endgroup$ – The Chaz 2.0 Oct 12 '11 at 18:21
$\begingroup$ @pic: actually, it's an iPhone. Please keep me out of such comments. $\endgroup$ – The Chaz 2.0 Aug 16 '12 at 0:42
For $e^x$ and limiting to fractions that are low multiples of $\frac{1}{3}$ ... one can approximate fairly well by knowing $3^n$ fairly well,
$$2.714 \approx 3^3/10$$
Using this you can then obtain results via,
$$e^n \approx 2.714^n \approx \left(\frac{3^{3}}{10}\right)^n = \frac{3^{3n}}{10^n}$$
Clearly $10^n$ isn't going to be an issue. Most mathematicians will probabaly be familiar up to $3^4=81$ by memory and calculating up to $3^6$ isn't too much trouble.
Examples: $$ e^2 \approx 729 / 100 \approx 7.3\\ e^2 = 7.4 $$
$$ e^{3} = e^2e^1 \approx 7.4 \times 2.7 \approx 3/4\times 27 \approx 81/4 \approx 20\\ e^3 = 20.09 $$ you could also get to near 20 by knowing that $3^6$ is 19000 and something
Finding the negative powers just flip the relation.
Alexander McFarlaneAlexander McFarlane
$\begingroup$ This is a method I use when calculating Poisson Processes and I don't want to get distracted by going on my laptop because when I go on my laptop I spend 30mins writing posts on Mathematics :) $\endgroup$ – Alexander McFarlane Apr 16 '16 at 13:20
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\begin{document}
\maketitle
\begin{abstract}
In this paper we resolve the degree-2 Abel map for nodal curves. Our results are based on a previous work of the authors reducing the problem of the resolution of the Abel map to a combinatorial problem via tropical geometry. As an application, we characterize when the (symmetrized) degree-2 Abel map is not injective, a property that, for a smooth curve, is equivalent to the curve being hyperelliptic. \end{abstract}
MSC (2020): 14H10, 14H40, 14T90.
Keywords: Algebraic curve, hyperelliptic curve, tropical curve, Abel map.
\section{Introduction}
This paper is dedicated to the construction of an explicit resolution of the degree-$2$ Abel-Jacobi map for a regular smoothing of a nodal curve. For a smooth curve $C$, the degree-$d$ Abel map is an important morphism taking a $d$-tuple of points on $C$, to the associated invertible sheaf on the curve (tensored with a fixed invertible sheaf on the curve). When the fixed invertible sheaf is $\mathcal O_C(dP_0)$ for a pont $P_0$ on $C$, the map is usually called Abel-Jacobi map. This map encodes many important properties of the curve. For instance, the degree-$2$ Abel map detects when the curve is hyperelliptic. More precisely, a smooth curve is hyperelliptic if and only the degree-$2$ Abel map is not injective, and in this case the curve is endowed with a $g^1_2$, which can be identified with the fiber of the Abel map (up to the action of the symmetric group). In this paper we investigate how we can extend the above description to singular curves.
We construct an explicit resolution of the degree-$2$ Abel-Jacobi map using the results in \cite{AAP}, where the general problem of resolving Abel maps is reduced to checking a certain combinatorial property of the tropical Abel map. More precisely, this translates into the problem of showing the existence of a compatibility between the polyhedral structures of the tropical Jacobian of a curve and the product of the relevant tropical curve. This is the result contained in \cite[Theorem A]{AAP}. In degree $2$, this yields a very explicit combinatorial condition describing how to blow up the domain of the geometric Abel map to get a resolution. This is summarized in Theorem \ref{thm:tropgeoAbel}.\par
Let $\pi\colon \mathcal C\to B$ be a regular smoothing of a curve $C$ with a section $\sigma$ through its smooth locus, and $\mu$ be a polarization on $\mathcal C$. We denote by $\overline{\mathcal{J}}_{\mu}^{\sigma}$ the Esteves compactified Jacobian parametrizing $(\sigma,\mu)$-quasistable torsion-free rank-$1$ sheaves on $\mathcal C$ (see \cite{EE01}). As usual, we write $\mathcal C^2:= \mathcal C\times_B\mathcal C$. Let $\mathcal L$ be an invertible sheaf on $\mathcal C/B$ of degree-$(k+2)$. We define the degree-$2$ Abel (rational) map $\alpha^2_{\mathcal L}$ as \begin{align*} \alpha^2_{\mathcal L}\colon \mathcal C^2 &\dashrightarrow \overline{\mathcal{J}}_{\mu}^{\sigma}\\
(Q_1,Q_2)&\longmapsto [\mathcal L|_{\pi^{-1}(Q_1)}(-Q_1-Q_2)]. \end{align*}
Our main result holds when $\mu$ is the trivial degree-$0$ polarization and $\mathcal L$ is the trivial sheaf $\mathcal O_{\mathcal C}$. An important ingredient to describe the resolution of the degree-$2$ Abel-Jacobi map is the notion of tail of a nodal curve. A subcurve of a nodal curve is a $\delta$-tail if the subcurve and its complementary curves are connected and intersect each other in $\delta$ nodes.
\begin{thm*}[Theorem \ref{thm:Abel2}]
Let $Z_1,\dots,Z_N$ be the 2-tails and the 3-tails of $C$ which do not contain $\sigma(0)$. Consider the sequence of blowups
\[ \widetilde\mathcal C^2_N\stackrel{\phi_N}{\longrightarrow}\cdots \stackrel{\phi_2}{\longrightarrow}\widetilde\mathcal C^2_1\stackrel{\phi_1}{\longrightarrow}\widetilde\mathcal C^2_0\stackrel{\phi_0}{\longrightarrow}\mathcal C^2
\]
where $\phi_0$ is the blowup of $\mathcal C^2$ along its diagonal subscheme and $\phi_i$ is the blowup of $\widetilde{\mathcal C}^2_{i-1}$ along the strict transform of the divisor $Z_i\times Z_i$ of $\mathcal C^2$ via $\phi_0\circ\cdots\circ \phi_{i-1}$. Then the rational map
\[ \alpha^2_{\mathcal O_\mathcal C}\circ\phi_0\circ\cdots\circ\phi_N\colon \widehat\mathcal C^2_N\dashrightarrow\overline{\mathcal J}^\sigma_\mu
\]
is a morphism, i.e., it is defined everywhere.
\end{thm*}
Next we investigate the relation between the degree-2 Abel map and hyperelliptic (nodal) curves. More precisely, we study when the (symmetrized) degree-2 Abel map is not injective. The upshot is that this happens exactly when the curve has a simple torsion-free rank-1 sheaf of degree 2 with non-negative degree over every component of the curve and at least two sections. We call a curve satisfying all these condition a \emph{pseudo-hyperelliptic} curve. It is easy to see that if a curve is hyperelliptic, then it is pseudo-hyperelliptic.
It is worth noticing that a variation of the condition of hyperelliptic curve was already given by Caporaso in \cite{Capohyper}. She introduced and study the notion of weakly-hyperelliptic curve, which is the condition of the existence of a balanced degree-2 invertible sheaf on a curve with at least 2 sections. Again, if a curve is hyperelliptic, then it is weakly-hyperelliptic. We study the relation between weakly-hyperelliptic and pseudo-hyperelliptic.
\begin{thm*}[Theorem \ref{thm:hyperelliptic}] Let $C$ be a curve with no separating nodes. The following properties hold. \begin{enumerate}
\item The curve $C$ is pseudo-hyperelliptic if and only if, for some (every) regular smoothing $\mathcal C\to B$ of $C$, the symmetrized degree-$2$ Abel map of $\mathcal C$ is not injective. \item If $C$ is stable and weakly-hyperelliptic, then $C$ is pseudo-hyperelliptic. \end{enumerate} \end{thm*}
\section{Preliminaries}
Throughout the paper, we will use the notations introduced in \cite[Sections 2 and 3]{AAMPJac} and \cite[Section 3]{AAP}. In this section we just recall some basic definitions and constructions.
Given a graph $\Gamma$, we denote by $V(\Gamma)$ and $E(\Gamma)$ the sets of vertices and edges of $\Gamma$. Given a subset $V\subset V(\Gamma)$, we set $V^c=V(\Gamma)\setminus V$. For an orientation $\overrightarrow \Gamma$ on $\Gamma$, we denote by $s(e)$ and $t(e)$ the source and target of an edge $e\in E(\Gamma)$. Given subsets $V,W\subset V(\Gamma)$, we let $E(V,W)$ be the set of edges of $\Gamma$ connecting a vertex of $V$ with a vertex of $W$. A \emph{refinement} of a graph $\Gamma$ is a graph obtained by inserting a non-negative number $n_e$ (depending on $e$) of vertices in the interior of each edge $e$ of $\Gamma$. If a vertex $v$ of the refinement is inserted in the interior of an edge $e$ of $\Gamma$, we say that $v$ is a vertex \emph{over} $e$.
A \textit{metric graph} is a pair $(\Gamma,\ell)$ where $\Gamma$ is a graph and $\ell$ is a function $\ell\colon E(\Gamma)\rightarrow\mathbb R_{>0}$, called the \textit{length function}. Let $(\Gamma,\ell)$ be a metric graph. If $\ora{\Gamma}$ is an orientation on $\Gamma,$ we define the \textit{tropical curve} $X$ associated to $(\ora{\Gamma},\ell)$ as \[X=\frac{\left(\bigcup_{e\in E(\ora{\Gamma})}I_{e}\cup V(\ora{\Gamma})\right)}{\sim}\] where $I_{e}=[0,\ell(e)]\times\{e\}$ and $\sim$ is the equivalence relation generated by $(0,e)\sim s(e)$ and $(\ell(e),e)\sim t(e).$ We usually just write $e$ to represent $I_e$ in $X$, and denote by $e^\circ$ the interior of $e$. We say that $(\Gamma,\ell)$ is a \emph{model} of the tropical curve $X$. We will identify tropical curves with isometric models.
Let $\Gamma$ be a graph and define $\ell\colon E(\Gamma)\to \mathbb{R}$ by $\ell(e)=1$ for every $e\in E(\Gamma)$. We denote by $X_{\Gamma}$ the tropical curve induced by the metric graph $(\Gamma,\ell)$.
Let $X$ be a tropical curve and $\Gamma$ be a graph.
A \textit{divisor} on $X$ (respectively, on $\Gamma$) is a function $\mathcal{D}\colon X\rightarrow\mathbb{Z}$ (respectively, $D\colon V(\Gamma)\to \mathbb{Z}$) such that $\mathcal{D}(p)\neq0$ only for finitely many points $p\in X.$ Given a divisor $\mathcal D$ on $X$, we define the \textit{support} of $\mathcal{D}$ as the set of points $p$ of $X$ such that $\mathcal{D}(p)\neq0$ and denote it by $\text{supp}(\mathcal{D})$. A \emph{polarization} on $X$ (respectively, on $\Gamma$) is a function $\mu\colon X\rightarrow \mathbb R$ (respectively, $\mu\colon V(\Gamma)\to \mathbb{R}$) such that $\mu(p)\ne 0$ only for finitely many points $p\in X$ and such that $\sum_{p\in X} \mu(p)$ (respectively, $\sum_{v\in V(v\in V(\Gamma)} \mu(v)$) is an integer, called the \emph{degree} of the polarization $\mu$.\par
Given a point $p_0$ in $X$, (respectively, a vertex $v_0\in V(\Gamma)$), a divisor $\mathcal{D}$ on $X$ (respectively, $D$ on $\Gamma$) is called $(p_0,\mu)$-quasistable (respectively, $(v_0,\mu)$-quasistable) if:
\[
\sum_{p\in Y}(\mathcal{D}(p)-\mu(p)) + \frac{\delta_Y}{2}\geq 0
\]
for every tropical subcurve $Y$ of $X$ (respectively, every subset $Y\subset V(\Gamma)$), where the inequality is strict if $p_0\in Y\neq X$. Here, $\delta_Y$ is the number of tangent direction outgoing from $Y$ in the case of a tropical curve (see \cite[Section 3.1]{AAMPJac} for the precise definition), while it is equal to $|E(Y,Y^c)|$ in the case of a graph.
Let $X$ be a tropical curve and $p_0$ be a point of $X$. Let $\mu$ be a polarization on $X$.
Recall that in an equivalence class of a divisor on a tropical curve there is just one $(p_0,\mu)$-quasistable divisor (see \cite[Theorem 5.6]{AAMPJac}). For a degree-$d$ divisor $\mathcal D$ on $X$, we denote by $\qs(\mathcal D)$ the unique $(p_0,\mu)$-quasistable divisor on $X$ which is equivalent to $\mathcal D$. Given an oriented model $(\Gamma,\ell)$ of $X$, for every edge $e\in E(\Gamma)$ and every real number $t\in [0,\ell(e)]$, we let by $p_{e,t}$ the point on $e$ at distance $t$ from the source of $e$.\par
Given a tropical curve $X$, we let $J^{\text{trop}}_{p_0,\mu}(X)$ be the tropical Jacobian parametrizing $(p_0,\mu)$-quasistable divisors on $X$. Recall that $J^{\text{trop}}_{p_0,\mu}(X)$ is homeomorphic to the usual tropical Jacobian (see \cite[Theorem 5.10]{AAMPJac}).
We set $X^2:=X\times X$. Given a divisor $\mathcal D^\dagger$ on $X$, we define the tropical Abel map \begin{align*}
\alpha_{2,\mathcal D^\dagger}^\text{trop} \colon X^2 &\to J^{\text{trop}}_{p_0,\mu}(X)\\ (p_1,p_2)&\longmapsto [\mathcal D^\dagger-p_1-p_2], \end{align*} where $[-]$ denotes the class of a divisor in the tropical Jacobian. Alternatively, the map $\alpha_{2,\mathcal D^\dagger}^\text{trop}$ takes $(p_1,p_2)$ to the unique $(p_0,\mu)$-quasistable divisor in the class of $\mathcal D^\dagger-p_1-p_2$.
\begin{Rem} \label{prop:quasiquasi} Let $X$ be a tropical curve with a point $p_0\in X$. Let $\Gamma$ be a model of $X$. Let $\mu$ be a polarization on $X$ induced by a polarization on $\Gamma$ and $\mathcal D$ a degree-$d$ divisor on $X$.
We let $\widehat{\Gamma}$ the minimal refinement of $\Gamma$ such that $\text{supp}(\mathcal D)\subset V(\widehat{\Gamma})$. We denote by $D$ the divisor on $\widehat{\Gamma}$ induced by $\mathcal D$. We call the pair $(\widehat{\Gamma},D)$ on $\widehat{\Gamma}$ the \emph{combinatorial type} of $\mathcal D$. By \cite[Proposition 5.3]{AAMPJac}, the degree-$d$ divisor $\mathcal D$ on $X$ is $(p_0,\mu)$-quasistable if and only if $\widehat{\Gamma}$ is obtained by inserting at most one vertex in the interior of each edge of $\Gamma$ and $D$ is $(p_0,\mu)$-quasistable on $\widehat{\Gamma}$. \end{Rem}
\section{Degree-2 Abel maps}\label{sec:degree2}
Let $C$ be a nodal curve over an algebraically closed filed $k$. A subcurve $Z$ of $C$ is a reduced union of components of $C$. Given a subcurve $Z$ of $C$, we let $Z^c:=\overline{C\setminus Z}$. Throughout this section we will fix a regular smoothing $\pi\colon\mathcal C\rightarrow B$ of a nodal curve $C$ with a section $\sigma\colon B\rightarrow \mathcal C$ of $\pi$ through its smooth locus. We denote by $\mathcal C^2:=\mathcal C\times_B\mathcal C$.
Let $\mu$ be a degree-$k$ polarization on $\mathcal C$. We denote by $\overline{\mathcal{J}}_{\mu}^{\sigma}$ the Esteves compactified Jacobian parametrizing $(\sigma,\mu)$-quasistable torsion-free rank-$1$ sheaves on the curves of the family $\pi$ (see \cite{EE01} for more details). Let $\mathcal L$ be an invertible sheaf on $\mathcal C/B$ of degree-$(k+2)$. As in \cite{AAJCMP}, we define the degree-$2$ Abel (rational) map $\alpha^2_{\mathcal L}$ as \begin{align*} \alpha^2_{\mathcal L}\colon \mathcal C^2 &\dashrightarrow \overline{\mathcal{J}}_{\mu}^{\sigma}\\
(Q_1,Q_2)&\longmapsto [\mathcal L|_{\pi^{-1}(\pi(Q_1))}(-Q_1-Q_2)]. \end{align*}
We let $\Gamma$ be the dual graph of $C$ and $X_{\Gamma}$ be the tropical curve induced by $\Gamma$ (with unitary lengths). Given an invertible sheaf $\mathcal L$ on $\mathcal C$, we denote by $D^{\dagger}_{\mathcal L}$ the divisor on $\Gamma$ given by the multidegree of $\mathcal L|_C$. We also let $\mathcal D^\dagger_\mathcal L$ be the divisor on $X_\Gamma$ induced by $D^{\dagger}_{\mathcal L}$.
Given a point $\mathcal N=(N_1,N_2)$ of $\mathcal C^2$, where $N_i$ is a node of $C$, we will consider the following two ways of blowing up $\mathcal C^2$ locally around $\mathcal N$. If $N_1\in Z_1\cap Z_1^c$ and $N_2\in Z_2\cap Z_2^c$ for subcurves $Z_1$ and $Z_2$ of $C$, we can consider the blowups $\phi\colon \widetilde{\mathcal \mathcal C}^2_\phi\rightarrow \mathcal C^2$ and $\phi'\colon \widetilde{\mathcal \mathcal C}^2_{\phi'}\rightarrow \mathcal C^2$ respectively along $Z_1\times Z_2$, or along $Z_1\times Z_2^c$. The first one is also equivalent to the blowup along $Z_1^c\times Z_2^c$ and the second one is equivalent to the blowup along $Z_1^c\times Z_2$. In both cases, the inverse image of $\mathcal N$ is isomorphic to $\mathbb P^1_k$. The situation is illustrated in Figure \ref{Fig:blowup}, where $\st_\phi$ and $\st_{\phi'}$ applied to a divisor of $\mathcal C^2$ denote the strict transform of this divisor. These blowups induce a dual picture on the product $X_\Gamma^2$: we illustrate the relation between these blowups and the dual picture in Figure \ref{fig:blowup-square1}.
\begin{figure}
\caption{The two types of blowup of $\mathcal C^2$ around $(N_1,N_2)$.}
\label{Fig:blowup}
\end{figure}
\begin{Thm}\label{thm:tropgeoAbel} Let $\pi\colon\mathcal C\rightarrow B$ be a regualar smoothing of a nodal curve $C$ with smooth components. Let $\sigma\colon B\rightarrow \mathcal C$ be a section of $\pi$ through its smooth locus. Let $\mu$ be a polarization of degree $k$ over the family and $\mathcal L$ be an invertible sheaf on $\mathcal C$ of degree $k+2$. Let $(N_1,N_2)$ be a point of $\mathcal C^2$, with $N_i$ a node of $C$. Let $Z_1$ and $Z_2$ be subcurves of $C$ such that $N_1\in Z_1\cap Z_1^c$ and $N_2\in Z_2\cap Z_2^c$. Let $e_1$ and $e_2$ be the edges in the dual graph $\Gamma$ of $C$ that correspond to $N_1$ and $N_2$, where $e_i$ is oriented from $Z_i$ to $Z_i^c$. Consider the divisor $\mathcal D_{x,y}=\mathcal D^\dagger_\mathcal L-p_{e_1,x}-p_{e_2,y}$ on $X_\Gamma$, for some $x,y\in [0,1]$.
\begin{enumerate}
\item If the combinatorial type of $\qs(\mathcal D_{x,y})$ is constant on the sets
\[
\{(x,y);\ 0<x<y<1\}\text{ and } \{(x,y);\ 0<y<x<1\},
\]
then the blowup of $\mathcal C^2$ along $Z_1\times Z_2$ resolves the Abel map $\alpha^2_\mathcal L$ locally around the point $(N_1,N_2)$.
\item If the combinatorial type of $\qs(\mathcal D_{x,y})$ is constant on the sets
\[
\{(x,y);\ 0<x<1-y<1\}\text{ and } \{(x,y);\ 0<1-y<x<1\},
\]
then the blowup of $\mathcal C^2$ along $Z_1\times Z_2^c$ resolves the Abel map $\alpha^2_\mathcal L$ locally around the point $(N_1,N_2)$. \begin{figure}
\caption{The sets $\{(x,y);\ 0<x<1-y<1\}$ and $\{(x,y);\ 0<1-y<x<1\}$ and the corresponding blow-up.}
\label{fig:blowup-square1}
\end{figure}
\item If the combinatorial type of $\qs(\mathcal D_{x,y})$ is constant on the set
\[
\{(x,y);\ 0<x,y<1\},
\]
then the Abel map $\alpha^2_\mathcal L$ is defined at the point $(N_1,N_2)$.
\end{enumerate}
\end{Thm} \begin{proof} Items (1) and (2) follow directly from \cite[Theorem 5.4]{AAP}.
Let us prove Item (3). Let $\mathcal N=(N_1,N_2)\in \mathcal C^2$. Let $\phi\colon \mathcal X\rightarrow \mathcal C^2$ and $\phi'\colon\mathcal Y\rightarrow \mathcal C^2$ be the blowups respectively along $Z_1\times Z_2$ and $Z_1\times Z_2^c$ (see Figure \ref{Fig:blowup}). By items (1) and (2), we know that $\alpha^2_{\mathcal L}\circ\phi$ and $\alpha^2_{\mathcal L}\circ\phi'$ are defined respectively over the inverse images $\phi^{-1}(\mathcal N)\cong \mathbb P^1_k$ and $\phi'^{-1}(\mathcal N)\cong \mathbb P^1_k$. Let $y_0$ be the distinguished point on $\phi^{-1}(\mathcal N)$ given by \[ y_0=\st_\phi(Z_1\times Z_2)\cap \st_\phi(Z_1\times Z_2^c)\cap \st_\phi(Z_1^c\times Z_2^c). \] Let $x_1,x_2$ be any two points on $\phi'^{-1}(\mathcal N)$. We know that $\alpha^2_\mathcal L\circ \phi'$ is defined at $x_1$ and $x_2$. For $i=1,2$, consider a map $\rho'_i\colon \Spec k[[t]]\rightarrow \mathcal Y$ such that $\rho'_i(0)=x_i$ and $\rho_i(\eta)$ is contained in $\st_{\phi'}(Z_1\times Z_2^c)$, as in Figure \ref{fig:blowups_phi}, where $0$ and $\eta$ are the special and generic points of $\Spec k[[t]]$, respectively. In particular, we have $\alpha^2_\mathcal L\circ \phi'(x_i)=\alpha^2_\mathcal L\circ\overline{\rho}_i(0)$, where $\overline{\rho}_i=\phi'\circ\rho'_i\colon \Spec(k[[t]])\rightarrow \mathcal C^2$. By construction, we can lift $\overline{\rho}_i$ to maps $\rho_i\colon \Spec k[[t]]\to \mathcal X$ such that $\rho_1(0)=\rho_2(0)=y_0$. By the same reasoning, we have $\alpha^2_\mathcal L\circ \phi(y_0)=\alpha^2_\mathcal L\circ\rho_i(0)$ for $i=1,2$. Then we get: \[ \alpha^2_{\mathcal L}\circ \phi'(x_1)=\alpha^2_{\mathcal L}\circ \phi(y_0)=\alpha^2_{\mathcal L}\circ \phi'(x_2). \] Hence $\alpha^2_{\mathcal L}\circ \phi'$ contracts all fibers of $\phi'$. Moreover, arguing as in the proof of \cite[Corollary III 11.4]{H}, we have $\phi'_*\mathcal O_{\mathcal Y}\cong \mathcal O_{\mathcal C^2}$, since $\phi'$ is birational and $\mathcal C^2$ is normal. Hence by the Rigidity Lemma (see \cite[Lemma 1.15, pag.12]{D}) the map $\alpha^2_{\mathcal L}\circ \phi'$ factors through $\phi'$, so $\alpha^2_{\mathcal L}$ is defined at $\mathcal N$.
\begin{figure}
\caption{The blowups $\phi$ and $\phi'$.}
\label{fig:blowups_phi}
\end{figure} \end{proof}
\section{The resolution of the degree-2 Abel map}
\subsection{Local resolutions}
Throughout this section we will fix a regular smoothing $\pi\colon\mathcal C\rightarrow B$ of a nodal curve $C$ with a section $\sigma\colon B\rightarrow \mathcal C$ of $\pi$ through its smooth locus.
We will perform blowups of $\mathcal C^2$ along divisors of type $Z\times Z$, where $Z$ is a subcurve of the special fiber $C$. Actually, we will restrict our attention to a special class of subcurves, called \emph{tails}.
\begin{Def}
A \emph{$\delta$-tail} of a nodal curve $C$ is a connected subcurve $Z$ such that $Z^c$ is connected and $|Z\cap Z^c|=\delta$. \end{Def}
\begin{Prop}\label{prop:1-tail}
Let $\mu$ be a polarization of degree $k$ and $\mathcal L$ be an invertible sheaf of degree $k+2$ over $\mathcal C/B$. Assume that the components of $C$ are smooth. Consider a point $\mathcal N=(N_1,N_2)\in\mathcal C^2$, where $N_1,N_2$ are nodes of $C$, with $N_1=Z\cap Z^{c}$ for a $1$-tail $Z$ of $C$. Then the degree-$2$ Abel map $\alpha^2_{\mathcal L}\colon\mathcal C^{2}\dashrightarrow \overline{\mathcal{J}}_{\mu}^\sigma$ is defined at $\mathcal N$.
\end{Prop}
\begin{proof}
Let $\Gamma$ be the dual graph of $C$ and $X=X_\Gamma$ the associated tropical curve with edges of unitary lengths.
Let $v_0$ be the vertex of $\Gamma$ corresponding to $P_0=\sigma(0)$, and $p_0\in X$ be the point corresponding to $v_0$. We let $e_1$ and $e_2$ be the edges of $\Gamma$ corresponding to $N_1$ and $N_2$. The tropical Abel map $\alpha^\text{trop}_{2,\mathcal D^\dagger_{\mathcal L}}\colon X^2\rightarrow J^\text{trop}_{p_0,\mu}$ takes a pair $(p_{e_1,t_1},p_{e_2,t_2})$, for real numbers $t_1,t_2\in (0,1)$, to the class of the divisor on $X$ given by: \begin{equation}\label{eq:alpha2} \alpha^\text{trop}_{2,\mathcal D^\dagger_{\mathcal L}}(p_{e_1,t_1},p_{e_2,t_2})=[\mathcal D^\dagger_\mathcal L-p_{e_1,t_1}-p_{e_2,t_2}]. \end{equation}
We define the divisor
$\mathcal P=p_{e_1,t_1}-p_{e_1,0}$ on $X$. Since $N_1=Z\cap Z^c$ for a $1$-tail $Z$ of $X$, we have that the graph obtained from $\Gamma$ by removing the edge $e_1$ is not connected. Hence the divisor $\mathcal P$ on $X$ is principal. So we can write: \[ \alpha^{\text{trop}}_{2,\mathcal D^\dagger_{\mathcal L}}(p_{e_1,t_1},p_{e_2,t_2})=[\mathcal{\widehat D}^\dagger-p_{e_2,t_2}], \] where $\mathcal{\widehat D}^\dagger=\mathcal D^\dagger_\mathcal L-p_{e_1,0}$ (which is a divisor on $X$ induced by a divisor on $\Gamma$). So we reduce ourselves to the case of the degree-$1$ Abel map. As explained in \cite[Lemma 5.10]{AAP} and in the proof of \cite[Theorem 5.8]{AAP}, the combinatorial type of the quasistable divisor on $X$ equivalent to $\mathcal{\widehat D}^\dagger-p_{e_1,t_2}$ is independent of $t_2$. Hence the combinatorial type of the quasistable divisor on $X$ equivalent to $\mathcal D^\dagger_\mathcal L-p_{e_1,t_1}-p_{e_2,t_2}$ is independent of the pairs $t_1,t_2\in (0,1)$. By Theorem \ref{thm:tropgeoAbel} (3), we deduce that the Abel map $\alpha^2_{\mathcal L}$ is already defined at $(N_1,N_2)$.
\end{proof}
\begin{Prop}\label{prop:2-tail}
Let $\mu$ be a polarization of degree $k$ and $\mathcal L$ an invertible sheaf of degree $k+2$ over $\mathcal C/B$. Assume that the components of $C$ are smooth.
Let $Z$ be a $2$-tail of $C$ and write $\{N_1,N_2\}=Z\cap Z^c$. Consider the point
\[
\mathcal N=(N_1,N_2)\in (Z\cap Z^c)\times (Z\cap Z^c)\subset \mathcal C^2.
\]
Let $\phi\colon\widetilde{\mathcal C}^2\rightarrow \mathcal C^2$ be the blowup of $\mathcal C^2$ with center $Z\times Z$. Then the rational map
\[
\widetilde{\alpha}^2_{\mathcal L}\colon \widetilde{\mathcal C}^2\stackrel{\phi}{\longrightarrow} \mathcal C^2\stackrel{\alpha^2_{\mathcal L}}{\dashrightarrow}\overline{\mathcal J}^\sigma_\mu
\] is defined along the rational curve $\phi^{-1}(\mathcal N)\cong\mathbb P^1_k$.
\end{Prop}
\begin{proof}
We can keep the set-up of Proposition \ref{prop:1-tail}. The tropical Abel map $\alpha^\text{trop}_{2,\mathcal D^\dagger_{\mathcal L}}\colon X^2\rightarrow J^\text{trop}_{p_0,\mu}$ is as in Equation \eqref{eq:alpha2}. Assume that $t_1>t_2$. We define the divisor on $X$: \[ \mathcal P=p_{e_1,t_1}-p_{e_1,t_1-t_2}+p_{e_2,t_2}-p_{e_2,0}. \] Since $\{N_1,N_2\}=Z\cap Z^c$ for a $2$-tail $Z$ of $C$, we have that the graph obtained from $\Gamma$ by removing the edges $e_1,e_2$ is not connected. Hence $\mathcal P$ is a principal divisor. Then we have \[ \alpha^{\text{trop}}_{2,\mathcal D^\dagger_{\mathcal L}}(p_{e_1,t_1},p_{e_2,t_2})=[\mathcal{\widehat D}^\dagger-p_{e_1,t}], \] where $t=t_1-t_2$ and $\mathcal{\widehat D}^\dagger=\mathcal D^\dagger_\mathcal L-p_{e_2,0}$ (which is a divisor induced by a divisor on $\Gamma$). So we reduce ourselves to the case of the degree-$1$ Abel map: as explained in \cite[Lemma 5.10]{AAP} and in the proof of \cite[Theorem 5.8]{AAP}, the combinatorial type of the quasistable divisor on $X$ equivalent to $\mathcal{\widehat D}^\dagger-p_{e_1,t}$ is independent of $t$. Hence the combinatorial type of the quasistable divisor on $X$ equivalent to $\mathcal D^\dagger_\mathcal L-p_{e_1,t_1}-p_{e_2,t_2}$ is independent of $(t_1,t_2)$ whenever $t_1>t_2$. A similar reasoning can be done for the case $t_1<t_2$. Hence, using Theorem \ref{thm:tropgeoAbel} (1), we conclude that the blowup along $Z\times Z$ gives rise to a resolution of $\alpha^2_{\mathcal L}$ locally around $\mathcal N=(N_1,N_2)$. \end{proof}
\begin{Prop}\label{prop:diagonal}
Let $\mu$ be a polarization of degree $k$ and $\mathcal L$ an invertible sheaf of degree $k+2$ over $\mathcal C/B$. Assume that the components of $C$ are smooth.
Consider the point
$\mathcal N=(N,N) \in \mathcal C^2$, for a node $N$ of $C$.
Let $\phi\colon\widetilde{\mathcal C}^2\rightarrow \mathcal C^2$ be the blowup of $\mathcal C^2$ with center the diagonal subscheme of $\mathcal C^2$. Then the rational map
\[
\widetilde{\alpha}^2_{\mathcal L}\colon \widetilde{\mathcal C}^2\stackrel{\phi}{\longrightarrow} \mathcal C^2\stackrel{\alpha^2_{\mathcal L}}{\dashrightarrow}\overline{\mathcal J}^\sigma_\mu
\] is defined along the rational curve $\phi^{-1}(\mathcal N)\cong\mathbb P^1_k$.
\end{Prop}
\begin{proof}
We can keep the set-up of Proposition \ref{prop:1-tail}. The tropical Abel map $\alpha^\text{trop}_{2,\mathcal D^\dagger_{\mathcal L}}\colon X^2\rightarrow J^\text{trop}_{p_0,\mu}$ is as in Equation \eqref{eq:alpha2}, where $e:=e_1=e_2$. Assume that $t_1+t_2<1$. We define the principal divisor on $X$: \[ \mathcal P=p_{e,0}-p_{e,t_1}-p_{e,t_2}+p_{e,t_1+t_2}. \] Then we have \[ \alpha^{\text{trop}}_{2,\mathcal D^\dagger_{\mathcal L}}(p_{e,t_1},p_{e,t_2})=[\mathcal{\widehat D}^\dagger-p_{e,t}], \] where $\mathcal{\widehat D}^\dagger=\mathcal D^\dagger_{\mathcal L}-p_{e,0}$ and $t=t_1+t_2$. So we reduce ourselves to the case of the degree-$1$ Abel map: as explained in \cite[Lemma 5.10]{AAP} and in the proof of \cite[Theorem 5.8]{AAP}, the combinatorial type of the quasistable divisor on $X$ equivalent to $\mathcal{\widehat D}^\dagger-p_{e,t}$ is independent of $t$. Hence the combinatorial type of the quasistable divisor on $X$ equivalent to $\mathcal D^\dagger_\mathcal L-p_{e,t_1}-p_{e,t_2}$ is independent of $(t_1,t_2)$ whenever $t_1+t_2<1$.\par
The reasoning is similar for $t_1+t_2>1$: we just consider
\[
\mathcal P=p_{e,1}-p_{e,t_1}-p_{e,t_2}+p_{e,t_1+t_2-1},
\]
$\mathcal{\widehat D}^\dagger=\mathcal D^\dagger_{\mathcal L}-p_{e,1}$ and $t=t_1+t_2-1$, so that
\[ \alpha^{\text{trop}}_{2,\mathcal D^\dagger_{\mathcal L}}(p_{e,t_1},p_{e,t_2})=[\mathcal{\widehat D}^\dagger-p_{e,t}]. \]
By Theorem \ref{thm:tropgeoAbel} (2), we deduce that the blowup along the diagonal subscheme of $\mathcal C^2$ gives rise to a resolution of $\alpha^2_{\mathcal L}\circ \phi$ locally around $\mathcal N=(N,N)$.
\end{proof}
\subsection{The resolution of the Abel-Jacobi map}
Our main goal is to give a complete resolution of the degree-$2$ Abel-Jacobi map of any nodal curve, namely the map taking a pair $(Q_1,Q_2)$ of points on a curve $C$ to $\mathcal O_C(2P_0-Q_1-Q_2)$ for a given smooth point $P_0$ of $C$. This is done in Theorem \ref{thm:Abel2}. Before, we need two results.
\begin{Lem}\label{lem:perfect}
Let $Z$ be a $\delta$-tail of a curve $C$. \begin{itemize}
\item[(1)] If $Z\cap Z^c\subset Z'$ for some tail $Z'$ of $C$, then either $Z\subset Z'$ or $Z^c\subset Z'$.
\item[(2)] If $|(Z\cap Z^c \cap Z'\cap (Z')^c|=\delta-1$ for some tail $Z'$ of $C$, then one of the following conditions holds:
\[
Z\subset Z', \;\; Z'\subset Z, \;\; Z^c\subset Z',\;\; Z'\subset Z^c.
\] \end{itemize}
\end{Lem}
\begin{proof}
See \cite[Lemma 2.4]{P}.
\end{proof}
\begin{Lem}
Let $P_0$ be a smooth point of $C$.
Let $\mathcal T=(Z_1,\dots,Z_h)$ be a sequence of tails of $C$, where $Z_i$ is a $k_i$-tail with $k_i\in\{2,3\}$ and $P_0\not\in Z_i$. Consider the sequence of blowups
\[ \phi_{\mathcal T}\colon\widetilde\mathcal C^2_h\stackrel{\phi_h}{\longrightarrow}\cdots \stackrel{\phi_3}{\longrightarrow}\widetilde\mathcal C^2_2\stackrel{\phi_2}{\longrightarrow}\widetilde\mathcal C^2_1\stackrel{\phi_1}{\longrightarrow}\widetilde\mathcal C^2_0\stackrel{\phi_0}{\longrightarrow}\mathcal C^2
\]
where $\phi_0$ is the blowup of $\mathcal C^2$ along its diagonal subscheme and $\phi_i$ is the blowup of $\widetilde{\mathcal C}^2_{i-1}$ along the strict transform of the divisor $Z_i\times Z_i$ of $\mathcal C^2$ via $\phi_1\circ\cdots\circ \phi_{i-1}$. Then $\phi_{\mathcal T}$ is independent of the ordering of the sequence $\mathcal T$. \end{Lem}
\begin{proof}
Assume that a permutation $\mathcal T'$ of $\mathcal T$ gives rise to a blowup $\phi_{\mathcal T'}$ different from $\phi_{\mathcal T}$. This implies that, locally at a point $\mathcal N=(N_1,N_2)$ with $N_1,N_2$ distinct nodes of $C$, the blowups $\phi_{\mathcal T}$ and $\phi_{\mathcal T'}$ are different.
We can assume that locally at $\mathcal N$, the blowup $\phi_{\mathcal T}$ has center $Z_i\times Z_i$ and the blowup $\phi_{\mathcal T'}$ has center $Z_j\times Z_j$, for $i,j\in\{1,\dots,h\}$, so that we have $Z_i\ne Z_j$ and
\[
\{N_1,N_2\}\subset Z_i\cap Z_i^c\cap Z_j\cap Z_j^c.
\]
By Lemma \ref{lem:perfect}, one of the following conditions holds:
\begin{equation}\label{eq:perfect}
Z_i\subset Z_j, \;\; Z_j\subset Z_i, \;\; Z^c_i\subset Z_j, \;\; Z_j\subset Z_i^c.
\end{equation}
Let $C_1$, $C'_1$, $C_2$, $C'_2$ be the components of $C$ such that $N_1\in C_1\cap C'_1$ and $N_2\in C_2\cap C'_2$. Since $\phi_{\mathcal T}$ and $\phi_{\mathcal T'}$ are different locally at $\mathcal N$, we can assume, without loss of generality, that $C_1\cup C_2\subset Z_i$ and $C_1\cup C_2'\subset Z_j$.
Then we have
\[
C_1\subset Z_i\cap Z_j, \;\; C_2\subset Z_i\cap Z_j^c, \;\; C'_2\subset Z_i^c\cap Z_j.
\]
On the other hand:
\begin{enumerate}
\item since $C_1\subset Z_i\cap Z_j$, it follows that $Z_j\not \subset Z_i^c$.
\item since $C_2\subset Z_i\cap Z_j^c$, it follows that $Z_i\not\subset Z_j$.
\item since $C_2'\subset Z_i^c\cap Z_j$, it follows that $Z_j\not\subset Z_i$.
\item since $P_0\in Z_i^c\setminus Z_j$, it follows that $Z_i^c\not\subset Z_j$.
\end{enumerate}
This contradicts Equation \eqref{eq:perfect}.
\end{proof}
\begin{Thm}[Degree-$2$ Abel-Jacobi map]\label{thm:Abel2}
Let $\pi\colon\mathcal C\rightarrow B$ be a regular smoothing of a nodal curve $C$. Let $\sigma\colon B\rightarrow \mathcal C$ be a section of $\pi$ through its smooth locus and $\mu$ be the trivial degree-$0$ polarization. Let $Z_1,\dots,Z_N$ be the 2-tails and the 3-tails of $C$ which do not contain $\sigma(0)$. Consider the sequence of blowups
\[ \widetilde\mathcal C^2_N\stackrel{\phi_N}{\longrightarrow}\cdots \stackrel{\phi_2}{\longrightarrow}\widetilde\mathcal C^2_1\stackrel{\phi_1}{\longrightarrow}\widetilde\mathcal C^2_0\stackrel{\phi_0}{\longrightarrow}\mathcal C^2
\]
where $\phi_0$ is the blowup of $\mathcal C^2$ along its diagonal subscheme and $\phi_i$ is the blowup of $\widetilde{\mathcal C}^2_{i-1}$ along the strict transform of the divisor $Z_i\times Z_i$ of $\mathcal C^2$ via $\phi_0\circ\cdots\circ \phi_{i-1}$. Then the rational map
\[ \alpha^2_{\mathcal O_\mathcal C}\circ\phi_0\circ\cdots\circ\phi_N\colon \widehat\mathcal C^2_N\dashrightarrow\overline{\mathcal J}^\sigma_\mu
\]
is a morphism, i.e., it is defined everywhere.
\end{Thm}
Before proving the theorem, we need to recall a result in \cite{P} describing how to convert the sheaf $\mathcal O_C(2P_0-Q_1-Q_2)$ into a $(\sigma,\mu)$-quasistable sheaf, where $P_0,Q_1,Q_2$ are smooth points of the nodal curve $C$. We will give the graph-theoretical equivalent of this result, which suits better with our purposes. More precisely, given a graph $\Gamma$ and vertices $v_0,v_1,v_2\in V(\Gamma)$, we will describe the $(v_0,\mu)$-quasistable divisor on $\Gamma$ equivalent to $2v_0-v_1-v_2$ (see Theorem \ref{thm:convert}).
Let $\Gamma$ be a graph. Given a subset $V\subset V(\Gamma)$, we denote by $\Gamma(V)$ the subgraph of $\Gamma$ whose set of vertices is $V$ and whose edges are the edges of $\Gamma$ connecting two (possibly coinciding) vertices of $V$.
\begin{Def}
A \emph{hemisphere} of $\Gamma$ is a subset $H\subset V(\Gamma)$ such that $\Gamma(H)$ and $\Gamma(H^c)$ are connected subgraphs of $\Gamma$. A $\delta$-hemisphere of $\Gamma$ is a hemisphere $H$ such that $|E(H,H^c)|=\delta$. \end{Def}
We denote by $\mathcal H_{\Gamma,\delta}$ the set of $\delta$-hemispheres of $\Gamma$. Given subsets $V,W\subset V(\Gamma)$, we define: \[
\mathcal H_{\Gamma,\delta}(V,W):=\{H \in \mathcal H_{\Gamma,\delta} | V\subset H^c \text{ and } W\subset H\}. \]
Let $S$ be a finite set. We say that a set $\mathcal H$ of subsets of $S$ is union-closed (respectively, intersection-closed) if $H_1\cup H_2\in \mathcal H$ (respectively, $H_1\cap H_2\in \mathcal H$) for every $H_1,H_2\in \mathcal H$. We note that every non-empty intersection-closed set has a unique minimal element and every non-empty union-closed set has a unique maximal element.
\begin{Prop} \label{prop:closed} Let $\Gamma$ be a graph and $v_0,v_1,v_2$ vertices of $\Gamma$. Then the sets $\mathcal H_{\Gamma,1}(v_0,v_1)$ and $\mathcal H_{\Gamma,2}(v_0,\{v_1,v_2\})$ are union-closed and intersection-closed. \end{Prop}
\begin{proof} See \cite[Lemma 4.3]{CE} and \cite[Section 3 and Proposition 3.1]{P}. \end{proof}
\begin{Def} Given subsets $V,W\subset V(\Gamma)$, we say that $W$ is \emph{$V$-free} if $E(V,V^c)\cap E(W,W^c)=\emptyset$. \end{Def}
\begin{Rem}\label{rem:H1H2} A $1$-hemisphere $H$ is $H'$-free for every $\delta$-hemisphere $H'\ne H, H^c$.
If $H_1,H_2\subset V(\Gamma)$ are $V$-free hemispheres, for some $V\subset V(\Gamma)$, then $H_1\cap H_2$ and $H_1\cup H_2$ are also $V$-free. \end{Rem}
\begin{Def} Let $\Gamma'$ be a subdivision of $\Gamma$. Let $V$ be a subset of $V(\Gamma')$. We say that an edge $e\in E(\Gamma)$ is \emph{fully contained} in $V$ if $V$ contains the vertices incident to $e$ and all the vertices over $e$. Note that when $\Gamma'=\Gamma$, this simply means that $e\in E(V,V)$. \end{Def}
\begin{Lem}\label{lem:intersect23}
Let $H_2$ and $H_3$ be a $2$-hemisphere and a $3$-hemisphere. Write $E(H_2,H_2^c)=\{f_1,f_2\}$ and $E(H_3,H_3^c)=\{e_1,e_2,e_3\}$. Assume that the intersection $H=H_2\cap H_3$ is non-empty and properly contained in $H_2$ and $H_3$. Assume that $H_2\cup H_3\neq V(\Gamma)$. Then, up to reordering the indices, one of the following properties hold
\begin{enumerate}
\item $H$ is a $2$-hemisphere such that $E(H,H^c)=\{f_1,e_1\}$, with $f_1$ fully contained in $H_3$ and $e_1$ fully contained in $H_2$, while $f_2$ is fully contained in $H_3^c$ and $e_2,e_3$ are fully contained in $H_2^c$.
\item $H$ is a $3$-hemisphere such that $E(H,H^c)=\{f_1,e_1,e_2\}$, with $f_1$ fully contained in $H_3$ and $e_1,e_2$ fully contained in $H_2$, while $f_2$ is fully contained in $H_3^c$ and $e_3$ is fully contained in $H_2^c$.
\end{enumerate} \end{Lem} \begin{proof} By the hypothesis, the sets $H$, $H_2\setminus H$, $H_3\setminus H$ and $H_2^c\cap H_3^c$ are nonempty and form a partition of $V(\Gamma)$. Since $H_3$ is connected, we have that $E(H,H_3\setminus H)$ is nonempty. However $E(H,H_3\setminus H)\subseteq E(H_2,H_2^c)=\{f_1,f_2\}$. We assume that $f_1\in E(H,H_3\setminus H)$. Arguing in a similar manner, using that $H_3^c$ is connected, we have that $f_2\in E(H_2\setminus H, H_2^c\cap H_3^c)$. Even more, we can conclude that $e_1\in E(H,H_2\setminus H)$ and $e_3\in E(H_3\setminus H,H_2^c\cap H_3^c)$ (see Figure \ref{fig:intersection_hemispheres}).
\begin{figure}
\caption{The edges $e_1,e_3,f_1,f_2$.}
\label{fig:intersection_hemispheres}
\end{figure}
Since $H_2$ is a $2$-hemisphere, we have that
\begin{align*}
\quad\quad\quad\quad\quad\quad\quad\quad E(H,H_3\setminus H)&=\{f_1\}, & E(H_2\setminus H,H_3\setminus H)&=\emptyset,\quad\quad\quad\quad\quad\quad\quad\quad \\
\quad\quad\quad\quad\quad\quad\quad\quad E(H,H_2^c\cap H_3^c)&=\emptyset, & \quad E(H_2\setminus H,H_2^c\cap H_3^c)&=\{f_2\}.\quad\quad\quad\quad\quad\quad\quad\quad
\end{align*}
Hence $E(H_3,H_3^c)=E(H,H_2\setminus H)\cup E(H_3\setminus H, H_2^c\cap H_3^c)$. Therefore, the edge $e_2$ only has two possibilities: it belongs to either $E(H_3\setminus H, H_2^c\cap H_3^c)$ or $E(H,H_2\setminus H)$. In the former case $H$ satisfies the conditions in item (1), while in the latter case it satisfies the conditions in item (2).
Note that the sets $H$, $H_3\setminus H$, $H_2\setminus H$ and $H_2^c\cap H_3^c$ are connected because $H_3=H\cup (H_3\setminus H)$ is connected and $H$ and $H_3\setminus H$ are connected by a single edge, hence each $H$ and $H_3\setminus H$ must be connected. The same reasoning holds for $H_2\setminus H$ and $H_2^c\cap H_3^c$, using the fact that $H_3^c$ is connected. Hence $H^c=(H_3\setminus H) \cup (H_2\setminus H) \cup (H_2^c\cap H_3^c)$ is connected, which means that $H$ is a hemisphere. \end{proof}
We let $H_{2,1}$ be the minimal element of $\mathcal H_{\Gamma,2}(v_0,\{v_1,v_2\})$ (which exists and is unique by Proposition \ref{prop:closed}). Define \[ \mathcal H^{\free}_{\Gamma,2}(v_0,\{v_1,v_2\})=\{H_{2,1},\dots,H_{2,m_2}\}, \] where $H_{2,i}$ is the minimal element of the set of hemispheres of $\mathcal H_{\Gamma,2}(v_0,\{v_1,v_2\})$ that are $H_{2,j}$-free for every $j<i\le m_2$ and containing $H_{2,i-1}$. The hemisphere $H_{2,i}$ exists and is unique since the set \[
\{H\in \mathcal H_{\Gamma,2}(v_0,\{v_1,v_2\})| H_{2,i-1}\subset H, H\text{ is } H_{2,j}\text{-free for $j=1,\ldots, i-1$}\} \] is intersection-closed by Proposition \ref{prop:closed} and Remark \ref{rem:H1H2}.
Notice that we have a sequence of nested $2$-hemispheres \[ H_{2,1}\subset H_{2,2}\subset \ldots \subset H_{2,m_2}. \]
We let $\mathcal H'_{\Gamma,3}(v_0,\{v_1,v_2\})$ be the subset of $\mathcal H_{\Gamma,3}(v_0,\{v_1,v_2\})$ of the hemispheres that are $H$-free for every $H\in \mathcal H_2^{\free}(v_0,\{v_1,v_2\})$.
\begin{Prop} The subset $\mathcal H'_{\Gamma,3}(v_0,\{v_1,v_2\})$ is intersection-closed. \end{Prop}
\begin{proof} See \cite[Proposition 3.5]{P}. \end{proof}
We let $H_{3,1}$ be the minimal element of $\mathcal H'_{\Gamma,3}(v_0,\{v_1,v_2\})$ and define \[ \mathcal H^{\free}_{\Gamma,3}(v_0,\{v_1,v_2\})=\{H_{3,1},\dots,H_{3,m_3}\}, \] where $H_{3,i}$ is the minimal element of the set of hemispheres $\mathcal H'_{\Gamma,3}(v_0,\{v_1,v_2\})$ that are $H_{3,j}$-free for every $j<i\le m_3$ and containing $H_{3,i-1}$. As before, we have that $\mathcal H_{\Gamma,3}^{\free}$ is well-defined. Notice that we have a sequence of nested $3$-hemispheres \[ H_{3,1}\subset H_{3,2}\subset\dots\subset H_{3,m_3}. \]
\begin{Rem}\label{rem:orientation}
Let $k=2,3$. Notice that we have a natural orientation on every edge $e\in E(H_{k,i}, H_{k,i}^c)$ such that $s(e)\in H_{k,i}$ and $t(e)\in H_{k,i}^c$.
Moreover, if $e\in E(H_{k,i}, H_{k,i}^c)$, then $t(e)\in H_{k,i+1}$. \end{Rem}
Finally we set: \begin{align*} \mathcal F_\Gamma(v_0,v_1,v_2)=&\ \mathcal H_{\Gamma,1}(v_0,v_1)\ \sqcup \ \mathcal H_{\Gamma,1}(v_0,v_2)\\ &\sqcup\ \mathcal H^{\free}_{\Gamma,2}(v_0,\{v_1,v_2\})\ \sqcup \ \mathcal H^{\free}_{\Gamma,3}(v_0,\{v_1,v_2\}). \end{align*} Notice that the same $1$-hemisphere could belong in both $\mathcal H_{\Gamma,1}(v_0,v_1)$ and $\mathcal H_{\Gamma,1}(v_0,v_2)$.
For every subset $V\subset V(\Gamma)$, we let $\textnormal{div}(V)$ be the principal divisor on $\Gamma$ given by: \[ \textnormal{div}(V)=\sum_{e\in E(V,V^c)}(s(e)-t(e)), \] where the orientation is chosen such that $s(e)\in V$ for every $e\in E(V,V^c)$. Let $v_0,v_1,v_2$ be vertices of $\Gamma$ and $\mu$ the trivial degree-$0$ polarization on $\Gamma$. The following result tells us how to find the $(v_0, \mu)$-quasistable divisor equivalent to the divisor $2v_0-v_1-v_2$.
\begin{Thm}\label{thm:convert} Let $\Gamma$ be a graph and $v_0,v_1,v_2$ vertices of $\Gamma$. Then \[ 2v_0-v_1-v_2-\sum_{V\in \mathcal F_\Gamma(v_0,v_1,v_2)}\textnormal{div}(V) \] is the $(v_0,\mu)$-quasistable divisor equivalent to $2v_0-v_1-v_2$. \end{Thm}
\begin{proof} See \cite[Theorem 5.3]{P}. \end{proof}
Before going on with the proof of Theorem \ref{thm:Abel2}, we need to introduce a divisor on a tropical curve $X$ attached to a hemisphere of its underlying graph.
\begin{Def} Let $X$ be a tropical curve and
$(\Gamma,\ell)$ be a model of $X$. Let $v_0$ be a vertex of $\Gamma$. For a hemisphere $H$ of $\Gamma$, consider the orientation on an edge $e\in E(H,H^c)$ from the vertex of $e$ contained in $H$ to the vertex of $e$ contained in $H^c$ (see Remark \ref{rem:orientation}).
We define the divisor \[ \mathcal P_H=\sum_{e\in E(H,H^c)} p_{e,0}-\sum_{e\in E(H,H^c)} p_{e,\ell(e)}. \]
\end{Def}
\begin{Rem}\label{rem:P_principal}
Notice that if $\ell(e)=\ell(e')$ for every $e,e'\in E(H,H^c)$, then $\mathcal P_H$ is a principal divisor on $X$. \end{Rem}
\begin{proof}[Proof of Theorem \ref{thm:Abel2}]
We can assume that all the components of $C$ are smooth. Indeed, the general case follows from the case in which the components of $C$ are smooth arguing as in the last part of the proof of \cite[Theorem 5.8]{AAP}, and using \cite[Theorem 1.3]{P}.
Since quasistability is an open property by \cite[Proposition 34]{EE01}, it is enough to check that the global blowup of $\mathcal C^2$ described in the statement is a blowup resolving the Abel map $\alpha^2_{\mathcal O_C}\colon \mathcal C^2\rightarrow \overline{\mathcal J}^\sigma_\mu$ locally around any point $\mathcal N=(N_1,N_2)$ of $\mathcal C^2$ for $N_i$ a node of $C$.
If $N_1=Z\cap Z^c$ for a $1$-tail $Z$ of $C$, the result follows from Proposition \ref{prop:1-tail}. If $N_1=N_2$, the result follows from Proposition \ref{prop:diagonal}. So we will assume, through the rest of the proof, that $N_1\neq N_2$ and neither $N_1$ nor $N_2$ disconnects $C$.
We will use Theorem \ref{thm:tropgeoAbel}. Let $\Gamma_0$ be the dual graph of $C$, and let
$X=X_{\Gamma_0}$, namely, the tropical curve whose underlying graph is $\Gamma_0$ with all unitary lengths.
Let $v_0$ be the vertex of $\Gamma_0$ corresponding to $P_0=\sigma(0)$, and $p_0\in X$ be the point corresponding to $v_0$.
We have $\mathcal D^\dagger_{\mathcal O_\mathcal C}=2p_0$. The tropical Abel map $\alpha^\text{trop}_{2,\mathcal D^\dagger_{\mathcal O_C}}\colon X^2\rightarrow J^\text{trop}_{p_0,\mu}$ takes a pair $(p_{e_1,t_1},p_{e_2,t_2})$, for edges $e_1,e_2\in E(\Gamma_0)$ and real numbers $t_1,t_2\in [0,1]$, to: \[ \alpha^\text{trop}_{2,\mathcal D^\dagger_{\mathcal O_C}}(p_{e_1,t_1},p_{e_2,t_2})=[2p_0-p_{e_1,t_1}-p_{e_2,t_2}]. \]
Let $(\Gamma,\ell)$ be the model of $X$ such that $\Gamma$ is the refinement of $\Gamma_0$ obtained by inserting vertices $v_{e_1},v_{e_2}$ in the interior of $e_1,e_2$, respectively, with $\ell([s(e_i),v_{e_i}])=t_i$.
We let $K_{2,1}$ be the minimal element of $\mathcal H^{\free}_{\Gamma,2}(v_0,\{v_{e_1},v_{e_2}\})$ and $K_{3,1}$ be the minimal element of $\mathcal H^{\free}_{\Gamma,3}(v_0,\{v_{e_1},v_{e_2}\})$.
We have three cases to consider.
\begin{enumerate}
\item We have that
$E(v_{e_i},\{v_{e_i}\}^c)\cap E(K_{2,1},K_{2,1}^c)\neq \emptyset$, for every $i=1,2$.
\item We have that
\[
E(v_{e_1},\{v_{e_1}\}^c)\cap E(K_{2,1},K_{2,1}^c)\neq \emptyset.
\]
\[
E(v_{e_2},\{v_{e_2}\}^c)\cap E(K_{2,1},K_{2,1}^c)= \emptyset
\]
We distinguish 3 subcases:
\begin{enumerate}
\item[(2.a)] There exists a $3$-hemisphere $K_3$ containing $v_{e_1}$ and $v_{e_2}$ and not containing $v_0$ such that $E(v_{e_i},\{v_{e_i}\}^c)\cap E(K_{3},K_{3}^c)\neq \emptyset$, for every $i=1,2$.
\item[(2.b)] Every $3$-hemisphere $K_3$ containing $v_{e_1}$ and $v_{e_2}$ and not containing $v_0$ satisfies the condition $E(v_{e_2},\{v_{e_2}\}^c)\cap E(K_{3},K_{3}^c)=\emptyset$ and there exists a $3$-hemisphere $K_3'$ such that $E(v_{e_1},\{v_{e_1}\}^c)\cap E(K_{3}',K_{3}'^c)\neq \emptyset$.
\item[(2.c)] Every $3$-hemisphere $K_3$ containing $v_{e_1}$ and $v_{e_2}$ and not containing $v_0$ satisfies the condition $E(v_{e_i},\{v_{e_i}\}^c)\cap E(K_{3},K_{3}^c)=\emptyset$, for every $i=1,2$.
\end{enumerate}
Notice that there are no other subcases to consider, because if the two conditions:
\[
E(v_{e_2},\{v_{e_2}\}^c)\cap E(K_{3},K_{3}^c)\neq \emptyset.
\]
\[
E(v_{e_1},\{v_{e_1}\}^c)\cap E(K_{3},K_{3}^c)= \emptyset
\]
hold for some $3$-hemisphere $K_3$, then, by Lemma \ref{lem:intersect23}, we have that $K_{2,1}\cap K_3$ is a $3$-hemisphere (because $K_{2,1}$ is minimal) and it would satisfy the condition in case (2.a).
\item We have
\[
E(v_{e_i},\{v_{e_i}\}^c)\cap E(K_{2,1},K_{2,1}^c)=\emptyset,\; \text{for every } i=1,2.
\]
We distinguish 3 subcases:
\begin{enumerate}
\item[(3.a)] We have that $E(v_{e_i},\{v_{e_i}\}^c)\cap E(K_{3,1},K_{3,1}^c)\neq \emptyset$, for every $i=1,2$.
\item[(3.b)] We have that
\[
E(v_{e_1},\{v_{e_1}\}^c)\cap E(K_{3,1},K_{3,1}^c)\neq \emptyset.
\]
\[
E(v_{e_2},\{v_{e_2}\}^c)\cap E(K_{3,1},K_{3,1}^c)= \emptyset\]
\item[(3.c)] We have that $E(v_{e_i},\{v_{e_i}\}^c)\cap E(K_{3,1},K_{3,1}^c)= \emptyset$, for every $i=1,2$.
\end{enumerate} \end{enumerate}
We discuss the above cases. Case (1) follows from Proposition \ref{prop:2-tail}.
Case (2.a). We assume that the orientation of $e_1$ and $e_2$ satisfies the condition $s(e_1),s(e_2)\in K_3$. Recall that $t_i=\ell([s(e_i),v_{e_i}])$.
We consider the refinement $\Gamma'$ of $\Gamma_0$ by adding two vertices over each edge. Of course, $\Gamma'$ is a refinement of $\Gamma$. We denote by $\psi$ the natural function
\[
\psi\colon E(\Gamma')\rightarrow E(\Gamma_0)
\]
taking an edge $e$ of $\Gamma'$ to the edge $f$ of $\Gamma_0$ if $e$ is obtained by subdividing $f$.
For $i=1,2$, we already have the vertex $v_{e_i}$ over the edge $e_i$, so we will only add another vertex $v'_{e_i}$. As illustrated in Figure \ref{fig:orderings}, if $t_1< t_2$, the vertices over $e_1$ will be ordered as follows: $s(e_1), v_{e_1}, v_{e_1}', t(e_1)$, while the vertices over $e_2$ are ordered as follows: $s(e_2), v'_{e_2}, v_{e_2}, t(e_2)$. On the other hand, if $t_1> t_2$, then the orderings become $s(e_1), v_{e_1}', v_{e_1}, t(e_1)$ and $s(e_2), v_{e_2}, v'_{e_2}, t(e_2)$.
\begin{figure}\label{fig:orderings}
\end{figure}
Throughout the proof of case (2a), we will assume that $t_1<t_2$, leaving to the reader the case $t_1>t_2$. Our goal will be to find a length function $\ell'$ on $\Gamma'$ so that $(\Gamma',\ell')$ is a model of $X$, and the divisors $\mathcal{P}_H$ for every $H\in\mathcal F_{\Gamma'}(v_0,v_{e_1},v_{e_2})$ are principal on $X$. This allows us to conclude the proof. Indeed, using Remark \ref{prop:quasiquasi} and Theorem \ref{thm:convert} we get that the divisor \begin{equation}\label{eq:graph} \mathcal{D}_{t_1,t_2}:=2p_0-p_{e_1,t_1}-p_{e_2,t_2}+\sum_{H\in \mathcal F_{\Gamma'}(v_0,v_{e_1},v_{e_2})}\mathcal P_H \end{equation} is $(p_0,\mu)$-quasistable.
Hence Theorem \ref{thm:tropgeoAbel} (1) tells us that the blowup illustrated on the left hand side of Figure \ref{Fig:blowup} with $Z_1=Z_2=\widehat Z$, where $\widehat Z$ is the $3$-tail of $C$ induced by $K_3\cap V(\Gamma)$, gives rise to a resolution of the Abel map $\alpha^2_{\mathcal O_\mathcal C}$ locally at $(N_1,N_2)$. This is the blowup locally around $(N_1,N_2)$ prescribed by the global blowup in the statement of Theorem \ref{thm:Abel2}.
We proceed with the construction of the length function $\ell'$. We write
\begin{equation}
\label{eq:H23_free}
\begin{aligned}
\mathcal H_{\Gamma',2}^{\free}(v_0,\{v_{e_1},v_{e_2}\})&=\{H_{2,1},\ldots, H_{2,{m_2}}\}\\
\mathcal H_{\Gamma',3}^{\free}(v_0,\{v_{e_1},v_{e_2}\})&=\{H_{3,1},\ldots, H_{3,m_3}\}.
\end{aligned}
\end{equation}
We define a sequence $f_1,f_2,\ldots, f_k$ of edges of $\Gamma_0$ as illustrated in Figure \ref{fig:fi_sequence_t1<t2_odd}, with $f_1=e_1$ and $f_2\in E(H_{2,1},H_{2,1}^c)$, and where the other edges are chosen as follows.
Assume that $t_1<t_2$. The edges of the sequence satisfy
\[
f_{2i+1},f_{2i+2}\in \psi(E(H_{2,{3i+1}},H^c_{2,{3i+1}})), \;\;
f_{2i+1},f_{2i+2}\in \psi(E(H_{2,{3i+2}},H_{2,{3i+2}}^c))
\]
\[f_{2i}, f_{2i+1}\in \psi(E(H_{2,{3i}},H^c_{2,{3i}})).
\]
Notice that if $k$ is odd with $k=2k'+1$, then
\[
\psi(E(H_{2,3i+1},H_{2,3i+1}^c))=\psi(E(H_{2,3i+2},H_{2,3i+2}^c))=\psi(E(H_{2,3i+3},H_{2,3i+3}^c))
\]
for every $i\geq k'$. If $k$ is even with $k=2k'$, then for every $i\geq k'$ we have
\[
\psi(E(H_{2,3i},H_{2,3i}^c))=\psi(E(H_{2,3i+1},H_{2,3i+1}^c))=\psi(E(H_{2,3i+2},H_{2,3i+2}^c)).
\]
Now we consider the $3$-hemispheres. If $\psi(E(H_{3,1},H_{3,1}^c))$ and $\psi(E(H_{2,i},H_{2,i}^c))$ are disjoint for every $i$, let us define a length function $\ell'$ on the set of edges of $\Gamma'$ so that $(\Gamma',\ell')$ is a model for $X$. We will assume that $k=2k'$ is even (see Figure \ref{fig:fi_sequence_t1<t2_even}, also see Figure \ref{fig:fi_sequence_t1>t2_even} for the case $t_1>t_2$), leaving to the reader to work out the other case (see Figure \ref{fig:fi_sequence_t1<t2_odd}). For every $e\in E(\Gamma')$, we define:
\[ \ell'(e)= \begin{cases} \begin{array}{ll}
\frac{1-t_1}{2} & \text{ if }e\in E(H_{2,{3i+1}},H_{2,3i+1}^c) \text{ or } e\in E(H_{2,3i+2},H_{2,3i+2}^c)\text{ for some } i<k'
\\
t_1 & \text{ if }\psi(e)=f_k\text{ and }e\notin E(H_{2,3k'-2},H_{2,3k'-2}^c)\cup E(H_{2,3k'-1},H_{2,3k'-1}^c)
\\
t_1 & \text{ if either } e\in E(H_{2,3i},H^c_{2,3i}) \text{ for some $i<k'$, or } t(e)=v_1\\
1/3 & \text{ otherwise}. \end{array} \end{cases} \]
For every edge $f\in E(\Gamma)$ we have that $\sum_{e\in \psi^{-1}(f)}\ell'(e)=1$. Indeed if $f=f_i$, then the sum of the lengths will be $t_1+\frac{1-t_1}{2}+\frac{1-t_1}{2}=1$, while if $f\notin\{f_1,\ldots, f_k\}$, then the 3 edges in $\psi^{-1}(f)$ will have length $1/3$. So $(\Gamma',\ell')$ is a model of $X$. Notice that the divisors $\mathcal{P}_{H_{2,i}}$ and $\mathcal{P}_{H_{3,i}}$ are principal divisors by Remark \ref{rem:P_principal}. This conclude the proof in this case.
\begin{figure}
\caption{Attributing lengths to the edges of $\Gamma'$ for $t_1<t_2$ and $k$ even. In this case, $v_{e_2}$ is contained in $H_{2,i}$ for every $i=1,\ldots, m_2$.}
\label{fig:fi_sequence_t1<t2_even}
\end{figure}
\begin{figure}
\caption{Attributing lengths to the edges of $\Gamma'$ for $t_1>t_2$ and $k$ even. In this case, $v_{e_2}$ is contained in $H_{2,i}$ for every $i=1,\ldots, m_2$.}
\label{fig:fi_sequence_t1>t2_even}
\end{figure}
We are left to consider the case in which $\psi(E(H_{3,1},H_{3,1}^c))$ and $\psi(E(H_{2,i},H_{2,i}^c))$ have a common edge for some $i$. In this case, this edge must be $f_k$.
We claim that $k$ is odd. First we prove that $H_{3,1}^c$ contains the vertices of $\Gamma'$ incident to $f_{2i}$ and the vertices over $f_{2i}$ for every $i$. Let us denote by $e_2,f, f_k$ the edges of $\psi(E(H_{3,1}, H_{3,1}^c))$.
The intersection $H_{3,1}\cap H_{2,1}$
cannot be a $2$-hemisphere, otherwise we would contradict the minimality of $H_{2,1}$. Indeed, the fact that
$E(v_{e_2},\{v_{e_2}\}^c)\cap E(H_{3,1},H_{3,1}^c)\neq \emptyset$ implies that
\[
E(v_{e_2},\{v_{e_2}\}^c)\cap E(H_{3,1}\cap H_{2,1},(H_{3,1}\cap H_{2,1})^c)\neq \emptyset,
\]
so $H_{3,1}\cap H_{2,1}\subsetneqq H_{2,1}$.
By Lemma \ref{lem:intersect23}, we see that $\psi(E(H_{3,1}\cap H_{2,1}, (H_{3,1}\cap H_{2,1})^c))=\{e_2, f, f_1\}$, with $e_2$ and $f$ fully contained in $H_{2,1}$ and $f_2$ fully contained in $H_{3,1}^c$.
We now iterate the reasoning. Intersecting $H_{3,1}\cap H_{2,3}$ (see Figure \ref{fig:fi_sequence_t1<t2_odd}), we must have that $e_2, f\in \psi(E(H_{3,1}\cap H_{2,3},(H_{3,1}\cap H_{2,3})^c))$, hence, by Lemma \ref{lem:intersect23}, $H_{3,1}\cap H_{2,3}$ is a $3$-hemisphere, and $f_3$ is fully contained in $H_{3,1}$ (as neither $f_2$ nor $f_k$ is fully contained in $H_{3,1}$). Considering $H_{3,1}\cap H_{2,4}$, we have that $f_4$ must be fully contained in $H_{3,1}^c$, and iterating this process we see that $f_{2i}$ is fully contained in $H_3^c$ for every $i=1,\ldots, \lfloor \frac{k}{2}\rfloor$. So $k$ must be odd and we write $k=2k'+1$.\par
As illustrated in Figure \ref{fig:fi_sequence_t1<t2_odd}, for every $e\in E(\Gamma')$, we define:
\[ \ell'(e)= \begin{cases} \begin{array}{ll}
\frac{1-t_1}{2} & \text{ if }e\in E(H_{2,3i+1},H_{2,3i+1}^c) \text{ or } E(H_{2,3i+2},H_{2,3i+2}^c)\text{ for some } i<k',
\\
t_1 & \text{ if } e\in E(H_{2,3i},H_{2,3i}^c) \text{ or } t(e)=v_1,\\
1-t_2 &\text{ if } e\in E(H_{3,3i+1},H_{2,3i+1}^c) \text{ for any } i,\\
t_2-t_1 &\text{ if } e\in E(H_{3,3i+2},H_{2,3i+2}^c) \text{ for any } i,\\
t_1 &\text{ if } e\in E(H_{3,{3i}},H^c_{2,{3i}}) \text{ for any } i,\\
1/3 & \text{ if } \psi(e)\notin \psi(E(H_{j,i},H^c_{j,i})) \text{ for any $j=2,3$ and $i$}. \end{array} \end{cases} \]
The remaining edges can be assigned lengths in a such way that $\sum_{e\in\psi^{-1}(f)}\ell'(e)=1$ for every $f\in E(\Gamma_0)$, so $(\Gamma',\ell')$ is a model of $X$. Again, by Remark \ref{rem:P_principal}, the divisors $\mathcal{P}_{H_{2,i}}$ and $\mathcal{P}_{H_{3,i}}$ are principal divisors, finishing the proof.
\begin{figure}
\caption{Attributing lengths to the edges of $\Gamma'$, for $t_1<t_2$ and $k$ odd. In this case, $v_{e_2}$ is contained in $H_{2,i}$ for every $i=1,\ldots, m_2$. }
\label{fig:fi_sequence_t1<t2_odd}
\end{figure}
\begin{figure}
\caption{Attributing lengths to the edges of $\Gamma'$ for $t_1>t_2$ and $k$ odd. In this case, $v_{e_2}$ is contained in $H_{2,i}$ for every $i=1,\ldots, m_2$.}
\label{fig:fi_sequence_t1>t2_odd}
\end{figure}
Case (2.b). Consider the refinement $\Gamma'$ of $\Gamma_0$ by adding one vertex over each edge. Notice that $\Gamma'$ is a refinement of $\Gamma$. Let $H_{2,i}$ and $H_{3,j}$ be defined as in Equation \eqref{eq:H23_free}. Let $k$ be the integer such that $|\psi(E(H_{2,i},H_{2,i}^c))\cap\psi(E(H_{2,i+1},H_{2,i+1}^c))|=1$ for every $i\leq k-1$, and \[ \psi(E(H_{2,k+2i+1},H_{2,k+2i+1}^c))=\psi(E(H_{2,k+2i+2},H_{2,k+2i+2}^c)), \text{ for } i\ge 0. \]
If $k$ is even, we define the length $\ell'$ on $\Gamma'$ as follows: \[ \ell'(e)= \begin{cases} \begin{array}{ll}
1-t_1 & \text{ if }e\in E(H_{2,2i+1},H_{2,2i+1}) \text{ with } i=0,\dots,\lfloor\frac{m_2-1}{2}\rfloor
\\
t_1 & \text{ if } e\in E(H_{2,2i},H_{2,2i}^c) \text{ with } i=1,\dots,\lfloor\frac{m_2}{2}\rfloor \text{ or } t(e)=v_1\\
1-t_1 &\text{ if } e\in E(H_{3,2i},H_{3,2i}^c) \text{ with } i=1,\dots,\lfloor\frac{m_3}{2}\rfloor\\
t_1 &\text{ if } e\in E(H_{3,2i+1},H_{3,2i+1}^c) \text{ with } i=0,\dots,\lfloor\frac{m_3-1}{2}\rfloor \\
1/2 & \text{ if } \psi(e)\notin \psi(E(H_{r,i},H_{r,i}^c) \text{ for any $r=2,3$ and $i$}. \end{array} \end{cases} \] The remaining edges can be assigned lengths in a such way that $\sum_{e\in\psi^{-1}(f)} \ell'(e)=1$ for every $f\in E(\Gamma_0)$, so $(\Gamma',\ell')$ is a model of $X$.
When $k$ is odd the situations is similar (see Figure \ref{fig:2b}): the unique difference is that we define $\ell'(e)=1-t_1$ for $e\in E(H_{3,2i+1},H_{3,2i+1}^c)$ and $\ell'(e)=t_1$ for $e\in E(H_{3,2i},H_{3,2i}^c)$. \begin{figure}
\caption{Attributing lengths to the edges of $\Gamma'$ for $k$ odd.}
\label{fig:2b}
\end{figure}
As in Case (2.a), we have that the combinatorial type of the divisor $\mathcal{D}_{t_1,t_2}$ defined in Equation \eqref{eq:graph} does not depend on $0<t_1,t_2<1$. Hence Theorem \ref{thm:tropgeoAbel} (3) ensures that the Abel map $\alpha^2_{\mathcal O_{\mathcal C}}$ is already defined at $(N_1,N_2)$, as given by the global blowup in the statement of Theorem \ref{thm:Abel2}.\\
Case (2.c). This case is the same as Case (2.b) except that $\psi(E(H_{2,i},H^c_{2,i})\cap \psi(E(H_{3,j},H^c_{3,j})=\emptyset$ for every $i=1,\ldots,m_2$ and $j=1,\ldots, m_3$. So we can freely assign lengths to the edges in $E(H_{3,j},H^c_{3,j})$. The conclusion is the same as in Case (2.b).
Case (3.a). This case follows the same steps in Case (2.a): the difference is that $k=0$ and the sequence of edges $f_1,\ldots, f_k$ is empty. The conclusion is the same as in Case (2.a).
Case (3.b). This case follows the same steps in Case (2.b): the difference is that $k=0$. The conclusion is the same as in Case (2.b).
Case (3.c). In this case, we do not have to further refine $\Gamma_0$ as $E(H,H^c)$ does not contain any edge incident to $v_{e_1}$ or $v_{e_2}$, for every $H$ in $\mathcal F_{\Gamma}(v_0,v_{e_1},v_{e_2})$. So $P_H$ is principal on $X$ for every $H\in \mathcal F_{\Gamma}(v_0,v_{e_1},v_{e_2})$.
As in the previous cases, using Remark \ref{prop:quasiquasi} and Theorem \ref{thm:convert}, the divisor \[ \mathcal{D}_{t_1,t_2}:=2p_0-p_{e_1,t_1}-p_{e_2,t_2}+\sum_{H\in \mathcal F_{\Gamma}(v_0,v_{e_1},v_{e_2})}\mathcal P_H \] on the tropical curve $X$ is $(p_0,\mu)$-quasistable and equivalent to $2p_0-p_{e_1,t_1}-p_{e_2,t_2}$. Since the combinatorial type of $\mathcal{D}_{t_1,t_2}$ is independent of $t_1$ and $t_2$, it follows from Theorem \ref{thm:tropgeoAbel} (3) that the Abel map $\alpha^2_{\mathcal O_{\mathcal C}}$ is already defined at $(N_1,N_2)$, as prescribed by the global blowup in the statement of Theorem \ref{thm:Abel2}. \end{proof}
Given a regular smoothing $f\colon\mathcal C\rightarrow B$ of a curve, consider the blowup $\widetilde \mathcal C\rightarrow \mathcal C$ giving rise to a resolution of the degree-2 Abel map $\alpha^2_{\mathcal O_{\mathcal C}}$, as in Theorem \ref{thm:Abel2}. Since the locus we are blowing up is invariant under the natural action of $S_2$ on $\mathcal{C}^2$, we can take the quotient \[ \Sym^2(\widetilde{\mathcal C})=\widetilde \mathcal C/S_2. \] We thus obtain a map: \begin{equation}\label{eq:beta2} \beta_2\colon \Sym^2(\widetilde{\mathcal C})\rightarrow \overline{\mathcal J}_\mu(\mathcal C) \end{equation} resolving the rational ``symmetrized" Abel map $\Sym(\mathcal C^2)\dashrightarrow \overline{\mathcal J}_\mu(\mathcal C)$.
\begin{Def}\label{def:pseudo} Let $C$ be a curve. We say that $C$ is \emph{pseudo-hyperelliptic} if it has a simple torsion-free rank-1 sheaf $I$ of degree 2 with non-negative degree over every component such that $h^0(C,I)\ge 2$. \end{Def}
Recall that a curve is weakly-hyperelliptic if it has a degree-$2$ balanced invertible sheaf (see \cite{Capohyper} for more details). If a stable curve is hyperelliptic, then it is weakly-hyperelliptic.
\begin{Thm}\label{thm:hyperelliptic} Let $C$ be a curve with no separating nodes. The following properties hold. \begin{enumerate}
\item $C$ is pseudo-hyperelliptic
if and only if, for some (every) regular smoothing $\mathcal C\to B$ of $C$, the map $\beta_2\colon \Sym^2(\mathcal C)\to \overline{\mathcal{J}}_{\mu}(\mathcal C)$ is not injective. \item if $C$ is stable and weakly-hyperelliptic, then $C$ is pseudo-hyperelliptic. \item if $C$ is stable and has a simple torsion-free rank-1 sheaf $I$ of degree 2 with non-negative degree over every component such that $h^0(C,I)\ge 2$, then $I$ is invertible. \end{enumerate} \end{Thm} \begin{proof}
If $C$ has a rational component $E$ such that $|E\cap E^c|\leq 2$ then it is easy to see that $C$ is pseudo-hyperelliptic and weakly-hyperellpitic, and that $\beta_2$ is not injective. So, we will assume that $C$ is stable.
Assume that $C$ is stable and pseudo-hyperelliptic. Let $I$ be a torsion free rank-1 sheaf satisfying the condition in Definition \ref{def:pseudo}. Let $\mathbb P(I):=\Proj(\Sym(I))\to C$ be the semistable modification of $C$ where we add a rational curve over the nodes of $C$ where $I$ is not locally free. We consider the invertible sheaf $L:=\mathcal O_{\mathbb P(I)}(1)$, so that we have $I=f_*(L)$ and $L$ has degree $1$ on the exceptional components (see \cite[Section 5]{EP}). Then, $L$ has non-negative degree on every component of $\mathbb{P}(I)$ and $h^0(\mathbb{P}(I),L)\geq 2$.
We will apply \cite[Theorem 5.9]{Capohyper} to $\mathbb{P}(I)$ and $L$. We have two cases. In the first case, there is a component $C_0$ of $\mathbb{P}(I)$ satisfying the following property. Let $Z_1,\dots Z_n$ be the connected components of $C_0^c$. Then \begin{equation} \label{eq:weakly_hyper}
h^0(L|_{C_0})\geq 2,\;\;\; L|_{C_0^c}=\mathcal{O}_{C_0^c}, \;\;\;L|_{C_0}=\mathcal{O}_{C_0}(C_0\cap Z_i),\;\;\; |C_0\cap Z_i|=2. \end{equation} This means that the component $C_0$ is not exceptional, because $L$ has degree $2$ on $C_0$. Moreover, $L$ has degree $0$ on every other component, which implies that $I$ is an invertible sheaf, hence $L=I$ and $\mathbb{P}(I)=C$. We can consider smooth points $q_1,q_2,q_1',q_2'$ of $C$ lying over $C_0$ such that \[
L|_{C_0}\cong\mathcal{O}_{C_0}(q_1+q_2)\cong\mathcal{O}_{C_0}(q_1'+q_2'). \] By Condition \eqref{eq:weakly_hyper}, we have that $L\cong\mathcal{O}_C(q_1+q_2)\cong\mathcal{O}_C(q'_1+q'_2)$, hence $\mathcal{O}_C(2p_0-q_1-q_2)\cong\mathcal{O}_C(2p_0-q_1'-q_2')$ which means that $\beta_2(q_1+q_2)=\beta_2(q_1'+q_2')$, where $\beta_2$ is the map in Equation \ref{eq:beta2} for some (every) regular smoothing of $C$.
In the second case, there are two components $C_1$ and $C_2$ of $\mathbb{P}(I)$ such that $(C_1,C_2)$ is a special $\mathcal B$-pair (in the sense of \cite[Definition 5.8]{Capohyper}). By \cite[Theorem 5.9]{Capohyper}, we have \[
\deg L_{C_1}=\deg L|_{C_2}=1,\;\;\; L|_{(C_1\cup C_2)^c}\cong \mathcal O_{(C_1\cup C_2)^c}. \] Notice that, if one between $C_1$ and $C_2$ is exceptional, then the other must be exceptional as well, and in particular this implies that $I$ is not simple, which is a contradiction. We deduce that $I$ is an invertible sheaf, hence $L=I$ and $\mathbb{P}(I)=C$. We can repeat the argument used in the first case, now taking $q_1,q_1'\in C_1$ and $q_2,q_2'\in C_2$. We leave the details to the reader. Notice that we proved (3) and the ``only if" part of (1).
Now assume that there is a regular smoothing $\mathcal C\rightarrow B$ of $C$ such that $\beta_2$ is not injective. We have different cases to consider.
In the first case, there are smooth points $q_1,q_2,q'_1,q'_2$ of $C$ such that $\beta_2(q_1+q_2)=\beta_2(q_1'+q_2')$. This means that there exists an invertible sheaf $T$ on $C$ of type $T=\mathcal O_{\mathcal C}(\sum a_i C_i)|_C$, where $a_i\in \mathbb Z$ and $C_i$ are the components of $C$, such that \[ \mathcal{O}_C(2p_0-q_1-q_2)\cong \mathcal{O}_C(2p_0-q_1'-q_2')\otimes T. \] We deduce that \[ \mathcal{O}_C(q_1'+q_2'-q_1-q_2)\cong T. \] Let $\Gamma$ be the dual graph of $C$. Notice that $\Gamma$ has no separating edge.
If $v_1,v_2,v_1',v_2'$ are the vertices of $\Gamma$ corresponding to the components containing the points $q_1,q_2, q_1',q_2'$, we have that $v_1'+v_2'-v_1-v_2$ is a principal divisor on $\Gamma$. Let $f\colon V(\Gamma)\to \mathbb{Z}$ be the rational function on $\Gamma$ such that $\textnormal{div}(f)=v_1'+v_2'-v_1-v_2$ (notice that $v'_i\neq v_j$ for every $i,j=1,2$ because $\Gamma$ has no separating edge). We denote by $Z$ the subcurve of $C$ corresponding to the vertices of $\Gamma$ where $f$ attains its minimum. In particular, $q_1,q_2\in Z$, $q_1',q_2'\in Z^c$ and $|Z\cap Z^c|=2$. Moreover we have $T|_Z=\mathcal{O}_Z(-Z\cap Z^c)$, which implies that \[
\mathcal{O}_Z(Z\cap Z^c)\otimes\mathcal O_C(-q_1-q_2)|_Z\cong (\mathcal{O}_C(q_1'+q_2'-q_1-q_2)\otimes T^{-1})|_{Z}\cong \mathcal{O}_C|_Z=\mathcal{O}_Z. \] Define $L:=\mathcal{O}_C(q_1+q_2)$. We see that $L$ satisfies $h^0(L,C)\geq 2$ (indeed $L$ has the trivial section that vanishes only over $q_1,q_2$ and a section that vanishes on the whole $Z^c$). Thus $C$ is pseudo-hyperellitic in the sense of Definition \ref{def:pseudo}.
In the second case, we have nodes $n,n'$ and smooth points $q,q'$ of $C$ such that $\beta_2(n+q)=\beta_2(n'+q')$. Let $\widetilde C$ be the semistable modification of $C$ obtained by adding an exceptional component over each node of $C$. Then, there exists a twister $T$ on $\widetilde C$ such that
\[
\mathcal{O}_{\widetilde{C}}(2p_0-\widetilde{n}-q)\cong \mathcal{O}_{\widetilde{C}}(2p_0-\widetilde{n}'-q')\otimes T
\]
where $\widetilde n$ and $\widetilde n'$ are any smooth points of $\widetilde C$ lying over the exceptional component over $n$ and $n'$.
Arguing as before, we see that $L:=\mathcal{O}_{\widetilde{C}}(\widetilde{n}+q)$ satisfies $h^0(L,\widetilde{C})\geq 2$, and hence $h^0(f_*(L),C)\geq2$. Thus $C$ is pseudo-hyperelliptic in the sense of Definition \ref{def:pseudo}.
In the third case, we have a node $n$ and smooth points $q,q'_1,q'_2$ such that $\beta_2(n+q)=\beta_2(q'_1+q'_2)$. This case is not possible, since the sheaf represented by $\beta_2(q'_1+q'_2)$ is invertible, while the one represented by $\beta_2(n+q)$ is not.
The remaining cases are the following ones: \begin{itemize}
\item $\beta(n_1+n_2)=\beta(q'_1+q'_2)$;
\item $\beta(n_1+n_2)=\beta(n'+q')$;
\item $\beta (n_1+n_2)=\beta (n_1'+n_2')$. \end{itemize} where $n_1$, $n_2$, $n'$, $n'_1$, $n'_2$ are nodes of $C$, and $q$, $q'_1$, $q'_2$ and smooth points. All these cases are done in an similar manner as the second case: first, we change $C$ by a suitable semistable modification $f\colon \widetilde{C}\to C$ and find a line bundle $L$ such that $h^0(L,\widetilde{C})\geq 2$. This implies that $h^0(f_*(L),C)\geq 2$ which proves that $C$ is pseudo-hyperelliptic in the sense of Definition \ref{def:pseudo}.
Finally, item (2) of the statement readily follows by \cite[Theorem 5.9]{Capohyper}. \end{proof}
\end{document} | arXiv |
So far, I have figured out that the lower limit on $z$ is $0$, and the lower limit on $y$ is also $0$. How would one go about calculating the upper limit on z and y as well as the limits for $x$?
I believe there is another question similar to this, but it does not explain how to calculate the plane boundaries, which is what I'm having trouble with.
Sorry for the crappy graph.
The body is located between $0$ and $1$ in $x$-direction, hence $$ \int_0^1\ldots\ dx. $$ Now for each $x\in[0,1]$ you get the section of the body with the plane at $x$ that is parallel to $yz$-plane. For $x=1$ you get the largest triangle, for $x=0$ you get one point (the origin), for $x$ somewhere in between you get a smaller proportional (similar) triangle (red in the picture below). Now the red triangle is located in $y$-direction between $0$ and some largest possible $y(x)$ - it is the integration limits for $dy$. At last, for each $y$ in the red triangle you can choose the integration limits for $z$ - from $0$ to the largest possible $z$ in the green segment. They may depend on both $x$ and $y$.
Not the answer you're looking for? Browse other questions tagged integration multivariable-calculus plane-curves or ask your own question.
Find a transformation from tetrahedron to cube in $R^3$ to calculate a triple integral?
Let B be the solid tetrahedron bounded by x = 0, y = 0, z = 0, and x+y+z = 1. | CommonCrawl |
\begin{document}
\title{\bf The Domination Number of Grids\thanks{This work has been
partially supported by the ANR Project GRATOS ANR-JCJC-00041-01.}} \author{Daniel Gon\c calves\thanks{Universit\'e Montpellier 2, CNRS, LIRMM
161 rue Ada,
34095 Montpellier Cedex, France.
Email: \texttt{\{daniel.goncalves | alexandre.pinlou | stephan.thomasse\}@lirmm.fr}}\addtocounter{footnote}{-1}
\and Alexandre Pinlou\footnotemark\ \ \thanks{Département Mathématiques
et Informatique Appliqués, Université Paul-Valéry, Montpellier 3,
Route de Mende, 34199 Montpellier Cedex 5, France.}\and Micha\"el Rao\thanks{CNRS, Laboratoire J.-V. Poncelet, Moscow, Russia.
Universit\'e Bordeaux - CNRS,
LABRI, 351, cours de la Lib\'eration 33405 Talence, France.
Email: \texttt{[email protected]}}\addtocounter{footnote}{-3} \and St\'ephan Thomass\'e\footnotemark }
\maketitle
\def\ \mbox{\rm{if}} \ {\ \mbox{\rm{if}} \ } \def\ \mbox{\rm{and}} \ {\ \mbox{\rm{and}} \ } \def\ \mbox{\rm{or}} \ {\ \mbox{\rm{or}} \ } \def\form#1#2{\left \lfloor \frac{(#1+2)(#2+2)}{5} \right \rfloor -4} \def\frceil#1#2{\lceil\frac{#1}{#2}\rceil} \def\frfloor#1#2{\lfloor\frac{#1}{#2}\rfloor} \def\ceil#1{\lceil#1\rceil} \def\fG#1#2{G_{#1,#2}} \def\gG#1#2{\gamma(G_{#1,#2})} \defG_{n,m}{G_{n,m}}
\def\fV#1#2{V_{#1,#2}} \defV_{n,m}{V_{n,m}}
\def\los#1{\ell(#1)} \def\ell_{n,m}{\ell_{n,m}} \defl_{i,f}{l_{i,f}}
\defC{C} \defM{M} \defM'{M'}
\defL{L} \defT{T}
\defb_{n,m}{b_{n,m}}
\def\fB#1#2{B_{#1,#2}} \defB_{n,m}{B_{n,m}}
\def\fBp#1#2{I(B_{#1,#2})} \defI(B_{n,m}){I(B_{n,m})}
\defw^{in}{w^{in}} \defw^{out}{w^{out}}
\def\mathcal{F}{\mathcal{F}}
\def\fP#1{P_{#1}} \defP_p{P_p} \defI(P_p){I(P_p)}
\def\fQ#1{Q_{#1}} \defQ_p{Q_p} \def\fR#1{R_{#1}} \defR_p{R_p} \def\fS#1{O_{#1}} \defO_p{O_p}
\def\fC#1{C_{#1}} \defO_p{O_p} \defI_p{I_p} \defC_q{C_q}
\deff_{n,m}{f_{n,m}} \deff_{m,n}{f_{m,n}}
\def\mathcal{W}{\mathcal{W}} \def\mathcal{D}{\mathcal{D}}
\def\mD^{w,w'}_p{\mathcal{D}^{w,w'}_p} \newcommand{\Dww}[1]{\mathcal{D}^{w,w'}_{#1}}
\begin{abstract} In this paper, we conclude the calculation of the
domination number of all $n\times m$ grid graphs. Indeed, we prove
Chang's conjecture saying that for every $16\le n\le m$,
$\gamma(G_{n,m})=\form{n}{m}$. \end{abstract}
\section{Introduction}
A {\it dominating set} in a graph $G$ is a subset of vertices $S$ such that every vertex in $V(G)\setminus S$ is a neighbour of some vertex of $S$. The {\it domination number} of $G$ is the minimum size of a dominating set of $G$. We denote it by $\gamma(G)$. This paper is devoted to the calculation of the domination number of complete grids.
The notation $[i]$ denotes the set $\{1,2,\ldots, i\}$. If $w$ is a word on the alphabet $A$, $w[i]$ is the $i$-th letter of $w$, and for every $a$ in $A$, $\vert w \vert_a$ denotes the number of occurrences of $a$ in $w$ (i.e. $\vert \{ i\in \{1,\ldots,\vert w\vert\} : w[i]=a\}\vert$). For a vertex $v$, $N[v]$ denotes the closed neighbourhood of $v$ (i.e. the set of neighbours of $v$ and $v$ itself). For a subset of vertices $S$ of a vertex set $V$ of a graph, we denote by $N[S] = \bigcup_{v\in S}N[v]$. Note that $D$ is a dominating set of $G$ if and only if $N[D] = V(G)$. Let $G_{n,m}$ be the $n\times m$ complete grid, i.e. the vertex set of $G_{n,m}$ is $V_{n,m}:=[n]\times [m]$, and two vertices $(i,j)$ and $(k,l)$ are adjacent if
$|k-i|+|l-j|=1$. The couple $(1,1)$ denotes the bottom-left vertex of the grid and the couple $(i,j)$ denotes the vertex of the $i$-th column and the $j$-th row. We will always assume in this paper that $n\leq m$. Let us illustrate our purpose by an example of a dominating set of the complete grid $\fG{24}{24}$ on Figure~\ref{fig:24x24}.
\begin{figure}
\caption{Example of a set of size 131 dominating the grid $\fG{24}{24}$}
\label{fig:24x24}
\end{figure}
The first results on the domination number of grids were obtained about 30 years ago with the exact values of $\gG2n$, $\gG3n$, and $\gG4n$ found by Jacobson and Kinch \cite{JK} in 1983. In 1993, Chang and Clark~\cite{CC} found those of $\gG5n$ and $\gG6n$. These results were obtained analytically. Chang~\cite{Chang} devoted his PhD thesis to study the domination number of grids; he conjectured that this invariant behaves well provided that $n$ is large enough. Specifically, Chang conjectured the following:
\begin{conjecture}[\cite{Chang}]\label{conj}
For every $16\le n \le m$, $$\gamma(G_{n,m}) = \form{n}{m}.$$ \end{conjecture}
Observe that for instance, this formula would give 131 for the domination number of the grid in Figure~\ref{fig:24x24}. To motivate his bound, Chang proposed some constructions of dominating sets achieving the upper bound:
\begin{lemma}[\cite{Chang}]\label{lem:cgang}
For every $8\le n\le m$, $$\gamma(G_{n,m}) \le \form{n}{m}$$ \end{lemma}
Later, some algorithms based on dynamic programming were designed to compute a lower bound of $\gG{n}{m}$. There were numerous intermediate results found for $\gG{n}{m}$ for small values of $n$ and $m$ (see \cite{CCH,HHH,SP} for details). In 1995, Hare, Hedetniemi and Hare \cite{HHH} gave a polynomial time algorithm to compute $\gamma(G_{n,m})$ when $n$ is fixed. Nevertheless, this algorithm is not usable in practice when $n$ hangs over $20$. Fisher \cite{Fisher} developed the idea of searching for periodicity in the dynamic programming algorithms and using this technique, he found the exact values of $\gG{n}{m}$ for all $n\leq 21$. We recall these values for the sake of completeness.
\begin{theorem}[\cite{Fisher}]
For all $n \le m$ and $n \le 21$, we have:
$$\gG{n}{m} = \left\{
\begin{array}{ll}
\frceil{m}{3} & \ \mbox{\rm{if}} \ n=1 \\*[0.1cm]
\frceil{m+1}{2} & \ \mbox{\rm{if}} \ n=2 \\*[0.1cm]
\frceil{3m+1}{4} & \ \mbox{\rm{if}} \ n=3 \\*[0.1cm]
m+1 & \ \mbox{\rm{if}} \ n=4 \ \mbox{\rm{and}} \ m=5,6,9\\*[0.1cm]
m & \ \mbox{\rm{if}} \ n=4 \ \mbox{\rm{and}} \ m\neq 5,6,9\\*[0.1cm]
\frceil{6m+4}{5} - 1 & \ \mbox{\rm{if}} \ n=5 \ \mbox{\rm{and}} \ m=7 \\*[0.1cm]
\frceil{6m+4}{5} & \ \mbox{\rm{if}} \ n=5 \ \mbox{\rm{and}} \ m\neq 7 \\*[0.1cm]
\frceil{10m+4}{7} & \ \mbox{\rm{if}} \ n=6 \\*[0.1cm]
\frceil{5m+1}{3} & \ \mbox{\rm{if}} \ n=7 \\*[0.1cm]
\frceil{15m+7}{8} & \ \mbox{\rm{if}} \ n=8 \\*[0.1cm]
\frceil{23m+10}{11} & \ \mbox{\rm{if}} \ n=9 \\*[0.1cm]
\frceil{30m+15}{13} - 1 & \ \mbox{\rm{if}} \ n=10 \ \mbox{\rm{and}} \ m\equiv_{13}10 \ \mbox{\rm{or}} \ m=13,16
\\*[0.1cm]
\frceil{30m+15}{13} & \ \mbox{\rm{if}} \ n=10 \ \mbox{\rm{and}} \ m\not\equiv_{13}10 \ \mbox{\rm{and}} \ m\neq13,16 \\*[0.1cm]
\frceil{38m+22}{15} - 1 & \ \mbox{\rm{if}} \ n=11 \ \mbox{\rm{and}} \ m=11,18,20,22,33 \\*[0.1cm]
\frceil{38m+22}{15} & \ \mbox{\rm{if}} \ n=11 \ \mbox{\rm{and}} \ m\neq11,18,20,22,33 \\*[0.1cm]
\frceil{80m+38}{29} & \ \mbox{\rm{if}} \ n=12 \\*[0.1cm]
\frceil{98m+54}{33} - 1 & \ \mbox{\rm{if}} \ n=13 \ \mbox{\rm{and}} \ m\equiv_{33}13,16,18,19 \\*[0.1cm]
\frceil{98m+54}{33} & \ \mbox{\rm{if}} \ n=13 \ \mbox{\rm{and}} \ m\not\equiv_{33}13,16,18,19 \\*[0.1cm]
\frceil{35m+20}{11} - 1 & \ \mbox{\rm{if}} \ n=14 \ \mbox{\rm{and}} \ m\equiv_{22}7\\*[0.1cm]
\frceil{35m+20}{11} & \ \mbox{\rm{if}} \ n=14 \ \mbox{\rm{and}} \ m\not\equiv_{22}7\\*[0.1cm]
\frceil{44m+28}{13} - 1 & \ \mbox{\rm{if}} \ n=15 \ \mbox{\rm{and}} \ m\equiv_{26}5\\*[0.1cm]
\frceil{44m+28}{13} & \ \mbox{\rm{if}} \ n=15 \ \mbox{\rm{and}} \ m\not\equiv_{26}5\\*[0.1cm]
\frfloor{(n+2)(m+2)}{5}-4 & \ \mbox{\rm{if}} \ n\ge16 \\*[0.1cm]
\end{array}\right.$$
\end{theorem}
Note that these values are obtained by specific formulas for every $1\leq n\leq 15$ and by the formula of Conjecture~\ref{conj} for every $16\le n \le 21$. This proves Chang's conjecture for all values $16\le n\le 21$.
In 2004, Conjecture~\ref{conj} has been confirmed up to an additive constant:
\begin{theorem}[Guichard \cite{Guichard}] For every $16\le n \le m$,
$$\gamma(G_{n,m}) \geq \left \lfloor \frac{(n+2)(m+2)}{5} \right
\rfloor -9.$$ \end{theorem}
In this paper, we prove Chang's conjecture, hence finishing the computation of $\gG{n}{m}$. We adapt Guichard's ideas to improve the additive constant from $-9$ to $-4$ when $24 \le n \le m$. Cases $n=22$ and $n=23$ can be proved in a couple of hours using Fisher's method (described in~\cite{Fisher}) on a modern computer. They can be also proved by a slight improvement of the technique presented in the next section.
\section{Values of $\gamma(G_{n,m})$ when $24 \le n \le m$}
Our method follows the idea of Guichard \cite{Guichard}. A slight improvement is enough to give the exact bound.
A vertex of the grid $G_{n,m}$ dominates at most 5 vertices (its four neighbours and itself). It is then clear that $\gG{n}{m} \ge \frac{n\times m}{5}$. The previous inequality would become an equality if there would be a dominating set $D$ such that every vertex of $G_{n,m}$ is dominated only once, and all vertices of $D$ are of degree 4 (i.e. dominates exactly 5 vertices); in this case, we would have
$5\times|D| - n\times m= 0$. This is clearly impossible (e.g. to dominate the corners of the grid, we need vertices of degree at most 3). Therefore, our goal is to find a dominating set $D$ of $G_{n,m}$ such that the difference $5\times |D| - n\times m$ is the smallest.
Let $S$ be a subset of $V(G_{n,m})$. The \emph{loss} of $S$ is $\los{S}=5 \times \vert S \vert - \vert N[S] \vert$.
\begin{proposition} \label{props}
The following properties of the loss function are straightforward:
\begin{enumerate}[(i)]
\item For every $S$, $\los{S} \ge 0$ (positivity), \label{props:1}
\item If $S_1\cap S_2=\emptyset$, then $\los{S_1\cup S_2} =
\los{S_1} + \los{S_2} + \vert N[S_1] \cap N[S_2] \vert$, \label{props:2}
\item If $S'\subseteq S$, then $\los{S'}\le \los{S}$ (increasing
function), \label{props:3}
\item If $S_1\cap S_2=\emptyset$, then $\los{S_1\cup S_2} \ge
\los{S_1} + \los{S_2}$ (super-additivity). \label{props:4}
\end{enumerate} \end{proposition}
Let us denote by $\ell_{n,m}$ the minimum of $\los{D}$ when $D$ is a dominating set of $G_{n,m}$.
\begin{lemma}\label{lemloss}
$\gamma(G_{n,m}) = \left \lceil \frac{n\times m + \ell_{n,m}}{5} \right
\rceil$ \end{lemma}
\begin{proof}
If $D$ is a dominating set of $G_{n,m}$, then $\los{D} = 5 \times \vert
D \vert - \vert N[D]\vert = 5 \times \vert
D \vert - n\times m$. Hence, by minimality of $\ell_{n,m}$ and
$\gamma(G_{n,m})$, we have $\ell_{n,m}= 5\times \gamma(G_{n,m}) - n\times m$. \end{proof}
Our aim is to get a lower bound for $\ell_{n,m}$. As the reader can observe in Figure~\ref{fig:24x24}, the loss is concentrated on the border of the grid. We now analyse more carefully the loss generated by the border of thickness $10$.
\begin{figure}
\caption{The graph $\fG{30}{40}$. The set $\fBp{30}{40}$ is the set
of vertices filled in black. The set $\fB{30}{40}$ is the set of
vertices filled in black or in gray.}
\label{fig:bnm}
\end{figure}
We define the border $B_{n,m} \subseteq V_{n,m}$ of $G_{n,m}$ as the set of vertices $(i,j)$ where $i\le 10$, or $j\le 10$, or $i>n-10$, or $j>m-10$ to which we add the four vertices $(11,11),(11,m-10),(n-10,11),(n-10,m-10)$. Given a subset $S\subseteq V$, let $I(S)$ be the \emph{internal vertices} of $S$, i.e. $I(S)= \{v\in S : N[v]\subseteq S\}$. These sets are illustrated in Figure~\ref{fig:bnm}. We will compute $b_{n,m} = \min_D \los{D}$, where $D$ is a subset of $B_{n,m}$ and dominates $I(B_{n,m})$, i.e. $D\subseteq B_{n,m}$ and $I(B_{n,m}) \subseteq N[D]$. Observe that this lower bound $b_{n,m}$ is a lower bound of $\ell_{n,m}$. Indeed, for every dominating set $D$ of $G_{n,m}$, the set $D':=D\cap B_{n,m}$ dominates $I(B_{n,m})$, hence $b_{n,m}\leq \los{D'}\leq \los{D}$. In the remainder, we will compute $b_{n,m}$ and we will show that $b_{n,m} = \ell_{n,m}$.
In the following, we split the border $B_{n,m}$ in four parts, $O_{m-12}, P_{n-12}, Q_{m-12}, R_{n-12}$, which are defined just below.
\begin{figure}
\caption{The set $\fP{19}$ (black and gray), the set of input vertices
(gray circles) and the set of output vertices (gray squares).}
\label{fig:ppcpa}
\end{figure}
For $p\ge 12$, let $P_p\subset B_{n,m}$ defined as follows : $P_p = ([10] \times \{12\}) \cup ([11] \times \{11\} ) \cup ([p] \times [10])$. We define the \emph{input vertices} of $P_p$ as $[10]\times\{12\}$ and the \emph{output vertices} of $P_p$ as $\{p\} \times [10]$. The set $P_p$, illustrated for $p=19$ in Figure~\ref{fig:ppcpa}, corresponds to the set of black and gray vertices. The input vertices are the gray circles, and the output vertices are the gray squares. Recall that in our drawing conventions, the vertex $(1,1)$ is the bottom-left vertex and hence the vertex $(i,j)$ is in the $i^{th}$ column from the left and in the $j^{th}$ row from the bottom.
\begin{figure}
\caption{The sets $O_{m-12}$, $P_{n-12}$, $Q_{m-12}$ and $R_{n-12}$.
}
\label{fig:pqrs}
\end{figure}
For $n,m\in \mathbb{N}^*$, let $f_{n,m}: [n]\times[m] \to [m]\times[n]$ be the bijection such that $f_{n,m}(i,j) = (j,n-i+1)$. It is clear that the set $B_{n,m}$ is the disjoint union of the following four sets depicted in Figure~\ref{fig:pqrs}: $\fP{n-12}$, $\fQ{m-12}=f_{n,m}(\fP{m-12})$, $\fR{n-12} = f_{m,n} \circ f_{n,m} (\fP{n-12})$ and $\fS{m-12}= f_{n,m}^{-1}(\fP{m-12})$. Similarly to $\fP{n-12}$, the sets $O_{m-12}$, $Q_{n-12}$ and $R_{m-12}$ have input and output vertices. For instance, the output vertices of $Q_{m-12}$ correspond in Figure~\ref{fig:ppcpa} to the white squares. Every set playing a symmetric role, we now focus on $P_{n-12}$.
Given a subset $S$ of $V_{n,m}$, let the labelling $\phi_S : V_{n,m} \to \{0,1,2\}$ be such that
$$\phi_S(i,j) = \left\{
\begin{array}{l}
0 \quad \mbox{if $(i,j)\in S$} \\
1 \quad \mbox{if $(i,j)\in N[S]\setminus S$} \\
2 \quad \mbox{otherwise}
\end{array}\right.$$
Note that $\phi_S$ is such that any two adjacent vertices in $G_{n,m}$ cannot be labelled~$0$ and~$2$.
Given $p\ge 12$ and a set $S\subseteq P_p$, the \emph{input word} (resp. \emph{output word}) of $S$ for $P_p$, denoted by $w^{in}(S)$ (resp. $w^{out}_p(S)$), is the ten letters word on the alphabet $\{0,1,2\}$ obtained by reading $\phi_{S}$ on the input vertices (resp. output vertices) of $P_p$. More precisely, its $i^{th}$ letter is $\phi_{S}(i,12)$ (resp. $\phi_{S}(p,i)$).
Similarly, $O_p$, $Q_p$ and $R_p$ have also input and output words. For example, the output word of $S\subseteq O_p$ for $O_p$ is $w^{out}_p(f_{n,m}(S))$.
According to the definition of $\phi$, the input and output words belong to the set $\mathcal{W}$ of ten letters words on $\{0,1,2\}$ which avoid $02$ and $20$. The number of $k$-digits trinary numbers without $02$ or $20$ is given by the following formula~\cite{Fisher}: \begin{equation}
\frac{(1+\sqrt{2})^{k+1}+(1-\sqrt{2})^{k+1}}{2}\label{eqn:trinary_words} \end{equation}
The size of $\mathcal{W}$ is therefore $|\mathcal{W}| = 8119$.
Given two words $w,w' \in \mathcal{W}$, we define $\mD^{w,w'}_p$ as the family of subsets $D$ of $\fP{p}$ such that: \begin{itemize} \item $D$ dominates $I(P_p)$, \item $w$ is the input word $w^{in}(D)$, \item $w'$ is the output word $w^{out}_p(D)$. \end{itemize}
A relevant information for our calculation will be to know, for two given words $w,w'\in \mathcal{W}$, the minimum loss over all losses $\los{D}$ where $D\in \mD^{w,w'}_p$. For this purpose, we introduce the $8119\times 8119$ square matrix $C_p$. For $w,w'\in \mathcal{W}$, let $C_p[w,w']=\min_{D\in\mD^{w,w'}_p} \los{D}$ where the minimum of the empty set is $+\infty$.
Let $w,w'\in \mathcal{W}$ be two given words. Due to the symmetry of $P_{12}$
with respect to the first diagonal (bottom-left to top-right) of the grid, if a vertex set $D$ belongs to $\Dww{12}$, then $D' = \{(j,i) | (i,j)\in D\}$ belongs to $\mathcal{D}^{w',w}_{12}$. Moreover, it is clear that, always due to the symmetry, $\los{D}=\los{D'}$. Therefore, we have $C_{12}[w,w'] = C_{12}[w',w]$ and thus $C_{12}$ is a symmetric matrix. Despite the size of $C_{12}$ and the size of $P_{12}$ (141 vertices), it is possible to compute $C_{12}$ in less than one hour by computer. Indeed, we choose a sequence of subsets $X_0=\emptyset, X_1, \ldots, X_{141}=P_{12}$ such that for every $i\in \{1,\ldots, 141\}$, $X_i\subseteq X_{i+1}$ and $X_{i+1} \setminus X_i = \{ x_{i+1} \}$. Moreover, we choose the sequence such that for every $i$, $X_i \setminus I(X_i)$ is at most $21$. This can be done for example by taking $x_{i+1}= \min \{ (x,y) : (x,y)\in P_{12} \setminus X_i \}$, where the order is the lexical order. For $i\in \{0,\ldots, 141\}$, we compute for every labeling $f \in \mathcal{F}_i$, where $\mathcal{F}_i$ is the set of functions $(X_i\setminus I(X_i)) \to \{0,1,2\}$, the minimal loss $l_{i,f}$ of a set $S\subseteq X_i$ which dominates
$I(X_i)$ and such that $\phi_S(v)=f(v)$ for every $v\in X_i \setminus I(X_i)$. Note that not every labeling is possible since two adjacent vertices cannot be labeled~$0$ and~$2$. The number of possible labellings can be computed using formula~(\ref{eqn:trinary_words}), and since $|X_i \setminus I(X_i)|$ can be covered by a path of at most $23$ vertices, this gives, in the worst case, that this number is less than $10^9$ and can be then processed by a computer. We compute inductively the sequence $(l_{i,f})_{f\in
\mathcal{F}_i}$ from the sequence $(l_{i-1,f})_{f\in \mathcal{F}_{i-1}}$ by dynamical programming, and $C$ is easily deduced from $(l_{141,f})_{f\in
\mathcal{F}_{141}}$.
In the following, our aim is to glue $P_{n-12}, Q_{m-12},R_{n-12},$ and $O_{m-12}$ together. For two consecutive parts of the border, say $P_{n-12}$ and $Q_{m-12}$, the output word of $Q_{m-12}$ should be compatible with the input word of $P_{n-12}$. Two words $w,w'$ of $\mathcal{W}$ are \emph{compatible} if the sum of their corresponding letters is at most 2, i.e. $w[i]+w'[i]\leq 2$ for all $i\in [9]$. Note that $w[10] + w'[10]$ should be greater than 2 since the corresponding vertices can be dominated by some vertices of $V_{n,m} \setminus B_{n,m}$.
Given two words $w,w'\in \mathcal{W}$, let $\ell(w,w')= \vert \{ i\in [10] : w[i]\ne 2 \text{ and } w'[i]=0 \} \vert + \vert \{ i\in [10] : w'[i]\ne 2 \text{ and } w[i]=0 \} \vert$.
\begin{lemma} \label{lem:compatible}Let $D$ be a dominating set of $G_{n,m}$ and let us denote
$D_P = D\cap P_{n-12}$ and $D_Q = D\cap Q_{m-12}$. Then $\ell(D\cap
(P_{n-12} \cup Q_{m-12}))= \ell(D_P) + \ell(D_Q) + \ell(w,w')$, where
$w=w^{in}(D_P)$ and $w'=w^{out}_q(f_{n,m}^{-1}(D_Q))$. Moreover, $w$ and $w'$
are compatible. \end{lemma}
\begin{proof}
By Proposition \ref{props}(\ref{props:2}), $\ell(D\cap (P_{n-12} \cup Q_{m-12}))=
\ell(D_P) + \ell(D_Q) + \vert N[D_P] \cap N[D_Q]\vert$. It suffice then to note that
$\ell(w,w')= \vert N[D_P] \cap N[D_Q]\vert$ to get $\ell(D\cap (P_{n-12} \cup Q_{m-12}))=
\ell(D_P) + \ell(D_Q) + \ell(w,w')$.
In what remains, we prove that $w$ and $w'$ are compatible. If
those two words were not compatible, there would exist an index $i\in[9]$
such that $w^{out}_{m-12}(f_{n,m}^{-1}(D_Q))[i] + w^{in}(D_P)[i] > 2$. Thus
at least one of these two letters should be a 2, and the other one
should not be 0.
Suppose that $w^{out}_{m-12}(f_{n,m}^{-1}(D_Q))[i] = 2$ and note that
this means that the vertex $(i,13)$ is not dominated by a vertex in
$D_Q$. Since $D$ is a dominating set of $G_{n,m}$, every output vertex of
$Q_{m-12}$ except $(10,13)$ (and every input vertex of $P_{n-12}$
except $(10,12)$) is dominated by a vertex of $D_Q$ or by a vertex of
$D_P$. Thus $(i,13)$ should be dominated by its unique neighbour in
$P_{n-12}$, $(i,12)$. This would imply that $(i,12)\in D$
contradicting the fact that $w^{in}(D_P)[i] \neq 0$.
Similarly, if $w^{in}(D_P)[i] =2$, the vertex $(i,12)$
is not dominated by a vertex in $D_P$, thus $(i,12)$ must be dominated
by the vertex $(i,13)\in D$, contradicting the fact that
$w^{out}_{m-12}(f_{n,m}^{-1}(D_Q))[i] \neq 0$. \end{proof}
Lemma~\ref{lem:compatible} is designed for the two consecutive parts $P_{n-12}$ and $Q_{m-12}$ of the border of $G_{n,m}$. Its easy to see that this extends to any pair of consecutive parts of the border, i.e. $Q_{m-12}$ and $R_{n-12}$, $R_{n-12}$ and $O_{m-12}$, $O_{m-12}$ and $P_{n-12}$.
We define the matrix $8119\times 8119$ square matrix $L$ which contains, for every pair of words $w,w'\in \mathcal{W}$, the value $\ell(w,w')$: $$L[w,w']= \begin{cases}
+\infty \text{ if $w$ and $w'$ are not compatible,}\\
\ell(w,w') \text{ otherwise.} \end{cases}$$ Note that $L$ is symmetric since $\ell(w,w') = \ell(w',w)$.
Let $\otimes$ be the matrix multiplication in $(\min,+)$ algebra, i.e. $C = A \otimes B$ is the matrix where for all $i,j$, $C[i,j] = \min_k A[i,k] + B[k,j]$.
Let $M_p=L \otimes C_p$ for $p\ge 12$.
By construction, $M_{n-12}[w,w']$ corresponds to the minimum possible loss $\los{D\cap P_{n-12}}$ of a dominating set $D\subseteq V_{n,m}$ that dominates $I(P_{n-12})$ and such that $w$ is the output word of $Q_{m-12}$ and $w'$ is the output word of $P_{n-12}$.
\begin{lemma}
\label{lem:bnm}
For all $24 \le n \le m$, we have
$$b_{n,m}\geq \min_{w_1,w_2,w_3,w_4\in \mathcal{W}} M_{n-12}[w_1,w_2] +
M_{m-12}[w_2,w_3] + M_{n-12}[w_3,w_4]+ M_{m-12}[w_4,w_1].$$ \end{lemma} \begin{proof} Consider a set $D \subseteq B_{n,m}$ which dominates
$I(B_{n,m})$ and achieving $\ell (D)=b_{n,m}$. Let $D_P= D\cap P_{n-12}$,
$D_Q= D\cap Q_{m-12}$, $D_R= D\cap R_{n-12}$ and $D_O= D\cap
O_{m-12}$. Let $w_P$ ($w_Q$, $w_R$ and $w_O$, respectively) be the
input word of $P_{n-12}$ ($Q_{m-12}$, $R_{n-12}$ and $O_{m-12}$), and
$w'_P$ ($w'_Q$, $w'_R$ and $w'_O$) be the output word of $P_{n-12}$
($Q_{m-12}$, $R_{n-12}$ and $O_{m-12}$). By definition of $C_p$,
the loss of $D_P$ is at least $C_{n-12}[w_P,w'_P]$. Similarly, we
have $\ell(D_Q)\geq C_{m-12}[w_Q,w'_Q]$, $\ell(D_R)\geq
C_{n-12}[w_R,w'_R]$ and $\ell(D_O)\geq C_{m-12}[w_O,w'_O]$.
By the definition of the loss:
\begin{align} \ell(D) = {} & b_{n,m} \nonumber\\ = {} & 5\times
\vert D \vert - \vert N[D]\vert \nonumber\\ = {} & \ell(D_O)
+\ell(D_P) +\ell(D_Q) +\ell(D_R) + L[w'_O,w_P] + L[w'_P,w_Q] +
L[w'_Q,w_R] + L[w'_R,w_O] \nonumber\\ & \text{~~ by
Lemma~\ref{lem:compatible} and since $N[D_P] \cap N[D_R] = N[D_Q] \cap
N[D_O] =\emptyset$} \nonumber\\ \ge {} & C_{m-12}[w_O,w'_O] +
C_{n-12}[w_P,w'_P] + C_{m-12}[w_Q,w'_Q] +
C_{n-12}[w_R,w'_R] \nonumber\\ & + L[w'_O,w_P] +
L[w'_P,w_Q] + L[w'_Q,w_R] + L[w'_R,w_O] \nonumber\\ \ge {}
& M_{m-12}[w_O,w_P] + M_{n-12}[w_P,w_Q] +
M_{m-12}[w_Q,w_R] + M_{n-12}[w_R,w_O] \nonumber \\
& \text{~~ since $w'_O$ and $w_P$ (resp. $w'_P$ and $w_Q$, $w'_Q$
and $w_R$, $w'_R$ and $w_O$) are compatibles.} \nonumber
\end{align} \end{proof}
According to Lemma~\ref{lem:bnm}, to bound $b_{n,m}$ it would be thus interesting to know $M_p$ for $p> 12$. It is why we introduce the following $8119\times 8119$ square matrix, $T$.
\begin{lemma}
There exists a matrix $T$ such that $C_{p+1}=C_p \otimes T$
for all $p\ge 12$. This matrix is defined as follows:
$$ T[w,w']=
\begin{cases}
+\infty \ \ \ \text{ if $\exists i\in [10]$ s.t. $w[i]=0$ and $w'[i]=2$}\\
+\infty \ \ \ \text{ if $\exists i\in [9]$ s.t. $w[i]=2$ and $w'[i]\ne 0$}\\
+\infty \ \ \ \text{ if $\exists i\in \{2,\ldots, 9\}$
s.t. $w'[i]=1$, $w[i]\ne 0$, $w'[i-1]\ne 0$ and $w'[i+1]\ne
0$}\\
+\infty \ \ \ \text{ if $w'[1]=1$, $w[1]\ne 0$ and $w'[2]\ne 0$}\\
+\infty \ \ \ \text{ if $w'[10]=1$, $w[10]\ne 0$ and $w'[9]\ne 0$}\\
3 \times \vert w'\vert_0 - \vert w\vert_2 - \vert w' \vert_1 +
\vert w\vert_0 - 1 \ \ \
\text{if $w'[10] = 0$} \\
3 \times \vert w'\vert_0 - \vert w\vert_2 - \vert w' \vert_1 +
\vert w\vert_0 \ \ \ \text{ otherwise.}
\end{cases}
$$ \end{lemma}
\begin{proof} Consider a set $S'\subseteq P_{p+1}$ dominating
$I(P_{p+1})$ and let $S =S' \cap P_p$. Let $w = w^{out}_p(S)$ and $w' =
w^{out}_{p+1}(S')$. Let $\Delta(S,S')= \los{S'} - \los{S}$. By
definition of the loss, $\Delta(S,S')= 5 \times |S'\setminus S| -
|N[S']\setminus N[S]|$. Let us compute $\Delta(S,S')$ in term of the
number of occurrences of $0$'s, $1$'s and $2$'s in the words $w$ and
$w'$. The set $S'\setminus S$ corresponds to the vertices $\{(p+1,i)
\mid i\in[10], w'[i] = 0\}$. The set $N[S']\setminus N[S]$ corresponds
to the vertices dominated by $S'$ but not dominated by $S$; these
vertices clearly belong to the columns $p$, $p+1$ and $p+2$. Since
$S'$ dominates $I(P_{p+1})$, those in the column $p$ are the vertices
$\{(p,i) \mid i\in[10], w[i]=2\}$. Those in the column $p+1$ are the
vertices $\{(p+1,i) \mid i\in[10], w'[i]\neq 2, w[i] \neq0\}$ and
possibly the vertex $(p+1,11)$ when $w'[10]=0$. Finally, those in the
column $p+2$ are the vertices $\{(p+2,i) \mid i\in[10], w'[i]=0\}$.
We then get:
$$
\Delta(S,S')=
\begin{cases}
3 \times \vert w'\vert_0 - \vert w \vert_2
- \vert w'\vert_1 + \vert w\vert_0 - 1 & \text{if $w'[10] = 0$} \\
3 \times \vert w'\vert_0 - \vert w \vert_2
- \vert w'\vert_1 + \vert w\vert_0 & \text{otherwise}
\end{cases}
$$
where $|w|_n$ denotes the number of occurrences of the letter $n$ in
the word $w$.
Thus $\Delta(S,S')$ only depends on the output words of $S$ and $S'$,
and we can denote this value by $\Delta(w,w')$. Note however that
there exist pairs of
words $(w,w')$ that could not be the output words of $S$ and $S'$;
there are three cases:
\begin{enumerate}[C{a}se 1.]
\item $w[i]=0$ and $w'[i]=2$ since the vertex
$(p+1,i)$ would be dominated by $(p,i)$ contradicting its label 2;
\item $w[i]=2$ and $w'[i]\ne 0$ for $i\in[9]$ since
$(p,i)$ would not be dominated contradict the fact that $S'$
dominates $I(P_{p+1})$;
\item $w'[i]=1$ and $w'[i-1] \neq 0$, $w'[i+1]\neq 0$, $w[i]\neq 0$
since $(p+1,i)$ would be dominated according to its label but none
of its neighbors belong to $S'$.
\end{enumerate}
For these forbidden cases, we set $\Delta(w,w') = +\infty$.
By definition, $C_{p+1}[w_i,w']$ is the minimum loss $\ell(S')$
of a set $S'\subseteq P_{p+1}$ that dominates $I(P_{p+1})$, with $w_i$
as input word and $w'$ as output word. It is clear that $S= S'\cap P_p$
dominates $I(P_{p})$ and has $w_i$ as input word. Let $w$ be its output
word and note that $C_{p+1}[w_i,w'] = \ell(S') = \ell(S) +
\Delta(w_i,w')$. The minimality of $\ell(S')$ implies the minimality of
$\ell(S)$ over the sets $X\in \mathcal{D}^{w_i,w'}_p$. Indeed, any set $X\in
\mathcal{D}^{w_i,w'}_p$ could be turned in a set of $X'\in \mathcal{D}^{w_i,w'}_{p+1}$ by
adding vertices of the $p+1^{th}$ column accordingly to $w'$. Thus
$$
C_{p+1}[w_i,w'] = C_{p}[w_i,w] + \Delta(w,w')
$$
which implies that
$$
C_{p+1}[w_i,w'] \geq \min_{w} C_{p}[w_i,w] + \Delta(w,w').
$$
On the other hand, for every word $w_o \in \mathcal{W}$ such that
$C_{p}[w_i,w_o]\neq +\infty$ and $\Delta(w_o,w')\neq
+\infty$, there is a set $S \in \mathcal{D}^{w_i,w_o}_p$, with $\ell(S) =
C_{p}[w_i,w_o]$, that can be turned in a set $S' \in
D^{w_i,w'}_{p+1}$ with $\ell(S') = C_{p}[w_i,w_o] +
\Delta(w_o,w')$. Thus
$$
C_{p+1}[w_i,w'] \leq \min_{w_o} C_{p}[w_i,w_o] + \Delta(w_o,w').
$$
This concludes the proof of the lemma. \end{proof}
By the definition of $M_p$, we have also $M_{p+1}=M_p \otimes T$. Note that $T$ is a sparse matrix: about $95.5 \%$ of its $8119^2$ entries are $+\infty$. Thus the multiplication by $T$ in the $(\min,+)$ algebra can be done in a reasonable amount of time by a trivial algorithm.
\begin{fact}\label{fact126} The computations give us that
$M_{126}=M_{125}+1$. Thus, since $(A + c)\otimes B =
(A\otimes B) + c$ for any matrices $A$, $B$ and any integer $c$, we
have that $M_{125+k}=M_{125}+k$ for every $k\in \mathbb{N}$. \end{fact}
Let us define $M'_p= \min_{k\in \mathbb{N}} ( M_{p+k} - k )$. Then, for all $q\ge p$, $M_q \ge M'_p + (q-p)$. By Fact~\ref{fact126}, $M'_p= \min_{k\in \{0,\dots 125-p\}} ( M_{p+k} - k )$
\begin{fact}\label{fact23} By computing $M'_{12}$, and $A'=
M'_{12} \otimes M'_{12}$, we obtain that $\min_{w_1,w_3} (A'
+ A'^T)[w_1,w_3] = 76$ (where $A^T$ is the transpose of $A$). \end{fact} This implies that $$ \min_{w_1,w_3} \ (\min_{w_2} M'_{12}[w_1,w_2]+M'_{12}[w_2,w_3]) \ + \ (\min_{w_4} M'_{12}[w_3,w_4]+ M'_{12}[w_4,w_1]) \ = \ 76 $$ $$ \min_{w_1,w_2,w_3,w_4} M'_{12}[w_1,w_2]+M'_{12}[w_2,w_3]+M'_{12}[w_3,w_4]+ M'_{12}[w_4,w_1] \ = \ 76. $$
\begin{theorem}
If $24 \le n \le m$, then $$\gamma(G_{n,m}) = \form{n}{m}.$$ \end{theorem}
\begin{proof} By Chang's construction \cite{CHHW}, $\gamma(G_{n,m}) \le
\form{n}{m}$. Let us now compute a lower bound for the loss of a
dominating set of $G_{n,m}$.
\begin{eqnarray*}
\ell_{n,m} &\ge& b_{n,m}\\
&\ge& \min_{w_1,w_2,w_3,w_4} M_{n-12}[w_1,w_2] +
M_{m-12}[w_2,w_3] + M_{n-12}[w_3,w_4]+
M_{m-12}[w_4,w_1]\\
& & \; \mbox{by Lemma~\ref{lem:bnm}} \\
&\ge& \min_{w_1,w_2,w_3,w_4} M'_{12}[w_1,w_2]+(n-12-12) +
M'_{12}[w_2,w_3]+ (m-12-12) + M'_{12}[w_3,w_4]\\&&
\phantom{\min_{w_1,w_2,w_3,w_4}}+(n-12-12) +
M'_{12}[w_4,w_1]+ (m-12-12)\\
&\ge& 2 \times (n+m -48) + \min_{w_1,w_2,w_3,w_4}
M'_{12}[w_1,w_2] + M'_{12}[w_2,w_3] +
M'_{12}[w_3,w_4] + M'_{12}[w_4,w_1]\\
&\ge& 2 \times (n+m -48) + 76\\
&\ge& 2\times(n+m)-20.
\end{eqnarray*}
Thus by Lemma~\ref{lemloss}, we have:
\begin{eqnarray*}
\gamma(G_{n,m}) &\ge& \left \lceil \frac{n\times m + 2\times(n+m)-20}{5}
\right \rceil\\
&\ge& \left \lceil \frac{(n+2)(m + 2) -4}{5}
\right \rceil - 4\\
&\ge& \form{n}{m}.
\end{eqnarray*} \end{proof}
\end{document} | arXiv |
›Basic Concepts
This overview describes the basic components of BoTorch and how they work together. For a high-level view of what BoTorch tries to achieve in more abstract terms, please see the Introduction.
Black-Box Optimization
At a high level, the problem underlying Bayesian Optimization (BayesOpt) is to maximize some expensive-to-evaluate black box function $f$. In other words, we do not have access to the functional form of $f$ and our only recourse is to evaluate $f$ at a sequence of test points, with the hope of determining a near-optimal value after a small number of evaluations. In many settings, the function values are not observed exactly, and only noisy observations are available
Bayesian Optimization is a general approach to adaptively select these test points (or batches of test points to be evaluated in parallel) that allows for a principled trade-off between evaluating $f$ in regions of good observed performance and regions of high uncertainty.
Bayesian Optimization
In order to optimize $f$ within a small number of evaluations, we need a way of extrapolating our belief about what $f$ looks like at points we have not yet evaluated. In Bayesian Optimization, this is referred to as the surrogate model. Importantly, the surrogate model should be able to quantify the uncertainty of its predictions in form of a posterior distribution over function values $f(x)$ at points $x$.
The surrogate model for $f$ is typically a Gaussian Process (GP), in which case the posterior distribution on any finite collection of points is a multivariate normal distribution. A GP is specified by a mean function $\mu(x)$ and a covariance kernel $k(x, x')$, from which a mean vector $(\mu(x_0), \ldots, \mu(x_k))$ and covariance matrix $\Sigma$ with $\Sigma_{ij} = k(x_i, x_j)$ can be computed for any set of points $(x_1, \ldots x_k)$. Using a GP surrogate model for $f$ means that we assume $(f(x_1), \ldots, f(x_k))$ is multivariate normal with a mean vector and covariance matrix determined by $\mu(x)$ and $k(x, x')$.
BoTorch provides first-class support for GPyTorch, a package for scalable GPs and Bayesian deep learning implemented in PyTorch.
While GPs have been a very successful modeling approach, BoTorch's support for MC-sampling based acquisition functions makes it straightforward to also use other model types. In particular, BoTorch makes no particular assumptions on what kind of model is being used, so long as is able to produce samples from a posterior over outputs given an input $x$. See Models for more details on models in BoTorch.
Posteriors represent the "belief" a model has about the function values at a point (or set of points), based on the data it has been trained with. That is, the posterior distribution over the outputs conditional on the data observed so far.
When using a GP model, the posterior is given explicitly as a multivariate Gaussian (fully parameterized by its mean and covariance matrix). In other cases, the posterior may be implicit in the model and not easily described by a small set of parameters.
BoTorch abstracts away from the particular form of the posterior by providing a simple Posterior API that only requires implementing an rsample() method for sampling from the posterior. For more details, please see Posteriors.
Acquisition functions are heuristics employed to evaluate the usefulness of one of more design points for achieving the objective of maximizing the underlying black box function.
Some of these acquisition functions have closed-form solutions under Gaussian posteriors, but many of them (especially when assessing the joint value of multiple points in parallel) do not. In the latter case, one can resort to using Monte-Carlo (MC) sampling in order to approximate the acquisition function.
BoTorch supports both analytic as well as (quasi-) Monte-Carlo based acquisition functions. It provides an AcquisitionFunction API that abstracts away from the particular type, so that optimization can be performed on the same objects. Please see Acquisition Functions for additional information.
Evaluating Monte-Carlo Acquisition Functions
The idea behind using Monte-Carlo sampling for evaluating acquisition functions is simple: instead of computing an (intractable) expectation over the posterior, we sample from the posterior and use the sample average as an approximation.
To give additional flexibility in the case of MC-based acquisition functions, BoTorch provides the option of transforming the output(s) of the model through an Objective module, which returns a one-dimensional output that is passed to the acquisition function. The MCAcquisitionFunction class defaults its objective to IdentityMCObjective, which simply returns the last dimension of the model output. Thus, for the standard use case of a single-output GP that directly models the black box function $f$, no special objective is required. For more details on the advanced features enabled by the Objective, see Objectives.
The Re-Parameterization Trick
The re-parameterization trick (see e.g. [1], [2]) can be used to write the posterior distribution as a deterministic transformation of an auxiliary random variable $\epsilon$. For example, a normally distributed random variable $X$ with mean $\mu$ and standard deviation $\sigma$ has the same distribution as $\mu + \sigma \epsilon$ where $\epsilon$ is a standard normal. Therefore, an expectation with respect to $X$ can be approximated using samples from $\epsilon$. In the case where $\mu$ and $\sigma$ are parameters of an optimization problem, MC approximations of the objective at different values of $\mu$ and $\sigma$ can be computed using a single set of "base samples." Importantly, a re-parameterization of this kind allows for back-propagation of gradients through the samples, which enables auto- differentiation of MC-based acquisition functions with respect to the candidate points.
In BoTorch, base samples are constructed using an MCSampler object, which provides an interface that allows for different sampling techniques. IIDNormalSampler utilizes independent standard normal draws, while SobolQMCNormalSampler uses quasi-random, low-discrepancy "Sobol" sequences as uniform samples which are then transformed to construct quasi-normal samples. Sobol sequences are more evenly distributed than i.i.d. uniform samples and tend to improve the convergence rate of MC estimates of integrals/expectations. We find that Sobol sequences substantially improve the performance of MC-based acquisition functions, and so SobolQMCNormalSampler is used by default. For more details, see Monte-Carlo Samplers.
D. P. Kingma, M. Welling. Auto-Encoding Variational Bayes. ICLR, 2013. ↩
D. J. Rezende, S. Mohamed, D. Wierstra. Stochastic Backpropagation and Approximate Inference in Deep Generative Models. ICML, 2014. ↩
← Getting StartedModels → | CommonCrawl |
An INTEGRAL/SPI view of reticulum II: Particle dark matter and primordial black holes limits in the MeV range
Final Accepted Version (PDF, 1Mb)
Siegert, Thomas
Boehm, Celine
Calore, Francesca
Diehl, Roland
Krause, Martin G. H.
Serpico, Pasquale D.
Vincent, Aaron C.
Reticulum II (Ret II) is a satellite galaxy of the Milky Way and presents a prime target to investigate the nature of dark matter (DM) because of its high mass-to-light ratio. We evaluate a dedicated INTEGRAL observation campaign data set to obtain $\gamma$-ray fluxes from Ret II and compare those with expectations from DM. Ret II is not detected in the $\gamma$-ray band 25--8000 keV, and we derive a flux limit of $\lesssim 10^{-8}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$. The previously reported 511 keV line is not seen, and we find a flux limit of $\lesssim 1.7 \times 10^{-4}\,\mathrm{ph\,cm^{-2}\,s^{-1}}$. We construct spectral models for primordial black hole (PBH) evaporation and annihilation/decay of particle DM, and subsequent annihilation of positrons produced in these processes. We exclude that the totality of DM in Ret II is made of a monochromatic distribution of PBHs of masses $\lesssim 8 \times 10^{15}\,\mathrm{g}$. Our limits on the velocity-averaged DM annihilation cross section into $e^+e^-$ are $\langle \sigma v \rangle \lesssim 5 \times 10^{-28} \left(m_{\rm DM} / \mathrm{MeV} \right)^{2.5}\,\mathrm{cm^3\,s^{-1}}$. We conclude that analysing isolated targets in the MeV $\gamma$-ray band can set strong bounds on DM properties without multi-year data sets of the entire Milky Way, and encourage follow-up observations of Ret II and other dwarf galaxies.
https://doi.org/10.1093/mnras/stac008 | CommonCrawl |
\begin{document}
\begin{abstract}
We prove functional inequalities on vector fields
$u : \R^d \to \R^d$ when $\R^d$ is equipped with a bounded
measure $e^{-\phi} \,\mathrm d x$ that satisfies a Poincar\'e
inequality, and study associated self-adjoint operators. The
{\sl weighted Korn inequality} compares the differential matrix
$D u$, once projected orthogonally to certain
finite-dimensional spaces, with its symmetric part $D^s u$ and,
in an improved form of the inequality, an additional term
$\nabla\phi\cdot u$. We also consider {\sl Poincar\'e-Korn
inequalities} for estimating a projection of $u$ by $D^s u$
and zeroth-order versions of these inequalities obtained using
the Witten-Laplace operator. The constants depend on geometric
properties of the potential $\phi$ and the estimates are
quantitative and constructive. These inequalities are motivated
by kinetic theory and related with the {\sl Korn inequality}
(1906) in mechanics, which compares $D u$ and $D^s u$ on a
bounded domain.
\end{abstract}
\maketitle \thispagestyle{empty} \vspace*{-0.5cm}
\section{Introduction and main results} \label{Sec:intro}
\subsection{The problem at hand}
Korn's inequality~\cite{Kor06,Kor08,Kor09} is a classical tool in continuum mechanics which asserts the control of the $L^2$ norm of the gradient of a vector field defined on a smooth bounded domain~$\Omega$ of $\R^d$ by the $L^2$ norm of its symmetric part: \begin{equation}\label{eq:KornOmega}
\|D u\|_{L^2(\Omega)}^2 \leq 2\,\|D^s
u\|_{L^2(\Omega)}^2,\quad \forall\,u \in C^2(\overline\Omega;
\R^d) \ \text{ such that } \ u=0 \text{ on } \partial\Omega\,. \end{equation} If $\mathfrak M$ (resp.~$\mathfrak M^s$) is the set of $d \times d$ real (resp.~symmetric) matrices, $D u$ (resp.~$D^s u$) is the differential of $u$ (resp.~its symmetric part) and takes values in $\mathfrak M$ (resp.~$\mathfrak M^s$). Written with cartesian coordinates, this means \begin{equation*}
(D^su)_{ij}=\frac12\( \partial_j u_i + \partial_i u_j\)
\quad\mbox{and}\quad
(D^au)_{ij}=\frac12\( \partial_j u_i - \partial_i u_j\). \end{equation*} We denote by $A^a \in \mathfrak M^a$ the skew-symmetric part of $A \in \mathfrak M$ and by $A^s=A-A^a$ its symmetric part. The original proof of~(\ref{eq:KornOmega}) in~\cite{Kor06} is instructive: it is enough to integrate over $\Omega$ the pointwise identities \begin{equation}\label{eq:Pointwise1}
|D^a u|^2-|D^s u|^2+(\nabla\cdot u)^2-\nabla \cdot
\Big[ u\,(\nabla\cdot u)-(u\cdot \nabla )\,u \Big]=0 \quad
\text{ and }\quad |D u|^2=|D^s u|^2+|D^a u|^2 \end{equation} and use the boundary condition to get
$2\,\|D^s u\|_{L^2(\Omega)}^2=\|D u\|_{L^2(\Omega)}^2+\|\nabla\cdot u\|_{L^2(\Omega)}^2$, where $\nabla\cdot u$ denotes the divergence of $u$ and $L^2(\Omega)$ is the $L^2$ norm for matrix-valued, vector-valued or real-valued functions.
The boundary condition $u=0$ is a severe restriction. In view of applications in kinetic theory, Desvillettes and Villani in~\cite{DV02} enlarge the set of possible vector fields to those satisfying only $u\cdot {\bf n}=0$ on the boundary, where ${\bf n}$ denotes the outward normal unit vector to $\partial\Omega$. The set of {\sl infinitesimal rotations} $$ {\mathcal R}:=\set{ R : x \in \R^d \mapsto A\,x \in \R^d \ \text{with} \ A\in\mathfrak M^a} $$ is a family of vector fields which plays a key role. If $\Omega$ has some rotational invariance, then some infinitesimal rotations satisfy the boundary condition while their symmetric differential is zero. Being invariant under the action of a group of rotations $t \to e^{tA}$ for a given $A \in \mathfrak M^a$ means that \begin{equation*} \forall\,t \in \R\,,\quad e^{tA}\,\Omega=\Omega \end{equation*} (here we suppose that $\Omega$ is invariant under rotations centred at $0$ without loss of generality). Taking the derivative with respect to $t$ shows that the set of {\sl infinitesimal
rotations preserving $\Omega$} is \begin{equation*}
{\mathcal R}_\Omega :=\big\{ R\in{\mathcal R}\,:\,\forall\,x\in\partial \Omega\,,\;
{\bf n}(x)\cdot R(x)=0 \big\}, \end{equation*} where we implicitly use the fact that skew-symmetric matrices generate the tangent space of the orthogonal group. In~\cite[Inequality~(38)]{DV02}, Desvillettes and Villani state the following {\sl Korn inequality} \begin{equation}\label{eq:KornOmega2}
\inf_{R\in{\mathcal R}_\Omega}\|D (u-R)\|^2_{L^2(\Omega)} \le C_\Omega\,\|D^s u\|^2_{L^2(\Omega)}\,,\quad \forall\,u \in C^2(\overline\Omega; \R^d) \ \text{ such that } \ u\cdot {\bf n}=0 \text{ on } \partial\Omega\,, \end{equation} which takes into account invariances by rotation. They obtain quantitative estimates on the constant $C_\Omega$. Inequality~\eqref{eq:KornOmega2} is an important ingredient in~\cite{Desvillettes-Villani-2005} to prove hypocoercivity for the Boltzmann equation in $\Omega$.
In this article, our aim is to establish similar {\sl
Korn and related inequalities}, with constructive constants, in the whole Euclidean space~$\R^d$ in presence of a {\sl confining
potential} $\phi : \R^d \to \R$, {\em i.e.}, in the $L^2$ space with reference measure $e^{-\phi(x)} \,\mathrm d x$. Our motivation comes from the hypocoercivity theory of kinetic operators with more than one microscopic invariant studied in~\cite{CDHMMShypo}, but the inequalities are of independent interest.
\subsection{Assumptions and notations}\label{subsec:notation}
We consider a potential \hbox{$\phi : \R^d \to \R$}, $d \ge 2$ satisfying the conditions: \\[4pt] \circled1 the measure $e^{-\phi(x)} \,\mathrm d x$ is a centred probability measure \begin{equation}
\label{hyp:intnorm} \tag{H1}
\int_{\R^d}e^{-\phi(x)} \,\mathrm d x=1\quad \mbox{and}\quad \int_{\R^d}
x\,e^{-\phi(x)} \,\mathrm d x=0\,, \end{equation} \\[4pt] \circled2 the potential $\phi$ is of class $C^2(\R^d;\R)$ and, for all ${\varepsilon}>0$, there exist a constant $C_{\varepsilon}$ such that \begin{equation}
\label{hyp:regularity} \tag{H2}
\begin{array}{c}
\forall\,x\in \R^d,\quad |D^2\phi(x)| \leq
{\varepsilon}\,|\nabla\phi(x)|^2+C_{\varepsilon}\,,
\end{array} \end{equation} \\[4pt] \circled3 the measure $e^{-\phi} \,\mathrm d x$ satisfies the Poincar\'e inequality with constant $C_P$: for all scalar functions~$f$ in the space ${\mathcal C}_c^\infty(\R^d;\R)$ of smooth functions with compact support, we have \begin{equation}
\label{hyp:poincarebasique} \tag{H3}
\int_{\R^d}\left|f(x)-\seq f\right|^2\,e^{-\phi(x)} \,\mathrm d x\leq C_{\mathrm{\scriptscriptstyle P}}
\int_{\R^d} |\nabla f(x)|^2\,e^{-\phi(x)} \,\mathrm d
x\,,\quad\mbox{where}\quad\seq f:=\int_{\R^d}
f(x)\,e^{-\phi(x)}\,\mathrm d x\,. \end{equation} Assumption~\eqref{hyp:intnorm} is a classical integrability condition on $\phi$. The fact that the center of mass $\int_{\R^d} x\,e^{-\phi(x)} \,\mathrm d x$ is finite is a consequence of~\eqref{hyp:poincarebasique} applied with $f(x)=x_i$, $i=1,\ldots,d$. There is no loss of generality in choosing
$\int_{\R^d} x\,e^{-\phi(x)} \,\mathrm d x=0$. Assumption~\eqref{hyp:regularity} is a regularity assumption at infinity which, in the language of operator theory and in a suitable functional framework, says that the multiplication operator by $|D^2 \phi(x)|$ is {\sl
infinitesimally bounded} by the multiplication operator by
$|\nabla\phi(x)|^2$ (see~\cite[Chapter~X]{RS75}). Here no growth assumption is made on
$|\nabla\phi|$. Assumption~\eqref{hyp:regularity} is satisfied for instance if \begin{equation*}
\sup_{x \in \R^d} \frac{ D^2
\phi(x)}{\sqrt{1+|\nabla\phi(x)|^2}} < \infty\quad
\textrm{or}\quad \lim_{|x| \rightarrow \infty} \frac{ D^2
\phi(x)}{1+|\nabla\phi(x)|^2}=0\,. \end{equation*} Assumption~\eqref{hyp:poincarebasique} can be interpreted as a {\sl measure concentration} property: it implies that \begin{equation}
\label{Concentration}
\int_{\R^d} |\nabla\phi(x)
|^2\,e^{-\phi(x)} \,\mathrm d
x<\infty\quad\mbox{and}\quad\forall\,k\in\N\,,\quad\int_{\R^d}
|x|^{2k}\,e^{-\phi(x)} \,\mathrm d x<\infty \end{equation} by~\eqref{hyp:regularity} for the first estimate and by an easy induction (see Remark~\ref{rem:concentration} in Appendix~\ref{Sec:Remarks}) for the second~one. Assumptions~\eqref{hyp:intnorm},~\eqref{hyp:regularity} and~\eqref{hyp:poincarebasique} are satisfied, for instance, if $\phi\in C^2(\R^d;\R)$ and either
$\phi(x)=\alpha\,|x|^\gamma+\beta$ with $\gamma \ge 1$ or
$\phi(x)=\alpha\,e^{|x|^2}+\beta$, for large values of $|x|$, where $\alpha>0$ and $\beta$ are two parameters. The three assumptions are also satisfied by the {\sl normalized Gaussian} defined~by \begin{equation}\label{Gaussian} \forall\,x \in \R^d,\quad
\phi(x)=\frac12\,|x|^2+\frac d2\,\ln(2\pi)\,. \end{equation} Associated with $\phi$ and thanks to~\eqref{hyp:regularity}, there exists two constants $C_\phi>8$ and $C_\phi'>C_\phi$ such that \begin{equation}
\label{eq:cphi}
\forall\,x \in \R^d,\quad 4\,\sqrt d\,|D^2 \phi(x)|
\leq |\nabla\phi(x)|^2+C_\phi -1
\quad \mbox{and}\quad
4\,|D^2 \phi(x)| \leq
C_\phi^{-1/2}\,\big(|\nabla\phi(x)|^2+C_\phi'\big)\,. \end{equation}
As in the case of a bounded domain, the set ${\mathcal R}$ of {\sl infinitesimal rotations} plays a key role in the study of Korn inequalities in the whole space. The symmetric differential applied to an infinitesimal rotation is zero. In our setting, the invariance under the action of a group of rotations $t \to e^{tA}$ for a given $A \in \mathfrak M^a$ means \begin{equation*}
\label{eq:defrotinv}
\forall\,t \in \R\,,\quad \forall\,x \in \R^d,\quad
\phi\(e^{tA}x\)=\phi(x)\,, \end{equation*} where, again, we implicitly use the assumption that the measure is centred. Differentiating the above identity with respect to $t$ yields that the set of {\sl infinitesimal rotations
preserving $\phi$} is \begin{equation*}
{\mathcal R}_\phi :=\big\{ R\in{\mathcal R}\,:\,\forall\,x\in\R^d,\;\nabla\phi(x)
\cdot R(x)=0 \big\}\,. \end{equation*} This set is a central geometric objet in our analysis. In the inequalities, the addition of a term involving $\nabla\phi\cdot R$ allows us to control the infinitesimal rotations for which $\phi$ is {\sl not} invariant, as we shall se later.
In this article, we adopt the following conventions. We denote by $|\cdot |$ the Euclidean norm in $\R$, $\R^d$ and~$\mathfrak M$, by $a\cdot b$ the scalar product of two vectors in $\R^d$ and by $A : B$ the scalar product of two matrices $A$ and~$B$ seen as vectors in $\R^{d^2}$. We denote by
$\|\cdot\|$ the $L^2$ norm corresponding to~$|\cdot |$ and weight $e^{-\phi} \,\mathrm d x$, and by $(\cdot,\cdot)$ the corresponding scalar product, that is, \begin{equation*}
(f,f)=\|f\|^2=\int_{\R^d}|f(x)|^2\,e^{-\phi} \,\mathrm d x \end{equation*} and we will refer to $L^2$ indifferently for functions with values in $\R$, $\R^d$ and~$\mathfrak M$. We shall use $\langle\cdot \rangle$ for the average (component by component) according to the measure $e^{-\phi} \,\mathrm d x$ of functions with values in $\R$, $\R^d$ and~$\mathfrak M$. We use the notation $\nabla$ for the gradient of scalar functions (with values in $\R^d$) and $D$ for the gradient of vector fields (with values in $\mathfrak M$). We denote by $H^1$ the space of functions $f$ or (when there is no ambiguity) vector fields $u$ such that respectively $f$ and $\nabla f$ or $u$ and $Du$ are in $L^2$. The space $H^{-1}$ is the dual of $H^1$ with respect to the $L^2$ scalar product. The weight function $\lfloor\nabla\phi\rceil$ is defined by $$
\lfloor\nabla\phi\rceil:=\sqrt{1+|\nabla\phi|^2}\,. $$
Let $\P$ be the orthogonal projection of vector-valued functions in $L^2$ onto ${\mathcal R}$, and $\P_\phi$ the orthogonal projection onto ${\mathcal R}_\phi$. We denote by ${\mathcal R}_\phi^\perp$ the orthogonal vector space to ${\mathcal R}_\phi$ in $L^2$ and ${\mathcal R}_\phi^c={\mathcal R} \cap {\mathcal R}_\phi^\bot$ the restriction of the orthogonal space to ${\mathcal R}_\phi$ in ${\mathcal R}$ or, in other words, ${\mathcal R}_\phi^c=\P\,{\mathcal R}_\phi^\bot$. For instance if $\phi$ has {\sl
no} invariance by any rotation $e^{tA}$ then ${\mathcal R}_\phi=\{ 0\}$, and if $\phi$ is radially symmetric then ${\mathcal R}_\phi={\mathcal R}$. Let $\mathfrak{P}$ be the orthogonal projection of matrix-valued functions in $L^2$ onto the set of constant antisymmetric matrices $\mathfrak M^a=D{\mathcal R}$. For all vector field $u \in H^1$, we have \begin{equation}
\label{ExplProj}
\mathfrak{P}(Du)=\seq{D^a u}. \end{equation} We also denote by $\mathfrak{P}_\phi$ the $L^2$ orthogonal projector onto $\mathfrak M_\phi:=D {\mathcal R}_\phi$ and by $\mathfrak M_\phi^c$ the orthogonal of $\mathfrak M_\phi$ in~$\mathfrak M^a$, {\sl i.e.}, $\mathfrak M_\phi^c=\mathfrak{P}\mathfrak M_\phi^\bot$. The projections are summarised in Figure~\ref{fig:diagram}. Note that $D {\mathcal R}_\phi^c$ and $\mathfrak M_\phi^c$ generically differ since the inner products underlying the two orthogonal decomposition are different.
\begin{figure}
\caption{Representation of the orthogonal
decompositions.}
\label{fig:diagram}
\end{figure}
One additional notation will be used throughout this paper: if $x$ and $y$ are two vectors in $\R^d$, we denote by $x \otimes y$ the matrix $(x_i\,y_j)_{1\le i,j\le d}$. Further details on $D {\mathcal R}_\phi^c$ and $\mathfrak M_\phi^c$ are collected in Appendix~\ref{Appendix:B2}.
\subsection{Main results}\label{Sec:Main}
All inequalities in this paper are quantitative with explicit estimates on the constants. The first result is the counterpart of~\eqref{eq:KornOmega2} in the whole Euclidean space, with some additional consequences based on Poincar\'e inequalities. The statement involves the whole set of {\sl infinitesimal rotations} ${\mathcal R}$.
\begin{theo}[Korn, Poincar\'e-Korn and strong Poincar\'e-Korn
inequalities]
\label{theo:KPK}
Suppose~\eqref{hyp:intnorm}--\eqref{hyp:regularity}--\eqref{hyp:poincarebasique}. Then
there are a Korn constant $C_{\mathrm{\scriptscriptstyle K}}$, a Poincar\'e-Korn constant
$C_{\mathrm{\scriptscriptstyle PK}} $ and a strong Poincar\'e-Korn constant $C_{\mathrm{\scriptscriptstyle SPK}}$ with
explicit bounds involving only $C_{\mathrm{\scriptscriptstyle P}}$, $C_\phi$ and $C_\phi'$ such
that, for all $u\in H^1$,
\begin{align}
\label{eq:WKfull}
& \inf_{R\in{\mathcal R}}\|D (u-R)\|^2=\|D u-\mathfrak{P} (Du)\|^2
\leq C_{\mathrm{\scriptscriptstyle K}}\,\|D^s u\|^2,\\
\label{eq:WPKfull}
& \inf_{R\in{\mathcal R}}\|u-\seq{u}-R\|^2=\|u-\seq{u}-\P (u)\|^2
\leq C_{\mathrm{\scriptscriptstyle PK}}\,\|D^s u\|^2,\\
\label{eq:WPKstrong}
& \big\|\lfloor\nabla\phi\rceil\,\big(u-\seq{u}-\P (u)\big)\big\|^2 \leq
C_{\mathrm{\scriptscriptstyle SPK}}\,\|D^s u\|^2.
\end{align}
Moreover in the Gaussian case~\eqref{Gaussian}, optimal
constants are $C_{\mathrm{\scriptscriptstyle K}}=4$, and $C_{\mathrm{\scriptscriptstyle PK}}=2$. \end{theo}
The terminology {\sl ``Korn constant''} refers to Korn's original results~\cite{Kor06,Kor08,Kor09} whereas {\sl
``Poincar\'e-Korn''} and {\sl ``strong Poincar\'e-Korn''} respectively refer to usual Poincar\'e inequalities and to strong Poincar\'e inequalities (see Proposition~\ref{prop:poincares}). For brevity, we shall speak generically of {\sl ``Korn-type inequalities''} or simply {\sl
``Korn inequalities''}. Explicit bounds for $C_{\mathrm{\scriptscriptstyle K}}$, $C_{\mathrm{\scriptscriptstyle PK}}$ and $C_{\mathrm{\scriptscriptstyle SPK}}$ will also be given later. The constant $C_{\mathrm{\scriptscriptstyle P}}$ is the optimal constant in~\eqref{hyp:poincarebasique} while $C_\phi$ and $C_\phi'$ refer to~\eqref{eq:cphi}. The minimum in the left-hand side of~\eqref{eq:WKfull} is explicit: according to~\eqref{ExplProj}, we have $$
\|D u-\mathfrak{P} (Du)\|^2=\|D u-\seq{D^a u}\|^2 \leq C_{\mathrm{\scriptscriptstyle K}}\,\|D^s u\|^2. $$ In~\eqref{eq:WPKfull} and~\eqref{eq:WPKstrong}, there is no simple expression for $\P(u)$ as for $\mathfrak{P}(Du)$. Note that the strong Poincar\'e inequality (see Proposition~\ref{prop:poincares} below) implies ${\mathcal R}\subset H^1$, so that the statement of Theorem~\ref{theo:KPK} makes sense.
The defaults of axisymmetry of the boundary of the domain were taken into account in~\cite{DV02}, in the bounded domain case {\sl without potential} (this case will be called the {\sl ``flat case''} from now on). Here the eventual non-axisymmetry arises from the potential. Measuring the default of axisymmetry motivates our introduction of the finite dimensional space ${\mathcal R}_\phi^c$ and of the {\sl rigidity of vector
fields} constant~$C_{\mathrm{\scriptscriptstyle RV}}$ defined by \begin{equation}
\label{eq:rigidityvect}
C_{\mathrm{\scriptscriptstyle RV}}^{-1}
:=\min_{ A\,x+b \: \in \: ({\mathcal R}_\phi^c \oplus \R^d)\setminus \set{0}}
\frac{\norm{\nabla\phi(x)\cdot (A\,x+b)}^2}{\norm{A\,x+b\,}^2} \end{equation} if ${\mathcal R}_\phi^c\neq\set{0}$, and of the {\sl rigidity of
differential} constant $C_{\mathrm{\scriptscriptstyle RD}}$ defined by \begin{equation}
\label{eq:rigiditydiff}
C_{\mathrm{\scriptscriptstyle RD}}^{-1} =\min_{ (A,b) \: \in \: (\mathfrak M_\phi^c\otimes\R^d)
\setminus
\set{(0,0)}}
\frac{\norm{\nabla\phi(x)\cdot (A\,x+b) }^2}{|A|^2+|b|^2} \end{equation} if $\mathfrak M_\phi^c\neq\set{0}$. We adopt the convention that $C_{\mathrm{\scriptscriptstyle RV}}=0$ (respectively $C_{\mathrm{\scriptscriptstyle RD}}=0$) if ${\mathcal R}_\phi^c=\set{0}$ (respectively $\mathfrak M_\phi^c=\set{0}$). Let us show that these constants are well-defined in $\R_+$. Since ${\mathcal R}$ and $\mathfrak M_\phi$ have a finite dimension the minima in~\eqref{eq:rigidityvect} and~\eqref{eq:rigiditydiff} exist. The first (respectively second) minimum is positive when ${\mathcal R}_\phi^c \neq\set{0}$ (respectively $\mathfrak M_\phi^c \neq\set{0}$). Indeed the linear maps ${\mathcal R}_\phi^c\oplus \R^d : A\,x+b \mapsto \nabla\phi(x)\cdot (A\,x+b) \in L^2$ and $\mathfrak M_\phi^c\oplus \R^d : (A,b) \mapsto \nabla\phi(x)\cdot (A\,x+b) \in L^2$ are injective: if $\nabla\phi(x)\cdot (A\,x+b)=0$ for all $x \in \R^d$, then by integration by parts \begin{equation*}
0=\int_{\R^d} \nabla\phi(x)\cdot (A\,x+b)\ b\cdot x\,
e^{-\phi(x)} \,\mathrm d x=|b|^2 \end{equation*} because $\seq{x}=0$, so that $b=0$, and as a consequence $A=0$ since ${\mathcal R}_\phi \cap {\mathcal R}_\phi^c=\set{0}$ and $\mathfrak M_\phi \cap \mathfrak M^c_\phi = \{0\}$. We can now state {\it
precised} Korn and Poincar\'e-Korn inequalities in which $u$ is also controlled on the space of infinitesimal rotation that do not leave $\phi$ invariant.
\begin{theo}[Precised Poincar\'e-Korn and Korn inequalities]\label{theo:PKPK} Suppose~\eqref{hyp:intnorm}--\eqref{hyp:regularity}--\eqref{hyp:poincarebasique}. Then there are a precised Korn constant $C_{\mathrm{\scriptscriptstyle K}}'$ and a precised Poincar\'e-Korn constant $C_{\mathrm{\scriptscriptstyle PK}}'$ such that, for all $u\in H^1$, \begin{align}
\label{eq:WPK} & \inf_{R\in{\mathcal R}_\phi}\|u-R\|^2=\|u-\P_\phi (u)\|^2 \leq C_{\mathrm{\scriptscriptstyle PK}}'\,\|D^s u\|^2+2\,C_{\mathrm{\scriptscriptstyle RV}}\,\|\nabla\phi\cdot u\|^2,\\
\label{eq:WK} & \inf_{R\in{\mathcal R}_\phi}\|D (u-R)\|^2=\|D u-\mathfrak{P}_\phi (Du)\|^2 \le C_{\mathrm{\scriptscriptstyle K}}'\,\|D^s u\|^2+2\,C_{\mathrm{\scriptscriptstyle RD}}\,\|\nabla\phi\cdot u\|^2. \end{align} Moreover the constants $C_{\mathrm{\scriptscriptstyle PK}}'$ and $C_{\mathrm{\scriptscriptstyle K}}'$ have explicit bounds depending only on the structural constants $C_{\mathrm{\scriptscriptstyle P}}$, $C_\phi$, $C_\phi'$, $C_{\mathrm{\scriptscriptstyle RD}}$ and $C_{\mathrm{\scriptscriptstyle RV}}$. \end{theo}
Explicit bounds for $C_{\mathrm{\scriptscriptstyle K}}'$ and $C_{\mathrm{\scriptscriptstyle PK}}'$ will be given in the proofs. For any $u\in H^1$, the strong Poincar\'e inequality implies $\nabla\phi\cdot u \in L^2$ thanks to~\eqref{Concentration}, so that the statement makes sense (see Proposition~\ref{prop:poincares} below and Remarks~\ref{rem:concentration} and~\ref{rem:integrability} in Appendix~\ref{Sec:Remarks}). The main difference with the flat case is the $\norm{\nabla\phi\cdot u}^2$ term in the right-hand side of the inequality whereas there is no additional term in~\eqref{eq:KornOmega2}. This cannot be avoided as shown by the following example. Let us consider an infinitesimal rotation $u=R$, with $R\neq 0$, in the case without invariance by any rotation $e^{tA}$, that is, ${\mathcal R}_\phi=\set{0}$. Then $u \in H^1$ and inequality~\eqref{eq:WK} reduces to $0 \neq \norm{R}^2 \leq 2\,C_{\mathrm{\scriptscriptstyle RV}}\,\norm{\nabla\phi\cdot R}^2$ since $D^s R=0$. This also shows that, compared to~\eqref{eq:WKfull}, an additional term is needed.
As often, the functional inequalities of Theorems~\ref{theo:KPK} and~\ref{theo:PKPK} are linked with spectral properties of nonnegative differential operators. By a simple integration by parts, the formal adjoint of $\nabla$ in $L^2$ equipped with the weight $e^{-\phi(x)} \,\mathrm d x$ is $\nabla^* u=-\,\nabla_{\!\phi}\cdot u$ for any smooth vector field $u$, where $\nabla_{\!\phi}\cdot u :=\nabla\cdot u- \nabla\phi\cdot u$. The first operator is the so-called {\sl Witten-Laplace operator on functions} $-\Delta_\phi$ (sometimes also called the {\sl Ornstein-Uhlenbeck operator}) which replaces the usual Laplacian in the flat case. It is associated with the quadratic form $f\mapsto\|\nabla f\|^2=\int_{\R^d} |\nabla f|^2\,e^{-\phi} \,\mathrm d x$ and defined by \begin{equation}\label{def:Deltaphi} -\Delta_\phi f :=-\,\nabla_{\!\phi}\cdot \nabla f=-\,\Delta f+\nabla\phi\cdot \nabla f\,. \end{equation} The operator $-\Delta_\phi$ is nonnegative and symmetric.
For convenience, we shall also denote by $-\Delta_\phi$ the operator acting coordinate by coordinate on vector fields, that is for any smooth vector field $u$, $(\Delta_\phi u)_i=\Delta_\phi u_i$, and similarly extend it to matrices. In the same spirit, we introduce various differential operators. The formal adjoint of $D^s$ is defined by $(D^s)^*\mathfrak F=-\,D^s_\phi\cdot\mathfrak F$ for any matrix-valued function $\mathfrak F$, so that $D^s_\phi:=D^s-D^s\phi$ acts on matrix-valued functions and takes value in a space of vector fields. Here $D^s\phi\cdot\mathfrak F:=\nabla\phi\cdot\mathfrak F^s$. Let us consider \begin{equation*}\label{eq:SWS}
-\Delta_S\,u:=-\,D^s_\phi\cdot D^su\quad\mbox{and}\quad
-\Delta_{S\phi}\,u:=-\,D^s_\phi\cdot D^s u+(\nabla\phi
\otimes \nabla\phi)\,u\,, \end{equation*} acting on a smooth vector field $u$. The differential operators
$-\Delta_S$ and $-\Delta_{S\phi}$ are associated respectively with $\|D^s u\|^2$ and $\|D^s u\|^2+\|\nabla\phi\cdot u\|^2$, which appear in the various Korn and Poincar\'e-Korn inequalities. Additional details have been collected in Appendix~\ref{Appendix:B4}.
\begin{theo}[Associated operators acting on vector fields]
\label{theo:ao}
Suppose~\eqref{hyp:intnorm}--\eqref{hyp:regularity}--\eqref{hyp:poincarebasique}. Then
the operators $-\Delta_\phi$, $-\Delta_S$, and
$-\Delta_{S\phi}$ are essentially self-adjoint on $L^2$. They
have a common domain ${\mathcal D}$, finite dimensional kernels
\begin{equation*}\label{kernel}
\ker(-\Delta_\phi)=\R^d,\quad
\ker(-\Delta_S)=\R^d\oplus{\mathcal R}\,,
\quad\ker(-\Delta_{S\phi})={\mathcal R}_\phi\,,
\end{equation*}
and positive spectral gaps. The spectral gap of $-\Delta_\phi$
is the Poincar\'e constant $C_{\mathrm{\scriptscriptstyle P}}$ while the spectral gaps of
$-\Delta_S$ and $-\Delta_{S\phi}$ are estimated respectively in
Theorems~\ref{theo:KPK}~and~\ref{theo:PKPK}. \end{theo}
A positive spectral gap means that the infimum of the restriction of the spectrum to $(0,+\infty)$ is positive. Our last main result is devoted to a Korn-type inequality valid for vector fields $u\in L^2$ while Theorems~\ref{theo:KPK} and~\ref{theo:PKPK} are limited to $u\in H^1$. We shall compose by inverse powers of the following positive operator \begin{equation}
\label{Lambda}
\Lambda:=-\,\Delta_\phi+\mathrm{Id} \end{equation} acting on functions, vector fields or matrices, {\sl coordinate
by coordinate}. By Theorem~\ref{theo:ao}, $\Lambda$ is essentially self-adjoint (we keep the same name for the unique self-adjoint extension), $\Lambda\ge\mathrm{Id}$, and $\Lambda^{-1/2}$ is one-to-one from $H^{-1}$ into~$L^2$ (see Propositions~\ref{prop:H1} and~\ref{prop:domain} for more details). In order to measure the possible non-axisymmetry of the potential $\phi$ in an $L^2$ setting, we introduce the {\sl
rigidity of vector fields} constant \begin{equation}\label{eq:rigidityzero} C_{\mathrm{\scriptscriptstyle RV0}}^{-1}:=\min_{ A\,x+b \: \in \: ({\mathcal R}_\phi^c \oplus
\R^d)\setminus \set{0}} \frac{\norm{\Lambda^{-1/2}\,
\nabla\phi(x)\cdot (A\,x+b)}^2}{\norm{A\,x+b}^2}\,. \end{equation} when ${\mathcal R}_\phi^c\neq\set{0}$ and, by convention, $C_{\mathrm{\scriptscriptstyle RV0}}:=0$ if ${\mathcal R}_\phi^c=\set{0}$. The proof that this constant $C_{\mathrm{\scriptscriptstyle RV0}}$ is well-defined in $\R_+$ is exactly similar to that for $C_{\mathrm{\scriptscriptstyle RV}}$.
\begin{theo}[Zeroth order Korn and Poincar\'e-Korn
inequalities]\label{theo:KPK0}
Suppose~\eqref{hyp:intnorm}--\eqref{hyp:regularity}--\eqref{hyp:poincarebasique}. Then
there are a zeroth order Korn constant $C_{\mathrm{\scriptscriptstyle K0}}$ and a zeroth
order Poincar\'e-Korn constant $C_{\mathrm{\scriptscriptstyle PK0}}$ with explicitly
computable bounds depending only on $\phi$ such that, for all
$u\in L^2$,
\begin{align}
\label{eq:WKZfull}
& \inf_{R\in{\mathcal R}}\|\Lambda^{-1/2}\,D (u-R)\,\|^2=
\big\|\Lambda^{-1/2}\,\big(D u-\mathfrak{P} (Du)\big)\big\|^2
\leq
C_{\mathrm{\scriptscriptstyle K0}}\,\|\Lambda^{-1/2}\,D^s u\|^2,\\
\label{eq:WPKZfull}
& \inf_{R\in{\mathcal R}}\|u-\seq{u}-R\,\|^2=
\norm{u-\seq{u}- \P(u)}^2 \le
C_{\mathrm{\scriptscriptstyle PK0}}\,\|\Lambda^{-1/2}\,D^s u\,\|^2.
\end{align}
As a consequence, there is a zeroth order precised
Poincar\'e-Korn constant $C_{\mathrm{\scriptscriptstyle PK0}}'$ such that, for all
$u \in L^2$,
\begin{align}
\label{eq:WPKzero}
& \inf_{R\in{\mathcal R}_\phi}\|u-R\,\|^2=\|u-\P_\phi (u)\|^2
\le C_{\mathrm{\scriptscriptstyle PK0}}'\,\|\,\Lambda^{-1/2}\,D^s u\,\|^2
+2\,C_{\mathrm{\scriptscriptstyle RV0}}\,\|\Lambda^{-1/2}\,\nabla\phi\cdot u\,\|^2.
\end{align} \end{theo}
Inequality~\eqref{eq:WPKzero} is a straightforward consequence of the Poincar\'e-Korn inequality~\eqref{eq:WPKZfull} and the existence of the rigidity constant $C_{\mathrm{\scriptscriptstyle RV0}}$.
\subsection{Main tools and considerations on the optimal cases
and optimal constants}
The paper relies on three main tools.
\noindent\circled1 {\sl Poincar\'e-Wirtinger and
Poincar\'e-Lions inequalities.} The proof of Theorems~\ref{theo:KPK} and~\ref{theo:PKPK} for vector fields relies on Poincar\'e-Wirtinger inequalities for {\sl scalar
functions}, which go as follows.
\begin{prop}
\label{prop:poincares}
Assume that~\eqref{hyp:intnorm},~\eqref{hyp:regularity}
and~\eqref{hyp:poincarebasique} hold, for some Poincar\'e
constant $C_{\mathrm{\scriptscriptstyle P}}$. Then there exists a {\sl strong Poincar\'e}
constant $C_{\mathrm{\scriptscriptstyle SP}}>0$ such that
\begin{equation}
\label{eq:strongpoincare}
\forall\,f\in H^1,\quad\|\lfloor\nabla\phi\rceil(f-\seq{f})\|^2 \le
C_{\mathrm{\scriptscriptstyle SP}}\,\|\nabla f\|^2
\end{equation}
with $C_{\mathrm{\scriptscriptstyle SP}} \leq C_\phi\,(1+C_{\mathrm{\scriptscriptstyle P}})$. With $\Lambda$ as
in~\eqref{Lambda}, there exists also a {\sl Poincar\'e-Lions}
constant $C_{\mathrm{\scriptscriptstyle PL}}>0$ such that
\begin{equation}
\label{eq:poincarelions}
\forall\,f\in L^2,\quad\norm{f-\seq{f}}^2 \leq C_{\mathrm{\scriptscriptstyle PL}}\,
\|\Lambda^{-1/2}\,\nabla f\|^2 \leq C_{\mathrm{\scriptscriptstyle PL}}\,\norm{f-\seq{f}}^2.
\end{equation} \end{prop}
Under the sole assumptions~\eqref{hyp:intnorm}--\eqref{hyp:regularity}--\eqref{hyp:poincarebasique}, inequalities~\eqref{eq:strongpoincare} and~\eqref{eq:poincarelions} are not completely standard. These inequalities are linked to the spectral properties of $-\Delta_\phi$, studied in Section~\ref{sec:wittenpoincare}, where elements of proofs of~\eqref{eq:strongpoincare} and~\eqref{eq:poincarelions} are also collected. An estimate of $C_{\mathrm{\scriptscriptstyle PL}}$ is given in~\eqref{CPL}.
\noindent\circled2 The {\sl Schwarz Theorem} allows us to write all components of the second-order differential of a vector field~$u$ thanks to its symmetric components using the identity \begin{equation}\label{Schwarz}
\forall\,i,j,k \in \set{1,\cdots, d}, \quad \partial_k
\left( D^a u \right)_{ij}=\partial_j \left( D^s u \right)_{ik}
-\partial_i \left( D^s u \right)_{jk}. \end{equation} This algebraic property is at the core of all Korn-type inequalities, it means that derivatives of $D^au$ are in the span of the derivatives of $D^s u$. Note that the Schwarz Theorem also implies $D^a\,\nabla=0$ which is central in the construction of the De Rham complex.
\noindent\circled3 The {\sl rigidity constants}, as defined in~\eqref{eq:rigidityvect},~\eqref{eq:rigiditydiff} and~\eqref{eq:rigidityzero}, measure the defects of axisymmetry of the potential $\phi$. See Appendix~\ref{subsec:rigidity} for a discussion.
Our method of proof can be summarised as follows: (i) we take care of the finite-dimensional parts ${\mathcal R}_\phi^c$ and $\mathfrak M_\phi^c$ thanks to the rigidity constants in \circled3, (ii) we apply twice the Poincar\'e inequality in \circled1, first in the form~\eqref{eq:strongpoincare} and second in the form~\eqref{eq:poincarelions}, so that we access second-order derivatives but remain at first order thanks to $\Lambda^{-1/2}$, (iii) we use the algebraic property in \circled2 to get rid of the derivatives of $D^a u$.
The infima in~\eqref{eq:WKfull},~\eqref{eq:WPKfull},~\eqref{eq:WPK},~\eqref{eq:WK},~\eqref{eq:WPKZfull} and~\eqref{eq:WPKzero} are achieved respectively at $\mathfrak{P}(Du)=\seq{D^a u}$ (see Section~\ref{Sec:Prf1}), $\seq u+\P(u)$, $\P_\phi(u)$, $\mathfrak{P}_\phi(Du)$, $\seq u+\P(u)$ and $\P_\phi(u)$ as a consequence of the definitions of the various orthogonal projections. In the Gaussian case~\eqref{Gaussian}, we have $D \P(u)=\mathfrak{P} (Du)$, but this relation is not true otherwise. The constants are estimated explicitely and a summary is provided in Appendix~\ref{Sec:Constants}.
\subsection{A brief review of the literature and a conjecture} \label{sec:review}
We refer to~\cite[Eq.~(13)]{MR0022750},~\cite[Chapter~3, Section~3.3]{MR0521262},~\cite[page~291]{MR936420}, and~\cite{MR1368384} for statements of the original Korn inequality which goes back to~\cite{Kor06,Kor08,Kor09} in a bounded domain with Dirichlet conditions, and to~\cite{MR3498171,MR3582590,MR3570353} for considerations on the best constant. There is a huge literature on applications to the Navier-Stokes equations and elasticity models, which is out of the scope of the present paper: see~\cite{MR3294348} for an introduction to Korn's inequality applied to these topics.
The case of Korn inequalities in bounded domains with {\sl
Neumann boundary conditions} was carried out in~\cite{DV02}, driven by applications in kinetic theory in~\cite{Desvillettes-Villani-2005}. The proof relates the Korn constant to the so-called {\sl Grad number}, which is further studied in~\cite{MR2542573} and related to other geometric bounds. The notion of Grad's number goes back to~\cite{Grad65} in a bounded domain and was used in~\cite{DV02}. We refer to Appendix~\ref{subsec:rigidity} for a more detailed discussion and how it relates to our rigidity constants.
In bounded domains, inequalities of type~\eqref{eq:WPKfull} are usually called {\sl Poincar\'e-Korn} estimates (see for instance~\cite[Section~1.3.1]{MR3294348}), and inequality~\eqref{eq:WK} is reminiscent of what is sometimes called the {\sl second Korn inequality:} see~\cite[Inequality~(7)]{MR631678} and~\cite[Theorem~2]{MR995908}.
To our knowledge, the only result in the whole space with a confinement potential is~\cite[Section~5]{Duan_2011} where the Korn inequality~\eqref{eq:WPKfull} is proved by compactness, under an additional growth condition on $\nabla\phi$. The original contributions of this paper are\\ \hspace*{12pt} (i) a proof of weighted Poincar\'e-Korn and Korn inequalities, under rather general conditions,\\ \hspace*{12pt} (ii) a constructive method which provides us with quantitative estimates on the constants,\\ \hspace*{12pt} (iii) some optimal constants in the Gaussian case.\\ Our method is likely adaptable to bounded domains and also to fractional inequalities in the spirit of~\cite{MR2746437}. Inspired by the properties of the Gaussian Poincar\'e inequality, {\sl e.g.}, in~\cite{Courtade_2020}, we finally make the following conjecture.
\begin{conj}[Optimal constants]
For a given $\phi$
satisfying~\eqref{hyp:intnorm}--\eqref{hyp:regularity}--\eqref{hyp:poincarebasique}
with $\int_{\R^d} x_i\,x_j\,e^{-\phi(x)} \,\mathrm d x=\delta_{ij}$ for
all $i,j \in \set{1,\cdots, d}$ and
$D^2 \phi \ge \text{{\normalfont Id}}$, one has $C_{\mathrm{\scriptscriptstyle PK}} \ge 2$ and
$C_{\mathrm{\scriptscriptstyle K}} \ge 4$, with equality in the normalized centred Gaussian
case~\eqref{Gaussian} and only in that case. \end{conj}
\subsection{Outline of the paper}
In Section~\ref{sec:Gaussian} we prove Theorem~\ref{theo:KPK} in the simple case of Gaussian potentials. This has a pedagogical interest but also an interest {\sl per se} as the method captures some (conjectured) optimal constants. Section~\ref{sec:wittenpoincare} is devoted to classical results on the Witten-Laplace operator on functions and a sketch of the proof of Poincar\'e inequalities under assumptions~\eqref{hyp:intnorm}--\eqref{hyp:regularity}--\eqref{hyp:poincarebasique} with some short quantitative proofs for which we lack of references. In Section~\ref{sec:general} we prove Theorems~\ref{theo:KPK} and~\ref{theo:PKPK} in the general case. Section~\ref{sec:ao} is devoted to the functional analysis of operators (Theorem~\ref{theo:ao}) associated with various quadratic forms under consideration. We prove Theorem~\ref{theo:KPK0} on zeroth order Korn inequalities in Section~\ref{sec:kornzero}. Appendix~\ref{Appendix:A} is devoted to generalizations, a discussion of the measure of the defects of axisymmetry by rigidity constants, and an elementary application of our main results to a simple kinetic equation with multiple conservations laws. For the convenience of the reader, some computational details are collected in Appendix~\ref{Appendix:B}.
\section{Proof of the Korn inequalities of Theorem~\ref{theo:KPK}
in the Gaussian case} \label{sec:Gaussian}
Inspired by the proof of~\eqref{eq:KornOmega}, we first prove inequalities~\eqref{eq:WKfull} and~\eqref{eq:WPKfull} of Theorem~\ref{theo:KPK} for the normalized Gaussian measure, and establish the optimality of the constants in that case. We begin with two useful identities valid for a general function $\phi\in\mathrm W^{2,\infty}_{\mathrm{loc}}(\R^d)$ and any $u \in C^1_c(\R^d;\R^d)$, \begin{eqnarray}
\label{eq:weakKorn}
&&\|D^a u\|^2+\|(\nabla-\nabla\phi)\cdot u\,\|^2
=\|D^s u\|^2+\int_{\R^d} D^2 \phi : u \otimes u\,
e^{-\phi} \,\mathrm d x\,,\\
\label{eq:weakKorn2}
&&\|D u\|^2 \le 2\,\|D^s u\|^2+\int_{\R^d}
D^2 \phi : u \otimes u\,e^{-\phi} \,\mathrm d x\,. \end{eqnarray} Identity~\eqref{eq:weakKorn} is obtained by a simple integration by parts, a commutation and the Schwarz Theorem (or~\eqref{eq:Pointwise1} integrated against $e^{-\phi}$), while~\eqref{eq:weakKorn2} follows from
$|D u|^2=|D^s u|^2+|D^a u|^2$.
In the remainder of this section, let us focus on the {\sl Gaussian case}~\eqref{Gaussian} such that \begin{equation*}
e^{-\phi(x)}=(2\pi)^{-d/2}\,e^{-\frac12|x|^2} \end{equation*} is the standard centred normalized Gaussian. This is the only Gaussian function satisfying hypotheses~\eqref{hyp:intnorm} with the additional normalization $\seq{D^2 \phi}=\mathrm{Id}$, and it satisfies~\eqref{hyp:regularity} with $\varepsilon=0$ and $C_\varepsilon=C_0=d$ and it satisfies~\eqref{hyp:poincarebasique} with $C_{\mathrm{\scriptscriptstyle P}}=1$. We first recall the following improved version of the Poincar\'e inequality.
\begin{lem}[Improved Poincar\'e inequality]
\label{lem:improvedpoincare}
Assume~\eqref{Gaussian}. Then for any $u\in H^1$ such that
$\langle u_i\,x_j \rangle=\langle u_i \rangle=0$ with
$i,j=1,\ldots,d$, there holds
\begin{equation}
\label{eq:improvedpoincare}
2\,\norm{u}^2 \le\norm{D u}^2.
\end{equation} \end{lem}
This result is standard: the operator $-\Delta_\phi$ reduces after conjugation by $e^{-\phi/2}$ to the harmonic oscillator
$P_\phi=-\,\Delta+|x|^2/4-d/2$ which has a discrete spectrum made of all nonnegative integers (see Section~\ref{sec:wittenpoincare} for the definition of the operator). The lowest eigenvalue is $0$ with multiplicity $1$ and the first positive eigenvalue is $1$ with multiplicity $d$ and eigenfunctions $x_j$, $j=1,2,\ldots,d$. Conditions on $u$ amount to the orthogonality condition to these two eigenspaces, so that $2$ corresponds to the next eigenvalue. The result follows from the spectral theorem (see for instance to~\cite[Lemma 2 of Chapter V and Chapter~8]{RS75} or~\cite{CFKS87}).
\begin{proof}[Proof of Theorem~\ref{theo:KPK} in the Gaussian
case]
Since
$D^s (\langle D^a u \rangle\,x)=D^s(\langle u
\rangle)=D(\langle u \rangle)=0$ and
$\P(\langle D^a u \rangle\,x)=\langle D^a u \rangle\,x$, it is
enough to prove the inequalities for a vector field $u \in H^1$
such that $\langle D^a u\rangle=0$ and
$\langle u \rangle=0$. We have to show that
\begin{equation}\label{GaussianRelations}
\norm{u}^2 \leq 2 \norm{D^s u}^2\quad \mbox{and}
\quad \norm{Du}^2 \leq 4 \norm{D^s u}^2.
\end{equation}
Let us define the {\sl corrected vector field} $v \in H^1$ by
\begin{equation*}
v(x) :=u(x)-B\,x\quad\mbox{where}\quad
B_{ij} :=\langle u_i\,x_j\,\rangle\,,
\end{equation*}
and note the elementary property (using that $e^{-\phi} \,\mathrm d x$
is Gaussian)
\begin{equation*}\label{eq:ippgaussian}
B_{ij}=\langle u_i\,x_j \rangle=
\int_{\R^d} u_i\,x_j\,e^{-\phi} \,\mathrm d x=
\int_{\R^d} \partial_j u_i\,e^{-\phi} \,\mathrm d x=\seq{D u}_{ij}.
\end{equation*}
This implies that the matrix $B$ is symmetric since
$\langle D^a u\rangle=0$ and that $v$ satisfies
$\langle v_i\,x_j \rangle=\langle v_i \rangle=0$ for all
$i,j=1, \ldots, d$. We can then apply the improved Poincar\'e
inequality~\eqref{eq:improvedpoincare} in
Lemma~\ref{lem:improvedpoincare} to $v$ and get
\begin{equation*}
2\left\|v \right\|^2 \le \left\|D v \right\|^2.
\end{equation*}
Using this together with~\eqref{eq:weakKorn2} and
$D^2 \phi=\text{Id}$, we obtain
$2\,\|v\|^2 \le\|D v\|^2 \le 2\,\|D^s
v\|^2+\|v\|^2$ which implies $\|v\|^2 \le 2\,\|D^s v\|^2$
and $\|D v\|^2 \le 4\,\|D^s v\|^2$,
{\em i.e.},~\eqref{GaussianRelations} written for $v$. Next we compute
\begin{align*}
&\|D u\|^2=\|D v+B\|^2=\|D^s v+B\|^2+\|D^a v\|^2 =
\|D v\|^2+|B|^2+2 \int_{\R^d} \left( D^s v : B \right)\,
e^{-\phi} \,\mathrm d x\,,\\
&\|D^s u\|^2=\|D^s v+B\|^2=\|D^s v\|^2+|B|^2 +
2 \int_{\R^d} \left( D^s v : B \right)\,e^{-\phi} \,\mathrm d x\,,\\
&\|u\|^2=\|v+B\,x\|^2=\|v\|^2+|B|^2 +
2 \int_{\R^d} \left( v\cdot B\,x \right)\,e^{-\phi} \,\mathrm d x=
\|v\|^2+|B|^2,
\end{align*}
where we used the fact that $(D^s u+B)$ and $D^a u$ are
orthogonal in $L^2$ and $\langle v_i\,x_j \rangle=0$. By an
integration by parts, we obtain
\begin{equation*}\label{eq:cross-nul}
\int_{\R^d} \left( D^s v : B \right)\,e^{-\phi} \,\mathrm d x=
\frac12 \sum_{i,j} B_{ij} \int_{\R^d} (\partial_i v_j+\partial_j
v_i)\,
e^{-\phi} \,\mathrm d x=
\frac12 \sum_{i,j} B_{ij} \int_{\R^d}
\left(x_i\,v_j+x_j\,v_i\right )\,e^{-\phi} \,\mathrm d x=0
\end{equation*}
using again $\langle v_i\,x_j \rangle=0$. Altogether, we deduce
that
\begin{align*}
\|D u\|^2=\|D v\|^2+|B|^2,\quad\|D^s u\|^2=\|D^s v\|^2+|B|^2,
\quad\|u\|^2=\|v\|^2+|B|^2.
\end{align*}
We deduce~\eqref{GaussianRelations} on $u$
from~\eqref{GaussianRelations} on $v$ and the last equations,
which proves~\eqref{eq:WKfull} with $C_{\mathrm{\scriptscriptstyle K}}\le4$
and~\eqref{eq:WPKfull} with $C_{\mathrm{\scriptscriptstyle PK}}\le2$. To
saturate~\eqref{GaussianRelations} it is enough to search for
$u=v$ with $B=0$. With
$u(x)=(1-x_2^2,x_1\,x_2,0,\ldots,0)^\perp$, an elementary
computation (see details in Appendix~\ref{Appendix:B5}) shows
that
$$
\seq u=0,\quad\|u\|^2=3\,,\quad\P(u)=0,\quad\mathfrak{P}
(Du)=0=\seq{D^au}\,,\quad\|D^su\|^2=\frac32\,,\quad\|D^au\|^2=\frac92\,,\quad\|Du\|^2=6\,.
$$
This completes the proof of~\eqref{eq:WKfull} with $C_{\mathrm{\scriptscriptstyle K}}=4$
and~\eqref{eq:WPKfull} with $C_{\mathrm{\scriptscriptstyle PK}}=2$.
It remains to establish~\eqref{eq:WPKstrong}. By expanding the
square $\int_{\R^d}|D(u\,e^{-\phi/2})|^2 \,\mathrm d x$ as
in~\cite[ineq.~(4)]{Dolbeault_2012}, we obtain after one
integration by parts that
$$
\int_{\R^d}|x|^2\,|u(x)|^2\,e^{-\phi(x)}\,\mathrm d x\le4\int_{\R^d}|D
u|^2\,e^{-\phi}\,\mathrm d x+2\,d\int_{\R^d}|u|^2\,e^{-\phi}\,\mathrm d x\,.
$$
Combined with~\eqref{eq:WKfull} and~\eqref{eq:WPKfull}, this
completes the proof with $C_{\mathrm{\scriptscriptstyle PK}}\leC_{\mathrm{\scriptscriptstyle SPK}}\le2\,(2\,d+9)$. \end{proof}
\section{The Witten-Laplace operator on scalar functions and Poincar\'e inequalities}\label{sec:wittenpoincare}
Here we consider the Poincar\'e inequalities of Proposition~\ref{prop:poincares} and some related properties of the Witten-Laplace operator $\Delta_\phi$, as defined in~\eqref{def:Deltaphi}, in the case of a general probability measure $e^{-\phi}\,\mathrm d x$ with a potential~$\phi$ such that assumptions~\eqref{hyp:intnorm}--\eqref{hyp:regularity}--\eqref{hyp:poincarebasique} are fulfilled. Some results of this section are classical and we claim no originality. For a general theory of self-adjoint operators, we refer for instance to~\cite{Sim78,RS75} and we refer to~\cite{Wit82,HS94,Sjo96,Joh00} or~\cite{HN05} for more details on Witten-Laplace operators. Proofs are given when we are not aware of any precise reference or when we look for explicit estimates.
\subsection{Two toolboxes and the proof of the strong Poincar\'e
inequality
(Proposition~\texorpdfstring{\ref{prop:poincares}}{Proposition5})}\label{Sec:Toolboxes}
For all functions $f$, $g\in {\mathcal C}_c^\infty(\R^d; \R)$, we have by integration by parts \begin{equation*}
\label{eq:ipp}
(-\Delta_\phi f, g)=-\,\sep{\nabla_{\!\phi}\cdot \nabla f, g}=
(\nabla f, \nabla g)\,, \end{equation*} so that $-\nabla_{\!\phi}=-\,\nabla+\nabla\phi$ is the formal adjoint of $\nabla$, $-\Delta_\phi$ is nonnegative and symmetric, and $\Lambda$, as defined by~\eqref{Lambda}, is symmetric. The Lax-Milgram theorem allows us to solve in $H^1$, equipped with the norm $f \mapsto (\norm{f}^2+\norm{\nabla f}^2)^{1/2}$, the problem $\Lambda f=\xi$ for any given $\xi \in H^{-1}$, and to build a self-adjoint extension of $\Lambda$ associated to the coercive bilinear form $(f,g) \mapsto (\nabla f, \nabla g)+(f,g)$. On the other hand, by the well-known change of function $f\mapsto e^{-\phi/2}f$, $\Lambda$ is conjugated~to \begin{equation*}
\label{eq:conjugaison}
P_\phi :=e^{-\phi/2}\,\Lambda\,e^{-\phi/2} =
-\,\Delta+\tfrac14\,|\nabla\phi|^2-\tfrac12\,\Delta \phi+1 \end{equation*} acting on the usual space $L^2(\,\mathrm d x)$. From~\eqref{hyp:regularity}, we get that
$|\nabla\phi|^2/4-\Delta \phi/2$ is bounded from below. From Kato's result~\cite{Kat72} (also see, {\em e.g.},~\cite[Theorem X-28]{RS75}), this implies that $\Lambda$ has a unique Friedrichs self-adjoint extension such that ${\mathcal C}_c^\infty(\R^d;\R)$ is dense in its domain w.r.t.~the graph norm, that is, $\Lambda$~is {\sl essentially self-adjoint}. For notational simplicity, we use the same name for the operator and for its extension. We denote by ${\mathcal D}(\Lambda)$ the domain of $\Lambda$.
Hence ${\mathcal C}_c^\infty(\R^d;\R)$ is a core for the self-adjoint operator $\Lambda \geq \mathrm{Id}$, which has a one-to-one operator extension from $H^1$ to $H^{-1}$. Tools of functional calculus and spectral analysis apply. This gives sense to $\Lambda^\sigma$
with domain ${\mathcal D}(\Lambda^\sigma)$ for all $\sigma\in \R$. For instance, ${\mathcal D}(\Lambda^{1/2})=H^1$ and $\Lambda^{1/2}$ has a bounded one-to-one operator extension from $L^2$ to $H^{-1}$ by duality. Recall that no specific growth, apart from the general condition~\eqref{hyp:regularity}, is assumed on $|\nabla\phi|$ at infinity in the computations of this section (see~\cite{HN04,HN05,MR913672} for other results without growth condition). Let us show that~\eqref{hyp:regularity} implies that $\lfloor\nabla\phi\rceil\,f$ is square integrable whenever $f\in H^1$, which allows to make sense of $\norm{\nabla\phi\cdot u}$ in inequalities~\eqref{eq:WPK} and~\eqref{eq:WK}.
\begin{prop}[$H^1$ toolbox]
\label{prop:H1}
Assume~\eqref{hyp:regularity}. Then the space $H^1$ is
\begin{equation*}
H^1 =
\set{f\in L^2 \ : \ \nabla f \in L^2\ \mbox{{\normalfont and}}
\ \lfloor\nabla\phi\rceil\,f \in L^2}
\end{equation*}
and for any $f \in L^2$, we have the inequalities
\begin{equation}\label{eq:toolboxH1}
\big\|\nabla\Lambda^{-1/2}\,f\big\|^2 \leq\|f\|^2,\quad
\big\|\lfloor\nabla\phi\rceil\,\Lambda^{-1/2}\,f\big\|^2 \leq C_\phi\,\|f\|^2,
\end{equation}
\begin{equation}\label{eq:toolboxH1adj}
\big\|\Lambda^{-1/2}\,\nabla f\big\|^2 \leq\|f\|^2,\quad
\big\|\Lambda^{-1/2}\,\lfloor\nabla\phi\rceil\,f\big\|^2 \leq C_\phi\,\|f\|^2.
\end{equation} \end{prop}
\begin{proof}
For all $f \in D(\Lambda)$ we have
$\norm{\nabla f}^2 \leq (\Lambda
f,f)=\|\Lambda^{1/2}\,f\|^2$. By density of $D(\Lambda)$ in
$H^1$ we get $\norm{\nabla f}^2 \leq\|\Lambda^{1/2}\,f\|^2$ for
all $f \in H^1$ and applying this inequality to
$\Lambda^{-1/2}\,f \in H^1$ proves the first inequality
in~\eqref{eq:toolboxH1}.
Let us note that
$0\leq|\nabla\phi|^2-4\,\sqrt
d\,|D^2\phi|+C_\phi-1\leq|\nabla\phi|^2-4\,\Delta\phi+C_\phi-1$
because $\Delta\phi\le\sqrt d\,|D^2\phi|$ and according
to~\eqref{eq:cphi}, so that
\begin{equation*}
\lfloor\nabla\phi\rceil^2 \leq
8\(\tfrac14\,|\nabla\phi|^2-\tfrac12\,\Delta \phi\)+C_\phi\,.
\end{equation*}
As a consequence, we get the operator inequality
$\lfloor\nabla\phi\rceil^2 \leq -\,8\,\Delta_\phi+C_\phi\,\mathrm{Id} \leq C_\phi\,\Lambda$
using the fact that the usual Laplacian $-\Delta$ is
nonnegative on $L^2(\,\mathrm d x)$ and $C_\phi\ge8$. This implies that,
for all $f \in D(\Lambda)$, we have that $\lfloor\nabla\phi\rceil\,f$ is in $L^2$
and
$\norm{\lfloor\nabla\phi\rceil\,f}^2 \leq C_\phi\,(\Lambda
f,f)=C_\phi\,\|\Lambda^{1/2}\,f\|^2$. By density of $D(\Lambda)$
in $H^1$, we~get
\begin{equation}\label{eq:phimajparnabla}
\forall\,f \in H^1,\quad\big\|\lfloor\nabla\phi\rceil\,f\big\|^2 \leq
C_\phi\,\big\|\Lambda^{1/2}\,f\big\|^2.
\end{equation}
For any $f\in L^2$, applying~\eqref{eq:phimajparnabla} to
$\Lambda^{-1/2}\,f \in H^1$ gives the second inequality
in~\eqref{eq:toolboxH1}. Inequalities
in~\eqref{eq:toolboxH1adj} are obtained
from~\eqref{eq:toolboxH1} by considering the adjoint operators. \end{proof}
\begin{proof}[Proof of the strong Poincar\'e
inequality~\eqref{eq:strongpoincare}] So far we did not
use~\eqref{hyp:poincarebasique} and its spectral
consequences. Using the density of ${\mathcal C}_c^\infty(\R^d,\R)$ in
$D(\Lambda)$ and~\eqref{hyp:poincarebasique}, we get that $0$
is an isolated eigenvalue of $-\Delta_\phi=\Lambda-1$ with
associated eigenspace
$\R$. Inequality~\eqref{eq:strongpoincare} follows
from~\eqref{eq:phimajparnabla} applied to $f-\seq{f}$
and~\eqref{hyp:poincarebasique}, with
$C_{\mathrm{\scriptscriptstyle SP}} \leq C_\phi\,(1+C_{\mathrm{\scriptscriptstyle P}})$. \end{proof}
The following toolbox is a key step in proof of the {\sl
Poincar\'e-Lions} inequality~\eqref{eq:poincarelions}.
\begin{prop}[$D(\Lambda)$-Toolbox]
\label{prop:domain}
Assume~\eqref{hyp:intnorm} and~\eqref{hyp:regularity}. Then
\begin{equation*}
{\mathcal D}(\Lambda)=\Big\{ f\in L^2\,:\,\big\|\lfloor\nabla\phi\rceil^2\,f\big\| +
\big\|\lfloor\nabla\phi\rceil\,\nabla f\big\|+\big\|D^2f\big\|<+\infty \Big\}\,
\end{equation*}
and there exists a positive constant $C_{\mathrm{\scriptscriptstyle B}}$ depending only on
$C_\phi$, $C_\phi'$ and $d$ such that, for any $f\in L^2$,
\begin{equation}\label{eq:toolbox}
\big\|D^2 \Lambda^{-1} f \big\|^2+
\big\|\lfloor\nabla\phi\rceil\,\nabla\Lambda^{-1} f \big\|^2+
\big\|\lfloor\nabla\phi\rceil^2\,\Lambda^{-1} f \big\|^2 \leq C_{\mathrm{\scriptscriptstyle B}}\,\|f\|^2,
\end{equation}
\begin{equation}\label{eq:toolboxadj}
\big\|\Lambda^{-1} D^2f \big\|^2+
\big\|\Lambda^{-1} \lfloor\nabla\phi\rceil\,\nabla f \big\|^2+
\big\|\Lambda^{-1} \lfloor\nabla\phi\rceil^2\,f \big\|^2 \leq C_{\mathrm{\scriptscriptstyle B}}\,\|f\|^2.
\end{equation} \end{prop}
\begin{proof} Inequality~\eqref{eq:toolboxadj} follows
from~\eqref{eq:toolbox} by duality
using~\eqref{hyp:regularity}.
Let us denote by $\mathcal S$ the subspace of $L^2$ such that
$D^2f$, $\lfloor\nabla\phi\rceil\,\nabla f$ and $\lfloor\nabla\phi\rceil^2\,f$ are square
integrable. It is elementary to check that
${\mathcal D}(\Lambda)\subset\mathcal S$. In order to prove that,
reciprocally, $\mathcal S\subset{\mathcal D}(\Lambda)$, let us argue by
density of ${\mathcal C}_c^\infty(\R^d;\R)$. For any
$f \in {\mathcal C}_c^\infty(\R^d;\R)$, let us prove that
$\xi=\Lambda f=-\,\Delta_\phi f+f$ is such that
\begin{equation}
\label{eq:Phi1estimate}
\big\|\lfloor\nabla\phi\rceil^2f\big\|^2+\big\|\lfloor\nabla\phi\rceil\,\nabla
f\big\|^2+\big\|D^2f\big\|^2\leq C_{\mathrm{\scriptscriptstyle B}}\,\|\xi\|^2
\end{equation}
for some explicit constant $C_{\mathrm{\scriptscriptstyle B}}$, so that
${\mathcal D}(\Lambda)=\mathcal S$ and~\eqref{eq:toolbox} directly
follow.
It follows from
$\norm{f}^2 \leq (\Lambda f,f) \leq \norm{\Lambda f}\,\norm{f}$
that $\|f\|\leq\|\xi\|$. Similarly, using~\eqref{eq:toolboxH1},
we have
\begin{equation}
\label{eq:Dphifxi}
\|\lfloor\nabla\phi\rceil\,f\|\leq
C_\phi^{1/2}\, \| \Lambda^{1/2} f\| = C_\phi^{1/2}\, (\Lambda
f,f)^{1/2} \le C_\phi^{1/2}\,\|\xi\|\,.
\end{equation}
Next, we estimate
$\big\|\lfloor\nabla\phi\rceil^2\,f\big\|$. Using~\eqref{eq:toolboxH1} and the
triangular inequality we have
\begin{equation*}
\|\lfloor\nabla\phi\rceil^2\,f\|=\|\lfloor\nabla\phi\rceil\,\lfloor\nabla\phi\rceil\,f\|\leq
C_\phi^{1/2}\,\|\Lambda^{1/2}\,(\lfloor\nabla\phi\rceil\,f)\|
\leq C_\phi^{1/2}\,\norm{\nabla(\lfloor\nabla\phi\rceil\,f)}
+C_\phi^{1/2}\,\norm{\lfloor\nabla\phi\rceil\,f}\,.
\end{equation*}
With $\nabla(\lfloor\nabla\phi\rceil\,f)=\lfloor\nabla\phi\rceil\,\nabla f+(\nabla\lfloor\nabla\phi\rceil)\,f$, we get
\begin{equation*}
\begin{split}
\|\lfloor\nabla\phi\rceil^2\,f\|
& \leq C_\phi^{1/2}\,\norm{\lfloor\nabla\phi\rceil\,\nabla f}+C_\phi^{1/2}\,
\norm{\lfloor\nabla\phi\rceil\,f}+C_\phi^{1/2}\,\norm{(\nabla\lfloor\nabla\phi\rceil)\,f}\\
& \leq C_\phi^{1/2}\,\norm{\lfloor\nabla\phi\rceil\,\nabla
f}+C_\phi\,\norm{\xi}+C_\phi^{1/2}\,\norm{(\nabla\lfloor\nabla\phi\rceil)\,f}
\end{split}
\end{equation*}
by~\eqref{eq:Dphifxi}. Estimate~\eqref{eq:cphi} yields
\begin{equation}
\label{Pointwise:nablaphiter}
\big|\nabla\lfloor\nabla\phi\rceil\big|\le\frac{|D^2\phi|\,|\nabla\phi|}{\lfloor\nabla\phi\rceil}\le
\tfrac14\,C_\phi^{-1/2}\,\lfloor\nabla\phi\rceil^2+\tfrac14\,C_\phi'\,C_\phi^{-1/2}
\end{equation}
and, as a consequence,
\begin{equation*}
\begin{split}
& \big\|\lfloor\nabla\phi\rceil^2\,f\big\|\leq C_\phi^{1/2}\,\norm{\lfloor\nabla\phi\rceil\,\nabla
f}+C_\phi\,\norm{\xi}+\tfrac14\,\norm{\lfloor\nabla\phi\rceil^2\,f}
+\tfrac14\,C_\phi' \norm{f}\,,
\end{split}
\end{equation*}
so that using $C_\phi' \geq C_\phi$ and
$\norm{f} \leq \norm{\xi}$, we get
\begin{equation}\label{DVDu}
\big\|\lfloor\nabla\phi\rceil^2\,f\big\|\leq \tfrac43\,C_\phi^{1/2}\,
\norm{\lfloor\nabla\phi\rceil\,\nabla f}+\tfrac53\,C_\phi' \norm{\xi}\,.
\end{equation}
Next we estimate $\big\|\lfloor\nabla\phi\rceil\,\nabla f\big\|^2$ by
\begin{equation*}
\label{Eqn:intermCite}
\begin{split}
\big\|\lfloor\nabla\phi\rceil\,\nabla f\big\|^2
&=( \lfloor\nabla\phi\rceil^2\,\nabla f, \nabla f)=
( \nabla (\lfloor\nabla\phi\rceil^2\,f), \nabla f)
- ( (\nabla\lfloor\nabla\phi\rceil^2)\,f, \nabla f)\\
&=( \lfloor\nabla\phi\rceil^2\,f, \xi-f)- ( (\nabla\lfloor\nabla\phi\rceil^2)\,f, \nabla f)\\
&\leq ( \lfloor\nabla\phi\rceil^2\,f, \xi)
- 2\,\big( (\nabla\lfloor\nabla\phi\rceil)\,f, \lfloor\nabla\phi\rceil\,\nabla f\big)\\
& \leq
\norm{\lfloor\nabla\phi\rceil^2\,f}\,\norm{\xi}
+2\,\norm{(\nabla\lfloor\nabla\phi\rceil)\,f}\,\norm{\lfloor\nabla\phi\rceil\,\nabla
f}\,.
\end{split}
\end{equation*}
Using~\eqref{Pointwise:nablaphiter} and
$\norm{f} \leq \norm{\xi}$, we get
\begin{equation}
\label{DVDubis}
\big\|\lfloor\nabla\phi\rceil\,\nabla f\big\|^2\leq
\norm{\lfloor\nabla\phi\rceil^2\,f}\,\norm{\xi}+\tfrac12\,C_\phi^{-1/2}\,
\norm{\lfloor\nabla\phi\rceil^2\,f} \,\norm{\lfloor\nabla\phi\rceil\,\nabla
f}+\tfrac12\,C_\phi'\,C_\phi^{-1/2}\,\norm{\xi}\,
\norm{\lfloor\nabla\phi\rceil\,\nabla
f}\,.
\end{equation}
With elementary estimates, we deduce
from~\eqref{DVDu}-\eqref{DVDubis} that
\begin{equation}
\label{Est:4}
\big\|\lfloor\nabla\phi\rceil\,\nabla f\big\|\le 9\,C_\phi'\,\|\xi\|\quad
\mbox{and}\quad \big\|\lfloor\nabla\phi\rceil^2f\big\|\leq 14\,C_\phi'\,\|\xi\|\,.
\end{equation}
Integrations by parts show that
\begin{align*}
\big\|D^2f\big\|^2
&={\textstyle \sum_{i,j}}(\partial_{ij}f, \partial_{ij}
f)={\textstyle \sum_{i,j}} \big(\partial_j f,
-\,\partial_{iij} f+\partial_{ij}f\,\partial_i \phi\big)\\
&={\textstyle \sum_{i,j}}\big(\partial_j f, \partial_j
(-\,\partial_{ii} f)\big)+\tfrac12\,{\textstyle
\sum_{i,j}}\(\partial_i( |\partial_j f|^2), \partial_i
\phi\) =\big(\nabla f,\nabla(-\Delta
f)\big)+\tfrac12\(|\nabla
f|^2,|\nabla\phi|^2-\Delta\phi\).
\end{align*}
Using the elementary estimates
\begin{align*}
&\big(\nabla f,\nabla(-\Delta f)\big)=(\Delta_\phi f,\Delta
f)=(f-\xi,\Delta f)\le(f,\Delta f)+\|\xi\|\,\|\Delta f\|\,,\\
&(f,\Delta f)=(f,\Delta_\phi f)
+(f,\nabla\phi\cdot\nabla f)=-\,\|\nabla f\|^2
+\tfrac12\, (\nabla f^2,\nabla\phi)
\le-\tfrac12\,(f^2,\Delta_\phi\phi)
=\tfrac12\(|f|^2,|\nabla\phi|^2-\Delta\phi\),
\end{align*}
and using~\eqref{eq:cphi} and $\lfloor\nabla\phi\rceil\ge1$ and the fact that
\begin{equation*}
|\nabla\phi|^2-\Delta\phi\le|\nabla\phi|^2+\sqrt
d\,|D^2\phi|\le|\nabla\phi|^2
+\tfrac14\(|\nabla\phi|^2+C_\phi-1\)
\le\tfrac14\(5\,\lfloor\nabla\phi\rceil^2+C_\phi-6\)\le\tfrac14\(C_\phi-1\)\lfloor\nabla\phi\rceil\,,
\end{equation*}
we obtain, using also~\eqref{Est:4}, the estimate
\begin{multline*}
\frac1d\,\|\Delta f\|^2\le\,\big\|D^2f\big\|^2\le\|\xi\|\,
\|\Delta f\|+\tfrac12\(|f|^2+|\nabla f|^2,|\nabla\phi|^2-
\Delta\phi\)\\
\le\|\xi\|\,\|\Delta
f\|+\tfrac18\(C_\phi-1\)\(\big\|\lfloor\nabla\phi\rceil^2f\big\|^2
+\big\|\lfloor\nabla\phi\rceil\,\nabla
f\big\|^2\)\le\|\xi\|\,\|\Delta f\|+C\,\|\xi\|^2
\end{multline*}
with $C=\tfrac18\,277\(C_\phi-1\){C_\phi'}^2$ because
$277=9^2+14^2$. As a straightforward consequence, we obtain
\begin{equation*}
\|\Delta
f\|\le\tfrac12\(d+\sqrt{d^2+4\,C}\)\|\xi\|
\quad\mbox{and}\quad\big\|D^2f\big\|^2\le\tfrac
Cd+\tfrac12\(d+\sqrt{d^2+4\,C}\)\|\xi\|^2.
\end{equation*}
With~\eqref{Est:4}, this completes the proof
of~\eqref{eq:Phi1estimate}. A detailed computation of $C_{\mathrm{\scriptscriptstyle B}}$ is
given in Appendix~\ref{Sec:Dlambda}. \end{proof}
\subsection{The Poincar\'e-Lions inequality (Proposition~\texorpdfstring{\ref{prop:poincares}}{Proposition5})}\label{Sec:PL}
We now focus on~\eqref{eq:poincarelions}. As a preliminary remark, note that this inequality is the counterpart in the whole space of the so-called {\sl Lions lemma} in the smooth bounded domain case $\Omega \subset \R^d$, which amounts to the existence of some $c_\Omega >0$ such that \begin{equation*}
\forall\,f\in L^2(\Omega)\,,\quad c_\Omega\,\|f - \left<f\right>\|^2_{L^2(\Omega)} \leq\|\nabla f\|^2_{H^{-1}(\Omega)} \le d\,\|f-\left<f\right>\|^2_{L^2(\Omega)} \end{equation*} (see for instance~\cite{MR0521262} and~\cite[Theorem 6.11.4]{Cia13}). This inequality belongs to the folklore in Hodge theory, see for instance~\cite{HS94} or~\cite{Joh00}, with variants involving the so-called {\sl Witten-Laplacian on one-forms}.
\begin{proof}[Proof of the Poincar\'e-Lions inequality~\eqref{eq:poincarelions}] First note that the right inequality directly follows from~\eqref{eq:toolboxH1adj} applied to $f-\seq{f}$. We focus on the left one. The spectral theorem implies for all $f \in D(\Lambda)$ with $\seq{f}=0$ \begin{equation}\label{eq:spectralthm}
(1+C_{\mathrm{\scriptscriptstyle P}})^{-1}\,\norm{f}^2 \leq \sep{(-\Delta_\phi)\,\Lambda^{-1} f, f}=\sep{\Lambda^{1/2}\,\nabla\Lambda^{-1} f, \Lambda^{-1/2}\,\nabla f}\le\|\Lambda^{1/2}\,\nabla\Lambda^{-1} f\|\,\|\Lambda^{-1/2}\,\nabla f\| \end{equation} because $1/(1+C_{\mathrm{\scriptscriptstyle P}})\le s/(s+1)$ for any $s\in[1/C_{\mathrm{\scriptscriptstyle P}},\infty)$. Let us prove that $\Lambda^{1/2}\,\nabla\Lambda^{-1}$ is a bounded operator. Using the commutator $[\Lambda,\nabla]=-\,D^2\phi\,\nabla$, we compute \begin{multline*} \Lambda^{1/2}\,\nabla\Lambda^{-1} =\Lambda^{-1/2}\,\Lambda \nabla\Lambda^{-1}= \Lambda^{-1/2}\,\nabla+\Lambda^{-1/2}\,[\Lambda, \nabla]\,\Lambda^{-1}\\ =\Lambda^{-1/2}\,\nabla-\Lambda^{-1/2}\,D^2 \phi\,\nabla\Lambda^{-1}=\Lambda^{-1/2}\,\nabla+\Lambda^{-1/2}\,\lfloor\nabla\phi\rceil\,\big(\lfloor\nabla\phi\rceil^{-1} D^2\phi\,\lfloor\nabla\phi\rceil^{-1}\big)\,\lfloor\nabla\phi\rceil\,\nabla\Lambda^{-1}\,. \end{multline*} {}From~\eqref{eq:toolboxH1adj}, we know that $\Lambda^{-1/2}\,\nabla$ and $\Lambda^{-1/2}\,\lfloor\nabla\phi\rceil$ are bounded respectively by $1$ and $\sqrt{C_\phi}$, from~\eqref{eq:toolbox} the operator $\lfloor\nabla\phi\rceil\,\nabla\Lambda^{-1}$ is bounded by $\sqrt{C_{\mathrm{\scriptscriptstyle B}}}$, and we have $$ \lfloor\nabla\phi\rceil^{-1} D^2\phi\,\lfloor\nabla\phi\rceil^{-1}\le\frac{C_\phi'}{4\,\sqrt{C_\phi}} $$ as a consequence of~\eqref{eq:cphi}. Altogether, $\Lambda^{1/2}\,\nabla\Lambda^{-1}$ is bounded and \begin{equation}\label{eq:estimatecpl1new}
\|\Lambda^{1/2}\,\nabla\Lambda^{-1}f\|\leq\(1+\tfrac14\,C_\phi'\,\sqrt{C_{\mathrm{\scriptscriptstyle B}}}\) \norm f\,, \end{equation} which completes the proof with \begin{equation}\label{CPL} C_{\mathrm{\scriptscriptstyle PL}}=(1+C_{\mathrm{\scriptscriptstyle P}})^2\,\big(1+\tfrac14\,C_\phi'\,\sqrt{C_{\mathrm{\scriptscriptstyle B}}}\big)^2. \end{equation} \end{proof}
\section{Proof of the Korn inequalities of
Theorems~\ref{theo:KPK} and~\ref{theo:PKPK} for general
potentials} \label{sec:general}
In this section, we assume that the potential satisfies~\eqref{hyp:intnorm},~\eqref{hyp:regularity} and~\eqref{hyp:poincarebasique}.
\subsection{Proof of Theorem~\ref{theo:KPK}} \label{Sec:Prf1}
As a preliminary remark, we recall that \begin{equation*}
\label{eq:pdudau}
\forall\,u \in H^1,\quad \mathfrak{P}(Du)=\seq{D^a u}. \end{equation*} Indeed $Du=(Du- \seq{D^a u})+\seq{D^a u}$ is an orthogonal decomposition because \begin{equation*}
(Du- \seq{D^a u}, \seq{D^a u})=
(D^a u- \seq{D^a u}, \seq{D^a u})
+(D^s u, \seq{D^a u})=
\seq{D^s u}:\seq{D^a u}=0 \end{equation*} and the uniqueness of this decomposition shows the result.
\noindent$\rhd$ {\sl Proof of~\eqref{eq:WKfull}}. Let us take $u\in H^1$ such that $\seq{u}=0$ and $\seq{D^a
u}=0$. Using the Poincar\'e-Lions inequality~\eqref{eq:poincarelions}, we have \begin{equation}
\label{eq:WKfull1}
\|Du\|^2=\|D^s u\|^2+\|D^a u\|^2 =
\|D^s u\|^2+\sum_{i,j=1}^d\|(D^a u)_{ij}\|^2
\leq\|D^s u\|^2+C_{\mathrm{\scriptscriptstyle PL}} \,
\|\Lambda^{-1/2}\, \nabla (D^a u)\|^2 \end{equation} with
$\|\Lambda^{-1/2}\,\nabla (D^a u)\|^2=\sum_{i,j=1}^d\|\Lambda^{-1/2}\,\nabla (D^a u)_{ij}\|^2$. The Schwarz Theorem as stated in~\eqref{Schwarz} gives \begin{equation}
\label{eq:WKfull2}
\|\Lambda^{-1/2}\,\nabla (D^a
u)\|^2\leq 2 \sum_{i,j,k=1}^d \sep{\|\Lambda^{-1/2}\,\partial_i
(D^s u)_{jk}\|^2+\|\Lambda^{-1/2}\,\partial_j (D^s
u)_{ik}\|^2} =4 \sum_{j,k=1}^d\|\Lambda^{-1/2}\,\nabla
(D^s u)_{jk}\|^2. \end{equation} The right-hand side of the Poincar\'e-Lions inequality~\eqref{eq:poincarelions} yields \begin{equation*}
\sum_{j,k=1}^d\|\Lambda^{-1/2}\,\nabla (D^s u)_{jk}\|^2
\leq \sum_{j,k=1}^d\|(D^s u)_{jk}\|^2=\|D^s u\|^2. \end{equation*}
Together with~\eqref{eq:WKfull1} and~\eqref{eq:WKfull2}, this gives $\|Du\|^2 \leq (1+4\,C_{\mathrm{\scriptscriptstyle PL}})\,\|D^s u\|^2$ so that we can take $C_{\mathrm{\scriptscriptstyle K}} \leq 1+4\,C_{\mathrm{\scriptscriptstyle PL}}$. This proves~\eqref{eq:WKfull} since $D^s {\mathcal R}=\set{0}$.
\noindent $\rhd$ {\sl Proof of~\eqref{eq:WPKfull}}. Let us take $u\in H^1$ such that $\seq{u}=0$ and $\P(u)=0$. By definition of $\P$, we have $ \norm{u}^2 \leq \norm{u- \mathfrak{P}(Du)\,x} $ since $x \mapsto \mathfrak{P}(Du)\,x$ is in ${\mathcal R}$. Applying~\eqref{hyp:poincarebasique} and~\eqref{eq:WKfull} gives \begin{equation*}
\norm{u}^2 \leq C_{\mathrm{\scriptscriptstyle P}} \norm{Du- \mathfrak{P}(Du)}^2
\leq C_{\mathrm{\scriptscriptstyle P}}\,C_{\mathrm{\scriptscriptstyle K}}\,\|D^s u\|^2 \end{equation*} This proves~\eqref{eq:WPKfull} with $C_{\mathrm{\scriptscriptstyle PK}} \leq C_{\mathrm{\scriptscriptstyle P}}\,C_{\mathrm{\scriptscriptstyle K}}$.
\noindent $\rhd$ {\sl Proof of~\eqref{eq:WPKstrong}}. Let us take $u\in H^1$ such that $\seq{u}=0$ and $\P(u)=0$. Applying the strong Poincar\'e inequality~\eqref{eq:strongpoincare} and the Korn inequality~\eqref{eq:WKfull} gives \begin{equation}
\label{eq:WPKstrong1}
\norm{\lfloor\nabla\phi\rceil\,u}^2 \leq C_{\mathrm{\scriptscriptstyle SP}} \norm{Du}^2=
C_{\mathrm{\scriptscriptstyle SP}} \sep{ \norm{Du-\mathfrak{P}(Du)}^2+\norm{\mathfrak{P}(Du)}^2}
\leq C_{\mathrm{\scriptscriptstyle SP}}\,C_{\mathrm{\scriptscriptstyle K}} \norm{D^s u}^2+C_{\mathrm{\scriptscriptstyle SP}} \norm{\mathfrak{P}(Du)}^2. \end{equation} An integration by parts, Jensen's inequality and the Cauchy-Schwarz inequality show that \begin{equation}\label{eq:fpbounded}
\norm{\mathfrak{P}(Du)}^2=|\seq{D^a u}|^2 =
\tfrac14\,\sum_{i,j=1}^d\abs{ \int_{\R^d} \big(\partial_j \phi\,u_i-
\partial_i \phi\,u_j\big)\,e^{-\phi} \,\mathrm d x }^2
\leq \norm{\nabla\phi}^2 \norm{u}^2. \end{equation}
An integration by parts, $\Delta\phi\le\sqrt d\,|D^2\phi|$ and~\eqref{eq:cphi} provide us with \begin{equation*}
\int_{\R^d} |\nabla\phi|^2\,e^{-\phi} \,\mathrm d x =
\int_{\R^d} \Delta \phi\,e^{-\phi} \,\mathrm d x \leq
\tfrac14\int_{\R^d} |\nabla\phi|^2\,e^{-\phi} \,\mathrm d x +
\tfrac14\(C_\phi-1\) \end{equation*} so that $\norm{\nabla\phi}^2 \leq 3\,C_\phi$ and we conclude that
$\norm{\mathfrak{P}(Du)}^2 \leq 3\,C_\phi\,\norm{u}^2 \leq 3\,C_\phi\,C_{\mathrm{\scriptscriptstyle PK}}\,\|D^s u\|^2$ by~\eqref{eq:WPKfull}. Inserting this estimate in~\eqref{eq:WPKstrong1} completes the proof of~\eqref{eq:WPKstrong} with $C_{\mathrm{\scriptscriptstyle SPK}} \leq C_{\mathrm{\scriptscriptstyle SP}}(C_{\mathrm{\scriptscriptstyle K}}+3\,C_\phi\,C_{\mathrm{\scriptscriptstyle PK}})$.\qed
\subsection{Proof of Theorem~\ref{theo:PKPK}} \label{Sec:Prf2}~
\noindent$\rhd$ {\sl Proof of~\eqref{eq:WPK}}. Since for any $R\in{\mathcal R}_\phi$, $D^sR=0$ and $\nabla\phi\cdot R=0$, we can consider $u\in H^1$ such that $\P_\phi(u)=0$ without loss of generality, so that $\P(u) \in {\mathcal R}_\phi^c$. According to~\eqref{eq:WPKfull} and by definition of the rigidity constant~$C_{\mathrm{\scriptscriptstyle RV}}$ in~\eqref{eq:rigidityvect}, we have \begin{multline*}
\norm{u}^2=\norm{u-\P(u)-\seq{u}}^2+\norm{\P(u)+\seq{u}}^2
\leq C_{\mathrm{\scriptscriptstyle PK}}\,\|D^s u\|^2 +
C_{\mathrm{\scriptscriptstyle RV}} \norm{\nabla\phi\cdot (\P(u)+\seq{u})}^2\\
\leq C_{\mathrm{\scriptscriptstyle PK}}\,\|D^s u\|^2+2\,C_{\mathrm{\scriptscriptstyle RV}} \norm{\nabla\phi\cdot
u}^2+2\,C_{\mathrm{\scriptscriptstyle RV}} \norm{\nabla\phi\cdot (u-\P(u)-\seq{u})}^2. \end{multline*} Applying then the strong Poincar\'e-Korn inequality~\eqref{eq:WPKstrong} gives \begin{equation*}
\norm{u}^2 \leq C_{\mathrm{\scriptscriptstyle PK}}\,\|D^s u\|^2+2\,C_{\mathrm{\scriptscriptstyle RV}}
\norm{\nabla\phi\cdot u }^2+2\,C_{\mathrm{\scriptscriptstyle RV}}\,C_{\mathrm{\scriptscriptstyle SPK}}\,\|D^s
u\|^2. \end{equation*} This completes the proof of~\eqref{eq:WPK} with $C_{\mathrm{\scriptscriptstyle PK}}' \leq C_{\mathrm{\scriptscriptstyle PK}}+2\,C_{\mathrm{\scriptscriptstyle RV}}\,C_{\mathrm{\scriptscriptstyle SPK}}$.
\noindent $\rhd$ {\sl Proof of~\eqref{eq:WK}}. Since for any $R\in{\mathcal R}_\phi$, $D^sR=0$ and $\nabla\phi\cdot R=0$, we can again consider $u\in H^1$ such that $\mathfrak{P}_\phi(Du)=0$ without loss of generality, so that $\mathfrak{P}(Du) \in \mathfrak M_\phi^c$. According to~\eqref{eq:WKfull} and by definition of the rigidity constant~$C_{\mathrm{\scriptscriptstyle RD}}$ in~\eqref{eq:rigiditydiff}, we have \begin{multline*}
\norm{Du}^2=\norm{Du-\mathfrak{P}(Du)}^2+\norm{\mathfrak{P}(Du)}^2 \leq
C_{\mathrm{\scriptscriptstyle K}}\,\|D^s u\|^2+C_{\mathrm{\scriptscriptstyle RD}} \norm{\nabla\phi\cdot
(\mathfrak{P}(Du)\,x+\seq{u})}^2 \\
\leq C_{\mathrm{\scriptscriptstyle K}}\,\|D^s u\|^2+2\,C_{\mathrm{\scriptscriptstyle RD}} \norm{\nabla\phi\cdot
u}^2+2\,C_{\mathrm{\scriptscriptstyle RD}} \norm{\nabla\phi\cdot (u-\mathfrak{P}(D
u)\,x-\seq{u})}^2. \end{multline*} Applying the strong Poincar\'e inequality~\eqref{eq:strongpoincare} gives \begin{equation*}
\norm{Du}^2 \leq C_{\mathrm{\scriptscriptstyle K}}\,\|D^s u\|^2+2\,C_{\mathrm{\scriptscriptstyle RD}}
\norm{\nabla\phi\cdot u }^2+2\,C_{\mathrm{\scriptscriptstyle RD}}\,C_{\mathrm{\scriptscriptstyle SP}}\|Du- \mathfrak{P}(Du)\|^2, \end{equation*} and by the Korn inequality~\eqref{eq:WKfull} again, \begin{equation*}
\norm{Du}^2 \leq C_{\mathrm{\scriptscriptstyle K}}\,\|D^s u\|^2+2\,C_{\mathrm{\scriptscriptstyle RD}}
\norm{\nabla\phi\cdot u }^2+2\,C_{\mathrm{\scriptscriptstyle RD}}\,C_{\mathrm{\scriptscriptstyle SP}}\,C_{\mathrm{\scriptscriptstyle K}}\,\|D^s
u\|^2, \end{equation*} This gives~\eqref{eq:WPK} with $C_{\mathrm{\scriptscriptstyle K}}' \leq C_{\mathrm{\scriptscriptstyle K}}( 1+2\,C_{\mathrm{\scriptscriptstyle RD}}\,C_{\mathrm{\scriptscriptstyle SP}})$, with $C_{\mathrm{\scriptscriptstyle SP}} \leq C_\phi\,(1+C_{\mathrm{\scriptscriptstyle P}})$ according to Proposition~\ref{prop:poincares}.\qed
\section{Operators on vector fields: proof of
Theorem~\texorpdfstring{\ref{theo:ao}}{Theorem3}} \label{sec:ao}
In this section we develop the functional analysis and the spectral theory of operators on vector fields, and prove Theorem~\ref{theo:ao}. All results on the tensorized operator $-\Delta_\phi$ on vector fields are direct consequences of the study of the corresponding scalar operator: from Section~\ref{sec:wittenpoincare}, we learn that $-\Delta_\phi$ is essentially self-adjoint and admits ${\mathcal C}_c^\infty(\R^d;\R^d)$ as a core, the domain of its unique self-adjoint extension is \begin{equation*}
D(-\Delta_\phi)=\Big\{u \in L^2\,:\,\forall\,j \in \set{
1,\cdots, d}, \;\|\lfloor\nabla\phi\rceil^2\,u_j\|^2+\big\|\lfloor\nabla\phi\rceil\,\nabla
u_j\big\|^2+\big\|D^2 u_j\big\|^2<\infty \Big\}\,, \end{equation*} and its kernel is $\ker(-\Delta_\phi)=\R^d$.
Let us deal with the other operators of Theorem~\ref{theo:ao}. Recall that the operator $-\Delta_S$ is defined on ${\mathcal C}_c^\infty(\R^d;\R^d)$ vector fields by $ -\Delta_S=-\,D^s_\phi\cdot D^s$. It is nonnegative and $\mathrm{Id} -\Delta_S$ has therefore a Friedrichs extension with domain included in $H^1_S$ defined as the completion of ${\mathcal C}_c^\infty(\R^d;\R^d)$ with respect to the norm given by
$u\mapsto\|u\|^2+\|D^s u\|^2$. On the other hand, a maximal self-adjoint extension of $\mathrm{Id} -\Delta_S$ can be built according to the Lax-Milgram Theorem and its domain is included in $H^1$. The Korn inequality~\eqref{eq:WKfull} implies that $H^1_S=H^1$ so that the two extensions coincide, since there is a unique extension for which the domain is contained in $H^1_S$ (\cite[Theorem~X.23]{RS75}), which is the case for the maximal one. We have proven that $-\Delta_S$ is essentially self-adjoint. From the Poincar\'e-Korn inequality~\eqref{eq:WPKfull}, we learn that $\ker (-\Delta_S)={\mathcal R}\oplus\R^d$ and that $\inf\big(\hbox{\rm Sp}(-\Delta_S) \cap(0,+\infty)\big) \geq C_{\mathrm{\scriptscriptstyle PK}}^{-1} >0$. This concludes the proof of Theorem~\ref{theo:ao} for $-\Delta_S$.
The same argument applies to $-\Delta_{S\phi}=-\Delta_{S}-\nabla\phi \otimes \nabla\phi $ using~\eqref{eq:WKfull}--\eqref{eq:WPKstrong} as we know from Proposition~\ref{prop:domain} that $ \nabla\phi \otimes \nabla\phi\,u \in L^2$ for all $u \in D$. This completes the proof of of Theorem~\ref{theo:ao}.
\begin{rem}
Note also that the alternative operator defined on smooth
vector fields by
$u\mapsto-\,D^s_\phi\cdot D^s u-\nabla(\nabla_{\!\phi}\cdot u)$ has
exactly the same properties as $-\Delta_S$ because
$\|D^s u\|^2+\|\nabla\phi\cdot u\|^2 \sim\|D^s
u\|^2+\|\nabla_{\!\phi}\cdot u\|^2$ where
$\nabla_{\!\phi} u :=\nabla\cdot u- \nabla\phi\cdot u$ and
$\nabla\cdot u=\mathrm{Tr}(D^su)$. \end{rem}
\section{Zeroth order Korn inequalities: proof of
Theorem~\ref{theo:KPK0}} \label{sec:kornzero}
In order to prove~\eqref{eq:WKZfull}, we use a new Poincar\'e-Lions-type inequality of order $-1$ and the Schwarz Lemma.
\subsection{A Poincar\'e-Lions inequality of order
\texorpdfstring{$-1$}{-1}} \label{Sec:6.1}
\begin{lem}
\label{LemPL-1}
There exists two positive constants $C_{\mathrm{\scriptscriptstyle LPL}}$ and $C_{\mathrm{\scriptscriptstyle RPL}}$ such
that, for all $f \in H^{-1}$, we have
\begin{equation}
\label{eq:poincarelions-1}
C_{\mathrm{\scriptscriptstyle LPL}}^{-1}\,\|\Lambda^{-1/2}(f-\seq{f})\|^2
\leq\|\Lambda^{-1}\,\nabla f\|^2 \leq
C_{\mathrm{\scriptscriptstyle RPL}}\,\|\Lambda^{-1/2}\,(f-\seq{f})\|^2.
\end{equation} \end{lem}
\begin{proof} We rely on the same strategy as for the proof of
the Poincar\'e-Lions inequality~\eqref{eq:poincarelions}. For
any $f \in H^{-1}$, the mean makes sense because
$\seq{f}=\Lambda^{-1/2}\,\seq{f}=\seq{\Lambda^{-1/2}\,f}$ as
$\Lambda=\mathrm{Id}$ when restricted on constants. We can therefore
take $\seq{f}=0$ w.l.o.g.~and apply the spectral theorem as
in~\eqref{eq:spectralthm}, for any $f \in D(\Lambda)$, to get
\begin{equation}
\label{eq:spectralthmbis}
(1+C_{\mathrm{\scriptscriptstyle P}})^{-1}\,\|\Lambda^{-1/2}\,f\|^2 \leq
\sep{(-\Delta_\phi)\,\Lambda^{-1}
\Lambda^{-1/2}\,f,\Lambda^{-1/2}\,f}=
\sep{\Lambda\nabla\Lambda^{-3/2}
(\Lambda^{-1/2}\,f), \Lambda^{-1}\nabla f},
\end{equation}
where we used that $-\Delta_\phi=-\,\nabla_{\!\phi}\cdot \nabla$ and
$\Lambda$ commute. In order to prove the left inequality
in~\eqref{eq:poincarelions-1}, it is sufficient to prove that
$\Lambda\nabla\Lambda^{-3/2}$ is a bounded operator. We work in
${\mathcal C}_c^\infty(\R^d;\R)$, which is a core for $\Lambda$, and
the conclusion follows by density in $L^2$. Let us write
\begin{equation*}
\Lambda\nabla\Lambda^{-3/2}=\nabla\Lambda^{-1/2}+[\Lambda,\nabla]
\, \Lambda^{-3/2}=\nabla\Lambda^{-1/2}-D^2\phi\,\nabla
\Lambda^{-3/2} = \nabla\Lambda^{-1/2}-D^2\phi\,\Lambda^{-1}\,
\big(\Lambda \nabla\Lambda^{-3/2}\big)\,.
\end{equation*}
By assumption~\eqref{hyp:regularity}, for all ${\varepsilon}>0$ and for
all $g \in {\mathcal C}^{\infty}(\R^d;\R)$, we know that
\begin{align*}
\|\Lambda \nabla\Lambda^{-3/2} g\|
& \leq\|\nabla\Lambda^{-1/2}\,g\|
+{\varepsilon}\,\|\lfloor\nabla\phi\rceil^2\,\nabla\Lambda^{-3/2}g\|
+C_{\varepsilon}\,\|\nabla\Lambda^{-3/2}g\|\\
& \leq\|\nabla\Lambda^{-1/2}\,g\|
+{\varepsilon}\,\big\|\lfloor\nabla\phi\rceil^2\,\Lambda^{-1}
\big(\Lambda \nabla\Lambda^{-3/2}g\big)\big\|
+C_{\varepsilon}\,\|\nabla\Lambda^{-1/2}\,(\Lambda^{-1}g)\|\,.
\end{align*}
The operators $\nabla\Lambda^{-1/2}$ and $\lfloor\nabla\phi\rceil^2\,\Lambda^{-1}$
are bounded respectively by $1$ and $\sqrt{C_{\mathrm{\scriptscriptstyle B}}}$ according
to~\eqref{eq:toolboxH1} and~\eqref{eq:toolbox}, and
$\Lambda^{-1}\le1$, so that
\begin{equation*}
\|\Lambda \nabla\Lambda^{-3/2} g\|
\leq (1+C_{\varepsilon})\,\|g\|+{\varepsilon}\, \sqrt{C_{\mathrm{\scriptscriptstyle B}}}\,
\|\Lambda \nabla\Lambda^{-3/2}g\|\,.
\end{equation*}
With the choice ${\varepsilon}=1/(2\,\sqrt{C_{\mathrm{\scriptscriptstyle B}}})$ and
$C_\phi'':=C_{\varepsilon}=C_{1/(2\,\sqrt{C_{\mathrm{\scriptscriptstyle B}}})}$
in~\eqref{hyp:regularity}, we obtain
\begin{equation*}
\|\Lambda \nabla\Lambda^{-3/2} g\|\leq 2\,(1+C_\phi'')\,\|g\|\,.
\end{equation*}
Coming back to~\eqref{eq:spectralthmbis} with
$g=\Lambda^{-1/2}\,f$, we obtain
\begin{equation*}
\|\Lambda^{-1/2}\,f\|\leq 2\,(1+C_{\mathrm{\scriptscriptstyle P}})\,(1+C_\phi'')\,
\|\Lambda^{-1}\,\nabla f\|\,,
\end{equation*}
so that $C_{\mathrm{\scriptscriptstyle LPL}} \leq 4\,(1+C_{\mathrm{\scriptscriptstyle P}})^2\,(1+C_\phi'')^2$ and the left
inequality is proven.
In order to prove the right inequality
in~\eqref{eq:poincarelions-1}, we notice that
$\Lambda^{-1}\,\nabla f
=\Lambda^{-1}\,\nabla\Lambda^{1/2}\,(\Lambda^{-1/2}\,f)$ and it
is therefore sufficient to prove that
$\Lambda^{-1}\nabla\Lambda^{1/2}$ is a bounded operator. This
is done as in~\eqref{eq:estimatecpl1new} by writing
\begin{equation*}
\Lambda^{-1}\nabla\Lambda^{1/2}=
\Lambda^{-1}\nabla\Lambda\,\Lambda^{-1/2}=
\nabla\Lambda^{-1/2}+\Lambda^{-1}
[\nabla,\Lambda]\,\Lambda^{-1/2}=
\nabla\Lambda^{-1/2}+\Lambda^{-1}\,D^2\phi\,
\big(\nabla\,\Lambda^{-1/2}\big)\,.
\end{equation*}
As in the proof of~\eqref{eq:estimatecpl1new}, we obtain for
any $f\in{\mathcal C}_c^\infty(\R^d;\R)$ and $g=\Lambda^{-1/2}\,f$ the
estimate
\begin{equation*}
\|\Lambda^{-1}\nabla f\|=\|\Lambda^{-1}\nabla\Lambda^{1/2}g\|
\leq\(1+\tfrac14\,C_\phi'\,{\textstyle\sqrt{C_{\mathrm{\scriptscriptstyle B}}/C_\phi}}\,\)
\|\nabla\,\Lambda^{-1/2}g\|\leq
\(1+\tfrac14\,C_\phi'\,{\textstyle\sqrt{C_{\mathrm{\scriptscriptstyle B}}/C_\phi}}\,\)
\|g\|\,,
\end{equation*}
using~\eqref{eq:cphi},~\eqref{eq:toolboxH1}
and~\eqref{eq:toolboxadj}. This concludes the proof with
$C_{\mathrm{\scriptscriptstyle RPL}}=\big(1+\tfrac14\,C_\phi'\,\sqrt{C_{\mathrm{\scriptscriptstyle B}}/C_\phi}\,\big)^2$. \end{proof}
\subsection{Proof of the Korn inequalities in Theorem~\ref{theo:KPK0}}
As a consequence of~\eqref{eq:fpbounded} the projection $u \mapsto \mathfrak{P}(Du)=\seq{D^a u}$ has a unique extension as a bounded operator on $L^2$ with norm bounded by $\|\nabla{\phi}\|$ since~$H^1$ is dense in $L^2$. We keep the same name for the extension and notice that \begin{equation*} \Lambda^{-1/2}\,\mathfrak{P}(Du)=\Lambda^{-1/2}\,\seq{D^a u}=\langle \Lambda^{-1/2}\,D^a u \rangle= \mathfrak{P}( \Lambda^{-1/2}\,Du)\,. \end{equation*}
\noindent$\rhd$ {\sl Proof of~\eqref{eq:WKZfull}}. Let us take $u\in L^2$ such that $\seq{u}=0$ and $\mathfrak{P}(Du)=\seq{D^a u}=0$. Using~\eqref{eq:poincarelions-1}, we have \begin{equation*}\label{eq:WKZfull1}
\|\Lambda^{-1/2}\,Du\|^2=\|\Lambda^{-1/2}\,D^s u\|^2+\|\Lambda^{-1/2}\,D^a u\|^2\leq\|\Lambda^{-1/2}\,D^s u\|^2+C_{\mathrm{\scriptscriptstyle LPL}}\,\|\Lambda^{-1}\,\nabla(D^a u)\|^2 \end{equation*}
where $\|\Lambda^{-1}\,\nabla(D^a u)\|^2=\sum_{i,j=1}^d\|\Lambda^{-1}\,\nabla (D^a u)_{ij}\|^2$. By the Schwarz Theorem~\eqref{Schwarz}, \begin{equation*}\label{eq:WKZfull3}
\|\Lambda^{-1}\,\nabla(D^a u)\|^2\leq 2 \sum_{i,j,k=1}^d\Big(\|\Lambda^{-1} \partial_i (D^s u)_{jk}\|^2+\|\Lambda^{-1} \partial_j (D^s u)_{ik}\|^2\Big)=4 \sum_{j,k=1}^d\|\Lambda^{-1}\,\nabla (D^s u)_{jk}\|^2, \end{equation*} and~\eqref{eq:poincarelions-1} yields \begin{equation*}
\sum_{i,j=1}^d\|\Lambda^{-1}\,\nabla (D^s u)_{ij}\|^2
\leq C_{\mathrm{\scriptscriptstyle RPL}}\sum_{j,k=1}^d\|\Lambda^{-1/2}\,(D^s
u)_{jk}\|^2 =C_{\mathrm{\scriptscriptstyle RPL}}\,\|\Lambda^{-1/2}\,D^s u\|^2. \end{equation*} Altogether, this proves \begin{equation*}
\label{eq:WKZfullbis}
\|\Lambda^{-1/2}\,\big(Du-\mathfrak{P}(Du)\big)\|^2 \leq (1+4\,C_{\mathrm{\scriptscriptstyle LPL}}\,
C_{\mathrm{\scriptscriptstyle RPL}} )\,\|\Lambda^{-1/2}\,D^s u\|^2 \end{equation*} and~\eqref{eq:WKZfull} follows with $C_{\mathrm{\scriptscriptstyle K0}}=1+4\,C_{\mathrm{\scriptscriptstyle LPL}}\,C_{\mathrm{\scriptscriptstyle RPL}}$.
\noindent$\rhd$ {\sl Proof of~\eqref{eq:WPKZfull}}. Let us take $u\in L^2$ such that $\seq{u}=0$ and $\P(u)=0$. By definition of $\P$,~\eqref{eq:poincarelions} and~\eqref{eq:WKZfull} we get \begin{equation*}
\norm{u}^2 \leq \norm{u- \mathfrak{P}(Du)\,x} \leq
C_{\mathrm{\scriptscriptstyle PL}}\,\norm{\Lambda^{-1/2}\,(Du- \mathfrak{P}(Du))}^2 \leq
C_{\mathrm{\scriptscriptstyle PL}}\,(1+4\,C_{\mathrm{\scriptscriptstyle LPL}}\,C_{\mathrm{\scriptscriptstyle RPL}})\,\|\Lambda^{-1/2}\,D^s u\|^2. \end{equation*} This proves~\eqref{eq:WPKZfull} with $C_{\mathrm{\scriptscriptstyle PK0}} \leq C_{\mathrm{\scriptscriptstyle PL}}\,(1+4\,C_{\mathrm{\scriptscriptstyle LPL}}\,C_{\mathrm{\scriptscriptstyle RPL}})$.
\noindent$\rhd$ {\sl Proof of~\eqref{eq:WPKzero}}. Let us consider $u\in L^2$ such that $\P_\phi(u)=0$ so that $\P(u) \in {\mathcal R}_\phi^c$. By~\eqref{eq:WPKZfull} and by definition~\eqref{eq:rigidityzero}, we have \begin{equation*} \begin{split}
\norm{u}^2
&=\norm{u-\P(u)-\seq{u}}^2+\norm{\P(u)+\seq{u}}^2\\
&\leq C_{\mathrm{\scriptscriptstyle PK0}}\,\|\Lambda^{-1/2}\,D^s
u\|^2+C_{\mathrm{\scriptscriptstyle RV0}}\,
\|\Lambda^{-1/2}\,\nabla\phi\cdot (\P(u)+\seq{u})\|^2\\
&\leq C_{\mathrm{\scriptscriptstyle PK0}}\,\|\Lambda^{-1/2}\,D^s
u\|^2+2\,C_{\mathrm{\scriptscriptstyle RV0}}\,\|\Lambda^{-1/2}\,(\nabla\phi\cdot
u)\|^2+2\,C_{\mathrm{\scriptscriptstyle RV0}}\,\|\Lambda^{-1/2}\,[\nabla\phi\cdot
(u-\P(u)-\seq{u})]\|^2. \end{split} \end{equation*} Inequalities~\eqref{eq:toolboxH1adj}, $\Lambda^{-1/2}\le1$ and~\eqref{eq:WPKZfull} yield \begin{equation*} \begin{split}
\norm{u}^2 & \leq C_{\mathrm{\scriptscriptstyle PK0}}\,\|\Lambda^{-1/2} D^s
u\|^2+2\,C_{\mathrm{\scriptscriptstyle RV0}}\,\|\Lambda^{-1/2} (\nabla\phi\cdot
u)\|^2+2\,C_{\mathrm{\scriptscriptstyle RV0}}\,C_\phi\,\|u-\P(u)-\seq{u}\|^2\\
& \leq C_{\mathrm{\scriptscriptstyle PK0}}\|\Lambda^{-1/2} D^s u\|^2+2\,C_{\mathrm{\scriptscriptstyle RV0}}
\norm{\Lambda^{-1/2} (\nabla\phi\cdot
u)}^2+2\,C_{\mathrm{\scriptscriptstyle RV0}}\,C_\phi\,C_{\mathrm{\scriptscriptstyle PK0}}\,\|\Lambda^{-1/2} D^s u\|^2. \end{split} \end{equation*} This proves~\eqref{eq:WPKzero} with $C_{\mathrm{\scriptscriptstyle PK0}}' \leq C_{\mathrm{\scriptscriptstyle PK0}}(1+2\,C_{\mathrm{\scriptscriptstyle RV0}}\,C_\phi)$, and also completes the proof of Theorem~\ref{theo:KPK0}.\qed
\appendix\section{Extensions, geometric observations and
motivation from kinetic theory} \label{Appendix:A}
\subsection{On the assumptions and some generalizations} \label{Sec:Remarks}
\begin{rem}
\label{rem:concentration}
The fact that the Poincar\'e
inequality~\eqref{hyp:poincarebasique} implies that
$e^{-\phi} \,\mathrm d x$ has an average $\seq{x}$ and a variance
$\langle |x|^2 \rangle$ is classical (see, {\em e.g.},~\cite[Corollary
3.2]{Led01}). Indeed~\eqref{hyp:poincarebasique} yields a
concentration property of $e^{-\phi(x)} \,\mathrm d x$ via a {\sl
concentration function}. A direct application of Fatou's
Lemma allows to extend~\eqref{hyp:poincarebasique} to the set
$W^{1,\infty}$ of uniformly Lipschitz functions, which includes
$x\mapsto x_j$ for all $j\in \set{ 1,\cdots , d}$. This
directly gives
$\int_{\R^d} |x|^2\,e^{-\phi(x)} \,\mathrm d x \leq d \,C_{\mathrm{\scriptscriptstyle P}}$ and the
integrability of $x$ w.r.t.~$e^{-\phi(x)} \,\mathrm d x$ follows by the
Cauchy-Schwarz inequality. By induction, we get
under~\eqref{hyp:poincarebasique} alone that the set of all
polynomial functions $\R[x]$ is included in $L^2$ and even
$H^1$. \end{rem}
\begin{rem}
\label{rem:integrability}
As a consequence of Remark~\ref{rem:concentration} and of the
strong Poincar\'e inequality~\eqref{eq:strongpoincare}, we
directly get that for all $i,j \in \set{1,\cdots d}$,
$\int_{\R^d} |x_i\,\partial_j \phi(x)|\,e^{-\phi(x)} \,\mathrm d x
<\infty$. It is indeed sufficient to apply the strong
Poincar\'e inequality to $x \mapsto x_j$, which is in
$H^1$. Note that this gives sense to all quantities of
Theorem~\ref{theo:KPK}, {\em e.g.}, $\norm{\nabla\phi\cdot R}$ for any
infinitesimal rotation $R$. \end{rem}
\begin{rem}
\label{rem:poincare}
There are many sufficient conditions for the Poincar\'e
inequality of
Assumption~\eqref{hyp:poincarebasique}. When~$\phi$ is
uniformly convex, it is shown in~\cite{Bakry-Emery} that
$C_{\mathrm P}$ is greater or equal than the convexity constant,
hence leading to fully explicit estimates in the two main
theorems. If we only assume that
$\lim_{|x| \mapsto\infty } |\nabla\phi(x)|=+\infty$,
then~$\Lambda$ is in fact an operator with compact resolvent,
hence with discrete spectrum, and~\eqref{hyp:poincarebasique}
follows. Another, less stringent, sufficient condition on
$\phi$ is
$\liminf_{|x| \to \infty} \big(\frac12
|\nabla\phi(x)|^2-\Delta\phi(x)\big) >c$ for some $c>0$ (it
implies the Poincar\'e inequality from the Persson-Agmon formula
of~\cite{Per60} or~\cite[Theorem 3.2]{Agm82}). We note that this
last assumption is satisfied by any regular function which
coincides with $x \mapsto \alpha\,|x|+\beta$ outside of a large
centred ball, where $\alpha$ and $\beta$ are normalization
constants. \end{rem}
\begin{rem}\label{rem:boundedcase}
Assumptions~\eqref{hyp:intnorm}--\eqref{hyp:regularity}--\eqref{hyp:poincarebasique}
may be satisfied in other geometries than the one of the whole
Euclidean space $\R^d$. In particular, given an open, smooth,
bounded and connected subset $\Omega$ of $\R^d$, we observe
that these hypotheses are satisfied by the potential
$\phi(x)=\exp\big(1/d^2(x,\partial\Omega)\big)$, where
$d(x,\partial\Omega)$ denotes the usual Euclidean distance from
$x$ to $\partial\Omega$. Here $\R^d$ is replaced by $\Omega$
equipped with the measure $e^{-\phi(x)} \,\mathrm d x$. It is an open
question to understand how our results could be extended to
usual boundary problems with potentials mimicking walls at the
boundary of $\Omega$. \end{rem}
\subsection{Rigidity constants and defects of axisymmetry} \label{subsec:rigidity}
In the proofs of Theorems~\ref{theo:KPK} and~\ref{theo:KPK0} (see also Appendix~\ref{Sec:Constants}), we used the two rigidity constants $C_{\mathrm{\scriptscriptstyle RV}}$ and $C_{\mathrm{\scriptscriptstyle RV0}}$ defined by~\eqref{eq:rigidityvect} and~\eqref{eq:rigidityzero}, depending on the level of regularity in each case, to measure the defects of axisymmetry (note that $C_{\mathrm{\scriptscriptstyle RV}} \leq C_{\mathrm{\scriptscriptstyle RV0}} $ since $\Lambda \geq \mathrm{Id}$). We used also $C_{\mathrm{\scriptscriptstyle RD}}$ defined in~\eqref{eq:rigiditydiff} but remark that $C_{\mathrm{\scriptscriptstyle RV}}$ and $C_{\mathrm{\scriptscriptstyle RD}}$ are not directly comparable either, because $\mathfrak M_\phi^c \neq D {\mathcal R}_\phi^c$. Other ways of measuring the default of axisymmetry of the potential $\phi$ can be considered.
\noindent \circled1 One can consider, again, a {\sl rigidity of vector
fields} constant, but this time defined alternatively by \begin{equation*}
C_{\mathrm{\scriptscriptstyle RVL}}^{-1}=\min_{ A\,x \in {\mathcal R}_\phi^c \setminus \set{0}}
\frac{\norm{\nabla\phi\cdot A\,x}^2}{\norm{A\,x}^2}\quad
\mbox{when}\quad {\mathcal R}_\phi^c \neq \set{0}\quad\mbox{and}
\quadC_{\mathrm{\scriptscriptstyle RVL}}=0\quad\mbox{otherwise}\,. \end{equation*} This leads to the {\sl modified Poincar\'e-Korn inequality} \begin{equation}
\label{eq:WPKL}
\inf_{A\,x\in{\mathcal R}_\phi}\|u- \seq{u}-A\,x\|^2=\|u- \seq{u}-
\P_\phi (u)\|^2 \leq C_{\mathrm{\scriptscriptstyle PKL}}'\,\|D^s u\|^2
+2\,C_{\mathrm{\scriptscriptstyle RVL}}\,\|\nabla \phi\cdot (u-\seq{u})\|^2 \end{equation} with an explicit bound for the constant $C_{\mathrm{\scriptscriptstyle PKL}}'$ using~\eqref{eq:WPKfull} and the method of proof of~\eqref{eq:WPK}. Once more, the existence of $C_{\mathrm{\scriptscriptstyle RVL}}$ follows from the injectivity of $A\,x \mapsto \nabla\phi\cdot A\,x$ on ${\mathcal R}_\phi^c$ and the fact that ${\mathcal R}_\phi^c$ is of finite dimension. The main advantage of this approach is to preserve a continuity property with respect to axisymmetry, which can be stated as follows: a small perturbation of a radial potential $\phi$ gives rise to a small constant $C_{\mathrm{\scriptscriptstyle RVL}}$, the limiting case being ${\mathcal R}_\phi^c=\set{0}$ and $C_{\mathrm{\scriptscriptstyle RVL}}=0$. The main drawback is that the symmetric operator associated to~\eqref{eq:WPKL} is neither local nor differential because of the term $\seq{u}$ which appears in the right-hand side of~\eqref{eq:WPKL}.
\noindent \circled2 In a bounded domain $\Omega\subset\R^d$, with flat metric ({\sl i.e.}, for a constant potential $\phi$) considered in~\cite{DV02}, the authors use {\sl Grad's number}. Let us explain how to adapt this method in our context under, {\em e.g.}, the additional condition \begin{equation*}
\lim_{|x| \rightarrow
\infty}\frac{D^2\phi(x)}{\lfloor\nabla\phi(x)\rceil^2}=0\,. \end{equation*} This property implies that the multiplication operator by $D^2 \phi$ is relatively compact with respect to $-\Delta_\phi$ acting on vector fields, with essential spectrum in $[C_{\mathrm{\scriptscriptstyle P}}, \infty)$. The spectrum in $[0,C_{\mathrm{\scriptscriptstyle P}})$ is then a pure point spectrum and the kernel is finite dimensional. For any antisymmetric matrix $A$, there exists an affine space $\mathfrak{V}_\antiSigma$ of functions $v\in H^1$ solving {\sl the Witten-Hodge
problem} \begin{equation*}\label{eq:uA}
\Divphi v=0\,,\quad D^a v=A\,. \end{equation*} The {\sl Witten-Hodge inequality} asserts that \begin{equation}\label{eq:IntroWHineq}
\inf_{v \in \mathfrak{V}_\antiSigma}\|D^s v\|^2 \le c_{\mathrm H}\,|A|^2 \end{equation} for some constant $c_{\mathrm H}\in (0,\infty)$. The reverse inequality amounts to the existence of {\sl Grad's number} such that \begin{equation*}\label{gradnumber}
C_{\mathrm G}^{-1} :=\inf_{A\in \mathfrak M_\phi^c,\,|A|=1,\,v \in \mathfrak{V}_\antiSigma}\;\|D^s v\|^2. \end{equation*} The existence of $C_{\mathrm G}$ as well as a quantitative positive lower bound could be establish using mass transport theory exactly as in~\cite{DV02}. Of course, $C_{\mathrm G}$ is well defined only when ${\mathcal R}_\phi^c\neq\{0\}$, {\em i.e.}, under the condition that $\phi$ is not radially symmetric. Inequality~\eqref{eq:IntroWHineq} is natural in differential geometry and more specifically in De Rham cohomology theory: we refer to~\cite{AS00,HN05} for further developments on this topic. In bounded domains, how to measure the symmetry defect by Grad's number in view of Korn type inequalities is at the core of~\cite{DV02} but has also been studied in~\cite{MR2542573}. This approach differs from ours. Using $C_{\mathrm G}$ and~\eqref{eq:WKfull}, the inequality~\eqref{eq:WPK} can be proved along a similar strategy as in~\cite{DV02}, although with different constants.
\subsection{An elementary application in kinetic theory} \label{subsec:toymodel}
The main motivation for this paper comes from {\sl kinetic
equations involving a confining potential} studied in~\cite{CDHMMShypo}. Also see~\cite[Section~2]{DV02} and~\cite{MR3815207,Duan_2011} for applications of Korn inequalities to kinetic equations. As an example, let us consider the linear relaxation model of {\sl BGK}-type \begin{equation}\label{eq:toykinintro}
\partial_tf+v\cdot\nabla_x f -\nabla_x \phi\cdot \nabla_v f
=\mathcal Lf :=G_f-f, \end{equation} where $f(t,x,v)$ is an unknown distribution function for a system of particles depending on time $t \ge 0$, position $x\in\R^d$ and velocity $v\in\R^d$, and where $G_f$ is defined by \begin{equation*}
G_f:=(\rho+{\bf u}\cdot v)\,\mu\quad\mbox{where}\quad \rho(t,x)
:=\int_{\R^d} f(t,x,v)\,\,\mathrm d v\,,\quad {\bf u}(t,x)
:=\int_{\R^d}v\,f(t,x,v)\,\,\mathrm d v\,. \end{equation*}
Here $\mu(v):=(2\,\pi)^{-d/2}\,e^{-|v|^2/2}$ while $\rho $ and ${\bf u}$ are respectively the macroscopic density and the average velocity associated with $f$. The collision kernel admits $d+1$ conserved moments, in the sense that
$\int_{\R^d}\mathcal Lf(t,x,v)\,\,\mathrm d v=0=\int_{\R^d}v_i\,\mathcal Lf(t,x,v)\,\,\mathrm d v$ for any $i=1,\ldots,d$ and $f \in L^1((1+|v|){\rm d} v)$.
A natural question is to look for equilibria of~\eqref{eq:toykinintro}. A quick glance at the equation shows that ${\mathcal M}(x,v) :=e^{-\phi(x)}\,\mu(v)$ is one of them. Korn inequalities provide us with a complete answer.
\begin{proposition}
Under
Assumptions~\eqref{hyp:intnorm}--\eqref{hyp:regularity}--\eqref{hyp:poincarebasique},
all equilibria of~\eqref{eq:toykinintro} in
$L^2(\mathcal M^{-1}\,\mathrm d x\,\mathrm d v)$ take the form
$f(x,v)=\big(( R(x)\cdot v)+c\big)\,{\mathcal M}$ for some
$R \in {\mathcal R}_\phi$ and $c \in \R$. \end{proposition}
\begin{proof}
Write $f=h\,\mathcal M$ with $h \in L^2(\mathcal M \,\mathrm d x\,\mathrm d v)$,
$\rho=r\,e^{-\phi}$, ${\bf u}=u\,e^{-\phi}$, so that
equation~\eqref{eq:toykinintro} reads
\begin{equation}
\label{eq:toykinintrobis}
\partial_th+v\cdot\nabla_x h -\nabla_x \phi\cdot \nabla_v h =
L(h) :=h-r-u\cdot v.
\end{equation}
The restriction of $L$ to $L^2(\mu \,\mathrm d v)$ is $L=-\,\Pi^\bot$
where $\Pi$ is the orthogonal projection onto
$\mathrm{Span} \{1, v_1, \ldots, v_d\}$. We compute
\begin{equation*}
\frac d{dt}\iint_{\R^d\times\R^d}|h|^2\,{\mathcal M} \,\mathrm d x\,\mathrm d v =
2\iint_{\R^d\times\R^d}(L h)\,h\,{\mathcal M} \,\mathrm d x\,\mathrm d v=
-\,2 \iint_{\R^d \times \R^d} \left| \Pi^\bot h \right|^2\,
{\mathcal M} \,\mathrm d x \,\mathrm d v
\end{equation*}
and deduce that any stationary solution
of~\eqref{eq:toykinintrobis} takes the form
$h(x,v)=r(x)+u(x)\cdot v $. Equation~\eqref{eq:toykinintrobis}
then reads
\begin{equation*}
v\cdot\nabla_x(r+u\cdot v)=\nabla_x\phi\cdot u.
\end{equation*}
Integrating the latter equation against respectively $1$, $v_i$
and $v_i\,v_j$ with $i \not=j$ in $L^2(\mu \,\mathrm d v)$ yields
\begin{equation*}
i)\quad\nabla_x\cdot u-\nabla_x \phi\cdot u=0,\quad
ii)\quad D^s u=0,\quad iii)\quad\nabla r=0.
\end{equation*}
From iii) we get that there exists $c \in \R$ such that
$r=c$. As for i), an integration by parts gives
\begin{equation*}
0=\int_{\R^d} (\nabla_x\cdot u-\nabla_x \phi\cdot u)
\seq{u}\cdot x\,e^{-\phi(x)} \,\mathrm d x=
-\,\int_{\R^d} u\cdot \nabla
\sep{ \seq{u}\cdot x}\,e^{-\phi(x)} \,\mathrm d x=
-\,\int_{\R^d} u\cdot \seq{u}\,e^{-\phi(x)} \,\mathrm d x=-\,\seq{u}^2,
\end{equation*}
so that $\seq{u}=0$. Note also that taking the trace in ii)
yields $\nabla_x\cdot u=0$ so that i) reads
$\nabla_x \phi\cdot u=0$. Using this and~\eqref{eq:WPKZfull} in
Theorem~\ref{theo:KPK0} shows that $u=R$ with
$R=\P_\phi(u)\in{\mathcal R}_\phi$. Hence $h=R(x)\cdot v+c$ and
$f(x,v)=\big(( R(x)\cdot v)+c\big)\,{\mathcal M}$. The reciprocal is
straightforward, which completes the proof. \end{proof}
\section{Additional details on computations} \label{Appendix:B}
\subsection{Functions, derivatives and projections} \label{Appendix:B2}
We denote by $f$ a generic scalar function on $\R^d$ and by $u:\R^d\to\R^d$ a generic vector field, so that \hbox{$\nabla f=(\partial_if)_{i=1}^d$} is a vector field and $Du=(\partial_ju_i)_{i,j=1}^d$ takes values in $\mathfrak M$. The symmetric and the antisymmetric differentials of $u$, respectively $D^su=\big((D^su)_{ij}\big)_{i,j=1}^d$ and $D^au=\big((D^au)_{ij}\big)_{i,j=1}^d$ are defined by \begin{equation*} (D^su)_{ij}:=\tfrac12\(\partial_ju_i+\partial_iu_j\)\quad\mbox{and}\quad (D^au)_{ij}:=\tfrac12\(\partial_ju_i-\partial_iu_j\) \end{equation*} so that $D^su+D^au=Du$.
The orthogonal projection $\P$ of vector-valued functions is defined as follows. Let $(A_{ij})_{1\le i<j\le d}$ be a basis of $\mathfrak M^a$ whose elements are \begin{equation*}
A_{ij}=\big((\delta_{ij}-\delta_{ji})\,\delta_{ki}\,\delta_{j\ell}
\big)_{k,\ell=1}^d \end{equation*} and $(R_{ij})_{1\le i<j\le d}$ the orthonormal (in the $L^2$ sense) basis of $\mathcal R$ given by \begin{equation*}
R_{ij}(x)=Z_{ij}^{-1}\,A_{ij}\,x \end{equation*} whose coordinates are all $0$ except the $i^{th}$ and the $j^{th}$ ones, with respective values $-x_j/Z_{ij}$ and $x_i/Z_{ij}$, {\sl i.e.}, \begin{equation*}
R_{ij}(x)^\perp=Z_{ij}^{-1}\,\big(0,\ldots,0,-x_j,0,
\ldots,0,x_i,0,\ldots,0\big)\,, \end{equation*} and where the normalization constant is $Z_{ij}=\(\int_{\R^d}(x_i^2+x_j^2)\,e^{-\phi(x)}\,\mathrm d x\)^{1/2}$. With these notations, $\P u$ is the vector field \begin{equation*}
x\mapsto\P u(x):=\sum_{1\le i<j\le d}\mathsf c_{ij}\,R_{ij}(x) \end{equation*} where the coefficients are computed, for all integers $i$, $j$ such that $1\le i<j\le d$, as \begin{equation*}
\textstyle\mathsf c_{ij}=
\int_{\R^d}u(x)\cdot R_{ij}(x)\,e^{-\phi(x)}\,\mathrm d x=
\frac1{Z_{ij}}\int_{\R^d}\(x_i\,u_j(x)-x_j\,u_i(x)\)\,
e^{-\phi(x)}\,\mathrm d x\,. \end{equation*}
The orthogonal projection $\mathfrak{P}$ of a matrix-valued function $\mathfrak F$ is defined as \begin{equation*}
\mathfrak{P}\,\mathfrak F:=\sum_{1\le i<j\le d}\mathsf d_{ij}\,A_{ij} \end{equation*} where the coefficients are computed, for all integers $i$, $j$ such that $1\le i<j\le d$, as \begin{equation*}
\textstyle\mathsf d_{ij}=
\frac12\int_{\R^d}\mathfrak F(x):A_{ij}\,e^{-\phi(x)}\,\mathrm d x=
\frac12\int_{\R^d}
\big(\mathfrak F_{ij}(x)-\mathfrak F_{ji}(x)\big)\,
e^{-\phi(x)}\,\mathrm d x\,. \end{equation*} As a a consequence, we deduce that $\mathfrak{P}\mathfrak F=\seq{\mathfrak F^a}$ and $\mathfrak{P}(Du) = \seq{D^a u}$ for any $u \in H^1$.
A matrix $A\in D{\mathcal R}_\phi^c$ is such that $A\in\mathfrak M^a$ and for any $B\in D{\mathcal R}_\phi\subset\mathfrak M^a$, \begin{equation*}
\textstyle0=\int_{\R^d}A\,x\cdot B\,x\,e^{-\phi(x)}\,\mathrm d x=
\sum_{i,j,k=1}^dA_{ij}\,B_{ik}
\int_{\R^d}x_j\,x_k\,e^{-\phi(x)}\,\mathrm d x\,. \end{equation*} A matrix $A\in\mathfrak M_\phi^c$ is such that $A\in\mathfrak M^a$ and for any $B\in D{\mathcal R}_\phi=\mathfrak M_\phi\subset\mathfrak M^a$, \begin{equation*}
\textstyle0=\int_{\R^d}A:B\,e^{-\phi(x)}\,\mathrm d x=
A:B=\sum_{i,j}^dA_{ij}\,B_{ij}. \end{equation*} Based on these two definitions, it is clear that $D{\mathcal R}_\phi^c$ and $\mathfrak M_\phi^c$ generically differ.
\subsection{Operators} \label{Appendix:B4}
Let us give some details on the differential operators
$-\Delta_\phi$ and $-\Delta_S$ associated respectively with the quadratic forms $f\mapsto\|\nabla f\|^2$ and
$u\mapsto\|D^s u\|^2$.
\noindent$\rhd$ Using $\nabla_{\!\phi} u := \nabla\cdot u- \nabla \phi\cdot u$, the {\sl
Witten-Laplace operator $\Delta_\phi$ on functions} is such that $\|\nabla f\|^2=(f,-\Delta_\phi f)$ and takes the form \begin{equation*}
\Delta_\phi f= e^\phi\,\nabla\cdot\(\nabla f\,e^{-\phi}\)
= \nabla_{\!\phi} \cdot \nabla f =
\Delta f - \nabla \phi \cdot \nabla f\,. \end{equation*}
\noindent$\rhd$ By definition of $D^su$ and using integration by parts, we have \begin{equation*}
\begin{array}{rl}
(-\Delta_S u,u)\kern-8pt
& =2\int_{\R^d}|D^su|^2\,e^{-\phi}\,\mathrm d
x=\frac12\sum_{i,j=1}^d
\int_{\R^d}\(\partial_iu_j+\partial_ju_i\)^2e^{-\phi}\,\mathrm d
x\\
&=-\frac12\sum_{i,j=1}^d\int_{\R^d}u_j\,
\partial_i\((\partial_iu_j+\partial_ju_i)\,e^{-\phi}\)\,\mathrm d
x
-\frac12\sum_{i,j=1}^d\int_{\R^d}u_i\,
\partial_j\((\partial_iu_j+\partial_ju_i)\,
e^{-\phi}\)\,\mathrm d x\\
&=-\sum_{i,j=1}^d\int_{\R^d}u_j\,
\partial_i\((\partial_iu_j+\partial_ju_i)\,
e^{-\phi}\)\,\mathrm d x\\
&=-\sum_{i=1}^d\int_{\R^d}\big(u_i\,\Delta
u_i+u_i\,\partial_{ij}u_j\big)\,e^{-\phi}\,\mathrm d
x
+\sum_{i=1}^d\int_{\R^d}u_i\,
\big((\nabla\phi\cdot\nabla)\,u_i
+2\,(D^su\,\nabla\phi)_i\big)\,
e^{-\phi}\,\mathrm d x\\
&=-\int_{\R^d}u\cdot\big(\Delta
u+\nabla(\nabla\cdot
u)-(\nabla\phi\cdot\nabla)\,u
-2\,D^su\,\nabla\phi\big)\,\mathrm d x\,.
\end{array} \end{equation*} so that $-\Delta_S$ is given, for an arbitrary vector field $u\in{\mathcal C}_c^\infty(\R^d;\R^d)$, by \begin{equation*}
-\Delta_S\,u=-\,D^s_\phi\cdot D^su=
-\big(\Delta u+\nabla(\nabla\cdot u)
-(\nabla\phi\cdot\nabla)\,u-2\,D^su\,\nabla\phi\big)\,. \end{equation*}
\subsection{Gaussian measure}\label{Appendix:B5}
In the normalized centred Gaussian case
$\phi(x)=\frac12\,|x|^2+\frac d2\,\ln(2\pi)$ corresponding to~\eqref{Gaussian}, the basic constants are \hbox{$C_{\mathrm{\scriptscriptstyle P}}=1$} (which is the optimal constant in the Gaussian Poincar\'e inequality), either $C_\phi=1+4\,d$ and $C_\phi'=4\,\sqrt{d\,(1+4\,d)}$ if $d\ge2$, or $C_\phi=8$ and $C_\phi'=8\,\sqrt2$ if $d=1$, as a limit case.
Let $u(x)=(1-x_2^2,x_1\,x_2,0,\ldots0)^\perp$. By elementary computations, we find that \begin{equation*}
Du=
\begin{pmatrix}
\begin{array}{cc}
0&-\,2\,x_2\\
x_2&x_1\end{array}&{\mathbf0}\\
{\mathbf0}&{\mathbf0}
\end{pmatrix},\quad
D^su=
\begin{pmatrix}
\begin{array}{cc}0&-\,\frac12\,x_2\\
-\,\frac12\,x_2&x_1\end{array}&{\mathbf0}\\
{\mathbf0}&{\mathbf0}
\end{pmatrix},\quad
D^au=
\begin{pmatrix}
\begin{array}{cc}0&-\,\frac32\,x_2\\
\frac32\,x_2&0\end{array}&{\mathbf0}\\
{\mathbf0}&{\mathbf0}
\end{pmatrix} \end{equation*} where $\mathbf0$ denotes $2\times(d-2)$, $(d-2)\times2$, and $(d-2)\times(d-2)$ null matrices. After integration against the normalized centred Gaussian measure, we have \begin{equation*}
\seq u=0,\quad\|u\|^2=3\,,\quad
\P(u)=0\,,\quad\mathfrak{P} (Du)=0=\seq{D^au}\,,\quad
\|D^su\|^2=\frac32\,,\quad\|D^au\|^2=\frac92\,,
\quad\|Du\|^2=6\,. \end{equation*} This proves that $C_{\mathrm{\scriptscriptstyle K}}=4$ in
$\|D u-\mathfrak{P} (Du)\|^2 \leq C_{\mathrm{\scriptscriptstyle K}}\,\|D^s u\|^2$ and $C_{\mathrm{\scriptscriptstyle PK}}=2$ in
$\|u-\seq{u} - \P (u)\|^2 \leq C_{\mathrm{\scriptscriptstyle PK}}\,\|D^s u\|^2$ are both optimal.
\subsection{Estimates for the
\texorpdfstring{$D(\Lambda)$}{Dlambda}-Toolbox and
consequences} \label{Sec:Dlambda}
Here we give some details on the computation of $C_{\mathrm{\scriptscriptstyle B}}$ in the proof of Proposition~\ref{prop:domain} in Section~\ref{Sec:Toolboxes}. Let $A=\big\|\lfloor\nabla\phi\rceil\,\nabla f\big\|$,
$B=\big\|\lfloor\nabla\phi\rceil^2\,f\big\|$, $Z=\|\xi\|$ and let ${c'}^2=C_\phi'\ge C_\phi=c^2$. Inequalities~\eqref{DVDu} and~\eqref{DVDubis} amount to \begin{equation*}
B\leq\tfrac43\,c\,A+\tfrac53\,{c'}^2\,Z\quad\mbox{and} \quad
A^2\leq B\,Z+\tfrac1{2\,c}\,A\,B+\tfrac{{c'}^2}{2\,c}\,A\,Z\,. \end{equation*} Taking the equality case in the first inequality, we find that \begin{equation*}
A^2-4\,A\,\big(c+{c'}^2/c\big)\,Z-5\,{c'}^2\,Z^2\le0 \end{equation*} which means that \begin{equation*}
\textstyle A\le\(2\,\sigma+\sqrt{4\,\sigma^2+5}\,\)c'\,Z
\quad\mbox{with}\quad\sigma=\frac c{c'}+\frac{c'}c\ge2\,. \end{equation*} On $[2,+\infty)$, the function $\sigma\mapsto\sqrt{4\,\sigma^2+5}/\sigma$ is monotone non-increasing, so that $\sqrt{4\,\sigma^2+5}\le\frac12\,\sqrt{21}\,\sigma$. Using the monotonicity of $c\mapsto c\,\sigma$ and $c\le c'$, we also have $c\,\sigma\le2\,c'$. As a consequence, we have \begin{align*}
&\textstyle A\le\(2+\tfrac12\,\sqrt{21}\,\)c\,\sigma\,
\frac{c'}c\,Z\le\(4+\sqrt{21}\,\)\frac{{c'}^2}c\,Z
\le 9\,\frac{{c'}^2}c\,Z\,,\\
&\textstyle B
\le\frac13\(4\,c\(2\,\sigma
+\sqrt{4\,\sigma^2+5}\,\)+5\,c'\)c'\,Z\le\(7+4\,\sqrt{7/3}\,\)
\le14\,{c'}^2\,Z\,, \end{align*} that is, the bounds~\eqref{Est:4}. Moreover, from
$\big\|D^2f\big\|^2\le\tfrac Cd+\tfrac12\(d+\sqrt{d^2+4\,C}\,\)\|\xi\|^2$ with $C=\tfrac{277}8\(C_\phi-1\){C_\phi'}^2$, we deduce that \begin{equation}
\label{Eq:CB}
C_{\mathrm{\scriptscriptstyle B}}=81\,\tfrac{{C_\phi'}^2}{C_\phi}
+196\,{C_\phi'}^2+\tfrac Cd+\tfrac12\(d+\sqrt{d^2+4\,C}\,\). \end{equation}
\subsection{Estimates on various constants} \label{Sec:Constants}
The constants $C_\phi$ and $C_\phi'$ appear in~\eqref{eq:cphi} as a consequence of~\eqref{hyp:regularity} while the Poincar\'e constant $C_{\mathrm{\scriptscriptstyle P}}$ follows from Assumption~\eqref{hyp:poincarebasique}. According to~\eqref{CPL}, the constant in the Poincar\'e-Lions inequality~\eqref{eq:poincarelions} is given with~$C_{\mathrm{\scriptscriptstyle B}}$ as in~\eqref{Eq:CB} by $C_{\mathrm{\scriptscriptstyle PL}}=(1+C_{\mathrm{\scriptscriptstyle P}})^2\,\big(1+C_\phi'\,\sqrt{C_{\mathrm{\scriptscriptstyle B}}}/4\big)^2$. From Proposition~\ref{prop:poincares}, we know that the strong Poincar\'e inequality~\eqref{eq:strongpoincare} holds for some $C_{\mathrm{\scriptscriptstyle SP}}\leqC_\phi\,(1+C_{\mathrm{\scriptscriptstyle P}})$. As for the other constants in Theorems~\ref{theo:KPK} and~\ref{theo:PKPK}, we learn from the proofs in Sections~\ref{Sec:Prf1} and~\ref{Sec:Prf2} that \begin{align*}
&C_{\mathrm{\scriptscriptstyle K}}\leq1+4\,C_{\mathrm{\scriptscriptstyle PL}}\,,\quadC_{\mathrm{\scriptscriptstyle PK}}\leqC_{\mathrm{\scriptscriptstyle P}}\,C_{\mathrm{\scriptscriptstyle K}}\,,
\quadC_{\mathrm{\scriptscriptstyle SPK}}\leqC_{\mathrm{\scriptscriptstyle SP}}(C_{\mathrm{\scriptscriptstyle K}}+3\,C_\phi\,C_{\mathrm{\scriptscriptstyle PK}})\,,\\
&C_{\mathrm{\scriptscriptstyle PK}}'\leqC_{\mathrm{\scriptscriptstyle PK}}+2\,C_{\mathrm{\scriptscriptstyle RV}}\,C_{\mathrm{\scriptscriptstyle SPK}}\quad\mbox{and}\quad
C_{\mathrm{\scriptscriptstyle K}}'\leqC_{\mathrm{\scriptscriptstyle K}}(1+2\,C_{\mathrm{\scriptscriptstyle RD}}\,C_{\mathrm{\scriptscriptstyle SP}})\,. \end{align*} In Section~\ref{Sec:6.1}, Lemma~\ref{LemPL-1}, using $C_\phi'':=C_{\varepsilon}$ as in~\eqref{hyp:regularity} with ${\varepsilon}=1/(2\,\sqrt{C_{\mathrm{\scriptscriptstyle B}}})$, the constants in~\eqref{eq:poincarelions-1} are \begin{equation*}
C_{\mathrm{\scriptscriptstyle LPL}} \leq 4\,(1+C_{\mathrm{\scriptscriptstyle P}})^2\,(1+C_\phi'')^2\quad\mbox{and}\quad
C_{\mathrm{\scriptscriptstyle RPL}}\leq
\(1+\tfrac14\,C_\phi'\,{\textstyle\sqrt{C_{\mathrm{\scriptscriptstyle B}}/C_\phi}}\,\)^2\,. \end{equation*} Finally, the constants in Theorem~\ref{theo:KPK0} are given by \begin{equation*}
C_{\mathrm{\scriptscriptstyle K0}}=1+4\,C_{\mathrm{\scriptscriptstyle LPL}}\,,\quadC_{\mathrm{\scriptscriptstyle PK0}} \leq C_{\mathrm{\scriptscriptstyle PL}}\,(1+4\,C_{\mathrm{\scriptscriptstyle LPL}})
\quad\mbox{and}\quadC_{\mathrm{\scriptscriptstyle PK0}}' \leq C_{\mathrm{\scriptscriptstyle PK0}}(1+2\,C_{\mathrm{\scriptscriptstyle RV0}}\,C_\phi). \end{equation*}
\section*{Acknowledgments} \noindent\quad \small This work has been partially supported by the Projects EFI (K.C., J.D., ANR-17-CE40-0030) and Kibord (K.C., J.D., S.M., ANR-13-BS01-0004) of the French National Research Agency (ANR). C.M.~and S.M.~acknowledge partial funding by the ERC grants MATKIT 2011-2016 and MAFRAN 2017-2022. Moreover C.M.~is very grateful for the hospitality at Universit\'e Paris-Dauphine.\\ \noindent{\scriptsize\copyright\,2020 by the authors. This paper
may be reproduced, in its entirety, for non-commercial
purposes.}
\begin{center}\rule{2cm}{0.5pt}\end{center}
\end{document} | arXiv |
Observation of Complex Time Structures in the Cosmic-Ray Electron and Positron Fluxes
We published high-statistics, precision measurements of the detailed time and energy dependence of the primary cosmic-ray electron flux and positron flux over 79 Bartels rotations from May 2011 to May 2017 in the energy range from 1 to 50 GeV.
These data allow comprehensive studies of the energy and charge-sign dependence of short-term effects on the time scale of months, related to solar activity, and long-term effects on the time scale of years, related to the 22-year cycle of the solar magnetic field.
Time-dependent structures in the energy spectra are expected from the solar modulation of interstellar cosmic rays when they enter the heliosphere. Solar modulation involves convective, diffusive, particle drift, and adiabatic energy loss processes. Only particle drift induces a dependence of solar modulation on the particle charge sign. Since electrons and positrons differ only in charge sign, their simultaneous measurement offers a unique way to study charge-sign dependent solar modulation effects.
For the first time, the charge-sign dependent modulation during solar maximum has been investigated in detail by leptons alone by AMS. Based on 23.5 million events, we published the observation of short-time structures on the time scale of months coincident in both the electron flux and the positron flux. The fluxes are shown in Figure 1 as a function of time for five characteristic energy bins. We find a clear evolution of the fluxes with time at low energies that gradually diminishes towards high energies. At the lowest energies, the amplitudes of both the electron flux and the positron flux change by a factor of 3. Both fluxes exhibit profound short- and long-term variations. The short-term variations occur simultaneously in both fluxes with approximately the same relative amplitude. Several prominent and distinct structures are observed. They are characterized by minima, visible in both the electron flux and the positron flux across the energy range below $E \lesssim 10$ GeV. These are marked by dashed vertical lines in Figure 1. At energies above 20 GeV, neither the electron flux nor the positron flux exhibits significant time dependence.
Figure 1. Fluxes of cosmic-ray positrons (red, left axis) and electrons (blue, right axis) as functions of time, for five of the 49 energy bins. The error bars are the statistical uncertainties. Prominent and distinct time structures visible in both the positron spectrum and the electron spectrum and at different energies are marked by dashed vertical lines.
The long-term time structure of the data in Figure 1 shows that the changes in relative amplitude are different for electrons and positrons. To quantify this effect, we use the $e^{+}/e^{-}$ flux ratio $R_e$, shown in Figure 2. In $R_e$, the important, newly discovered short-term variations in the fluxes largely cancel, and a clear overall long-term trend appears. At low energies, $R_e$ is flat at first, then smoothly increases after the time of the solar magnetic field reversal, to reach a plateau at a higher amplitude.
We use a model independent approach to extract the energy dependence of the quantities that characterize the observed transition in $R_{e}$. With a set of four parameters, the 3871 independent $R_{e}$ measurements as a function of energy and time can be described well with a logistic function:
\begin{equation} \label{eq:1} R_e(t,\, E) =R_0(E)\times \left[1+\frac{C(E)}{{\rm exp}\left(-\frac{t-t_{1/2}(E)}{\Delta t(E)/\Delta} + 1\right)}\right]\end{equation}
At a given energy $E$, the time dependence is related to three parameters in the function: the amplitude of the transition $C$, the midpoint of the transition $t_{1/2}$, and the duration of the transition $\Delta t$. We choose $\Delta = 4.39$, such that $\Delta t$ is the time it takes for the transition to proceed from 10% to 90% of the change in magnitude. The behavior of the logistic function is illustrated in Figure 2 using the fit to data in the energy range [1.01 – 1.22] GeV.
Figure 2. Illustration of the logistic function parameters in describing the time and energy dependence of $R_{e}$, using the fit in the energy bin [1.01 – 1.22] GeV as an example. The fit result is shown by the red curve. The period without well-defined polarity is marked by the shaded area. Our choice for the effective time of the reversal of the solar magnetic field $t_{\rm rev}$ is marked by a black dashed vertical line. The fit result for the midpoint of the transition $t_{1/2}$ is marked by a red dashed vertical line. The width of the red horizontal bar indicates the duration of the transition $\Delta t$. It takes time $\Delta t$ for the transition to proceed from 10% to 90% of the change in magnitude.
The results of fitting for several energy bins from 1 to 21 GeV are shown in Figure 3. We obtain $\chi^2/{\rm d.o.f} \approx 1$ for all the fits.
Figure 3. The ratio $R_e$ of the positron flux to the electron flux as a function of time. The error bars are statistical. The best-fit parametrization of a logistic function is shown by red curves. The polarity of the heliospheric magnetic field is denoted by $A < 0$ and $A > 0$. The period without well-defined polarity is marked by the shaded area.
The parameters $t_{1/2}$ and Δt can only be determined at low energies, where the amplitude of the transition is large, see Figure 3. The energy dependences of these parameters are shown in Figure 4. As seen, the transition duration Δt is independent of energy (Figure 4a), and we obtain a value of $830\pm 30$ days. The midpoint of the transition, $t_{1/2} - t_{\rm rev}$, shows an energy dependent delay relative to the reversal and changes by $260\pm 30$ days from 1 to 6 GeV, as seen in Figure 4b.
Figure 4. Results of the logistic function fit to the ratio $R_e$ as a function of energy (blue circles): (a) $\Delta t$ and the best fit constant value of 830 days (red line), (b) $t_{1/2} - t_{\rm rev}$ with its parametrization (red curve), (c) amplitude C with a dashed line at zero to guide the eye.
To study the amplitude C in Figure 4c, we have fixed $\Delta t$ to its average value of 830 days and we use the value of $t_{1/2}$ calculated from Eq. (\ref{eq:1}) for energies above 6 GeV. At high energies, the fit result for the amplitude depends only weakly on the choice of the values for $\Delta t$ and $t_{1/2}$. As seen in Figure 4c, the amplitude $C$ is close to 1 at $E=1$ GeV and decreases smoothly with energy. This is in qualitative agreement with the expectation from solar modulation models including drift effects. Above 20 GeV, the amplitude is consistent with zero.
In conclusion, for the first time, the charge-sign dependent modulation during solar maximum has been investigated in detail by leptons alone. We observe prominent, distinct, and coincident structures in both the positron flux and the electron flux on a time scale of months. These structures are not visible in the $e^+/e^-$ flux ratio. We also observe the existence of a long-term feature in the $e^+/e^-$ flux ratio, namely, a smooth transition from one value to another, after the polarity reversal of the solar magnetic field. The duration of the transition is measured to be $830\pm 30$ days, independent of energy. The transition magnitude is decreasing as a function of energy. The midpoint of the transition relative to the polarity reversal of the solar magnetic field changes by $260\pm 30$ days from 1 to 6 GeV. These high-statistics, precision data on positrons and electrons provide accurate input to the understanding of solar modulation.
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Observation of Complex Time Structures in the Cosmic-Ray Electron and Positron Fluxes with the Alpha Magnetic Spectrometer on the International Space Station | CommonCrawl |
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